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+# Adaptive Risk Minimization: Learning to Adapt to Domain Shift
+
+Marvin Zhang⇤1, Henrik Marklund⇤2, Nikita Dhawan⇤1, Abhishek Gupta1, Sergey Levine1, Chelsea Finn2
+
+1 UC Berkeley, 2 Stanford University
+
+# Abstract
+
+A fundamental assumption of most machine learning algorithms is that the training and test data are drawn from the same underlying distribution. However, this assumption is violated in almost all practical applications: machine learning systems are regularly tested under distribution shift, due to changing temporal correlations, atypical end users, or other factors. In this work, we consider the problem setting of domain generalization, where the training data are structured into domains and there may be multiple test time shifts, corresponding to new domains or domain distributions. Most prior methods aim to learn a single robust model or invariant feature space that performs well on all domains. In contrast, we aim to learn models that adapt at test time to domain shift using unlabeled test points. Our primary contribution is to introduce the framework of adaptive risk minimization (ARM), in which models are directly optimized for effective adaptation to shift by learning to adapt on the training domains. Compared to prior methods for robustness, invariance, and adaptation, ARM methods provide performance gains of $1 - 4 \%$ test accuracy on a number of image classification problems exhibiting domain shift.
+
+# 1 Introduction
+
+The standard assumption in empirical risk minimization (ERM) is that the data distribution at test time will match the training distribution. When this assumption does not hold, i.e., when there is distribution shift, the performance of standard ERM methods can deteriorate significantly [54, 38].
+
+As an example which we study quantitatively in Section 5, consider a handwriting classification model that, after training on data from past users, is deployed to new end users. Each new user represents a new test distribution that differs from the training distribution. Thus, each test setting involves dealing with shift. In Figure 1, we visualize a batch of 50 examples from a test user, and we highlight an ambiguous example which may be either a “2” (written with a loop) or an “a” (in the double-storey style) depending on the user’s handwriting. Due to the biases in the training data, an ERM trained model incorrectly classifies this example as “2”. However, we can see that the batch of images from this test user contains other examples of “2” (written without loops) and “a” (also double-storey) from this user. Can we somehow leverage this unlabeled data to better handle test shifts caused by new users?
+
+![](images/eba7790167eea1016adf846184d756ac9c24d137f32ef1c5973b0041f9f7837b.jpg)  
+Figure 1: An example of ambiguous data points in handwriting classification, evaluated quantitatively in Section 5.
+
+Any framework that aims to address this question must use additional assumptions beyond the ERM setting, and many such frameworks have been proposed $\pmb { \Vert 5 4 \Vert }$ . One commonly used assumption within several frameworks, such as domain generalization [7, 23], is that the training data are provided in domains and distributions at test time will represent new domains. The example above neatly fits this description if we equate users with domains – we would be assuming that the training data are organized by users and that the model will be tested separately on new users, and these are reasonable assumptions. Constructing training domains in practice is generally accomplished by using meta-data, which exists for many commonly used datasets. Thus, this domain assumption is applicable for a wide range of realistic distribution shift problems (see, e.g., Koh et al. [35]).
+
+However, prior benchmarks for domain generalization and similar settings typically center around invariances – i.e., in these benchmarks, there is a consistent input-output relationship across all domains, and the goal is to learn this relationship while ignoring the spurious correlations within the domains (see, e.g., Gulrajani and Lopez-Paz $[ \overbar { 1 2 3 } ] \cdot$ ). Thus, prior methods aim for generalization to shifts by discovering this relationship, through techniques such as robust optimization and learning an invariant feature space [41, 3, 60]. These methods are appealing in that they make minimal assumptions about the information provided at test time – in particular, they do not require test labels, and the learned model can be immediately applied to predict on a single point. Nevertheless, these methods also have limitations, such as in dealing with problems where the input-output relationship varies across domains, e.g., the handwriting classification example above.
+
+In this paper, we instead focus on methods that aim to adapt at test time to domain shift. To do so, we study problems in which it is both feasible and helpful (and perhaps even necessary) to assume access to a batch or stream of inputs at test time. Leveraging this test assumption does not require labels for any test data and is feasible in many practical setups. For example, for handwriting classification, we do not access only single handwritten characters from an end user, but rather collections of characters such as sentences or paragraphs. Unlabeled adaptation has been shown empirically to be useful for distribution shift problems $[ \dot { \overline { { 6 9 } } } , \overline { { 6 3 } } , \overline { { 7 5 } } ]$ , such as for dealing with image corruptions $\mathbb { \left[ \left. 2 5 \right] \right. }$ . Taking inspiration from these findings, we propose and evaluate on a number of problems, detailed in Section 5, for which adaptation is beneficial in dealing with domain shift.
+
+Our main contribution is to introduce the framework of adaptive risk minimization (ARM), which proposes the following objective: optimize the model such that it can maximally leverage the unlabeled adaptation phase to handle domain shift. To do so, we instantiate a set of methods that, given a set of training domains, meta-learns a model that is adaptable to these domains. These methods are straightforward extensions of existing meta-learning approaches, thereby demonstrating that tools from the meta-learning toolkit can be readily adapted to tackle domain shift. Our experiments in Section 5 test on several image classification problems, derived from benchmarks for federated learning [9] and image classifier robustness $\mathbb { \lVert 2 5 \rVert }$ , in which training and test domains share structure that can be leveraged for improved performance. These testbeds are also a contribution of our work, as we believe these problems can supplement existing benchmarks which, almost exclusively, are designed with invariance in mind $[ \sqrt { 3 } , \sqrt { 5 3 } , \sqrt { 2 3 } ]$ . We also evaluate on the WILDS suite of distribution shift problems $\begin{array} { r l } {  { \bigl [ \bigl | 3 5 \bigr | \bigr ] } } & { { } } \end{array}$ , which have been curated to faithfully represent important real world problems. Empirically, we demonstrate that the proposed ARM methods, by leveraging meta-training and test time adaptation, are often able to outperform prior state-of-the-art methods by $1 \%$ test accuracy.
+
+# 2 Related Work
+
+A number of prior works have studied distribution shift in various forms $[ [ 5 4 ]$ . In this section, we review prior work in domain generalization, group robustness, meta-learning, and adaptation.
+
+Invariance and robustness to domains. As discussed above, a number of frameworks leverage training domains to address test time shift. The terminology in prior work is scattered and, depending on the application, includes terms such as “groups”, “datasets”, “subpopulations”, and “users”; in this work, we adopt the term “domains” which we believe is an appropriate unifying term. A number of testbeds for this problem setting have been proposed for image classification, including generalizing to new datasets $\sharp$ , new image types $\pm \boxed { 1 3 9 } \boxed { 5 3 } \cdots$ , and underrepresented demographics $\bar { \mathbb { E O } } \mathbb { I }$ .
+
+Prior benchmarks typically assume the existence of a consistent input-output relationship across domains that is learnable by the specified model, thus motivating methods such as learning an invariant feature space [41, 44, 3] or optimizing for worst case group performance $\pmb { \mathbb { B O } } \pmb { \mathbb { G O } } \mathbf { j }$ . In particular, methods for domain generalization – sometimes referred to as multi-source domain adaptation $\lVert \rVert$ or zero shot domain adaptation $\left[ \left[ 7 8 \right] \right]$ – have largely focused on learning invariant features [19, 67, 41, 44, 53]. Gulrajani and Lopez-Paz $\pmb { \left. \pmb { \left. \mathscr { Z } 3 \right. } \right. }$ provide a comprehensive survey of domain generalization benchmarks and find that, surprisingly, ERM is competitive with the state of the art across all the benchmarks considered. In Appendix C, we discuss this finding as well as the performance of an ARM method on this benchmark suite. In Section 5, we identify different problems for which adaptation is helpful, and we find that, on these problems, ARM methods consistently outperform ERM and other non adaptive methods for robustness and invariance.
+
+Meta-learning. Meta-learning [62, 6, 71, 27] has been most extensively studied in the context of few shot labeled adaptation [61, 74, 55, 18, 65]. Our aim is not to address few-shot recognition problems, nor to propose a novel meta-learning algorithm, but rather to extend meta-learning paradigms to problems requiring unlabeled adaptation, with the primary goal of tackling distribution shift. This aim differs from previous work in meta-learning for domain generalization $[ \bar { 4 0 } , \bar { 1 } 5 ]$ , which seek to metatrain models for non adaptive generalization performance. We discuss in Section 4 how paradigms such as contextual meta-learning [20, 57] can be readily extended using the ARM framework.
+
+Some other meta-learning methods adapt using both labeled and unlabeled data, either in the semi supervised learning setting [56, 81, 42] or the transductive learning setting [51, 46, 2, 29]. These works all assume access to labeled data for adaptation, whereas we propose methods and problems for purely unlabeled adaptation. Prior works in meta-learning for unlabeled adaptation include Yu et al. $\mathbf { [ 8 0 ] }$ , who adapt a policy to imitate human demonstrations in the context of robotic learning; Metz et al. $\lVert \rVert \mathbf { 4 8 } \rVert$ , who meta-learn an update rule for unsupervised representation learning, though they still require labels to learn a predictive model; and Alet et al. [1], who meta-learn adaptive models based on task specific unsupervised objectives. Unlike these prior works, we propose a general framework for tackling distribution shift problems by meta-learning unsupervised adaptation strategies. This framework simplifies the extension of meta-learning paradigms to these problems, encapsulates previous approaches such as the gradient based meta-learning approach of Yu et al. $\pmb { \| 8 0 \| }$ , and sheds light on how to improve existing strategies such as adaptation via batch normalization [43].
+
+Adaptation to shift. Unlabeled adaptation has primarily been studied separately from meta-learning. Domain adaptation is a prominent framework that assumes access to test examples at training time $\boxed { 1 3 } , \boxed { 7 6 }$ , similar to transductive learning $\mathbb { | \overline { {  { | Z 3 \| } } } | }$ . As such, most domain adaptation methods consider the problem setting where there is a single test distribution $\lVert 6 4 \rVert , \lVert 4 \rVert , \lVert 2 2 \rVert , \dot { \lVert 1 9 \rVert } , \lVert 7 2 \rVert , \lVert 0 \rVert$ , and some of these methods are difficult to apply to problems where there are multiple test distributions. Certain domain adaptation methods have also been applied in the domain generalization setting, such as methods for learning invariant features [19, 67, 41], and we compare to these methods in Section 5.
+
+Adaptive methods for domain generalization include Muandet et al. $\mathbb { \oplus 2 }$ and Kumagai and Iwata $\pmb { \Vert 3 7 } \Vert$ who propose a method similar to one of the ARM methods described below. We compare to a version of this method in Appendix E. Blanchard et al. [7] and Blanchard et al. $\pmb { \mathbb { B } } \|$ provide a theoretic study of domain generalization and establish favorable generalization bounds for models that can adapt to domain shift at test time. We summarize some of these results in $\mathsf { S e c t i o n } \ 3 .$ In comparison, our work establishes a framework that makes explicit the connection between adaptation to domain shift and meta-learning, allowing us to devise new methods in a straightforward and principled manner. These methods are amenable to expressive models such as deep neural networks, which enables us to propose and evaluate on problems with raw image observations.
+
+Test time adaptation has also been studied for dealing with label shift $\mathbb { \lVert 5 9 \rVert \lVert 5 \rVert \lVert 6 6 \rVert }$ and crafting favorable inductive biases for the domain of interest. For image classification, techniques such as normalizing via the test inputs $\mathbb { E 3 }$ and optimizing self-supervised surrogate losses $\lVert \boldsymbol { 6 9 } \rVert$ have proven effective for adapting to image corruptions $\pmb {  \pmb {  \bar { 2 } 5  } }$ . We compare to these prior methods in Section 5 and empirically demonstrate the advantage of using training domains to learn how to adapt.
+
+# 3 Preliminaries and Notation
+
+In this section, we discuss the domain generalization problem setting and formally describe adaptive models. In Section 4, we discuss how adaptive models can be meta-trained via the ARM objective and approach, and we instantiate ARM methods which we empirically evaluate in Section 5.
+
+Let $\mathbf { x } \in \mathcal { X }$ and $y \in \mathcal { V }$ represent the input and output, respectively. We can formalize the domain generalization problem setting using the following data generation process $\textcircled { 8 } \textcircled { 1 8 }$ : first, a joint data distribution $p _ { \mathbf { x } y }$ is sampled from a set of distributions $\mathcal { P } _ { \mathbf { x } y }$ , and then some data points are sampled from $p _ { \mathbf { x } y } \underline { { \mathbb { I } } }$ We refer to each $p _ { \mathbf { x } y }$ as a domain, e.g., a particular dataset or user, thus $\mathcal { P } _ { \mathbf { x } y }$ represents the set of all possible domains. We assume that the training dataset is composed of data from $S$ runs of this generative process, organized by domain. An equivalent characterization which we will use for clarity is that, within the training set, there are $S$ domains, and each data point $( \mathbf { x } ^ { ( i ) } , y ^ { ( i ) } )$ is annotated with a domain label $z ^ { ( i ) }$ . Each $z ^ { ( i ) }$ is an integer that takes on a value between 1 and $S$ , indicating which $p _ { \mathbf { x } y }$ generated the $i$ -th training point (though, of course, we do not have access to, or knowledge of, $p _ { \mathbf { x } y }$ itself). At test time, there may be multiple evaluation settings, where each setting is considered separately and contains only unlabeled data sampled via a new run of the same generative process. This data may represent, e.g., a new dataset or user, and the test domains are likely to be distinct from the training domains when $| \mathcal { P } _ { { \bf x } y } |$ is large or infinite.
+
+Our formal goal is to optimize for expected performance, e.g., classification accuracy, at test time. To do so, let us first consider predictive models of the form $f : \mathcal { X } \times \mathcal { P } _ { \mathbf { x } }  \mathcal { Y }$ , where the model $f$ takes in not just an input $\mathbf { x }$ but also the marginal input distribution $p _ { \mathbf { x } } \in \mathcal { P } _ { \mathbf { x } }$ that $\mathbf { x }$ was sampled from. We refer to $f$ as an adaptive model, as it has the opportunity to use $p _ { \mathbf { x } }$ to adapt its predictions on $\mathbf { x }$ . The underlying assumption is that $p _ { \mathbf { x } }$ provides information about $p _ { y | \mathbf { x } }$ , i.e., $p _ { \mathbf { x } }$ is used as a surrogate input in place of $p _ { \mathbf { x } y }$ . In the worst case, if $p _ { \mathbf { x } }$ and $p _ { y | \mathbf { x } }$ are sampled independently, then the model does not benefit at all from knowing $p _ { \mathbf { x } }$ . In many problems, however, we expect knowledge about $p _ { \mathbf { x } }$ to be useful, e.g., for resolving ambiguity as in the handwriting classification example in Section 1.
+
+Theoretically, when $p _ { \mathbf { x } }$ provides information about $p _ { y | \mathbf { x } }$ , and when training and test domains are drawn from the same distribution over $\mathcal { P } _ { \mathbf { x } y }$ , we can establish favorable generalization bounds for the expected performance of $f$ in adapting to domain shift at test time. We can formalize this as follows. First, define a prediction model to be a non adaptive model of the form $g : \mathcal { X }  \mathcal { Y }$ , and define the risk for a prediction model $g$ and loss function $\ell$ , under a data distribution $p _ { \mathbf { x } y }$ , as
+
+$$
+\mathcal { R } ( g , p _ { \mathbf { x } y } ) \triangleq \mathbb { E } _ { p _ { \mathbf { x } y } } \left[ \ell ( g ( \mathbf { x } ) , y ) \right] .
+$$
+
+Further, define the Bayes optimal risk for $\ell$ under $p _ { \mathbf { x } y }$ as
+
+$$
+\begin{array} { r } { \mathcal { R } ^ { \star } ( p _ { \mathbf { x } y } ) \triangleq \underset { g } { \operatorname* { m i n } } \mathcal { R } ( g , p _ { \mathbf { x } y } ) . } \end{array}
+$$
+
+Let $\mu$ denote the distribution on $\mathcal { P } _ { \mathbf { x } y }$ from which training and test domains $p _ { \mathbf { x } y }$ are sampled. To avoid overlapping terms, define the adaptive risk for an adaptive model $f$ and $\ell$ , under $\mu$ , to be
+
+$$
+\begin{array} { r } { \mathcal { E } ( f , \mu ) \triangleq \mathbb { E } _ { \mu } \left[ \mathbb { E } _ { p _ { \mathbf { x } _ { y } } } \left[ \ell ( f ( \mathbf { x } , p _ { \mathbf { x } } ) , y ) \right] \right] . } \end{array}
+$$
+
+We state the following result from Blanchard et al. $\pmb { \mathbb { B } } ] \mathbf  l$ , which details a condition on $\mu$ under which $\mathcal { E }$ is a strongly principled objective for learning adaptive models.
+
+Lemma 9 from Blanchard et al. [8]. Let $f ^ { \star }$ denote a minimizer of $\mathcal { E }$ for the given $\mu$ . If $\mu$ is a distribution on $\mathcal { P } _ { \mathbf { x } y }$ such that $\mu$ -almost surely it holds that $p _ { y | \mathbf { x } } = M ( p _ { \mathbf { x } } )$ for some deterministic mapping $M$ , then for $\mu$ -almost all $p _ { \mathbf { x } y }$ , we have
+
+$$
+\begin{array} { r } { \mathcal { R } \big ( f ^ { \star } \big ( \cdot , p _ { \mathbf { x } } \big ) , p _ { \mathbf { x } y } \big ) = \mathcal { R } ^ { \star } \big ( p _ { \mathbf { x } y } \big ) \implies \mathcal { E } \big ( f ^ { \star } , \mu \big ) = \mathbb { E } _ { \mu } \left[ \mathcal { R } ^ { \star } ( p _ { \mathbf { x } y } ) \right] . } \end{array}
+$$
+
+In other words, an adaptive model which minimizes the adaptive risk $\mathcal { E }$ coincides with a Bayes optimal decision function for $p _ { \mathbf { x } y }$ , for $\mu$ -almost all domains $p _ { \mathbf { x } y }$ .
+
+Remark. The required condition on $\mu -$ that $p _ { y | \mathbf { x } }$ is determined by $p _ { \mathbf { x } }$ – holds if, and only if, an expert (or oracle) is able to correctly label inputs from a given domain provided only information about the input distribution. This condition holds for the testbeds proposed in this paper, those in Gulrajani and Lopez-Paz $\mathbb { \left. 2 3 \right. }$ , and those in WILDS $\mathbb { \lVert 3 5 \rVert }$ . The condition does not hold for, e.g., standard few shot learning testbeds, where it is possible for two domains with identical input distributions to shuffle their label orderings differently $\pmb { \hat { \mathbb { Z } } } \pmb { \mathbb { \| } }$ . Thus, these problems are outside the scope of this work.
+
+This result provides strong justification for learning adaptive models $f$ by minimizing the adaptive risk $\mathcal { E }$ . However, a practical instantiation of this approach requires some approximations. First, we do not know and cannot input $p _ { \mathbf { x } }$ to $f$ in most cases. Instead, we instantiate $f$ such that it takes in a batch of inputs $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K }$ , all from the same domain, where $K$ can vary. $f$ makes predictions on the whole batch, which also serves as an empirical approximation (i.e., a histogram) $\hat { p } _ { \bf x }$ of $p _ { \mathbf { x } } \mathinner { \| { \boldsymbol { \mathrm { B } } } \| }$ . In our exposition, we will assume that a batch of unlabeled points is available at test time for adaptation. However, we also experiment in $\mathtt { S e c t i o n } 5$ with the streaming setting where the test inputs are observed one at a time and adaptation occurs incrementally.
+
+Notice that, if we instead passed in an approximation $\hat { p } _ { { \bf x } y }$ of $p _ { \mathbf { x } y }$ to the model, such as a batch of labeled data $( { \bf x } _ { 1 } , y _ { 1 } ) , \dots , ( { \bf x } _ { K } , y _ { K } )$ , then this setup would resemble the standard few shot metalearning problem $\pmb { \Vert 7 4 \Vert }$ . Formally, a meta-learning model takes in both an input $\mathbf { x }$ and $\hat { p } _ { { \bf x } y }$ , which approximates the distribution that $\mathbf { x }$ was sampled from and thus can be used to adapt the prediction on $\mathbf { x }$ . Compared to our problem setting, the meta-learning formalism can tackle a wider range of problems but also requires more restrictive assumptions, specifically, labels at test time via $\hat { p } _ { { \bf x } y }$ . Transductive meta-learning methods further assume that, in addition to $\hat { p } _ { { \bf x } y }$ , a full batch of inputs $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K }$ is passed into the model, which allows for better estimation of the input distribution $p _ { \mathbf { x } }$ [51, 46, 29]. The model then makes predictions on this entire batch. In meta-learning terminology, $\hat { p } _ { { \bf x } y }$ and $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K }$ are often referred to as the support and query, respectively. Therefore, another interpretation of the adaptive models that we study in this work is that they resemble transductive meta-learning models, but they are given only the unlabeled query and not the labeled support set.
+
+In the next section, we expand on this connection to develop the ARM framework, which then allows us to bring forward tools from meta-learning to tackle domain shift problems.
+
+# 4 Adaptive Risk Minimization
+
+In this section, we formally describe the ARM framework, which defines an objective for training adaptive models to tackle domain shift. Furthermore, we propose a general meta-learning algorithm as well as specific methods for optimizing the ARM objective. In Section 5, we test these ARM methods on problems for which unlabeled adaptation can be leveraged for better test performance.
+
+# 4.1 Devising the ARM objective
+
+We wish to learn an adaptive model $f : \mathcal { X } ^ { K } \to \mathcal { Y } ^ { K }$ to tackle domain shift. As noted, meta-learning methods for labeled adaptation study a similar form of model, and a common approach in many of these methods is to define $f$ such that it is composed of two parts: first, a learner which ingests the data and produces parameters, and second, a prediction model which uses these parameters to make predictions $\boxed { 7 4 } , \boxed { 1 8 } \vert$ . We will follow a similar strategy which, as we will discuss in subsection 4.2, allows us to easily extend and design meta-learning methods towards our goal.
+
+In particular, we will decompose the model $f$ into two modules: a standard prediction model $g ( \cdot ; \theta ) : \mathcal { X } \to \mathcal { Y } _ { }$ , that is parameterized by $\theta \in \Theta$ and predicts $y$ given $\mathbf { x }$ , and an adaptation model $h ( \cdot , \cdot ; \phi ) : \ \Theta \times \mathcal { X } ^ { K }  \Theta$ , which is parameterized by $\phi$ . $h$ takes in the prediction model parameters $\theta$ and $K$ unlabeled data points and uses the $K$ points to produce adapted parameters $\theta ^ { \prime }$ . This is analogous to the learner in meta-learning, however, $h$ adapts the model parameters using only unlabeled data. We defer the discussion of how to instantiate $h$ to subsection 4.2.2
+
+The ARM objective is to optimize $\phi$ and $\theta$ such that $h$ can adapt $g$ using unlabeled data sampled according to a particular domain $z$ . This can be expressed as the optimization problem
+
+$$
+\operatorname* { m i n } _ { \theta , \phi } \hat { \mathcal { E } } ( \theta , \phi ) = \mathbb { E } _ { p _ { z } } \left[ \mathbb { E } _ { p _ { \mathbf { x } \cdot p } | z } \left[ \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \ell ( g ( \mathbf { x } _ { k } ; \theta ^ { \prime } ) , y _ { k } ) \right] \right] , \mathrm { ~ w h e r e ~ } \theta ^ { \prime } = h ( \theta , \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K } ; \phi ) .
+$$
+
+Note that $\hat { \mathcal { E } }$ is the empirical form of the adaptive risk in $\mathrm { E q u a t i o n } 1$ for the form of $f$ we have defined. Mimicking the generative process from Section 3 that we assume generated the training data, $p _ { z }$ is a categorical distribution over $\{ 1 , \ldots , S \}$ which places uniform probability mass on each training domain, and $p _ { \mathbf { x } y | z }$ assigns uniform probability to only the training points within a particular domain. As we have established theoretically, we expect the trained models to perform well at test time if the test domains are sampled independently and identically – i.e., from the same distribution over $\mathcal { P } _ { \mathbf { x } y } -$ as the training domains. In practice, similar to how meta-learned few shot classification models are evaluated on new and unseen meta-test classes $\textcircled { 7 4 } , \textcircled { 1 8 } \textcircled { }$ , we empirically show in Section 5 that the trained models can generalize to test domains that are not sampled identically to the training domains.
+
+# 4.2 Optimizing the ARM objective
+
+Algorithm 1 presents a general meta-learning approach for optimizing the ARM objective. As described above, $h$ outputs updated parameters $\theta ^ { \prime }$ using an unlabeled batch of data (line 5). This mimics the adaptation procedure at test time, where we do not assume access to labels (lines 7-8). However, the training update itself does rely on the labels (line 6). We assume that $h$ is differentiable with respect to its input $\theta$ and $\phi$ , thus we use gradient updates on both $\theta$ and $\phi$ to optimize for post adaptation performance on a mini batch of data sampled according to a particular domain $z$ . In practice, we also sample mini batches of domains, rather than just one domain (as written in line 3), to provide a better gradient signal for optimizing $\phi$ and $\theta$
+
+# Algorithm 1 Meta-Learning for ARM
+
+// Training procedure
+
+Require: # training steps $T$ , batch size $K$ , learning rate $\eta$
+
+1: Initialize: $\theta , \phi$   
+2: for $t = 1 , \dots , T$ do   
+3: Sample $z$ uniformly from training domains   
+4: Sample $( \mathbf { x } _ { k } , y _ { k } ) \sim p ( \cdot , \cdot | z )$ for $k = 1 , \ldots , K$   
+5: $\theta ^ { \prime } \gets h ( \theta , \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K } ; \phi )$   
+6: $\begin{array} { r } { ( \theta , \phi )  ( \theta , \phi ) - \eta \nabla _ { ( \theta , \phi ) } \sum _ { k = 1 } ^ { K } \ell ( g ( \mathbf { x } _ { k } ; \theta ^ { \prime } ) , y _ { k } ) } \end{array}$
+
+// Test time adaptation procedure
+
+Require: $\theta , \phi$ , test batch $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K }$ 7: $\theta ^ { \prime } \gets h ( \theta , \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K } ; \phi )$ 8: $\hat { y } _ { k } \gets g ( \mathbf x _ { k } ; \theta ^ { \prime } )$ for $k = 1 , \ldots , K$
+
+Together, Equation 2 and Algorithm 1 shed light on a number of ways to devise methods for solving the ARM problem. First, we can extend meta-learning paradigms to the ARM problem setting, and any paradigm in which the adaptation model $h$ can be augmented to operate on unlabeled data is readily applicable. As an example, we propose the ARM-CML method, which is inspired by recent works in contextual meta-learning (CML) [20, 57]. Second, we can enhance prior unlabeled adaptation methods by incorporating a meta-training phase that allows the model to better leverage the adaptation. To this end, we propose the ARM-BN method, based on the general approach of adapting using batch normalization (BN) statistics of the test inputs [43, 63, 33, 50]. Third, we can incorporate existing methods for meta-learning unlabeled adaptation to solve domain shift problems. We demonstrate this by proposing the ARM-LL method, which is based on the robotic imitation learning method from Yu et al. $\pmb { \Vert 8 0 \Vert }$ which adapts via a learned loss (LL). All of these methods are straightforward extensions of existing meta-learning and adaptation methods, and this is intentional – we aim to show how existing tools can be readily adapted to tackle domain generalization problems. We summarize the methods here and refer the reader to Appendix B for complete details.
+
+ARM-CML. In ARM-CML, the parameters $\phi$ of $h$ define the weights of a context network $\mathrm { f } _ { \mathrm { c o n t } } ( \cdot ; \boldsymbol { \phi } ) : \mathcal { X } \to \mathbb { R } ^ { D }$ , parameterized by the adaptation model parameters $\phi$ . We also instantiate the model with a prediction network $\operatorname { f } _ { \mathrm { p r e d } } ( \cdot , \cdot ; \boldsymbol { \theta } ) \dot { : } \mathcal { X } \times \mathbb { R } ^ { D }  \dot { \mathcal { V } }$ , parameterized by $\theta$ . When given a mini batch of inputs, $\mathrm { f _ { c o n t } }$ processes each example $\mathbf { x } _ { k }$ in the mini batch separately and outputs $\mathbf { c } _ { k } \in \mathbb { R } ^ { D }$ for $k = 1 , \ldots , K$ , which are averaged together into a context $\begin{array} { r } { { \mathbf c } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } { \mathbf c } _ { k } } \end{array}$ . $D$ is a hyperparameter, and in our experiments, we choose to be the dimensionality of $\mathbf { x }$ , such that we can concatenate each image $\mathbf { x } _ { k }$ and the context c along the channel dimension to produce the input to $\mathrm { f _ { p r e d } }$ . In other words, $\mathrm { f _ { p r e d } }$ processes each $\mathbf { x } _ { k }$ separately to produce an estimate of the output $\hat { y } _ { k }$ , but it additionally receives c as input. In this way, $\mathbf { f } _ { \mathrm { c o n t } }$ can provide information about the entire batch of $K$ unlabeled data points to $\mathrm { f _ { p r e d } }$ for predicting the correct outputs.
+
+Note that the difference between ARM-CML and prior contextual meta-learning approaches is that, in prior approaches, the context network processes both inputs and outputs to produce each $\mathbf { c } _ { k }$ ARM-CML is designed for the domain generalization setting in which we do not assume access to labels at test time, thus we meta-train for unlabeled adaptation performance at training time.
+
+ARM-BN. ARM-BN is a particularly simple method that is applicable for any model $g$ that has BN layers [31] . Practically, training $g$ via ARM-BN follows the same protocol as Ioffe and Szegedy [31] except for two key differences: first, the training batches are sampled from a single domain, rather than from the entire dataset, and second, the normalization statistics are recomputed at test time rather than using a training running average. As noted, this second difference has been explored by several works as a method for test time adaptation, but the first difference is novel to ARM-BN. Following $\mathbb { E } \mathrm { { g o r i t h m } 1 } ,$ ARM-BN defines a meta-training procedure in which $g$ learns to adapt – i.e., compute normalization statistics – using batches of training points sampled from the same domain. We empirically show in Section 5 that, for problems where BN adaptation already has a favorable inductive bias, such as for image classification, further introducing meta-training boosts its performance. We believe that other test time adaptation methods, such as those based on optimizing surrogate losses [69, 75], may similarly benefit from their corresponding meta-training procedures.
+
+At a high level, ARM-BN operates in a similar fashion to ARM-CML, thus we group these methods together into the umbrella of contextual approaches, shown in Figure 2 (top). The interpretation of ARM-BN through the contextual approach is that $h$ replaces the running statistics used by standard BN with statistics computed on the batch of inputs, which then serves as the context c. Thus, for ARM-BN, there is no context network, and $h$ has no parameters beyond the model parameters $\theta$ involved in computing BN statistics. The model $g$ is again specified via a prediction network $\mathrm { f _ { p r e d } }$ , which must have BN layers. BN typically tracks a running average of the first and second moments of the activations in these layers, which are then used at test time. ARM-BN defines $h$ such that it swaps out these moments for the moments computed via the activations on the test batch, thus giving us adapted parameters $\theta ^ { \prime }$ if we view the moments as part of the model parameters. This method is remarkably simple, and in deep learning libraries such as PyTorch $\mathbb { \left. \boldsymbol { \bar { 5 2 } } \right. }$ , implementing ARM-BN involves changing a single line of code. However, as shown in Section 5, this method also performs very well empirically, and the adaptation effectiveness is further boosted by meta-training.
+
+![](images/4ad1ef1b3fb03fae82f6ea01d66756eaf3c7dbbf0305e9efb4474eae17a454b3.jpg)  
+Figure 2: In the contextual approach (top), $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K }$ are summarized into a context c, and we propose two methods for this summarization, either through a separate context network or using batch normalization activations in the model itself. c can then be used by the model to infer additional information about the input distribution. In the gradient based approach (bottom), an unlabeled loss function $\mathcal { L }$ is used for gradient updates to the model parameters, in order to produce parameters that are specialized to the test inputs and can produce more accurate predictions.
+
+ARM-LL. ARM-LL, depicted in Figure 2 (bottom), follows the gradient based meta-learning paradigm $\mathbb { \lVert 1 8 \rVert }$ and learns parameters $\theta$ that are amenable to gradient updates on a loss function in order to quickly adapt to a new problem. In other words, $h$ produces $\theta ^ { \prime } \bar { = } \theta - \alpha \nabla _ { \theta } \mathcal { L } ( \theta , \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { K } ; \phi )$ , where $\alpha$ is a hyperparameter. Note that the loss function $\mathcal { L }$ used in the gradient updates is different from the original supervised loss function $\ell$ , in that it operates on only the inputs $\mathbf { x }$ , rather than the input output pairs that $\ell$ receives. We follow the general implementation of this approach proposed in Yu et al. $\pmb { \| 8 0 \| }$ . We define $g$ to produce output features $\mathbf { o } \in \mathbb { R } ^ { | \mathcal { V } | }$ that are used as logits when making predictions. We then define the unlabeled loss function $\mathcal { L }$ to be the composition of $g$ and a loss network $\mathsf { f } _ { \mathrm { l o s s } } ( \cdot ; \boldsymbol { \phi } ) : \mathbb { R } ^ { | \mathcal { V } | }  \mathbb { R }$ , which takes in the output features from $g$ and produces a scalar. We use the $\ell _ { 2 }$ -norm of these scalars across the batch of inputs as the loss for updating $\theta$ . In other words, $h ( \theta , \mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { K } ; \phi ) = \theta - \alpha \nabla _ { \theta } \| \mathbf { v } \| _ { 2 }$ , where $\mathbf { v } = [ \mathbf { f } _ { \mathrm { l o s s } } ( g ( \mathbf { x } _ { 1 } ; \theta ) ; \phi ) , \dots , \mathbf { f } _ { \mathrm { l o s s } } ( g ( \mathbf { x } _ { K } ; \theta ) ; \phi ) ]$ .
+
+# 5 Experiments
+
+Our experiments are designed to answer the following questions:
+
+1. Do ARM methods learn models that can leverage unlabeled adaptation to tackle domain shift?   
+2. How do ARM methods compare to prior methods for robustness, invariance, and adaptation?   
+3. Can models trained via ARM methods adapt successfully in the streaming test setting?
+
+# 5.1 Evaluation domains and protocol
+
+We propose four image classification problems, which we present below and describe in full detail in Appendix D. We also present results on datasets from the WILDS benchmark [35] in subsection 5.4.
+
+We believe that the problems we propose in this paper can supplement existing benchmarks for domain shift, which, as discussed above, are designed to test invariances. A key characteristic of the problems presented here is the potential for adaptation to improve test performance, and this differs from prior benchmarks such as the problems compiled by DomainBed [23]. In Appendix C, we compare our testbeds to DomainBed and group robustness benchmarks, and we briefly discuss the results in Gulrajani and Lopez-Paz $\pmb { \left. \pmb { \left. \bar { 2 . 3 } \right. } \right. }$ , which also evaluate ARM-CML.
+
+Rotated MNIST. We study a modified version of MNIST where images are rotated in 10 degree increments, from 0 to 130 degrees. We treat each rotation as a separate domain, i.e., a different value of $z$ . We use only 108 training data points for each of the 2 smallest domains (120 and 130 degrees), and 324 points each for rotations 90 to 110, whereas the overall training set contains 32292 points. In this setting, we hypothesize that adaptation can specialize the model to specific domains, in particular the rare domains in the training set. For each test evaluation, we generate images from the MNIST test set with a certain rotation. We measure both worst case and average accuracy across domains.
+
+Federated Extended MNIST (FEMNIST). The extended MNIST (EMNIST) dataset consists of images of handwritten uppercase and lowercase letters, in addition to digits [12]. FEMNIST is the same dataset, but it also provides the meta-data of which user generated each data point [9]. We treat each user as a domain. We measure each method’s worst case and average accuracy across 35 test users, which are held out and thus disjoint from the training users. As discussed in Section 1, adaptation may help for this problem for specializing the model and resolving ambiguous data points.
+
+Corrupted image datasets. CIFAR-10-C and Tiny ImageNet-C $\lVert 2 5 \rVert$ augment the CIFAR-10 [36] and Tiny ImageNet test sets with common image corruptions that vary in type and severity. The original goal of these augmented test sets was to benchmark how well methods could handle these corruptions without access to any corruptions during training $\mathbb { \left[ \left[ 2 5 \right] \right] }$ . Thus, successful methods for these problems typically have relied on domain knowledge and heuristics designed specifically for image classification. For example, prior work has shown that carefully designed test time adaptation procedures are effective for these problems [69, 63, 75]. One possible reason for this phenomenon is that convolutional networks are biased toward texture $\scriptstyle { \left[ \left[ 2 1 \right] \right] }$ , which is distorted by corruptions, thus adaptation can help the model recover its performance for each corruption type.
+
+We study whether meta-training for adaptation performance can improve upon these results. To do so, we modify the protocol from Hendrycks and Dietterich $\mathbb { \left[ \left[ 2 5 \right] \right] }$ to fit into the ARM problem setting by applying a set of 56 corruptions to the training data, and we define each corruption to be a domain. We use a disjoint set of 22 corruptions for the test data, which are mostly of different types from the training corruptions (thus, not sampled identically), and we measure worst case and average accuracy across the test corruptions. This modification allows us to study, for both ARM and prior methods, whether seeing corruptions at training time can help the model deal with new corruptions at test time.
+
+# 5.2 Comparisons and ablations
+
+We compare the ARM methods against several prior methods designed for robustness, invariance, and adaptation. We describe the comparisons here and provide additional details in Appendix D.
+
+Test time adaptation. We evaluate the general approach of using test batches to compute BN statistics [43, 63, 33, 50], which we term BN adaptation. We also compare to test time training (TTT) $\mathbb { \left| \overline { { 6 9 } } \right| }$ which adapts the model at test time using a self-supervised rotation prediction loss. These methods have previously achieved strong results for image classification, likely because they constitute favorable inductive biases for improving on the true classification task [69].
+
+Ablations. We also include ablations of the ARM-CML and ARM-LL methods, which sample training batches of unlabeled examples uniformly from the entire training set, rather than sampling from a single domain.3 These “context ablation” and “learned loss ablation” are similar to test time adaptation methods in that they do not require training domains, thus they allow us to evaluate whether or not meta-training on domain shifts is important for improved performance.
+
+Group robustness and invariance. Sagawa et al. $\left[ \left[ 6 0 \right] \right]$ recently proposed a state-of-the-art method for group robustness, and we refer to this approach as distributionally robust neural networks (DRNN). Their work also evaluates a strong upweighting (UW) baseline that samples uniformly from each group, and so we also evaluate this approach in our experiments. Additionally, we compare to
+
+Table 1: Worst case (WC) and average (Avg) top 1 accuracy on all testbeds, where means and standard errors are reported across three separate runs of each method. Horizontal lines separate methods that make use of (from top to bottom): neither, training domains, test batches, or both. ARM methods consistently achieve greater robustness, measured by WC, and Avg performance compared to prior methods. $^ { * } \mathrm { U W }$ is identical to ERM for CIFAR-10-C and Tiny ImageNet-C.
+
+These results have been updated from an earlier version of the paper, primarily for CIFAR10-C, due to significant refactoring of the code, additional hyperparameter tuning for both the ARM methods and the prior methods, and efforts to standardize results across the authors’ different computing environments and library versions. These results are reproducible from the publicly available code: https://github.com/henrikmarklund/arm.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">MNIST</td><td colspan="2">FEMNIST</td><td colspan="2">CIFAR-10-C</td><td colspan="2">Tiny ImageNet-C</td></tr><tr><td>WC</td><td>Avg</td><td>WC</td><td>Avg</td><td>wC</td><td>Avg</td><td>wC</td><td>Avg</td></tr><tr><td>ERM</td><td>74.5 ± 1.4</td><td>93.6±0.4</td><td>62.4±0.4</td><td>79.1± 0.3</td><td>54.1± 0.3</td><td>70.4± 0.1</td><td>20.3±0.5</td><td>41.9 ± 0.1</td></tr><tr><td>UW*</td><td>80.3 ±1.2</td><td>95.1±0.1</td><td>65.7 ± 0.7</td><td>80.3±0.6</td><td></td><td></td><td></td><td></td></tr><tr><td>DRNN</td><td>79.9 ± 0.7</td><td>94.9 ±0.1</td><td>57.5 ± 1.7</td><td>76.5 ± 1.2</td><td>49.3± 0.9</td><td>65.7 ± 0.5</td><td>14.2 ± 0.2</td><td>31.6 ±1.0</td></tr><tr><td>DANN</td><td>78.8±0.8</td><td>94.9 ± 0.1</td><td>65.4 ± 1.0</td><td>81.7 ± 0.3</td><td>53.9±2.2</td><td>69.8±0.3</td><td>20.4± 0.7</td><td>40.9±0.2</td></tr><tr><td>MMD</td><td>82.4±0.9</td><td>95.3± 0.3</td><td>62.4±0.7</td><td>79.8 ±0.4</td><td>52.2 ±0.3</td><td>69.5 ±0.1</td><td>19.7 ±0.2</td><td>40.1±0.1</td></tr><tr><td>BN adaptation</td><td>78.0±0.3</td><td>94.4± 0.1</td><td>65.7 ± 1.5</td><td>80.0 ±0.5</td><td>60.6±0.3</td><td>70.9 ± 0.1</td><td>26.5 ± 0.3</td><td>42.8 ±0.0</td></tr><tr><td>TTT</td><td>81.1±0.3</td><td>95.4±0.1</td><td>68.6 ±0.4</td><td>84.2±0.1</td><td>61.5 ± 0.3</td><td>71.7 ± 0.5</td><td>27.6± 0.5</td><td>37.7 ± 0.3</td></tr><tr><td>CML ablation</td><td>63.5 ± 1.8</td><td>90.1 ±0.2</td><td>61.8 ±0.8</td><td>81.6 ± 0.5</td><td>58.8 ±0.1</td><td>69.6±0.2</td><td>26.3±0.6</td><td>42.5 ± 0.1</td></tr><tr><td>LL ablation</td><td>79.9 ± 1.1</td><td>95.0± 0.3</td><td>64.1 ± 1.6</td><td>80.8 ±0.2</td><td>60.9 ± 0.4</td><td>71.3 ± 0.0</td><td>21.6 ± 2.1</td><td>32.6 ± 3.2</td></tr><tr><td>ARM-CML</td><td>88.0±0.8</td><td>96.3±0.4</td><td>70.9 ± 1.4</td><td>86.4��0.3</td><td>61.2 ± 0.4</td><td>70.3±0.2</td><td>29.1± 0.4</td><td>43.3 ±0.1</td></tr><tr><td>ARM-BN</td><td>83.3 ±0.5</td><td>95.6 ±0.1</td><td>64.5 ±3.2</td><td>83.2 ± 0.5</td><td>61.7 ± 0.3</td><td>72.4±0.3</td><td>28.3± 0.3</td><td>43.3± 0.1</td></tr><tr><td>ARM-LL</td><td>88.9±0.8</td><td>96.9±0.2</td><td>67.0±0.9</td><td>84.3±0.7</td><td>61.2±0.7</td><td>72.5± 0.4</td><td>25.4± 0.1</td><td>35.7 ±0.4</td></tr></table>
+
+domain adversarial neural networks (DANN) $\mathbb { \lVert 1 9 \rVert }$ and maximum mean discrepancy (MMD) feature learning [41], two state-of-the-art methods for adversarial learning of invariant predictive features. For the WILDS datasets, we include the numbers reported in Koh et al. $\pmb { \Vert 3 5 \Vert }$ for DRNN and two other invariance methods, correlation alignment (CORAL) $ { \mathbb { I } }  { \mathbb { I } }$ and invariant risk minimization (IRM) [3].
+
+Robustness and invariance methods assume access to training domains but not test batches, whereas adaptation methods assume the opposite. Thus, at a high level, we can view the comparisons to these methods as evaluating the importance of each of these assumptions for the specified problems.
+
+# 5.3 Quantitative evaluation and comparisons
+
+The results for the four proposed benchmarks are presented in Table 1. The best results, stratified by classes of methods, are bolded, with the single best result across all methods underlined. Across all of these problems, ARM methods increase both worst case and average accuracy compared to all other methods. ARM-CML performs well across all tasks, and despite its simplicity, ARM-BN achieves the best performance overall on the corrupted image testbeds, demonstrating the effectiveness of metatraining on top of an already strong adaptation procedure. BN adaptation and TTT are the strongest prior methods, as these adaptation procedures constitute inductive biases that are generally well suited for image classification. However, ARM methods are comparatively less reliant on favorable inductive biases and consistently attain better results. In general, we observe poor performance from robustness methods, varying performance from invariance methods, strong performance from adaptation methods, and the strongest performance from ARM methods.
+
+When we cannot access a batch of test points all at once, and instead the points are observed in a streaming fashion, we can augment the proposed ARM methods to perform sequential model updates. For example, ARM-CML and ARM-BN can update their average context and normalization statistics, respectively, after observing each new test point. In Figure 3, we study this test setting for the Tiny ImageNet-C problem. We see that both models trained with ARM-CML and ARM-BN are able to achieve near their original worst case and average accuracy within observing 50 data points, well before the training batch size of 100. This result demonstrates that ARM methods are applicable for problems where test points must be observed one at a time, provided that the model is permitted to adapt using each point. We describe in detail how each ARM method can be applied to the streaming setting in Appendix B, and we provide streaming results on rotated MNIST in Appendix E.
+
+![](images/a266c73291c4902cc98f6f0a173936b65146e6d4cf5179e742872fa1480eb8c1.jpg)  
+Figure 3: In the streaming setting, ARM methods reach strong performance on Tiny ImageNet-C after fewer than 50 data points, despite using training batch sizes of 100. This highlights that the trained models are able to adapt successfully in the standard streaming evaluation setting.
+
+Table 2: Results on the WILDS image testbeds. Different methods are best suited for different problems, motivating the need for a wide range of methods. ARM-BN struggles on FMoW but performs well on the other datasets, in particular RxRx1.   
+
+<table><tr><td></td><td colspan="2">iWildCam</td><td>Camelyon17</td><td>RxRx1</td><td colspan="2">FMoW</td><td colspan="2">PovertyMap</td></tr><tr><td>Method</td><td>Acc</td><td>Macro F1</td><td>Acc</td><td>Acc</td><td>WC Acc</td><td>Avg Acc</td><td>WC Pearson r</td><td>Pearson r</td></tr><tr><td>ERM</td><td>71.6 ± 2.5</td><td>31.0 ±1.3</td><td>70.3±6.4</td><td>29.9 ± 0.4</td><td>32.3±1.25</td><td>53.0±0.55</td><td>0.45±0.06</td><td>0.78±0.04</td></tr><tr><td>DRNN</td><td>72.7±2.0</td><td>23.9 ± 2.1</td><td>68.4± 7.3</td><td>23.0±0.3</td><td>30.8 ±0.81</td><td>52.1±0.5</td><td>0.39 ±0.06</td><td>0.75±0.07</td></tr><tr><td>CORAL</td><td>73.3±4.3</td><td>32.8±0.1</td><td>59.5± 7.7</td><td>28.4±0.3</td><td>31.7 ±1.24</td><td>50.5 ±0.36</td><td>0.44±0.06</td><td>0.78±0.05</td></tr><tr><td>IRM</td><td>59.8 ±3.7</td><td>15.1 ± 4.9</td><td>64.2± 8.1</td><td>8.2 ±1.1</td><td>30.0±1.37</td><td>50.8 ±0.13</td><td>0.43 ± 0.07</td><td>0.77± 0.05</td></tr><tr><td>BN adaptation</td><td>46.4± 1.0</td><td>13.8±0.3</td><td>88.6±1.4</td><td>20.0±0.2</td><td>30.2±0.26</td><td>51.6 ± 0.16</td><td>0.39 ±0.17</td><td>0.82±0.06</td></tr><tr><td>ARM-BN</td><td>70.3±2.4</td><td>23.2 ±2.7</td><td>87.2±0.9</td><td>31.2 ±0.1</td><td>24.6 ±0.04</td><td>42.0±0.21</td><td>0.49±0.21</td><td>0.84±0.05</td></tr></table>
+
+# 5.4 WILDS results
+
+Finally, we present results on the WILDS benchmark $\pmb { \Vert 3 5 \Vert }$ t Table 2. We evaluate BN adaptation and ARM-BN on these testbeds. We see that, on these real world distribution shift problems, different methods perform well for different problems. CORAL, a method for invariance $\mathbf { \widehat { \mathbf { \phi } } } [ \mathbf { \overline { { 6 7 } } } ] \mathbf { \xi }$ , performs best on the iWildCam animal classification problem [5], whereas no methods outperform ERM by a significant margin on the FMoW [11] or PovertyMap $\textcircled { 1 7 9 }$ satellite imagery problems. ARM-BN performs particularly poorly on the FMoW problem. However, it performs well on PovertyMap and significantly improves performance on the RxRx1 $\mathbb { \ m }$ problem of treatment classification from medical images. On the other medical imagery problem of Camelyon17 [4] tumor identification, adaptation in general boosts performance dramatically. These results indicate the need to consider a wide range of tools, including meta-learning and adaptation, for combating distribution shift.
+
+# 6 Discussion and Future Work
+
+We presented adaptive risk minimization (ARM), a framework and problem formulation for learning models that can adapt in the face of domain shift at test time using only a batch of unlabeled test examples. We devised an algorithm and instantiated a set of methods for optimizing the ARM objective that meta-learns models that are adaptable to different domains of training data. Empirically, we observed that ARM methods consistently improve performance in terms of both average and worst case metrics, as compared to a number of prior approaches for handling domain shift.
+
+Though we provided contextual meta-learning as a concrete example, a number of other meta-learning paradigms would also be interesting to extend to the ARM setting. For example, few shot generative modeling objectives would be a natural fit for unlabeled adaptation $\boxed { 1 6 } \boxed { 2 6 } \boxed { 7 7 }$ . Another exciting direction for future work is to explore the problem setting where domains are not provided at training time. As discussed in $\underline { { \mathrm { ~ A p p e n d i x ~ } } } \mathrm { ~ \bar { E } ~ }$ in this setting, we can instead construct domains via unsupervised learning techniques. Similar to Hsu et al. $\pmb { \left. 2 8 \right. }$ , one promising approach is to generate a diverse set of domains in order to learn generally effective adaptation strategies. Robustness and invariance methods cannot be used easily with multiple different groupings, learned or otherwise, as techniques such as group weighted loss functions $\dot { \left\| 6 0 \right\| }$ and domain classifiers $\mathbb { \lVert 1 9 \rVert }$ are not immediately extendable to this setup. Thus, ARM methods may be uniquely suited to be paired with domain learning.
+
+# Acknowledgments and Disclosure of Funding
+
+MZ thanks Matt Johnson and Sharad Vikram for helpful discussions and was supported by an NDSEG fellowship. HM is funded by a scholarship from the Dr. Tech. Marcus Wallenberg Foundation for Education in International Industrial Entrepreneurship. AG was supported by an NSF graduate research fellowship. CF is a CIFAR Fellow in the Learning in Machines and Brains program. This research was supported by the DARPA Assured Autonomy and Learning with Less Labels programs.
+
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parse/train/-zgb2v8vV_w/-zgb2v8vV_w_content_list.json

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+    {
+        "type": "text",
+        "text": "Adaptive Risk Minimization: Learning to Adapt to Domain Shift ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Marvin Zhang⇤1, Henrik Marklund⇤2, Nikita Dhawan⇤1, Abhishek Gupta1, Sergey Levine1, Chelsea Finn2 ",
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+    {
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+        "text": "1 UC Berkeley, 2 Stanford University ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "A fundamental assumption of most machine learning algorithms is that the training and test data are drawn from the same underlying distribution. However, this assumption is violated in almost all practical applications: machine learning systems are regularly tested under distribution shift, due to changing temporal correlations, atypical end users, or other factors. In this work, we consider the problem setting of domain generalization, where the training data are structured into domains and there may be multiple test time shifts, corresponding to new domains or domain distributions. Most prior methods aim to learn a single robust model or invariant feature space that performs well on all domains. In contrast, we aim to learn models that adapt at test time to domain shift using unlabeled test points. Our primary contribution is to introduce the framework of adaptive risk minimization (ARM), in which models are directly optimized for effective adaptation to shift by learning to adapt on the training domains. Compared to prior methods for robustness, invariance, and adaptation, ARM methods provide performance gains of $1 - 4 \\%$ test accuracy on a number of image classification problems exhibiting domain shift. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "The standard assumption in empirical risk minimization (ERM) is that the data distribution at test time will match the training distribution. When this assumption does not hold, i.e., when there is distribution shift, the performance of standard ERM methods can deteriorate significantly [54, 38]. ",
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+        "text": "As an example which we study quantitatively in Section 5, consider a handwriting classification model that, after training on data from past users, is deployed to new end users. Each new user represents a new test distribution that differs from the training distribution. Thus, each test setting involves dealing with shift. In Figure 1, we visualize a batch of 50 examples from a test user, and we highlight an ambiguous example which may be either a “2” (written with a loop) or an “a” (in the double-storey style) depending on the user’s handwriting. Due to the biases in the training data, an ERM trained model incorrectly classifies this example as “2”. However, we can see that the batch of images from this test user contains other examples of “2” (written without loops) and “a” (also double-storey) from this user. Can we somehow leverage this unlabeled data to better handle test shifts caused by new users? ",
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+        "image_caption": [
+            "Figure 1: An example of ambiguous data points in handwriting classification, evaluated quantitatively in Section 5. "
+        ],
+        "image_footnote": [],
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+        "text": "Any framework that aims to address this question must use additional assumptions beyond the ERM setting, and many such frameworks have been proposed $\\pmb { \\Vert 5 4 \\Vert }$ . One commonly used assumption within several frameworks, such as domain generalization [7, 23], is that the training data are provided in domains and distributions at test time will represent new domains. The example above neatly fits this description if we equate users with domains – we would be assuming that the training data are organized by users and that the model will be tested separately on new users, and these are reasonable assumptions. Constructing training domains in practice is generally accomplished by using meta-data, which exists for many commonly used datasets. Thus, this domain assumption is applicable for a wide range of realistic distribution shift problems (see, e.g., Koh et al. [35]). ",
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+    {
+        "type": "text",
+        "text": "However, prior benchmarks for domain generalization and similar settings typically center around invariances – i.e., in these benchmarks, there is a consistent input-output relationship across all domains, and the goal is to learn this relationship while ignoring the spurious correlations within the domains (see, e.g., Gulrajani and Lopez-Paz $[ \\overbar { 1 2 3 } ] \\cdot$ ). Thus, prior methods aim for generalization to shifts by discovering this relationship, through techniques such as robust optimization and learning an invariant feature space [41, 3, 60]. These methods are appealing in that they make minimal assumptions about the information provided at test time – in particular, they do not require test labels, and the learned model can be immediately applied to predict on a single point. Nevertheless, these methods also have limitations, such as in dealing with problems where the input-output relationship varies across domains, e.g., the handwriting classification example above. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we instead focus on methods that aim to adapt at test time to domain shift. To do so, we study problems in which it is both feasible and helpful (and perhaps even necessary) to assume access to a batch or stream of inputs at test time. Leveraging this test assumption does not require labels for any test data and is feasible in many practical setups. For example, for handwriting classification, we do not access only single handwritten characters from an end user, but rather collections of characters such as sentences or paragraphs. Unlabeled adaptation has been shown empirically to be useful for distribution shift problems $[ \\dot { \\overline { { 6 9 } } } , \\overline { { 6 3 } } , \\overline { { 7 5 } } ]$ , such as for dealing with image corruptions $\\mathbb { \\left[ \\left. 2 5 \\right] \\right. }$ . Taking inspiration from these findings, we propose and evaluate on a number of problems, detailed in Section 5, for which adaptation is beneficial in dealing with domain shift. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our main contribution is to introduce the framework of adaptive risk minimization (ARM), which proposes the following objective: optimize the model such that it can maximally leverage the unlabeled adaptation phase to handle domain shift. To do so, we instantiate a set of methods that, given a set of training domains, meta-learns a model that is adaptable to these domains. These methods are straightforward extensions of existing meta-learning approaches, thereby demonstrating that tools from the meta-learning toolkit can be readily adapted to tackle domain shift. Our experiments in Section 5 test on several image classification problems, derived from benchmarks for federated learning [9] and image classifier robustness $\\mathbb { \\lVert 2 5 \\rVert }$ , in which training and test domains share structure that can be leveraged for improved performance. These testbeds are also a contribution of our work, as we believe these problems can supplement existing benchmarks which, almost exclusively, are designed with invariance in mind $[ \\sqrt { 3 } , \\sqrt { 5 3 } , \\sqrt { 2 3 } ]$ . We also evaluate on the WILDS suite of distribution shift problems $\\begin{array} { r l } {  { \\bigl [ \\bigl | 3 5 \\bigr | \\bigr ] } } & { { } } \\end{array}$ , which have been curated to faithfully represent important real world problems. Empirically, we demonstrate that the proposed ARM methods, by leveraging meta-training and test time adaptation, are often able to outperform prior state-of-the-art methods by $1 \\%$ test accuracy. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "A number of prior works have studied distribution shift in various forms $[ [ 5 4 ]$ . In this section, we review prior work in domain generalization, group robustness, meta-learning, and adaptation. ",
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+        "type": "text",
+        "text": "Invariance and robustness to domains. As discussed above, a number of frameworks leverage training domains to address test time shift. The terminology in prior work is scattered and, depending on the application, includes terms such as “groups”, “datasets”, “subpopulations”, and “users”; in this work, we adopt the term “domains” which we believe is an appropriate unifying term. A number of testbeds for this problem setting have been proposed for image classification, including generalizing to new datasets $\\sharp$ , new image types $\\pm \\boxed { 1 3 9 } \\boxed { 5 3 } \\cdots$ , and underrepresented demographics $\\bar { \\mathbb { E O } } \\mathbb { I }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prior benchmarks typically assume the existence of a consistent input-output relationship across domains that is learnable by the specified model, thus motivating methods such as learning an invariant feature space [41, 44, 3] or optimizing for worst case group performance $\\pmb { \\mathbb { B O } } \\pmb { \\mathbb { G O } } \\mathbf { j }$ . In particular, methods for domain generalization – sometimes referred to as multi-source domain adaptation $\\lVert \\rVert$ or zero shot domain adaptation $\\left[ \\left[ 7 8 \\right] \\right]$ – have largely focused on learning invariant features [19, 67, 41, 44, 53]. Gulrajani and Lopez-Paz $\\pmb { \\left. \\pmb { \\left. \\mathscr { Z } 3 \\right. } \\right. }$ provide a comprehensive survey of domain generalization benchmarks and find that, surprisingly, ERM is competitive with the state of the art across all the benchmarks considered. In Appendix C, we discuss this finding as well as the performance of an ARM method on this benchmark suite. In Section 5, we identify different problems for which adaptation is helpful, and we find that, on these problems, ARM methods consistently outperform ERM and other non adaptive methods for robustness and invariance. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Meta-learning. Meta-learning [62, 6, 71, 27] has been most extensively studied in the context of few shot labeled adaptation [61, 74, 55, 18, 65]. Our aim is not to address few-shot recognition problems, nor to propose a novel meta-learning algorithm, but rather to extend meta-learning paradigms to problems requiring unlabeled adaptation, with the primary goal of tackling distribution shift. This aim differs from previous work in meta-learning for domain generalization $[ \\bar { 4 0 } , \\bar { 1 } 5 ]$ , which seek to metatrain models for non adaptive generalization performance. We discuss in Section 4 how paradigms such as contextual meta-learning [20, 57] can be readily extended using the ARM framework. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Some other meta-learning methods adapt using both labeled and unlabeled data, either in the semi supervised learning setting [56, 81, 42] or the transductive learning setting [51, 46, 2, 29]. These works all assume access to labeled data for adaptation, whereas we propose methods and problems for purely unlabeled adaptation. Prior works in meta-learning for unlabeled adaptation include Yu et al. $\\mathbf { [ 8 0 ] }$ , who adapt a policy to imitate human demonstrations in the context of robotic learning; Metz et al. $\\lVert \\rVert \\mathbf { 4 8 } \\rVert$ , who meta-learn an update rule for unsupervised representation learning, though they still require labels to learn a predictive model; and Alet et al. [1], who meta-learn adaptive models based on task specific unsupervised objectives. Unlike these prior works, we propose a general framework for tackling distribution shift problems by meta-learning unsupervised adaptation strategies. This framework simplifies the extension of meta-learning paradigms to these problems, encapsulates previous approaches such as the gradient based meta-learning approach of Yu et al. $\\pmb { \\| 8 0 \\| }$ , and sheds light on how to improve existing strategies such as adaptation via batch normalization [43]. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Adaptation to shift. Unlabeled adaptation has primarily been studied separately from meta-learning. Domain adaptation is a prominent framework that assumes access to test examples at training time $\\boxed { 1 3 } , \\boxed { 7 6 }$ , similar to transductive learning $\\mathbb { | \\overline { {  { | Z 3 \\| } } } | }$ . As such, most domain adaptation methods consider the problem setting where there is a single test distribution $\\lVert 6 4 \\rVert , \\lVert 4 \\rVert , \\lVert 2 2 \\rVert , \\dot { \\lVert 1 9 \\rVert } , \\lVert 7 2 \\rVert , \\lVert 0 \\rVert$ , and some of these methods are difficult to apply to problems where there are multiple test distributions. Certain domain adaptation methods have also been applied in the domain generalization setting, such as methods for learning invariant features [19, 67, 41], and we compare to these methods in Section 5. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Adaptive methods for domain generalization include Muandet et al. $\\mathbb { \\oplus 2 }$ and Kumagai and Iwata $\\pmb { \\Vert 3 7 } \\Vert$ who propose a method similar to one of the ARM methods described below. We compare to a version of this method in Appendix E. Blanchard et al. [7] and Blanchard et al. $\\pmb { \\mathbb { B } } \\|$ provide a theoretic study of domain generalization and establish favorable generalization bounds for models that can adapt to domain shift at test time. We summarize some of these results in $\\mathsf { S e c t i o n } \\ 3 .$ In comparison, our work establishes a framework that makes explicit the connection between adaptation to domain shift and meta-learning, allowing us to devise new methods in a straightforward and principled manner. These methods are amenable to expressive models such as deep neural networks, which enables us to propose and evaluate on problems with raw image observations. ",
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+    {
+        "type": "text",
+        "text": "Test time adaptation has also been studied for dealing with label shift $\\mathbb { \\lVert 5 9 \\rVert \\lVert 5 \\rVert \\lVert 6 6 \\rVert }$ and crafting favorable inductive biases for the domain of interest. For image classification, techniques such as normalizing via the test inputs $\\mathbb { E 3 }$ and optimizing self-supervised surrogate losses $\\lVert \\boldsymbol { 6 9 } \\rVert$ have proven effective for adapting to image corruptions $\\pmb {  \\pmb {  \\bar { 2 } 5  } }$ . We compare to these prior methods in Section 5 and empirically demonstrate the advantage of using training domains to learn how to adapt. ",
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+    {
+        "type": "text",
+        "text": "3 Preliminaries and Notation ",
+        "text_level": 1,
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+        "text": "In this section, we discuss the domain generalization problem setting and formally describe adaptive models. In Section 4, we discuss how adaptive models can be meta-trained via the ARM objective and approach, and we instantiate ARM methods which we empirically evaluate in Section 5. ",
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+        "type": "text",
+        "text": "Let $\\mathbf { x } \\in \\mathcal { X }$ and $y \\in \\mathcal { V }$ represent the input and output, respectively. We can formalize the domain generalization problem setting using the following data generation process $\\textcircled { 8 } \\textcircled { 1 8 }$ : first, a joint data distribution $p _ { \\mathbf { x } y }$ is sampled from a set of distributions $\\mathcal { P } _ { \\mathbf { x } y }$ , and then some data points are sampled from $p _ { \\mathbf { x } y } \\underline { { \\mathbb { I } } }$ We refer to each $p _ { \\mathbf { x } y }$ as a domain, e.g., a particular dataset or user, thus $\\mathcal { P } _ { \\mathbf { x } y }$ represents the set of all possible domains. We assume that the training dataset is composed of data from $S$ runs of this generative process, organized by domain. An equivalent characterization which we will use for clarity is that, within the training set, there are $S$ domains, and each data point $( \\mathbf { x } ^ { ( i ) } , y ^ { ( i ) } )$ is annotated with a domain label $z ^ { ( i ) }$ . Each $z ^ { ( i ) }$ is an integer that takes on a value between 1 and $S$ , indicating which $p _ { \\mathbf { x } y }$ generated the $i$ -th training point (though, of course, we do not have access to, or knowledge of, $p _ { \\mathbf { x } y }$ itself). At test time, there may be multiple evaluation settings, where each setting is considered separately and contains only unlabeled data sampled via a new run of the same generative process. This data may represent, e.g., a new dataset or user, and the test domains are likely to be distinct from the training domains when $| \\mathcal { P } _ { { \\bf x } y } |$ is large or infinite. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our formal goal is to optimize for expected performance, e.g., classification accuracy, at test time. To do so, let us first consider predictive models of the form $f : \\mathcal { X } \\times \\mathcal { P } _ { \\mathbf { x } }  \\mathcal { Y }$ , where the model $f$ takes in not just an input $\\mathbf { x }$ but also the marginal input distribution $p _ { \\mathbf { x } } \\in \\mathcal { P } _ { \\mathbf { x } }$ that $\\mathbf { x }$ was sampled from. We refer to $f$ as an adaptive model, as it has the opportunity to use $p _ { \\mathbf { x } }$ to adapt its predictions on $\\mathbf { x }$ . The underlying assumption is that $p _ { \\mathbf { x } }$ provides information about $p _ { y | \\mathbf { x } }$ , i.e., $p _ { \\mathbf { x } }$ is used as a surrogate input in place of $p _ { \\mathbf { x } y }$ . In the worst case, if $p _ { \\mathbf { x } }$ and $p _ { y | \\mathbf { x } }$ are sampled independently, then the model does not benefit at all from knowing $p _ { \\mathbf { x } }$ . In many problems, however, we expect knowledge about $p _ { \\mathbf { x } }$ to be useful, e.g., for resolving ambiguity as in the handwriting classification example in Section 1. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Theoretically, when $p _ { \\mathbf { x } }$ provides information about $p _ { y | \\mathbf { x } }$ , and when training and test domains are drawn from the same distribution over $\\mathcal { P } _ { \\mathbf { x } y }$ , we can establish favorable generalization bounds for the expected performance of $f$ in adapting to domain shift at test time. We can formalize this as follows. First, define a prediction model to be a non adaptive model of the form $g : \\mathcal { X }  \\mathcal { Y }$ , and define the risk for a prediction model $g$ and loss function $\\ell$ , under a data distribution $p _ { \\mathbf { x } y }$ , as ",
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+        "type": "equation",
+        "img_path": "images/6c98e0e8479e0a863e5f86e0338c69278663b14fc2ecaeaecbed903bbfc4bf9b.jpg",
+        "text": "$$\n\\mathcal { R } ( g , p _ { \\mathbf { x } y } ) \\triangleq \\mathbb { E } _ { p _ { \\mathbf { x } y } } \\left[ \\ell ( g ( \\mathbf { x } ) , y ) \\right] .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Further, define the Bayes optimal risk for $\\ell$ under $p _ { \\mathbf { x } y }$ as ",
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+        "type": "equation",
+        "img_path": "images/38b06aaf98c8bcb6ae0f84d464dab95d1a9eebc34841ac6fbc841bf294fc27c2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { R } ^ { \\star } ( p _ { \\mathbf { x } y } ) \\triangleq \\underset { g } { \\operatorname* { m i n } } \\mathcal { R } ( g , p _ { \\mathbf { x } y } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $\\mu$ denote the distribution on $\\mathcal { P } _ { \\mathbf { x } y }$ from which training and test domains $p _ { \\mathbf { x } y }$ are sampled. To avoid overlapping terms, define the adaptive risk for an adaptive model $f$ and $\\ell$ , under $\\mu$ , to be ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/13eb1b719928b8100843f52c1b6f48b0ed3c7b8506789521ab58de95f64b5f21.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { E } ( f , \\mu ) \\triangleq \\mathbb { E } _ { \\mu } \\left[ \\mathbb { E } _ { p _ { \\mathbf { x } _ { y } } } \\left[ \\ell ( f ( \\mathbf { x } , p _ { \\mathbf { x } } ) , y ) \\right] \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We state the following result from Blanchard et al. $\\pmb { \\mathbb { B } } ] \\mathbf  l$ , which details a condition on $\\mu$ under which $\\mathcal { E }$ is a strongly principled objective for learning adaptive models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lemma 9 from Blanchard et al. [8]. Let $f ^ { \\star }$ denote a minimizer of $\\mathcal { E }$ for the given $\\mu$ . If $\\mu$ is a distribution on $\\mathcal { P } _ { \\mathbf { x } y }$ such that $\\mu$ -almost surely it holds that $p _ { y | \\mathbf { x } } = M ( p _ { \\mathbf { x } } )$ for some deterministic mapping $M$ , then for $\\mu$ -almost all $p _ { \\mathbf { x } y }$ , we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/63991dc40dbe5a93043db9e911643f892f6c5b24dd7de807b4883a08960b35b6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { R } \\big ( f ^ { \\star } \\big ( \\cdot , p _ { \\mathbf { x } } \\big ) , p _ { \\mathbf { x } y } \\big ) = \\mathcal { R } ^ { \\star } \\big ( p _ { \\mathbf { x } y } \\big ) \\implies \\mathcal { E } \\big ( f ^ { \\star } , \\mu \\big ) = \\mathbb { E } _ { \\mu } \\left[ \\mathcal { R } ^ { \\star } ( p _ { \\mathbf { x } y } ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In other words, an adaptive model which minimizes the adaptive risk $\\mathcal { E }$ coincides with a Bayes optimal decision function for $p _ { \\mathbf { x } y }$ , for $\\mu$ -almost all domains $p _ { \\mathbf { x } y }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark. The required condition on $\\mu -$ that $p _ { y | \\mathbf { x } }$ is determined by $p _ { \\mathbf { x } }$ – holds if, and only if, an expert (or oracle) is able to correctly label inputs from a given domain provided only information about the input distribution. This condition holds for the testbeds proposed in this paper, those in Gulrajani and Lopez-Paz $\\mathbb { \\left. 2 3 \\right. }$ , and those in WILDS $\\mathbb { \\lVert 3 5 \\rVert }$ . The condition does not hold for, e.g., standard few shot learning testbeds, where it is possible for two domains with identical input distributions to shuffle their label orderings differently $\\pmb { \\hat { \\mathbb { Z } } } \\pmb { \\mathbb { \\| } }$ . Thus, these problems are outside the scope of this work. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "This result provides strong justification for learning adaptive models $f$ by minimizing the adaptive risk $\\mathcal { E }$ . However, a practical instantiation of this approach requires some approximations. First, we do not know and cannot input $p _ { \\mathbf { x } }$ to $f$ in most cases. Instead, we instantiate $f$ such that it takes in a batch of inputs $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K }$ , all from the same domain, where $K$ can vary. $f$ makes predictions on the whole batch, which also serves as an empirical approximation (i.e., a histogram) $\\hat { p } _ { \\bf x }$ of $p _ { \\mathbf { x } } \\mathinner { \\| { \\boldsymbol { \\mathrm { B } } } \\| }$ . In our exposition, we will assume that a batch of unlabeled points is available at test time for adaptation. However, we also experiment in $\\mathtt { S e c t i o n } 5$ with the streaming setting where the test inputs are observed one at a time and adaptation occurs incrementally. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Notice that, if we instead passed in an approximation $\\hat { p } _ { { \\bf x } y }$ of $p _ { \\mathbf { x } y }$ to the model, such as a batch of labeled data $( { \\bf x } _ { 1 } , y _ { 1 } ) , \\dots , ( { \\bf x } _ { K } , y _ { K } )$ , then this setup would resemble the standard few shot metalearning problem $\\pmb { \\Vert 7 4 \\Vert }$ . Formally, a meta-learning model takes in both an input $\\mathbf { x }$ and $\\hat { p } _ { { \\bf x } y }$ , which approximates the distribution that $\\mathbf { x }$ was sampled from and thus can be used to adapt the prediction on $\\mathbf { x }$ . Compared to our problem setting, the meta-learning formalism can tackle a wider range of problems but also requires more restrictive assumptions, specifically, labels at test time via $\\hat { p } _ { { \\bf x } y }$ . Transductive meta-learning methods further assume that, in addition to $\\hat { p } _ { { \\bf x } y }$ , a full batch of inputs $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K }$ is passed into the model, which allows for better estimation of the input distribution $p _ { \\mathbf { x } }$ [51, 46, 29]. The model then makes predictions on this entire batch. In meta-learning terminology, $\\hat { p } _ { { \\bf x } y }$ and $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K }$ are often referred to as the support and query, respectively. Therefore, another interpretation of the adaptive models that we study in this work is that they resemble transductive meta-learning models, but they are given only the unlabeled query and not the labeled support set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the next section, we expand on this connection to develop the ARM framework, which then allows us to bring forward tools from meta-learning to tackle domain shift problems. ",
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+    {
+        "type": "text",
+        "text": "4 Adaptive Risk Minimization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In this section, we formally describe the ARM framework, which defines an objective for training adaptive models to tackle domain shift. Furthermore, we propose a general meta-learning algorithm as well as specific methods for optimizing the ARM objective. In Section 5, we test these ARM methods on problems for which unlabeled adaptation can be leveraged for better test performance. ",
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+    {
+        "type": "text",
+        "text": "4.1 Devising the ARM objective ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We wish to learn an adaptive model $f : \\mathcal { X } ^ { K } \\to \\mathcal { Y } ^ { K }$ to tackle domain shift. As noted, meta-learning methods for labeled adaptation study a similar form of model, and a common approach in many of these methods is to define $f$ such that it is composed of two parts: first, a learner which ingests the data and produces parameters, and second, a prediction model which uses these parameters to make predictions $\\boxed { 7 4 } , \\boxed { 1 8 } \\vert$ . We will follow a similar strategy which, as we will discuss in subsection 4.2, allows us to easily extend and design meta-learning methods towards our goal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In particular, we will decompose the model $f$ into two modules: a standard prediction model $g ( \\cdot ; \\theta ) : \\mathcal { X } \\to \\mathcal { Y } _ { }$ , that is parameterized by $\\theta \\in \\Theta$ and predicts $y$ given $\\mathbf { x }$ , and an adaptation model $h ( \\cdot , \\cdot ; \\phi ) : \\ \\Theta \\times \\mathcal { X } ^ { K }  \\Theta$ , which is parameterized by $\\phi$ . $h$ takes in the prediction model parameters $\\theta$ and $K$ unlabeled data points and uses the $K$ points to produce adapted parameters $\\theta ^ { \\prime }$ . This is analogous to the learner in meta-learning, however, $h$ adapts the model parameters using only unlabeled data. We defer the discussion of how to instantiate $h$ to subsection 4.2.2 ",
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+    {
+        "type": "text",
+        "text": "The ARM objective is to optimize $\\phi$ and $\\theta$ such that $h$ can adapt $g$ using unlabeled data sampled according to a particular domain $z$ . This can be expressed as the optimization problem ",
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+    {
+        "type": "equation",
+        "img_path": "images/082258130afc6e80b153202408aee336f2207f0ed01d624e300ec79b022b5f98.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta , \\phi } \\hat { \\mathcal { E } } ( \\theta , \\phi ) = \\mathbb { E } _ { p _ { z } } \\left[ \\mathbb { E } _ { p _ { \\mathbf { x } \\cdot p } | z } \\left[ \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\ell ( g ( \\mathbf { x } _ { k } ; \\theta ^ { \\prime } ) , y _ { k } ) \\right] \\right] , \\mathrm { ~ w h e r e ~ } \\theta ^ { \\prime } = h ( \\theta , \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K } ; \\phi ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\hat { \\mathcal { E } }$ is the empirical form of the adaptive risk in $\\mathrm { E q u a t i o n } 1$ for the form of $f$ we have defined. Mimicking the generative process from Section 3 that we assume generated the training data, $p _ { z }$ is a categorical distribution over $\\{ 1 , \\ldots , S \\}$ which places uniform probability mass on each training domain, and $p _ { \\mathbf { x } y | z }$ assigns uniform probability to only the training points within a particular domain. As we have established theoretically, we expect the trained models to perform well at test time if the test domains are sampled independently and identically – i.e., from the same distribution over $\\mathcal { P } _ { \\mathbf { x } y } -$ as the training domains. In practice, similar to how meta-learned few shot classification models are evaluated on new and unseen meta-test classes $\\textcircled { 7 4 } , \\textcircled { 1 8 } \\textcircled { }$ , we empirically show in Section 5 that the trained models can generalize to test domains that are not sampled identically to the training domains. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 Optimizing the ARM objective ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1 presents a general meta-learning approach for optimizing the ARM objective. As described above, $h$ outputs updated parameters $\\theta ^ { \\prime }$ using an unlabeled batch of data (line 5). This mimics the adaptation procedure at test time, where we do not assume access to labels (lines 7-8). However, the training update itself does rely on the labels (line 6). We assume that $h$ is differentiable with respect to its input $\\theta$ and $\\phi$ , thus we use gradient updates on both $\\theta$ and $\\phi$ to optimize for post adaptation performance on a mini batch of data sampled according to a particular domain $z$ . In practice, we also sample mini batches of domains, rather than just one domain (as written in line 3), to provide a better gradient signal for optimizing $\\phi$ and $\\theta$ ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Meta-Learning for ARM ",
+        "text_level": 1,
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "// Training procedure ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: # training steps $T$ , batch size $K$ , learning rate $\\eta$ ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "1: Initialize: $\\theta , \\phi$   \n2: for $t = 1 , \\dots , T$ do   \n3: Sample $z$ uniformly from training domains   \n4: Sample $( \\mathbf { x } _ { k } , y _ { k } ) \\sim p ( \\cdot , \\cdot | z )$ for $k = 1 , \\ldots , K$   \n5: $\\theta ^ { \\prime } \\gets h ( \\theta , \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K } ; \\phi )$   \n6: $\\begin{array} { r } { ( \\theta , \\phi )  ( \\theta , \\phi ) - \\eta \\nabla _ { ( \\theta , \\phi ) } \\sum _ { k = 1 } ^ { K } \\ell ( g ( \\mathbf { x } _ { k } ; \\theta ^ { \\prime } ) , y _ { k } ) } \\end{array}$ ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "// Test time adaptation procedure ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Require: $\\theta , \\phi$ , test batch $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K }$ 7: $\\theta ^ { \\prime } \\gets h ( \\theta , \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K } ; \\phi )$ 8: $\\hat { y } _ { k } \\gets g ( \\mathbf x _ { k } ; \\theta ^ { \\prime } )$ for $k = 1 , \\ldots , K$ ",
+        "bbox": [
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+        "type": "text",
+        "text": "Together, Equation 2 and Algorithm 1 shed light on a number of ways to devise methods for solving the ARM problem. First, we can extend meta-learning paradigms to the ARM problem setting, and any paradigm in which the adaptation model $h$ can be augmented to operate on unlabeled data is readily applicable. As an example, we propose the ARM-CML method, which is inspired by recent works in contextual meta-learning (CML) [20, 57]. Second, we can enhance prior unlabeled adaptation methods by incorporating a meta-training phase that allows the model to better leverage the adaptation. To this end, we propose the ARM-BN method, based on the general approach of adapting using batch normalization (BN) statistics of the test inputs [43, 63, 33, 50]. Third, we can incorporate existing methods for meta-learning unlabeled adaptation to solve domain shift problems. We demonstrate this by proposing the ARM-LL method, which is based on the robotic imitation learning method from Yu et al. $\\pmb { \\Vert 8 0 \\Vert }$ which adapts via a learned loss (LL). All of these methods are straightforward extensions of existing meta-learning and adaptation methods, and this is intentional – we aim to show how existing tools can be readily adapted to tackle domain generalization problems. We summarize the methods here and refer the reader to Appendix B for complete details. ",
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+        "type": "text",
+        "text": "ARM-CML. In ARM-CML, the parameters $\\phi$ of $h$ define the weights of a context network $\\mathrm { f } _ { \\mathrm { c o n t } } ( \\cdot ; \\boldsymbol { \\phi } ) : \\mathcal { X } \\to \\mathbb { R } ^ { D }$ , parameterized by the adaptation model parameters $\\phi$ . We also instantiate the model with a prediction network $\\operatorname { f } _ { \\mathrm { p r e d } } ( \\cdot , \\cdot ; \\boldsymbol { \\theta } ) \\dot { : } \\mathcal { X } \\times \\mathbb { R } ^ { D }  \\dot { \\mathcal { V } }$ , parameterized by $\\theta$ . When given a mini batch of inputs, $\\mathrm { f _ { c o n t } }$ processes each example $\\mathbf { x } _ { k }$ in the mini batch separately and outputs $\\mathbf { c } _ { k } \\in \\mathbb { R } ^ { D }$ for $k = 1 , \\ldots , K$ , which are averaged together into a context $\\begin{array} { r } { { \\mathbf c } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } { \\mathbf c } _ { k } } \\end{array}$ . $D$ is a hyperparameter, and in our experiments, we choose to be the dimensionality of $\\mathbf { x }$ , such that we can concatenate each image $\\mathbf { x } _ { k }$ and the context c along the channel dimension to produce the input to $\\mathrm { f _ { p r e d } }$ . In other words, $\\mathrm { f _ { p r e d } }$ processes each $\\mathbf { x } _ { k }$ separately to produce an estimate of the output $\\hat { y } _ { k }$ , but it additionally receives c as input. In this way, $\\mathbf { f } _ { \\mathrm { c o n t } }$ can provide information about the entire batch of $K$ unlabeled data points to $\\mathrm { f _ { p r e d } }$ for predicting the correct outputs. ",
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+        "text": "Note that the difference between ARM-CML and prior contextual meta-learning approaches is that, in prior approaches, the context network processes both inputs and outputs to produce each $\\mathbf { c } _ { k }$ ARM-CML is designed for the domain generalization setting in which we do not assume access to labels at test time, thus we meta-train for unlabeled adaptation performance at training time. ",
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+        "text": "ARM-BN. ARM-BN is a particularly simple method that is applicable for any model $g$ that has BN layers [31] . Practically, training $g$ via ARM-BN follows the same protocol as Ioffe and Szegedy [31] except for two key differences: first, the training batches are sampled from a single domain, rather than from the entire dataset, and second, the normalization statistics are recomputed at test time rather than using a training running average. As noted, this second difference has been explored by several works as a method for test time adaptation, but the first difference is novel to ARM-BN. Following $\\mathbb { E } \\mathrm { { g o r i t h m } 1 } ,$ ARM-BN defines a meta-training procedure in which $g$ learns to adapt – i.e., compute normalization statistics – using batches of training points sampled from the same domain. We empirically show in Section 5 that, for problems where BN adaptation already has a favorable inductive bias, such as for image classification, further introducing meta-training boosts its performance. We believe that other test time adaptation methods, such as those based on optimizing surrogate losses [69, 75], may similarly benefit from their corresponding meta-training procedures. ",
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+        "text": "At a high level, ARM-BN operates in a similar fashion to ARM-CML, thus we group these methods together into the umbrella of contextual approaches, shown in Figure 2 (top). The interpretation of ARM-BN through the contextual approach is that $h$ replaces the running statistics used by standard BN with statistics computed on the batch of inputs, which then serves as the context c. Thus, for ARM-BN, there is no context network, and $h$ has no parameters beyond the model parameters $\\theta$ involved in computing BN statistics. The model $g$ is again specified via a prediction network $\\mathrm { f _ { p r e d } }$ , which must have BN layers. BN typically tracks a running average of the first and second moments of the activations in these layers, which are then used at test time. ARM-BN defines $h$ such that it swaps out these moments for the moments computed via the activations on the test batch, thus giving us adapted parameters $\\theta ^ { \\prime }$ if we view the moments as part of the model parameters. This method is remarkably simple, and in deep learning libraries such as PyTorch $\\mathbb { \\left. \\boldsymbol { \\bar { 5 2 } } \\right. }$ , implementing ARM-BN involves changing a single line of code. However, as shown in Section 5, this method also performs very well empirically, and the adaptation effectiveness is further boosted by meta-training. ",
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+        "image_caption": [
+            "Figure 2: In the contextual approach (top), $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K }$ are summarized into a context c, and we propose two methods for this summarization, either through a separate context network or using batch normalization activations in the model itself. c can then be used by the model to infer additional information about the input distribution. In the gradient based approach (bottom), an unlabeled loss function $\\mathcal { L }$ is used for gradient updates to the model parameters, in order to produce parameters that are specialized to the test inputs and can produce more accurate predictions. "
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+        "text": "ARM-LL. ARM-LL, depicted in Figure 2 (bottom), follows the gradient based meta-learning paradigm $\\mathbb { \\lVert 1 8 \\rVert }$ and learns parameters $\\theta$ that are amenable to gradient updates on a loss function in order to quickly adapt to a new problem. In other words, $h$ produces $\\theta ^ { \\prime } \\bar { = } \\theta - \\alpha \\nabla _ { \\theta } \\mathcal { L } ( \\theta , \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { K } ; \\phi )$ , where $\\alpha$ is a hyperparameter. Note that the loss function $\\mathcal { L }$ used in the gradient updates is different from the original supervised loss function $\\ell$ , in that it operates on only the inputs $\\mathbf { x }$ , rather than the input output pairs that $\\ell$ receives. We follow the general implementation of this approach proposed in Yu et al. $\\pmb { \\| 8 0 \\| }$ . We define $g$ to produce output features $\\mathbf { o } \\in \\mathbb { R } ^ { | \\mathcal { V } | }$ that are used as logits when making predictions. We then define the unlabeled loss function $\\mathcal { L }$ to be the composition of $g$ and a loss network $\\mathsf { f } _ { \\mathrm { l o s s } } ( \\cdot ; \\boldsymbol { \\phi } ) : \\mathbb { R } ^ { | \\mathcal { V } | }  \\mathbb { R }$ , which takes in the output features from $g$ and produces a scalar. We use the $\\ell _ { 2 }$ -norm of these scalars across the batch of inputs as the loss for updating $\\theta$ . In other words, $h ( \\theta , \\mathbf { x } _ { 1 } , . . . , \\mathbf { x } _ { K } ; \\phi ) = \\theta - \\alpha \\nabla _ { \\theta } \\| \\mathbf { v } \\| _ { 2 }$ , where $\\mathbf { v } = [ \\mathbf { f } _ { \\mathrm { l o s s } } ( g ( \\mathbf { x } _ { 1 } ; \\theta ) ; \\phi ) , \\dots , \\mathbf { f } _ { \\mathrm { l o s s } } ( g ( \\mathbf { x } _ { K } ; \\theta ) ; \\phi ) ]$ . ",
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+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our experiments are designed to answer the following questions: ",
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+        "text": "1. Do ARM methods learn models that can leverage unlabeled adaptation to tackle domain shift?   \n2. How do ARM methods compare to prior methods for robustness, invariance, and adaptation?   \n3. Can models trained via ARM methods adapt successfully in the streaming test setting? ",
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+        "type": "text",
+        "text": "5.1 Evaluation domains and protocol ",
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+        "text": "We propose four image classification problems, which we present below and describe in full detail in Appendix D. We also present results on datasets from the WILDS benchmark [35] in subsection 5.4. ",
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+        "type": "text",
+        "text": "We believe that the problems we propose in this paper can supplement existing benchmarks for domain shift, which, as discussed above, are designed to test invariances. A key characteristic of the problems presented here is the potential for adaptation to improve test performance, and this differs from prior benchmarks such as the problems compiled by DomainBed [23]. In Appendix C, we compare our testbeds to DomainBed and group robustness benchmarks, and we briefly discuss the results in Gulrajani and Lopez-Paz $\\pmb { \\left. \\pmb { \\left. \\bar { 2 . 3 } \\right. } \\right. }$ , which also evaluate ARM-CML. ",
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+        "text": "Rotated MNIST. We study a modified version of MNIST where images are rotated in 10 degree increments, from 0 to 130 degrees. We treat each rotation as a separate domain, i.e., a different value of $z$ . We use only 108 training data points for each of the 2 smallest domains (120 and 130 degrees), and 324 points each for rotations 90 to 110, whereas the overall training set contains 32292 points. In this setting, we hypothesize that adaptation can specialize the model to specific domains, in particular the rare domains in the training set. For each test evaluation, we generate images from the MNIST test set with a certain rotation. We measure both worst case and average accuracy across domains. ",
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+        "type": "text",
+        "text": "Federated Extended MNIST (FEMNIST). The extended MNIST (EMNIST) dataset consists of images of handwritten uppercase and lowercase letters, in addition to digits [12]. FEMNIST is the same dataset, but it also provides the meta-data of which user generated each data point [9]. We treat each user as a domain. We measure each method’s worst case and average accuracy across 35 test users, which are held out and thus disjoint from the training users. As discussed in Section 1, adaptation may help for this problem for specializing the model and resolving ambiguous data points. ",
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+        "type": "text",
+        "text": "Corrupted image datasets. CIFAR-10-C and Tiny ImageNet-C $\\lVert 2 5 \\rVert$ augment the CIFAR-10 [36] and Tiny ImageNet test sets with common image corruptions that vary in type and severity. The original goal of these augmented test sets was to benchmark how well methods could handle these corruptions without access to any corruptions during training $\\mathbb { \\left[ \\left[ 2 5 \\right] \\right] }$ . Thus, successful methods for these problems typically have relied on domain knowledge and heuristics designed specifically for image classification. For example, prior work has shown that carefully designed test time adaptation procedures are effective for these problems [69, 63, 75]. One possible reason for this phenomenon is that convolutional networks are biased toward texture $\\scriptstyle { \\left[ \\left[ 2 1 \\right] \\right] }$ , which is distorted by corruptions, thus adaptation can help the model recover its performance for each corruption type. ",
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+        "type": "text",
+        "text": "We study whether meta-training for adaptation performance can improve upon these results. To do so, we modify the protocol from Hendrycks and Dietterich $\\mathbb { \\left[ \\left[ 2 5 \\right] \\right] }$ to fit into the ARM problem setting by applying a set of 56 corruptions to the training data, and we define each corruption to be a domain. We use a disjoint set of 22 corruptions for the test data, which are mostly of different types from the training corruptions (thus, not sampled identically), and we measure worst case and average accuracy across the test corruptions. This modification allows us to study, for both ARM and prior methods, whether seeing corruptions at training time can help the model deal with new corruptions at test time. ",
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+        "type": "text",
+        "text": "5.2 Comparisons and ablations ",
+        "text_level": 1,
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+        "text": "We compare the ARM methods against several prior methods designed for robustness, invariance, and adaptation. We describe the comparisons here and provide additional details in Appendix D. ",
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+        "type": "text",
+        "text": "Test time adaptation. We evaluate the general approach of using test batches to compute BN statistics [43, 63, 33, 50], which we term BN adaptation. We also compare to test time training (TTT) $\\mathbb { \\left| \\overline { { 6 9 } } \\right| }$ which adapts the model at test time using a self-supervised rotation prediction loss. These methods have previously achieved strong results for image classification, likely because they constitute favorable inductive biases for improving on the true classification task [69]. ",
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+        "text": "Ablations. We also include ablations of the ARM-CML and ARM-LL methods, which sample training batches of unlabeled examples uniformly from the entire training set, rather than sampling from a single domain.3 These “context ablation” and “learned loss ablation” are similar to test time adaptation methods in that they do not require training domains, thus they allow us to evaluate whether or not meta-training on domain shifts is important for improved performance. ",
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+        "type": "text",
+        "text": "Group robustness and invariance. Sagawa et al. $\\left[ \\left[ 6 0 \\right] \\right]$ recently proposed a state-of-the-art method for group robustness, and we refer to this approach as distributionally robust neural networks (DRNN). Their work also evaluates a strong upweighting (UW) baseline that samples uniformly from each group, and so we also evaluate this approach in our experiments. Additionally, we compare to ",
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+        "type": "text",
+        "text": "Table 1: Worst case (WC) and average (Avg) top 1 accuracy on all testbeds, where means and standard errors are reported across three separate runs of each method. Horizontal lines separate methods that make use of (from top to bottom): neither, training domains, test batches, or both. ARM methods consistently achieve greater robustness, measured by WC, and Avg performance compared to prior methods. $^ { * } \\mathrm { U W }$ is identical to ERM for CIFAR-10-C and Tiny ImageNet-C. ",
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+    {
+        "type": "text",
+        "text": "These results have been updated from an earlier version of the paper, primarily for CIFAR10-C, due to significant refactoring of the code, additional hyperparameter tuning for both the ARM methods and the prior methods, and efforts to standardize results across the authors’ different computing environments and library versions. These results are reproducible from the publicly available code: https://github.com/henrikmarklund/arm. ",
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+    {
+        "type": "table",
+        "img_path": "images/b1a364d63cca9cdf72507533fd7288236cadacde948e5c0d58b3be485ca3f7b6.jpg",
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+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">MNIST</td><td colspan=\"2\">FEMNIST</td><td colspan=\"2\">CIFAR-10-C</td><td colspan=\"2\">Tiny ImageNet-C</td></tr><tr><td>WC</td><td>Avg</td><td>WC</td><td>Avg</td><td>wC</td><td>Avg</td><td>wC</td><td>Avg</td></tr><tr><td>ERM</td><td>74.5 ± 1.4</td><td>93.6±0.4</td><td>62.4±0.4</td><td>79.1± 0.3</td><td>54.1± 0.3</td><td>70.4± 0.1</td><td>20.3±0.5</td><td>41.9 ± 0.1</td></tr><tr><td>UW*</td><td>80.3 ±1.2</td><td>95.1±0.1</td><td>65.7 ± 0.7</td><td>80.3±0.6</td><td></td><td></td><td></td><td></td></tr><tr><td>DRNN</td><td>79.9 ± 0.7</td><td>94.9 ±0.1</td><td>57.5 ± 1.7</td><td>76.5 ± 1.2</td><td>49.3± 0.9</td><td>65.7 ± 0.5</td><td>14.2 ± 0.2</td><td>31.6 ±1.0</td></tr><tr><td>DANN</td><td>78.8±0.8</td><td>94.9 ± 0.1</td><td>65.4 ± 1.0</td><td>81.7 ± 0.3</td><td>53.9±2.2</td><td>69.8±0.3</td><td>20.4± 0.7</td><td>40.9±0.2</td></tr><tr><td>MMD</td><td>82.4±0.9</td><td>95.3± 0.3</td><td>62.4±0.7</td><td>79.8 ±0.4</td><td>52.2 ±0.3</td><td>69.5 ±0.1</td><td>19.7 ±0.2</td><td>40.1±0.1</td></tr><tr><td>BN adaptation</td><td>78.0±0.3</td><td>94.4± 0.1</td><td>65.7 ± 1.5</td><td>80.0 ±0.5</td><td>60.6±0.3</td><td>70.9 ± 0.1</td><td>26.5 ± 0.3</td><td>42.8 ±0.0</td></tr><tr><td>TTT</td><td>81.1±0.3</td><td>95.4±0.1</td><td>68.6 ±0.4</td><td>84.2±0.1</td><td>61.5 ± 0.3</td><td>71.7 ± 0.5</td><td>27.6± 0.5</td><td>37.7 ± 0.3</td></tr><tr><td>CML ablation</td><td>63.5 ± 1.8</td><td>90.1 ±0.2</td><td>61.8 ±0.8</td><td>81.6 ± 0.5</td><td>58.8 ±0.1</td><td>69.6±0.2</td><td>26.3±0.6</td><td>42.5 ± 0.1</td></tr><tr><td>LL ablation</td><td>79.9 ± 1.1</td><td>95.0± 0.3</td><td>64.1 ± 1.6</td><td>80.8 ±0.2</td><td>60.9 ± 0.4</td><td>71.3 ± 0.0</td><td>21.6 ± 2.1</td><td>32.6 ± 3.2</td></tr><tr><td>ARM-CML</td><td>88.0±0.8</td><td>96.3±0.4</td><td>70.9 ± 1.4</td><td>86.4±0.3</td><td>61.2 ± 0.4</td><td>70.3±0.2</td><td>29.1± 0.4</td><td>43.3 ±0.1</td></tr><tr><td>ARM-BN</td><td>83.3 ±0.5</td><td>95.6 ±0.1</td><td>64.5 ±3.2</td><td>83.2 ± 0.5</td><td>61.7 ± 0.3</td><td>72.4±0.3</td><td>28.3± 0.3</td><td>43.3± 0.1</td></tr><tr><td>ARM-LL</td><td>88.9±0.8</td><td>96.9±0.2</td><td>67.0±0.9</td><td>84.3±0.7</td><td>61.2±0.7</td><td>72.5± 0.4</td><td>25.4± 0.1</td><td>35.7 ±0.4</td></tr></table>",
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+        "type": "text",
+        "text": "domain adversarial neural networks (DANN) $\\mathbb { \\lVert 1 9 \\rVert }$ and maximum mean discrepancy (MMD) feature learning [41], two state-of-the-art methods for adversarial learning of invariant predictive features. For the WILDS datasets, we include the numbers reported in Koh et al. $\\pmb { \\Vert 3 5 \\Vert }$ for DRNN and two other invariance methods, correlation alignment (CORAL) $ { \\mathbb { I } }  { \\mathbb { I } }$ and invariant risk minimization (IRM) [3]. ",
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+        "type": "text",
+        "text": "Robustness and invariance methods assume access to training domains but not test batches, whereas adaptation methods assume the opposite. Thus, at a high level, we can view the comparisons to these methods as evaluating the importance of each of these assumptions for the specified problems. ",
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+        "text": "5.3 Quantitative evaluation and comparisons ",
+        "text_level": 1,
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+        "text": "The results for the four proposed benchmarks are presented in Table 1. The best results, stratified by classes of methods, are bolded, with the single best result across all methods underlined. Across all of these problems, ARM methods increase both worst case and average accuracy compared to all other methods. ARM-CML performs well across all tasks, and despite its simplicity, ARM-BN achieves the best performance overall on the corrupted image testbeds, demonstrating the effectiveness of metatraining on top of an already strong adaptation procedure. BN adaptation and TTT are the strongest prior methods, as these adaptation procedures constitute inductive biases that are generally well suited for image classification. However, ARM methods are comparatively less reliant on favorable inductive biases and consistently attain better results. In general, we observe poor performance from robustness methods, varying performance from invariance methods, strong performance from adaptation methods, and the strongest performance from ARM methods. ",
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+        "type": "text",
+        "text": "When we cannot access a batch of test points all at once, and instead the points are observed in a streaming fashion, we can augment the proposed ARM methods to perform sequential model updates. For example, ARM-CML and ARM-BN can update their average context and normalization statistics, respectively, after observing each new test point. In Figure 3, we study this test setting for the Tiny ImageNet-C problem. We see that both models trained with ARM-CML and ARM-BN are able to achieve near their original worst case and average accuracy within observing 50 data points, well before the training batch size of 100. This result demonstrates that ARM methods are applicable for problems where test points must be observed one at a time, provided that the model is permitted to adapt using each point. We describe in detail how each ARM method can be applied to the streaming setting in Appendix B, and we provide streaming results on rotated MNIST in Appendix E. ",
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+        "img_path": "images/a266c73291c4902cc98f6f0a173936b65146e6d4cf5179e742872fa1480eb8c1.jpg",
+        "image_caption": [
+            "Figure 3: In the streaming setting, ARM methods reach strong performance on Tiny ImageNet-C after fewer than 50 data points, despite using training batch sizes of 100. This highlights that the trained models are able to adapt successfully in the standard streaming evaluation setting. "
+        ],
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+        "table_caption": [
+            "Table 2: Results on the WILDS image testbeds. Different methods are best suited for different problems, motivating the need for a wide range of methods. ARM-BN struggles on FMoW but performs well on the other datasets, in particular RxRx1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">iWildCam</td><td>Camelyon17</td><td>RxRx1</td><td colspan=\"2\">FMoW</td><td colspan=\"2\">PovertyMap</td></tr><tr><td>Method</td><td>Acc</td><td>Macro F1</td><td>Acc</td><td>Acc</td><td>WC Acc</td><td>Avg Acc</td><td>WC Pearson r</td><td>Pearson r</td></tr><tr><td>ERM</td><td>71.6 ± 2.5</td><td>31.0 ±1.3</td><td>70.3±6.4</td><td>29.9 ± 0.4</td><td>32.3±1.25</td><td>53.0±0.55</td><td>0.45±0.06</td><td>0.78±0.04</td></tr><tr><td>DRNN</td><td>72.7±2.0</td><td>23.9 ± 2.1</td><td>68.4± 7.3</td><td>23.0±0.3</td><td>30.8 ±0.81</td><td>52.1±0.5</td><td>0.39 ±0.06</td><td>0.75±0.07</td></tr><tr><td>CORAL</td><td>73.3±4.3</td><td>32.8±0.1</td><td>59.5± 7.7</td><td>28.4±0.3</td><td>31.7 ±1.24</td><td>50.5 ±0.36</td><td>0.44±0.06</td><td>0.78±0.05</td></tr><tr><td>IRM</td><td>59.8 ±3.7</td><td>15.1 ± 4.9</td><td>64.2± 8.1</td><td>8.2 ±1.1</td><td>30.0±1.37</td><td>50.8 ±0.13</td><td>0.43 ± 0.07</td><td>0.77± 0.05</td></tr><tr><td>BN adaptation</td><td>46.4± 1.0</td><td>13.8±0.3</td><td>88.6±1.4</td><td>20.0±0.2</td><td>30.2±0.26</td><td>51.6 ± 0.16</td><td>0.39 ±0.17</td><td>0.82±0.06</td></tr><tr><td>ARM-BN</td><td>70.3±2.4</td><td>23.2 ±2.7</td><td>87.2±0.9</td><td>31.2 ±0.1</td><td>24.6 ±0.04</td><td>42.0±0.21</td><td>0.49±0.21</td><td>0.84±0.05</td></tr></table>",
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+        "type": "text",
+        "text": "5.4 WILDS results ",
+        "text_level": 1,
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+        "text": "Finally, we present results on the WILDS benchmark $\\pmb { \\Vert 3 5 \\Vert }$ t Table 2. We evaluate BN adaptation and ARM-BN on these testbeds. We see that, on these real world distribution shift problems, different methods perform well for different problems. CORAL, a method for invariance $\\mathbf { \\widehat { \\mathbf { \\phi } } } [ \\mathbf { \\overline { { 6 7 } } } ] \\mathbf { \\xi }$ , performs best on the iWildCam animal classification problem [5], whereas no methods outperform ERM by a significant margin on the FMoW [11] or PovertyMap $\\textcircled { 1 7 9 }$ satellite imagery problems. ARM-BN performs particularly poorly on the FMoW problem. However, it performs well on PovertyMap and significantly improves performance on the RxRx1 $\\mathbb { \\ m }$ problem of treatment classification from medical images. On the other medical imagery problem of Camelyon17 [4] tumor identification, adaptation in general boosts performance dramatically. These results indicate the need to consider a wide range of tools, including meta-learning and adaptation, for combating distribution shift. ",
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+        "type": "text",
+        "text": "6 Discussion and Future Work ",
+        "text_level": 1,
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+        "text": "We presented adaptive risk minimization (ARM), a framework and problem formulation for learning models that can adapt in the face of domain shift at test time using only a batch of unlabeled test examples. We devised an algorithm and instantiated a set of methods for optimizing the ARM objective that meta-learns models that are adaptable to different domains of training data. Empirically, we observed that ARM methods consistently improve performance in terms of both average and worst case metrics, as compared to a number of prior approaches for handling domain shift. ",
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+    {
+        "type": "text",
+        "text": "Though we provided contextual meta-learning as a concrete example, a number of other meta-learning paradigms would also be interesting to extend to the ARM setting. For example, few shot generative modeling objectives would be a natural fit for unlabeled adaptation $\\boxed { 1 6 } \\boxed { 2 6 } \\boxed { 7 7 }$ . Another exciting direction for future work is to explore the problem setting where domains are not provided at training time. As discussed in $\\underline { { \\mathrm { ~ A p p e n d i x ~ } } } \\mathrm { ~ \\bar { E } ~ }$ in this setting, we can instead construct domains via unsupervised learning techniques. Similar to Hsu et al. $\\pmb { \\left. 2 8 \\right. }$ , one promising approach is to generate a diverse set of domains in order to learn generally effective adaptation strategies. Robustness and invariance methods cannot be used easily with multiple different groupings, learned or otherwise, as techniques such as group weighted loss functions $\\dot { \\left\\| 6 0 \\right\\| }$ and domain classifiers $\\mathbb { \\lVert 1 9 \\rVert }$ are not immediately extendable to this setup. Thus, ARM methods may be uniquely suited to be paired with domain learning. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "MZ thanks Matt Johnson and Sharad Vikram for helpful discussions and was supported by an NDSEG fellowship. HM is funded by a scholarship from the Dr. Tech. Marcus Wallenberg Foundation for Education in International Industrial Entrepreneurship. AG was supported by an NSF graduate research fellowship. CF is a CIFAR Fellow in the Learning in Machines and Brains program. This research was supported by the DARPA Assured Autonomy and Learning with Less Labels programs. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+    },
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parse/train/0CDKgyYaxC8/0CDKgyYaxC8.md

@@ -0,0 +1,234 @@
+# Evaluating Gradient Inversion Attacks and Defenses in Federated Learning
+
+Yangsibo Huang   
+Princeton University   
+Princeton, NJ 08540   
+yangsibo@princeton.edu
+
+Samyak Gupta Princeton University Princeton, NJ 08540 samyakg@cs.princeton.edu
+
+Zhao Song Adobe Research San Jose, CA 95110 zsong@adobe.com
+
+Kai Li Princeton University Princeton, NJ 08540 li@cs.princeton.edu
+
+Sanjeev Arora Princeton University Princeton, NJ 08540 arora@cs.princeton.edu
+
+# Abstract
+
+Gradient inversion attack (or input recovery from gradient) is an emerging threat to the security and privacy preservation of Federated learning, whereby malicious eavesdroppers or participants in the protocol can recover (partially) the clients’ private data. This paper evaluates existing attacks and defenses. We find that some attacks make strong assumptions about the setup. Relaxing such assumptions can substantially weaken these attacks. We then evaluate the benefits of three proposed defense mechanisms against gradient inversion attacks. We show the trade-offs of privacy leakage and data utility of these defense methods, and find that combining them in an appropriate manner makes the attack less effective, even under the original strong assumptions. We also estimate the computation cost of end-to-end recovery of a single image under each evaluated defense. Our findings suggest that the state-of-the-art attacks can currently be defended against with minor data utility loss, as summarized in a list of potential strategies.
+
+# 1 Introduction
+
+Federated learning [McMahan et al., 2016, Kairouz et al., 2021] is a framework that allows multiple clients in a distributed environment to collaboratively train a neural network model at a central server, without moving their data to the central server. At every training step, each client computes a model update —i.e., gradient— on its local data using the latest copy of the global model, and then sends the gradient to the central server. The server aggregates these updates (typically by averaging) to construct a global model, and then sends the new model parameters to all clients. By allowing clients to participate in training without directly sharing their data, such protocols align better with data privacy regulations such as Health Insurance Portability and Accountability Act (HIPPA) [Act, 1996], California Consumer Privacy Act (CCPA) [Legislature, 2018], and General Data Protection Regulation [Commission, 2018].
+
+While sharing gradients was thought to leak little information about the client’s private data, recent papers [Zhu et al., 2019, Zhao et al., 2020, Geiping et al., 2020, Yin et al., 2021] developed a “gradient inversion attack” by which an attacker eavesdropping on a client’s communications with the server can begin to reconstruct the client’s private data. The attacker can also be a malicious participant in the Federated Learning scheme, including a honest-but-curious server who wishes to reconstruct private data of clients, or a honest-but-curious client who wishes to reconstruct private data of other clients. These attacks have been shown to work with batch sizes only up to 100 but even so they have created doubts about the level of privacy ensured in Federated Learning. The current paper seeks to evaluate the risks and suggest ways to minimize them.
+
+Several defenses against gradient inversion attacks have been proposed. These include perturbing gradients [Zhu et al., 2019, Wei et al., 2020] and using transformation for training data that clients can apply on the fly [Zhang et al., 2018a, Huang et al., 2020]. More traditional cryptographic ideas including secure aggregation [Bonawitz et al., 2016] or homomorphic encryption [Phong et al., 2018] for the gradients can also be used and presumably stop any eavesdropping attacks completely. They will not be studied here due to their special setups and overhead.
+
+We are not aware of a prior systematic evaluation of the level of risk arising from current attacks and the level of security provided by various defenses, as well as the trade-off (if any) between test accuracy, computation overhead, and privacy risks.
+
+The paper makes two main contributions. First, we draw attention to two strong assumptions that a current gradient inversion attack [Geiping et al., 2020] implicitly makes. We show that by nullifying these assumptions, the performance of the attack drops significantly and can only work for lowresolution images. The findings are explored in Section 3 and already imply some more secure configurations in Federated Learning (Section 6).
+
+Second, we summarize various defenses (Section 4) and systematically evaluate (Section 5) some of their performance of defending against a state-of-the-art gradient inversion attack, and present their data utility and privacy leakage trade-offs. We estimate the computation cost of end-to-end recovery of a single image under each evaluated defense. We also experimentally demonstrate the feasibility and effectiveness of combined defenses. Our findings are summarized as strategies to further improve Federated Learning’s security against gradient inversion attacks (Section 6).
+
+In Appendix B, we provide theoretical insights for mechanism of each evaluated defense.
+
+# 2 Gradient Inversion Attacks
+
+Previous studies have shown the feasibility of recovering input from gradient (i.e. gradient inversion) for image classification tasks, by formulating it as an optimization problem: given a neural network with parameters $\theta$ , and the gradient $\nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } )$ computed with a private data batch $( x ^ { * } , y ^ { * } ) \in$ $\mathbb { R } ^ { b \times d } \times \mathbb { R } ^ { b }$ $( b , d$ being the batch size, image size), the attacker tries to recover $x \in \mathbb { R } ^ { b \times d }$ , an approximation of $x ^ { * }$ :
+
+$$
+\arg \operatorname* { m i n } _ { x } \mathcal { L } _ { \mathrm { g r a d } } ( x ; \theta , \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } ) ) + \alpha \mathcal { R } _ { \mathrm { a u x } } ( x )
+$$
+
+The optimization goal consists of two parts: $\mathcal { L } _ { \mathrm { g r a d } } ( x ; \theta , \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } ) )$ enforces matching of the gradient of recovered batch $x$ with the provided gradients $\mathcal { L } _ { \boldsymbol { \theta } } ( x ^ { * } , y ^ { * } )$ , and $\mathcal { R } _ { \mathrm { a u x } } ( x )$ regularizes the recovered image based on image prior(s).
+
+[Phong et al., 2017] brings theoretical insights on this task by proving that such reconstruction is possible with a single-layer neural network. [Zhu et al., 2019] is the first to show that accurate pixellevel reconstruction is practical for a maximum batch size of 8. Their formulation uses $\ell _ { 2 }$ -distance as $\mathcal { L } _ { \mathrm { g r a d } } ( \cdot , \cdot )$ but no regularization term $\mathcal { R } _ { \mathrm { a u x } } ( x )$ . The approach works for low-resolution CIFAR datasets [Krizhevsky et al., 2009], with simple neural networks with sigmoid activations, but cannot scale up to high-resolution images, or larger models with ReLU activations. A follow-up [Zhao et al., 2020] proposes a simple approach to extract the ground-truth labels from the gradient, which improves the attack but still cannot overcome its limitations.
+
+With a careful choice of $\mathcal { L } _ { \mathrm { g r a d } }$ and $\mathcal { R } _ { \mathrm { a u x } } ( x )$ , [Geiping et al., 2020] substantially improves the attack and succeeds in recovering a single ImageNet [Deng et al., 2009] image from gradient: their approach uses cosine distance as $\mathcal { L } _ { \mathrm { g r a d } }$ , and the total variation as $\mathcal { R } _ { \mathrm { a u x } } ( x )$ . Their approach is able to reconstruct low-resolution images with a maximum batch size of 100 or a single high-resolution image. Based on [Geiping et al., 2020], [Wei et al., 2020] analyzes how different configurations in the training may affect attack effectiveness.
+
+A more recent work [Yin et al., 2021] further improves the attack on high-resolution images, by introducing to $\mathcal { R } _ { \mathrm { a u x } } ( x )$ a new image prior term based on batch normalization [Ioffe and Szegedy, 2015] statistics, and a regularization term which enforces consistency across multiple attack trials.
+
+An orthogonal line of work [Zhu and Blaschko, 2021] proposes to formulate the gradient inversion attack as a closed-form recursive procedure, instead of an optimization problem. However, their implementation can recover only low-resolution images under the setting where batch size $= 1$ .
+
+# 3 Strong Assumptions Made by SOTA Attacks
+
+# 3.1 The state-of-the-art attacks
+
+Two recent attacks [Geiping et al., 2020, Yin et al., 2021] achieve best recovery results. Our analysis focuses on the former as the implementation of the latter is not available at the time of writing this paper. We plan to include the analysis for the latter attack in the final version of this paper if its implementation becomes available.
+
+[Geiping et al., 2020]’s attack optimizes the following objective function:
+
+$$
+\arg \operatorname* { m i n } _ { x } 1 - \frac {  \nabla _ { \theta } \mathcal { L } _ { \theta } ( x , y ) , \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } )  } { \| \nabla _ { \theta } \mathcal { L } _ { \theta } ( x , y ) \| \| \| \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } ) \| } + \alpha _ { \mathrm { T V } } \mathcal { R } _ { \mathrm { T V } } ( x )
+$$
+
+where $\langle \cdot , \cdot \rangle$ is the inner-product between vectors, and $\mathcal { R } _ { \mathrm { T V } } ( \cdot )$ is the total variation of images.
+
+We notice that Geiping et al. has made two strong assumptions (Section 3.2). Changing setups to invalidate those assumptions will substantially weaken the attacks (Section 3.3). We also summarize whether other attacks have made similar assumptions in Table 1.
+
+# 3.2 Strong assumptions
+
+We find that previous gradient inversion attacks have made different assumptions about whether the attacker knows Batch normalization statistics or private labels, as shown in Table 3.2. Note that [Geiping et al., 2020]’s attack makes both strong assumptions.
+
+Assumption 1: Knowing BatchNorm statistics. Batch normalization (BatchNorm) [Ioffe and Szegedy, 2015] is a technique for training neural networks that normalizes the inputs to a layer for every mini-batch. It behaves differently during training and evaluation. Assume the model has $L$ batch normalization layers. Given $x ^ { * }$ , a batch of input images, we use $x _ { l } ^ { * }$ to denote the input features to the $l$ -th BatchNorm layer, where $l \in [ L ]$ . During training, the $l$ -th BatchNorm layer normalizes $x _ { l } ^ { \ast }$ based on the batch’s mean $\mathrm { m e a n } ( x _ { l } ^ { \ast } )$ and variance $\mathrm { v a r } ( x _ { l } ^ { * } )$ , and keeps a running estimate of mean and variance of all training data points, denoted by $\mu _ { l }$ and $\sigma _ { l } ^ { 2 }$ . During inference, $\{ \mu _ { l } \} _ { l = 1 } ^ { L }$ and $\{ \sigma _ { l } ^ { 2 } \} _ { l = 1 } ^ { L }$ are used to normalize test images. In the following descriptions, we leave out $\{ \cdot \} _ { l = 1 } ^ { L }$ for simplicity (i.e. use $\mu , \sigma ^ { 2 }$ to denote $\{ \mu _ { l } \} _ { l = 1 } ^ { L } , \{ \sigma _ { l } ^ { 2 } \} _ { l = 1 } ^ { L }$ , and ${ \mathrm { m e a n } } ( x ^ { * } )$ , $\mathrm { v a r } ( x ^ { * } )$ to denote $\{ \mathrm { m e a n } ( x _ { l } ^ { * } ) \} _ { l = 1 } ^ { L }$ , $\{ \mathrm { v a r } ( x _ { l } ^ { * } ) \} _ { l = 1 } ^ { L } )$ .
+
+We notice that [Geiping et al., 2020]’s implementation1 assumes that BatchNorm statistics of the private batch, i.e., $\mathrm { m e a n } ( x ^ { * } )$ , $\mathrm { v a r } ( x ^ { * } )$ , are jointly provided with the gradient. Knowing BatchNorm statistics would enable the attacker to apply the same batch normalization used by the private batch on his recovered batch, to achieve a better reconstruction. This implicitly increases the power of the attacker, as sharing private BatchNorm statistics are not necessary in Federated learning [Andreux et al., 2020, Li et al., 2021].
+
+Note that this assumption may be realistic in some settings: 1) the neural network is shallow, thus does not require using BatchNorm layers, or 2) the neural network is deep, but adapts approaches that normalize batch inputs with a fixed mean and variance (as alternative to BatchNorm), e.g. Fixup initialization [Zhang et al., 2019].
+
+Assumption 2: Knowing or able to infer private labels. Private labels are not intended to be shared in Federated learning, but knowing them would improve the attack. [Zhao et al., 2020] finds that label information of a single private image can be inferred from the gradient (see Section 3.3 for details). Based on this, [Geiping et al., 2020] assumes the attacker knows private labels (see remark at the end of Section 4 in their paper). However, this assumption may not hold true when multiple images in a batch share the same label, as we will show in the next section.
+
+<table><tr><td> Assumptions</td><td></td><td></td><td>[[Zhu et al.,2019] [Zhao et al.,2020] [Geiping et al.,2020][Yin et al.,2021]</td><td></td></tr><tr><td>Knowing BN statistics</td><td>N/A+</td><td>N/A</td><td>Yes</td><td>Yes*</td></tr><tr><td>Knowing private labels</td><td>No</td><td>No</td><td>Yes</td><td>Not</td></tr></table>
+
+Table 1: Assumptions of gradient inversion attacks. †: its evaluation uses a simple model without a BatchNorm layer; $^ \ddag$ : it proposes a method to infer private labels, which works when images in a batch have unique labels (see Section 3.3); ∗: although the paper discusses a setting where BatchNorm statistics are unknown, its main results assume knowing BatchNorm statistics.
+
+# 3.3 Re-evaluation under relaxed assumptions
+
+We re-evaluate the performance of the gradient inversion attack in settings where two assumptions above are relaxed. For each relaxation, we re-design the attack (if needed) based on the knowledge that the attacker has.
+
+Relaxation 1: Not knowing BatchNorm statistics. We refer to the previous threat model as $\mathrm { B N } _ { \mathrm { e x a c t } }$ , where the attacker knows exact BatchNorm statistics of the private batch. We consider a more realistic threat model where these statistics are not exposed, and re-design the attack based on it.
+
+Threat model. In each training step, the client normalizes its private batch $x ^ { * }$ using the batch’s mean $\mathrm { m e a n } ( x ^ { * } )$ and variance $\mathrm { v a r } ( x ^ { * } )$ , keeps the running estimate of mean and variance locally as in [Li et al., 2021], and shares the gradient. The client releases the final aggregated mean $\mu$ , and aggregated variance $\sigma ^ { 2 }$ of all training data points at the end of training. Same as before, the attacker has access to the model and the gradient during training.
+
+Re-design A: $\mathrm { B N } _ { \mathrm { p r o x y } }$ , attacker naively uses $\mu$ and $\sigma ^ { 2 }$ . A simple idea is that the attacker uses $( \mu , \sigma ^ { 2 } )$ as the proxy for $( \mathbf { \dot { m } e a n } ( x ^ { * } ) , \mathbf { v a r } ( x ^ { * } ) )$ , and uses them to normalize $x$ , his guesses of the private batch. Other operations of the gradient inversion attack remain the same as before. However, Figure 1.d and 1.h show poor-quality reconstruction with this re-design.
+
+$R e$ -design $B$ : $\mathrm { B N } _ { \mathrm { i n f e r } }$ , attacker infers $( \mathrm { m e a n } ( x ^ { * } )$ , $\mathrm { v a r } ( x ^ { * } ) )$ based on $( \mu , \sigma ^ { 2 } )$ . A more reasonable attacker will try to infer $( \mathrm { m e a n } ( x ^ { * } ) , \mathrm { v a r } ( x ^ { * } ) )$ while updating $x$ , his guesses of the private batch, and uses $( \mathrm { m e a n } ( x ) , \mathrm { v a r } ( x ) )$ to normalize the batch. In this case, $( \mu , \sigma ^ { 2 } )$ could be used as a prior of BatchNorm statistics to regularize the recovery, as suggested in [Yin et al., 2021]:
+
+$$
+\arg \operatorname* { m i n } _ { x } 1 - \frac { \langle \nabla _ { \theta } \mathcal { L } _ { \theta } ( x , y ) , \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } ) \rangle } { \| \nabla _ { \theta } \mathcal { L } _ { \theta } ( x , y ) \| \| \nabla _ { \theta } \mathcal { L } _ { \theta } ( x ^ { * } , y ^ { * } ) \| } + \alpha _ { \mathrm { T V } } \mathcal { R } _ { \mathrm { T V } } ( x ) + \alpha _ { \mathrm { B N } } \mathcal { R } _ { \mathrm { B N } } ( x )
+$$
+
+We tune $\alpha _ { \mathrm { B N } }$ and present the best result in Figure 1.c and 1.g (see results of different $\alpha _ { \mathrm { B N } }$ ’s in Appendix A). As shown, for a batch of low-resolution images, $\mathrm { B N } _ { \mathrm { i n f e r } }$ gives a much better reconstruction result than $\mathrm { B N } _ { \mathrm { p r o x y } }$ , but still cannot recover some details of the private batch when compared with $\mathrm { B N } _ { \mathrm { e x a c t } }$ . The result for a single high-resolution image is worse: the attacker fails to return a recognizable reconstruction with $\mathrm { B N } _ { \mathrm { i n f e r } }$ . This suggests not having access to BatchNorm statistics of the private batch already weakens the state-of-the-art gradient inversion attack.
+
+![](images/3533a0e819a283dcb807f533392d098ad73413f94673729a43792161630be7cb.jpg)  
+Figure 1: Attacking a batch of 16 low-resolution images from CIFAR-10 (a-d) and a single highresolution image from ImageNet (e-h) with different knowledge of BatchNorm statistics. Attack is weakened when BatchNorm statistics are not available (c, d versus b, and g, h versus f). See Appendix A for more examples and quantitative results.
+
+![](images/e45dac0ee12484d36497b1950c3819cc8ca401abbb36add8f1dfda23b0743b84.jpg)  
+Figure 2: Attack is weakened when private labels are not available. (a) shows that for CIFAR-10, when the batch size is large, many images in the batch belong to the same class, which essentially weakens label restoration [Zhao et al., 2020, Yin et al., 2021]. (b) visualizes a reconstructed batch of 16 images with and without private labels known. The quality of the reconstruction drops without knowledge of private labels.
+
+Relaxation 2: Not knowing private labels. [Zhao et al., 2020] notes that label information of a single private image can be computed analytically from the gradients of the layer immediately before the output layer. [Yin et al., 2021] further extends this method to support recovery of labels for a batch of images. However, if multiple images in the private batch belong to a same label, neither approach can tell how many images belong to that label, let alone which subset of images belong to that label. Figure 2a demonstrates that with CIFAR-10, for batches of various sizes it is possible for many of the training samples to be have the same label, and the distribution of labels is not uniform - and hence, inferring labels becomes harder and the attack would be weakened. In Figure 2b we evaluate the worst-case for an attacker in this setting by comparing recoveries where the batch labels are simultaneously reconstructed alongside the training samples.
+
+# 4 Defenses Against the Gradient Inversion Attack
+
+Several defense ideas have been proposed to mitigate the risks of gradient inversion.
+
+# 4.1 Encrypt gradients
+
+Cryptography-based approaches encrypt gradient to prevent gradient inversion. [Bonawitz et al., 2016] presents a secure aggregation protocol for Federated learning by computing sum of gradient vectors based on secret sharing [Shamir, 1979]. [Phong et al., 2018] proposes using homomorphic encryption to encrypt the gradients before sending. These approaches require special setup and can be costly to implement.
+
+Moreover, with secure aggregation protocol, an honest-but-curious server can still launch the gradient inversion attack on the summed gradient vector. Similarly, an honest-but-curious client can launch the gradient inversion attack on the model returned by the server to reconstruct other clients’ private data, even with homomorphic encryption.
+
+As alternatives, two other types of defensive mechanisms have been proposed to mitigate the risks of attacks on plain-text gradient.
+
+# 4.2 Perturbing gradients
+
+Gradient pruning. When proposing the first practical gradient inversion attack, [Zhu et al., 2019] also suggests a defense by setting gradients of small magnitudes to zero (i.e. gradient pruning). Based on their attack, they demonstrate that pruning more than $7 0 \%$ of the gradients would make the recovered images no longer visually recognizable. However, the suggested prune ratio is determined based on weaker attacks, and may not remain safe against the state-of-the-art attack.
+
+Adding noise to gradient. Motivated by DPSGD [Abadi et al., 2016] which adds noise to gradients to achieve differential privacy [Dwork, 2009, Dwork and Roth, 2014], [Zhu et al., 2019, Wei et al.,
+
+2020] also suggests defending by adding Gaussian or Laplacian noise to gradient. They show that a successful defense requires adding high noise level such that its accuracy drops by more than $3 0 \%$ with CIFAR-10 tasks. Recent works [Papernot et al., 2020, Tramèr and Boneh, 2021] suggest using better pre-training techniques and a large batch size (e.g. 4, 096) to achieve a better accuracy for DPSGD training.
+
+Since most DPSGD implementations for natural image classification tasks [Abadi et al., 2016, Papernot et al., 2020, Tramèr and Boneh, 2021] use a pre-training and fine-tuning pipeline, it is hard to fairly compare with other defense methods that can directly apply when training the model from scratch. Thus, we leave the comparison with DPSGD to future work.
+
+# 4.3 Weak encryption of inputs (i.e. encoding inputs)
+
+MixUp. MixUp data augmentation [Zhang et al., 2018a] trains neural networks on composite images created via linear combination of image pairs. It has been shown to improve the generalization of the neural network and stabilizes the training. Recent work also suggests that MixUp increases the model’s robustness to adversarial examples [Pang et al., 2020, Lamb et al., 2019].
+
+InstaHide. Inspired by MixUp, [Huang et al., 2020] proposes InstaHide as a light-weight instanceencoding scheme for private distributed learning. To encode an image $x \in \mathbb { R } ^ { d }$ from a private dataset, InstaHide first picks $k - 1$ other images $s _ { 2 } , s _ { 3 } , \ldots , s _ { k }$ from that private dataset, or a large public dataset, and $k$ random nonnegative coefficients $\{ \lambda _ { i } \} _ { i = 1 } ^ { k }$ that sum to 1, and creates a composite image $\textstyle \lambda _ { 1 } x + \sum _ { i = 2 } ^ { k } \lambda _ { i } s _ { i }$ ( $k$ is typically small, e.g., 4). A composite label is also created using the same set of coefficients.2 Then it adds another layer of security: pick a random sign-flipping pattern $\sigma \in \{ - 1 , 1 \} ^ { d }$ and output the encryption $\begin{array} { r } { \tilde { x } = \sigma \circ ( \lambda _ { 1 } x + \sum _ { i = 2 } ^ { k } \lambda _ { i } s _ { i } ) } \end{array}$ , where $\circ$ is coordinate-wise multiplication of vectors. The neural network is then trained on encoded images, which look like random pixel vectors to the human eye and yet lead to good classification accuracy $\mathit { \Theta } _ { \left( < \mathrm { 6 \% } \right. }$ accuracy loss on CIFAR-10, CIFAR-100, and ImageNet).
+
+Recently, [Carlini et al., 2020] gives an attack to recover private images of a small dataset, when the InstaHide encodings are revealed to the attacker (not in a Federated learning setting). Their first step is to train a neural network on a public dataset for similarity annotation, to infer whether a pair of InstaHide encodings contain the same private image. With the inferred similarities of all pairs of encodings, the attacker then runs a combinatorial algorithm (cubic time in size of private dataset) to cluster all encodings based on their original private images, and finally uses a regression algorithm (with the help of composite labels) to recover the private images.
+
+Neither [Huang et al., 2020] or [Carlini et al., 2020] has evaluated their defense or attack in the Federated learning setting, where the attacker observes gradients of the encoded images instead of original encoded images. This necessitates the systematic evaluation in our next section.
+
+# 5 Evaluation of defenses
+
+The main goal of our experiments is to understand the trade-offs between data utility (accuracy) and securely defending the state-of-the-art gradient inversion attack even in its strongest setting, without any relaxation of its implicit assumptions. Specifically, we grant the attacker the knowledge of 1) BatchNorm statistics of the private batch, and 2) labels of the private batch.
+
+We vary key parameters for each defense, and evaluate their performance in terms of the test accuracy, computation overhead, and privacy risks (Section 5.2). We then investigate the feasibility of combining defenses (Section 5.3). We also estimate the computation cost of end-to-end recovery of a single image under evaluated defenses (Section 5.4).
+
+As the feasibility of the state-of-the-art attack [Geiping et al., 2020] on a batch of high-resolution images remain elusive when its implicit assumptions no longer hold (see Figure 1), we focus on the evaluation with low-resolution in trying to understand whether current attacks can be mitigated.
+
+# 5.1 Experimental setup
+
+Key parameters of defenses. We evaluate following defenses on CIFAR-10 dataset [Krizhevsky et al., 2009] with ResNet-18 architecture [He et al., 2016].
+
+• GradPrune (gradient pruning): gradient pruning set gradients of small magnitudes to zero. We vary the pruning ratio $p$ in $\{ 0 . 5 , 0 . 7 , 0 . 9 , 0 . 9 5 , 0 . 9 9 , 0 . 9 9 9 \}$ .   
+• MixUp: MixUp encodes a private image by linearly combining it with $k - 1$ other images from the training set. Following [Huang et al., 2020], we vary $k$ in $\{ 4 , 6 \}$ , and set the upper bound of a single coefficient to 0.65 (coefficients sum to 1).   
+• Intra-InstaHide: InstaHide [Huang et al., 2020] proposes two versions: Inter-InstaHide and Intra-InstaHide. The only difference is that at the mixup step, Inter-Instahide mixes up an image with images from a public dataset, whereas Intra-InstaHide only mixes with private images. Both versions apply a random sign flipping pattern on each mixed image. We evaluate Intra-InstaHide in our experiments, which is a weaker version of InstaHide. Similar to the evaluation of MixUp, we vary $k$ in $\{ 4 , 6 \}$ , and set the upper bound of a single coefficient to 0.65. Note that InstaHide flips signs of pixels in the image, which destroys the total variation prior. However, the absolute value of adjacent pixels should still be close. Therefore, for the InstaHide defense, we apply the total variation regularizer on $| x |$ , i.e. taking absolute value of each pixel in the reconstruction. We train the ResNet-18 architecture on CIFAR-10 using different defenses, and launch the attack.   
+We provide more details of the experiments in Appendix A.
+
+The attack. We use a subset of 50 CIFAR-10 images to evaluate the attack performance. Note that attacking MixUp and InstaHide involves another step to decode private images from the encoded images. We apply [Carlini et al., 2020]’s attack here as the decode step, where the attacker needs to eavesdrop $T$ epochs of training, instead of a single training step. We set $T = 2 0$ in our evaluation. We also grant the attacker the strongest power for the decode step to evaluate the upper bound of privacy leakage. Given a MixUp or Intra-InstaHide image which encodes $k$ private images, we assume the attacker knows:
+
+1. The indices of $k$ images in the private dataset. In a realistic scenario, the attacker of [Carlini et al., 2020] would need to train a neural network to detect similarity of encodings, and run a combinatorial algorithm to solve an approximation of this mapping.   
+2. The mixing coefficients for each of the $k$ private image. In real Federated learning, this information is not available.
+
+Hyper-parameters of the attack. The attack minimize the objective function given in Eq.3. We search $\alpha _ { \mathrm { T V } }$ in $\{ 0 , 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 \}$ for all defenses, and apply the best choice for each defense: 0.05 for GradPrune, 0.1 for MixUp, and 0.01 for Intra-InstaHide. We apply $\alpha _ { \mathrm { B N } } = 0 . 0 0 1$ for all defenses after searching it in $\{ 0 , 0 . 0 0 0 5 , 0 . 0 0 1 , 0 . 0 1 , 0 . 0 5 , 0 . 0 1 \}$ . We optimize the attack for 10, 000 iterations using Adam [Kingma and Ba, 2015], with initial learning rate 0.1. We decay the learning rate by a factor of 0.1 at $3 / \bar { 8 } , 5 / 8 , 7 / 8$ of the optimization.
+
+Batch size of the attack. [Zhu et al., 2019, Geiping et al., 2020] have shown that a small batch size is important for the success of the attack. We intentionally evaluate the attack with three small batch sizes to test the upper bound of privacy leakage, including the minimum (and unrealistic) batch size 1, and two small but realistic batch sizes, 16 and 32.
+
+Metrics for reconstruction quality. We visualize reconstructions obtained under different defenses. Following [Yin et al., 2021], we also use the learned perceptual image patch similarity (LPIPS) score [Zhang et al., 2018b] to measure mismatch between reconstruction and original images: higher values suggest more mismatch (less privacy leakage).
+
+![](images/cbc88a741e67fa86b91e694e9db69bdd604257e78a1dc37ece158a4c167d3c24.jpg)  
+Figure 3: Reconstruction results under different defenses with batch size being 1, 16 and 32. When batch size is 32, combining gradient pruning and Intra-InstaHide makes the reconstruction almost unrecognizable (the last column). See Figure 7 in Appendix A for the full version.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">None ，</td><td colspan="6">GradPrune (p)</td><td colspan="2">MixUp (k)</td><td colspan="2">Intra-InstaHide (k)</td><td colspan="2">GradPrune (p= 0.9) + MixUp| + Intra-InstaHide</td></tr><tr><td>0.5</td><td>0.7</td><td>0.9</td><td>0.95</td><td>0.99</td><td>|0.999 |</td><td>4</td><td>6</td><td>4 6</td><td>k=4</td><td>k=4</td></tr><tr><td>Parameter Test Acc.</td><td>93.37|</td><td colspan="2">[93.19 |93.01 90.57|</td><td colspan="2">|89.92 | 88.61</td><td colspan="3">|83.58|92.31|9</td><td colspan="2">90.04 88.20</td><td>91.37</td><td>86.10</td></tr><tr><td>Time (train)</td><td>1×</td><td colspan="2"></td><td colspan="3">1.04×</td><td colspan="2">|90.41 1.06×</td><td colspan="2">1.06×</td><td colspan="2">1.10×</td></tr><tr><td></td><td colspan="14">Attack batch size = 1</td></tr><tr><td rowspan="2">Avg.LPIPS↓ BestLPIPS (LPIPS std.) 0.16</td><td rowspan="2">0.19 0.19 0.02 0.02 0.05</td><td rowspan="2">0.22 0.14 0.22</td><td rowspan="2">0.35</td><td rowspan="2">0.42 0.32</td><td rowspan="2">0.52 0.52 0.36</td><td rowspan="2"></td><td rowspan="2">0.34 0.12</td><td rowspan="2">0.46</td><td rowspan="2"></td><td rowspan="2">0.61</td><td rowspan="2">0.41</td><td rowspan="2">0.60 0.43</td></tr><tr><td>0.58 0.41 0.42 0.06</td></tr><tr><td colspan="10">0.25 0.17 0.16 0.13 0.11 0.08 0.06 0.08 0.07</td><td>0.09</td><td>0.21 0.07</td><td></td><td>0.09</td></tr><tr><td>Avg.LPIPS ↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td>Attack batch size = 16</td><td></td><td></td><td></td><td>0.46</td><td></td><td></td></tr><tr><td rowspan="2">Best LPIPS↓ (LPIPS std.)</td><td>0.45 0.18</td><td>0.46 0.19</td><td>0.47 0.19</td><td>0.51 0.31</td><td>0.55 0.43</td><td>0.58 0.61 0.47 0.51</td><td>0.34 0.11</td><td></td><td>0.31 0.62 0.13 0.41</td><td>0.63 0.44</td><td>0.22</td><td></td><td>0.68 0.54</td></tr><tr><td>0.12</td><td>0.12</td><td>0.11</td><td>0.07</td><td>0.05</td><td>0.04</td><td>0.03</td><td>0.09</td><td>0.09</td><td>0.08</td><td>0.08</td><td>0.10</td><td>0.07</td></tr><tr><td colspan="10">Attack batch size = 32</td><td colspan="3"></td><td></td><td></td></tr><tr><td>Avg.LPIPS↓ BestLPIPS↓</td><td>0.45 0.18</td><td>0.46 0.18</td><td>0.48 0.22</td><td>0.52 0.31</td><td>0.54 0.43</td><td>0.58</td><td>0.63</td><td>0.50</td><td>0.49 0.28</td><td>0.69 0.56</td><td>0.69 0.56</td><td>0.62</td><td>0.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>0.48</td><td>0.54</td><td>0.31</td><td></td><td></td><td></td><td>0.37</td><td>0.65</td></tr><tr><td>(LPIPS std.)</td><td>0.11</td><td>0.11</td><td></td><td>0.07</td><td>0.05</td><td>0.04</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.07</td><td>0.10</td><td></td></tr><tr><td></td><td></td><td></td><td>0.09</td><td></td><td></td><td></td><td></td><td>0.10</td><td></td><td>0.06</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.04</td><td></td><td>0.10</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/48913c3f139daf90cfae8821a03044d6c82c1975b98b52cda2675a66ace6a03b.jpg)  
+Table 2: Utility-security trade-off of different defenses. We train the ResNet-18 model on the whole CIFAR-10 dataset, and report the averaged test accuracy and running time of 5 independent runs. We evaluate the attack on a subset of 50 CIFAR-10 images, and report the LPIPS score $\downarrow \colon$ lower values suggest more privacy leakage). We mark the least-leakage defense measured by the metric in green.
+
+# 5.2 Performance of defense methods
+
+We summarize the performance of each defense in Table 2, and visualize reconstructed images in Figure 3. We report the averaged and the best results for the metric of reconstruction quality, as a proxy for average-case and worst-case privacy leakage.
+
+No defense. Without any defense, when batch size is 1, the attack can recover images well from the gradient. Increasing the batch size makes it difficult to recover well, but the recovered images are visually similar to the originals (see Figure 3).
+
+Gradient pruning (GradPrune). Figure 3 shows that as the pruning ratio $p$ increases, there are more artifacts in the reconstructions. However, the reconstructions are still recognizable even when the pruning ratio $p = 0 . 9$ , thus the previous suggestion of using $p = 0 . 7$ by [Zhu et al., 2019] is no longer safe against the state-of-the-art attack. Our results suggest that, for CIFAR-10, defending the strongest attack with gradient pruning may require the pruning ratio $p \geq 0 . 9 9 9$ . As a trade-off, such a high pruning ratio would introduce an accuracy loss of around $1 0 \%$ (see Table 2).
+
+MixUp. MixUp introduces a small computational overhead to training. MixUp with $k = 4$ only has a minor impact $( \sim 2 \% )$ on test accuracy, but it is not sufficient to defend the gradient inversion attack (see Figure 3). Increasing $k$ from 4 to 6 slightly reduces the leakage, however, the reconstruction is still highly recognizable. This suggests that MixUp alone may not be a practical defense against the state-of-the-art gradient inversion attack.
+
+Intra-InstaHide. Intra-InstaHide with $k = 4$ incurs an extra ${ \sim } 2 \%$ accuracy loss compared with MixUp, but it achieves better defense performance: when batch size is 32, there are obvious artifacts and color shift in the reconstruction (see Figure 3). However, with batch size 32, Intra-InstaHide alone also cannot defend the state-of-the-art gradient inversion, as structures of private images are still vaguely identifiable in reconstructions.
+
+Appendix A provides the whole reconstructed dataset under MixUp and Intra-InstaHide.
+
+# 5.3 Performance of combined defenses
+
+We notice that two types of defenses (i.e perturbing gradient and encoding inputs) are complementary to each other, which motivates an evaluation of combining gradient pruning with MixUp or IntraInstaHide.
+
+As shown in Figure 3, when the batch size is 32, combining Intra-InstaHide $k = 4$ ) with gradient pruning $\gamma = 0 . 9 ,$ ) makes the reconstruction almost unrecognizable. The combined defense yields a higher LPIPS score than using gradient pruning with $p = 0 . 9 9 9$ , but introduces a smaller accuracy loss ${ \sim } 7 \%$ compared with a no-defense pipeline).
+
+Note that our evaluation uses the strongest attack and relatively small batch sizes. As shown in Appendix A, invalidating assumptions in Section 3 or increasing the batch size may hinder the attack with an even weaker defense (e.g. with a lower $p$ , or smaller $k$ ), which gives a better accuracy.
+
+# 5.4 Time estimate for end-to-end recovery of a single image
+
+Table 3 shows time estimates for the end-to-end recovery of a single image in a Federated learning setting with GradPrune or InstaHide defense. We do not estimate for MixUp since it has been shown to be a weak defense (see Section 5.2).
+
+Our time estimates consider three fairly small dataset sizes. The largest size in our estimate is a small fraction of a dataset of ImageNet scale. We consider a client holds a dataset of $N$ private images and participates in Federated learning, which trains a ResNet-18 model with batch size $b = 1 2 8$ . Assumes that the resolution of the client’s data is $3 2 \times 3 2 \times 3$ . If the attacker uses a single NVIDIA GeForce RTX 2080 Ti GPU as his computation resource, and runs gradient inversion with 10,000 iterations of optimization, then $t$ , the running time for attacking a single batch is ${ \sim } 0 . 2 5$ GPU hours (batch size $b$ has little impact on the attack’s running time, but a larger $b$ makes the attack less effective).
+
+Non-defended and gradient pruning. Recovering a single image in a non-defended pipeline (or a pipeline that applies gradient pruning alone as the defense) only requires the attacker to invert gradient of a single step of training, which takes time $t$ .
+
+InstaHide. When InstaHide is applied, the current attack [Carlini et al., 2020] suggests that recovering a single image would involve recovering the whole dataset first. As discussed in Section 4, Carlini et al.’s attack consists of two steps: 1) recover InstaHide images from gradient of $T$ epochs. This would take $( N T / b ) \times t$ GPU hours. 2) Run the decode attack [Carlini et al., 2020] on InstaHide images to recover the private dataset, which involves:
+
+2a Train a neural network to detect similarity in recovered InstaHide images. Assume that training the network requires at least $n$ recovered InstaHide images, then collecting these images by running gradient inversion would take $( n / b ) \times t$ GPU hours. The training takes $1 0 \mathrm { G P U }$ hours according to [Carlini et al., 2020], so training the similarity network would take $( n / b ) \times t + 1 0$ GPU hours in total.
+
+<table><tr><td>Size of client&#x27;s dataset (N)</td><td>No defense</td><td>GradPrune</td><td>InstaHide</td></tr><tr><td>5,000</td><td rowspan="3">0.25</td><td rowspan="3">0.25</td><td>934.48</td></tr><tr><td>50,000</td><td>46.579.01 (~ 5.5 GPU years)</td></tr><tr><td>500,000</td><td>4,215.524.32 (~ 493.4 GPU years)</td></tr></table>
+
+Table 3: Time estimates (NVIDIA GeForce RTX 2080 Ti GPU hours) of recovering a single image from the client’s dataset using the state-of-the-art gradient inversion attack [Geiping et al., 2020] under different defenses. We assume image resolution of the client’s data is $3 2 \times 3 2 \times 3$ .
+
+2b Run the combinotorial algorithm to recover original images. Running time of this step has been shown to be at least quadratic in $m$ , the number of InstaHide encodings [Chen et al., 2021]. This step takes $1 / 6$ GPU hours with $m = 5 \times 1 0 ^ { 3 }$ . Therefore for $m = N T$ , the running time is at least $\begin{array} { r } { 1 / 6 \times ( \frac { N T } { 5 \times 1 0 ^ { 3 } } ) ^ { 2 } } \end{array}$ GPU hours.
+
+In total, an attack on InstaHide in this real-world setting would take $( N T / b ) \times t + ( n / b ) \times t +$ $\begin{array} { r } { 1 0 + 1 / 6 \times ( \frac { N T } { 5 \times 1 0 ^ { 3 } } ) ^ { 2 } } \end{array}$ GPU hours. We use $T = 5 0$ (used by [Carlini et al., 2020]), $n = 1 0 , 0 0 0$ and give estimate in Table 3. As shown, when InstaHide is applied on a small dataset $( N = 5 , 0 0 0 )$ ), the end-to-end recovery of a single image takes $> 3$ , $0 0 0 \times$ longer than in a no-defense pipeline or GradPrune pipeline; when InstaHide is applied on a larger dataset $( N = 5 0 0 , 0 0 0 )$ , the computation cost for end-to-end recovery is enormous.
+
+# 6 Conclusions
+
+This paper first points out that some state-of-the-art gradient inversion attacks have made strong assumptions about knowing BatchNorm statistics and private labels. Relaxing such assumptions can significantly weaken these attacks.
+
+The paper then reports the performance of a set of proposed defenses against gradient inversion attacks, and estimates the computation cost of an end-to-end recovery of a single image in different dataset sizes. Our evaluation shows that InstaHide without mixing with data from a public dataset combined with gradient pruning can defend the state-of-the-art attack, and the estimated time to recover a single image in a medium-size client dataset (e.g. of 500,000 images) is enormous.
+
+Based on our evaluation of the attack by [Geiping et al., 2020] and multiple defenses for plain-text gradients, we have the following observations:
+
+• Using BatchNorm layers in your deep net but don’t share BatchNorm statistics of the private batch during Federated learning weakens the attack. We have demonstrated in Section 3 that exposing BatchNorm statistics to the attacker significantly improves the quality of gradient inversion. So a more secure configuration of Federated Learning would be to use BatchNorm layers, but do not share BatchNorm statistics in training, which has been shown feasible in [Andreux et al., 2020, Li et al., 2021].   
+Using a large batch size weakens the attack; a batch size smaller than 32 is not safe. We have shown that a larger batch size hinders the attack by making it harder to guess the private labels (Section 3) and to recover the private images even with correct private labels (Section 5). Our experiments suggest that even with some weak defenses applied, a batch size smaller than 32 is not safe against the strongest gradient inversion attack.   
+Combining multiple defenses may achieve a better utility-privacy trade-off. In our experiment, for a batch size of 32, combining InstaHide $k = 4$ ) with gradient pruning $( p = 0 . 9 )$ ) achieves the best utility-privacy trade-off, by making the reconstruction almost unrecognizable at a cost of ${ \sim } 7 \%$ accuracy loss (using InstaHide also makes the end-to-end recovery of a single image more computationally expensive). Best parameters would vary for different deep learning tasks, but we strongly encourage Federated learning participants to explore the possibility of combining multiple defensive mechanisms, instead of only using one of them.
+
+We hope to extend our work by including evaluation of defenses for high-resolution images, the attack by [Yin et al., 2021] (when its implementation becomes available), and more defense mechanisms including those rely on adding noise to gradients.

parse/train/0CDKgyYaxC8/0CDKgyYaxC8_content_list.json

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+    {
+        "type": "text",
+        "text": "Evaluating Gradient Inversion Attacks and Defenses in Federated Learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yangsibo Huang   \nPrinceton University   \nPrinceton, NJ 08540   \nyangsibo@princeton.edu ",
+        "bbox": [
+            199,
+            227,
+            390,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Samyak Gupta Princeton University Princeton, NJ 08540 samyakg@cs.princeton.edu ",
+        "bbox": [
+            423,
+            227,
+            630,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhao Song Adobe Research San Jose, CA 95110 zsong@adobe.com ",
+        "bbox": [
+            663,
+            227,
+            799,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kai Li Princeton University Princeton, NJ 08540 li@cs.princeton.edu ",
+        "bbox": [
+            269,
+            303,
+            434,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sanjeev Arora Princeton University Princeton, NJ 08540 arora@cs.princeton.edu ",
+        "bbox": [
+            537,
+            304,
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+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            395,
+            535,
+            411
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gradient inversion attack (or input recovery from gradient) is an emerging threat to the security and privacy preservation of Federated learning, whereby malicious eavesdroppers or participants in the protocol can recover (partially) the clients’ private data. This paper evaluates existing attacks and defenses. We find that some attacks make strong assumptions about the setup. Relaxing such assumptions can substantially weaken these attacks. We then evaluate the benefits of three proposed defense mechanisms against gradient inversion attacks. We show the trade-offs of privacy leakage and data utility of these defense methods, and find that combining them in an appropriate manner makes the attack less effective, even under the original strong assumptions. We also estimate the computation cost of end-to-end recovery of a single image under each evaluated defense. Our findings suggest that the state-of-the-art attacks can currently be defended against with minor data utility loss, as summarized in a list of potential strategies. ",
+        "bbox": [
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+            425,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            628,
+            310,
+            646
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated learning [McMahan et al., 2016, Kairouz et al., 2021] is a framework that allows multiple clients in a distributed environment to collaboratively train a neural network model at a central server, without moving their data to the central server. At every training step, each client computes a model update —i.e., gradient— on its local data using the latest copy of the global model, and then sends the gradient to the central server. The server aggregates these updates (typically by averaging) to construct a global model, and then sends the new model parameters to all clients. By allowing clients to participate in training without directly sharing their data, such protocols align better with data privacy regulations such as Health Insurance Portability and Accountability Act (HIPPA) [Act, 1996], California Consumer Privacy Act (CCPA) [Legislature, 2018], and General Data Protection Regulation [Commission, 2018]. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While sharing gradients was thought to leak little information about the client’s private data, recent papers [Zhu et al., 2019, Zhao et al., 2020, Geiping et al., 2020, Yin et al., 2021] developed a “gradient inversion attack” by which an attacker eavesdropping on a client’s communications with the server can begin to reconstruct the client’s private data. The attacker can also be a malicious participant in the Federated Learning scheme, including a honest-but-curious server who wishes to reconstruct private data of clients, or a honest-but-curious client who wishes to reconstruct private data of other clients. These attacks have been shown to work with batch sizes only up to 100 but even so they have created doubts about the level of privacy ensured in Federated Learning. The current paper seeks to evaluate the risks and suggest ways to minimize them. ",
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+        "page_idx": 0
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+        "type": "text",
+        "text": "",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several defenses against gradient inversion attacks have been proposed. These include perturbing gradients [Zhu et al., 2019, Wei et al., 2020] and using transformation for training data that clients can apply on the fly [Zhang et al., 2018a, Huang et al., 2020]. More traditional cryptographic ideas including secure aggregation [Bonawitz et al., 2016] or homomorphic encryption [Phong et al., 2018] for the gradients can also be used and presumably stop any eavesdropping attacks completely. They will not be studied here due to their special setups and overhead. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We are not aware of a prior systematic evaluation of the level of risk arising from current attacks and the level of security provided by various defenses, as well as the trade-off (if any) between test accuracy, computation overhead, and privacy risks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The paper makes two main contributions. First, we draw attention to two strong assumptions that a current gradient inversion attack [Geiping et al., 2020] implicitly makes. We show that by nullifying these assumptions, the performance of the attack drops significantly and can only work for lowresolution images. The findings are explored in Section 3 and already imply some more secure configurations in Federated Learning (Section 6). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Second, we summarize various defenses (Section 4) and systematically evaluate (Section 5) some of their performance of defending against a state-of-the-art gradient inversion attack, and present their data utility and privacy leakage trade-offs. We estimate the computation cost of end-to-end recovery of a single image under each evaluated defense. We also experimentally demonstrate the feasibility and effectiveness of combined defenses. Our findings are summarized as strategies to further improve Federated Learning’s security against gradient inversion attacks (Section 6). ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In Appendix B, we provide theoretical insights for mechanism of each evaluated defense. ",
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+    {
+        "type": "text",
+        "text": "2 Gradient Inversion Attacks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Previous studies have shown the feasibility of recovering input from gradient (i.e. gradient inversion) for image classification tasks, by formulating it as an optimization problem: given a neural network with parameters $\\theta$ , and the gradient $\\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } )$ computed with a private data batch $( x ^ { * } , y ^ { * } ) \\in$ $\\mathbb { R } ^ { b \\times d } \\times \\mathbb { R } ^ { b }$ $( b , d$ being the batch size, image size), the attacker tries to recover $x \\in \\mathbb { R } ^ { b \\times d }$ , an approximation of $x ^ { * }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/445b6f8320f11c9f4d95e1b472998887c0674dc2aa94943f29fbceb73454ce5f.jpg",
+        "text": "$$\n\\arg \\operatorname* { m i n } _ { x } \\mathcal { L } _ { \\mathrm { g r a d } } ( x ; \\theta , \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } ) ) + \\alpha \\mathcal { R } _ { \\mathrm { a u x } } ( x )\n$$",
+        "text_format": "latex",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The optimization goal consists of two parts: $\\mathcal { L } _ { \\mathrm { g r a d } } ( x ; \\theta , \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } ) )$ enforces matching of the gradient of recovered batch $x$ with the provided gradients $\\mathcal { L } _ { \\boldsymbol { \\theta } } ( x ^ { * } , y ^ { * } )$ , and $\\mathcal { R } _ { \\mathrm { a u x } } ( x )$ regularizes the recovered image based on image prior(s). ",
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+    {
+        "type": "text",
+        "text": "[Phong et al., 2017] brings theoretical insights on this task by proving that such reconstruction is possible with a single-layer neural network. [Zhu et al., 2019] is the first to show that accurate pixellevel reconstruction is practical for a maximum batch size of 8. Their formulation uses $\\ell _ { 2 }$ -distance as $\\mathcal { L } _ { \\mathrm { g r a d } } ( \\cdot , \\cdot )$ but no regularization term $\\mathcal { R } _ { \\mathrm { a u x } } ( x )$ . The approach works for low-resolution CIFAR datasets [Krizhevsky et al., 2009], with simple neural networks with sigmoid activations, but cannot scale up to high-resolution images, or larger models with ReLU activations. A follow-up [Zhao et al., 2020] proposes a simple approach to extract the ground-truth labels from the gradient, which improves the attack but still cannot overcome its limitations. ",
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+    {
+        "type": "text",
+        "text": "With a careful choice of $\\mathcal { L } _ { \\mathrm { g r a d } }$ and $\\mathcal { R } _ { \\mathrm { a u x } } ( x )$ , [Geiping et al., 2020] substantially improves the attack and succeeds in recovering a single ImageNet [Deng et al., 2009] image from gradient: their approach uses cosine distance as $\\mathcal { L } _ { \\mathrm { g r a d } }$ , and the total variation as $\\mathcal { R } _ { \\mathrm { a u x } } ( x )$ . Their approach is able to reconstruct low-resolution images with a maximum batch size of 100 or a single high-resolution image. Based on [Geiping et al., 2020], [Wei et al., 2020] analyzes how different configurations in the training may affect attack effectiveness. ",
+        "bbox": [
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+        "type": "text",
+        "text": "A more recent work [Yin et al., 2021] further improves the attack on high-resolution images, by introducing to $\\mathcal { R } _ { \\mathrm { a u x } } ( x )$ a new image prior term based on batch normalization [Ioffe and Szegedy, 2015] statistics, and a regularization term which enforces consistency across multiple attack trials. ",
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+    {
+        "type": "text",
+        "text": "An orthogonal line of work [Zhu and Blaschko, 2021] proposes to formulate the gradient inversion attack as a closed-form recursive procedure, instead of an optimization problem. However, their implementation can recover only low-resolution images under the setting where batch size $= 1$ . ",
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+    {
+        "type": "text",
+        "text": "3 Strong Assumptions Made by SOTA Attacks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "3.1 The state-of-the-art attacks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Two recent attacks [Geiping et al., 2020, Yin et al., 2021] achieve best recovery results. Our analysis focuses on the former as the implementation of the latter is not available at the time of writing this paper. We plan to include the analysis for the latter attack in the final version of this paper if its implementation becomes available. ",
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+        "type": "text",
+        "text": "[Geiping et al., 2020]’s attack optimizes the following objective function: ",
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+        "type": "equation",
+        "img_path": "images/f5afe4529a02a50d4d6ef877384663abbb0f8d32ca58c94f4a2a3521d5dce154.jpg",
+        "text": "$$\n\\arg \\operatorname* { m i n } _ { x } 1 - \\frac {  \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x , y ) , \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } )  } { \\| \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x , y ) \\| \\| \\| \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } ) \\| } + \\alpha _ { \\mathrm { T V } } \\mathcal { R } _ { \\mathrm { T V } } ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\langle \\cdot , \\cdot \\rangle$ is the inner-product between vectors, and $\\mathcal { R } _ { \\mathrm { T V } } ( \\cdot )$ is the total variation of images. ",
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+        "type": "text",
+        "text": "We notice that Geiping et al. has made two strong assumptions (Section 3.2). Changing setups to invalidate those assumptions will substantially weaken the attacks (Section 3.3). We also summarize whether other attacks have made similar assumptions in Table 1. ",
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+    {
+        "type": "text",
+        "text": "3.2 Strong assumptions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We find that previous gradient inversion attacks have made different assumptions about whether the attacker knows Batch normalization statistics or private labels, as shown in Table 3.2. Note that [Geiping et al., 2020]’s attack makes both strong assumptions. ",
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+        "type": "text",
+        "text": "Assumption 1: Knowing BatchNorm statistics. Batch normalization (BatchNorm) [Ioffe and Szegedy, 2015] is a technique for training neural networks that normalizes the inputs to a layer for every mini-batch. It behaves differently during training and evaluation. Assume the model has $L$ batch normalization layers. Given $x ^ { * }$ , a batch of input images, we use $x _ { l } ^ { * }$ to denote the input features to the $l$ -th BatchNorm layer, where $l \\in [ L ]$ . During training, the $l$ -th BatchNorm layer normalizes $x _ { l } ^ { \\ast }$ based on the batch’s mean $\\mathrm { m e a n } ( x _ { l } ^ { \\ast } )$ and variance $\\mathrm { v a r } ( x _ { l } ^ { * } )$ , and keeps a running estimate of mean and variance of all training data points, denoted by $\\mu _ { l }$ and $\\sigma _ { l } ^ { 2 }$ . During inference, $\\{ \\mu _ { l } \\} _ { l = 1 } ^ { L }$ and $\\{ \\sigma _ { l } ^ { 2 } \\} _ { l = 1 } ^ { L }$ are used to normalize test images. In the following descriptions, we leave out $\\{ \\cdot \\} _ { l = 1 } ^ { L }$ for simplicity (i.e. use $\\mu , \\sigma ^ { 2 }$ to denote $\\{ \\mu _ { l } \\} _ { l = 1 } ^ { L } , \\{ \\sigma _ { l } ^ { 2 } \\} _ { l = 1 } ^ { L }$ , and ${ \\mathrm { m e a n } } ( x ^ { * } )$ , $\\mathrm { v a r } ( x ^ { * } )$ to denote $\\{ \\mathrm { m e a n } ( x _ { l } ^ { * } ) \\} _ { l = 1 } ^ { L }$ , $\\{ \\mathrm { v a r } ( x _ { l } ^ { * } ) \\} _ { l = 1 } ^ { L } )$ . ",
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+    {
+        "type": "text",
+        "text": "We notice that [Geiping et al., 2020]’s implementation1 assumes that BatchNorm statistics of the private batch, i.e., $\\mathrm { m e a n } ( x ^ { * } )$ , $\\mathrm { v a r } ( x ^ { * } )$ , are jointly provided with the gradient. Knowing BatchNorm statistics would enable the attacker to apply the same batch normalization used by the private batch on his recovered batch, to achieve a better reconstruction. This implicitly increases the power of the attacker, as sharing private BatchNorm statistics are not necessary in Federated learning [Andreux et al., 2020, Li et al., 2021]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that this assumption may be realistic in some settings: 1) the neural network is shallow, thus does not require using BatchNorm layers, or 2) the neural network is deep, but adapts approaches that normalize batch inputs with a fixed mean and variance (as alternative to BatchNorm), e.g. Fixup initialization [Zhang et al., 2019]. ",
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+    {
+        "type": "text",
+        "text": "Assumption 2: Knowing or able to infer private labels. Private labels are not intended to be shared in Federated learning, but knowing them would improve the attack. [Zhao et al., 2020] finds that label information of a single private image can be inferred from the gradient (see Section 3.3 for details). Based on this, [Geiping et al., 2020] assumes the attacker knows private labels (see remark at the end of Section 4 in their paper). However, this assumption may not hold true when multiple images in a batch share the same label, as we will show in the next section. ",
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+    {
+        "type": "table",
+        "img_path": "images/0f6c59c4f5b6410a35730c60645b555ca324c3ec0d3bdd34f88d8cccfb75fdfc.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td> Assumptions</td><td></td><td></td><td>[[Zhu et al.,2019] [Zhao et al.,2020] [Geiping et al.,2020][Yin et al.,2021]</td><td></td></tr><tr><td>Knowing BN statistics</td><td>N/A+</td><td>N/A</td><td>Yes</td><td>Yes*</td></tr><tr><td>Knowing private labels</td><td>No</td><td>No</td><td>Yes</td><td>Not</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Assumptions of gradient inversion attacks. †: its evaluation uses a simple model without a BatchNorm layer; $^ \\ddag$ : it proposes a method to infer private labels, which works when images in a batch have unique labels (see Section 3.3); ∗: although the paper discusses a setting where BatchNorm statistics are unknown, its main results assume knowing BatchNorm statistics. ",
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+    {
+        "type": "text",
+        "text": "3.3 Re-evaluation under relaxed assumptions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We re-evaluate the performance of the gradient inversion attack in settings where two assumptions above are relaxed. For each relaxation, we re-design the attack (if needed) based on the knowledge that the attacker has. ",
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+    {
+        "type": "text",
+        "text": "Relaxation 1: Not knowing BatchNorm statistics. We refer to the previous threat model as $\\mathrm { B N } _ { \\mathrm { e x a c t } }$ , where the attacker knows exact BatchNorm statistics of the private batch. We consider a more realistic threat model where these statistics are not exposed, and re-design the attack based on it. ",
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+    {
+        "type": "text",
+        "text": "Threat model. In each training step, the client normalizes its private batch $x ^ { * }$ using the batch’s mean $\\mathrm { m e a n } ( x ^ { * } )$ and variance $\\mathrm { v a r } ( x ^ { * } )$ , keeps the running estimate of mean and variance locally as in [Li et al., 2021], and shares the gradient. The client releases the final aggregated mean $\\mu$ , and aggregated variance $\\sigma ^ { 2 }$ of all training data points at the end of training. Same as before, the attacker has access to the model and the gradient during training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Re-design A: $\\mathrm { B N } _ { \\mathrm { p r o x y } }$ , attacker naively uses $\\mu$ and $\\sigma ^ { 2 }$ . A simple idea is that the attacker uses $( \\mu , \\sigma ^ { 2 } )$ as the proxy for $( \\mathbf { \\dot { m } e a n } ( x ^ { * } ) , \\mathbf { v a r } ( x ^ { * } ) )$ , and uses them to normalize $x$ , his guesses of the private batch. Other operations of the gradient inversion attack remain the same as before. However, Figure 1.d and 1.h show poor-quality reconstruction with this re-design. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$R e$ -design $B$ : $\\mathrm { B N } _ { \\mathrm { i n f e r } }$ , attacker infers $( \\mathrm { m e a n } ( x ^ { * } )$ , $\\mathrm { v a r } ( x ^ { * } ) )$ based on $( \\mu , \\sigma ^ { 2 } )$ . A more reasonable attacker will try to infer $( \\mathrm { m e a n } ( x ^ { * } ) , \\mathrm { v a r } ( x ^ { * } ) )$ while updating $x$ , his guesses of the private batch, and uses $( \\mathrm { m e a n } ( x ) , \\mathrm { v a r } ( x ) )$ to normalize the batch. In this case, $( \\mu , \\sigma ^ { 2 } )$ could be used as a prior of BatchNorm statistics to regularize the recovery, as suggested in [Yin et al., 2021]: ",
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+    {
+        "type": "equation",
+        "img_path": "images/b725de587b248caed9a1845b4fff32c72d1f1933a170d0c313762bc4c34761ba.jpg",
+        "text": "$$\n\\arg \\operatorname* { m i n } _ { x } 1 - \\frac { \\langle \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x , y ) , \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } ) \\rangle } { \\| \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x , y ) \\| \\| \\nabla _ { \\theta } \\mathcal { L } _ { \\theta } ( x ^ { * } , y ^ { * } ) \\| } + \\alpha _ { \\mathrm { T V } } \\mathcal { R } _ { \\mathrm { T V } } ( x ) + \\alpha _ { \\mathrm { B N } } \\mathcal { R } _ { \\mathrm { B N } } ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We tune $\\alpha _ { \\mathrm { B N } }$ and present the best result in Figure 1.c and 1.g (see results of different $\\alpha _ { \\mathrm { B N } }$ ’s in Appendix A). As shown, for a batch of low-resolution images, $\\mathrm { B N } _ { \\mathrm { i n f e r } }$ gives a much better reconstruction result than $\\mathrm { B N } _ { \\mathrm { p r o x y } }$ , but still cannot recover some details of the private batch when compared with $\\mathrm { B N } _ { \\mathrm { e x a c t } }$ . The result for a single high-resolution image is worse: the attacker fails to return a recognizable reconstruction with $\\mathrm { B N } _ { \\mathrm { i n f e r } }$ . This suggests not having access to BatchNorm statistics of the private batch already weakens the state-of-the-art gradient inversion attack. ",
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+    {
+        "type": "image",
+        "img_path": "images/3533a0e819a283dcb807f533392d098ad73413f94673729a43792161630be7cb.jpg",
+        "image_caption": [
+            "Figure 1: Attacking a batch of 16 low-resolution images from CIFAR-10 (a-d) and a single highresolution image from ImageNet (e-h) with different knowledge of BatchNorm statistics. Attack is weakened when BatchNorm statistics are not available (c, d versus b, and g, h versus f). See Appendix A for more examples and quantitative results. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/e45dac0ee12484d36497b1950c3819cc8ca401abbb36add8f1dfda23b0743b84.jpg",
+        "image_caption": [
+            "Figure 2: Attack is weakened when private labels are not available. (a) shows that for CIFAR-10, when the batch size is large, many images in the batch belong to the same class, which essentially weakens label restoration [Zhao et al., 2020, Yin et al., 2021]. (b) visualizes a reconstructed batch of 16 images with and without private labels known. The quality of the reconstruction drops without knowledge of private labels. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Relaxation 2: Not knowing private labels. [Zhao et al., 2020] notes that label information of a single private image can be computed analytically from the gradients of the layer immediately before the output layer. [Yin et al., 2021] further extends this method to support recovery of labels for a batch of images. However, if multiple images in the private batch belong to a same label, neither approach can tell how many images belong to that label, let alone which subset of images belong to that label. Figure 2a demonstrates that with CIFAR-10, for batches of various sizes it is possible for many of the training samples to be have the same label, and the distribution of labels is not uniform - and hence, inferring labels becomes harder and the attack would be weakened. In Figure 2b we evaluate the worst-case for an attacker in this setting by comparing recoveries where the batch labels are simultaneously reconstructed alongside the training samples. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4 Defenses Against the Gradient Inversion Attack ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Several defense ideas have been proposed to mitigate the risks of gradient inversion. ",
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+    {
+        "type": "text",
+        "text": "4.1 Encrypt gradients ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Cryptography-based approaches encrypt gradient to prevent gradient inversion. [Bonawitz et al., 2016] presents a secure aggregation protocol for Federated learning by computing sum of gradient vectors based on secret sharing [Shamir, 1979]. [Phong et al., 2018] proposes using homomorphic encryption to encrypt the gradients before sending. These approaches require special setup and can be costly to implement. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Moreover, with secure aggregation protocol, an honest-but-curious server can still launch the gradient inversion attack on the summed gradient vector. Similarly, an honest-but-curious client can launch the gradient inversion attack on the model returned by the server to reconstruct other clients’ private data, even with homomorphic encryption. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "As alternatives, two other types of defensive mechanisms have been proposed to mitigate the risks of attacks on plain-text gradient. ",
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+    {
+        "type": "text",
+        "text": "4.2 Perturbing gradients ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Gradient pruning. When proposing the first practical gradient inversion attack, [Zhu et al., 2019] also suggests a defense by setting gradients of small magnitudes to zero (i.e. gradient pruning). Based on their attack, they demonstrate that pruning more than $7 0 \\%$ of the gradients would make the recovered images no longer visually recognizable. However, the suggested prune ratio is determined based on weaker attacks, and may not remain safe against the state-of-the-art attack. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Adding noise to gradient. Motivated by DPSGD [Abadi et al., 2016] which adds noise to gradients to achieve differential privacy [Dwork, 2009, Dwork and Roth, 2014], [Zhu et al., 2019, Wei et al., ",
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+    {
+        "type": "text",
+        "text": "2020] also suggests defending by adding Gaussian or Laplacian noise to gradient. They show that a successful defense requires adding high noise level such that its accuracy drops by more than $3 0 \\%$ with CIFAR-10 tasks. Recent works [Papernot et al., 2020, Tramèr and Boneh, 2021] suggest using better pre-training techniques and a large batch size (e.g. 4, 096) to achieve a better accuracy for DPSGD training. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Since most DPSGD implementations for natural image classification tasks [Abadi et al., 2016, Papernot et al., 2020, Tramèr and Boneh, 2021] use a pre-training and fine-tuning pipeline, it is hard to fairly compare with other defense methods that can directly apply when training the model from scratch. Thus, we leave the comparison with DPSGD to future work. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "4.3 Weak encryption of inputs (i.e. encoding inputs) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "MixUp. MixUp data augmentation [Zhang et al., 2018a] trains neural networks on composite images created via linear combination of image pairs. It has been shown to improve the generalization of the neural network and stabilizes the training. Recent work also suggests that MixUp increases the model’s robustness to adversarial examples [Pang et al., 2020, Lamb et al., 2019]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "InstaHide. Inspired by MixUp, [Huang et al., 2020] proposes InstaHide as a light-weight instanceencoding scheme for private distributed learning. To encode an image $x \\in \\mathbb { R } ^ { d }$ from a private dataset, InstaHide first picks $k - 1$ other images $s _ { 2 } , s _ { 3 } , \\ldots , s _ { k }$ from that private dataset, or a large public dataset, and $k$ random nonnegative coefficients $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { k }$ that sum to 1, and creates a composite image $\\textstyle \\lambda _ { 1 } x + \\sum _ { i = 2 } ^ { k } \\lambda _ { i } s _ { i }$ ( $k$ is typically small, e.g., 4). A composite label is also created using the same set of coefficients.2 Then it adds another layer of security: pick a random sign-flipping pattern $\\sigma \\in \\{ - 1 , 1 \\} ^ { d }$ and output the encryption $\\begin{array} { r } { \\tilde { x } = \\sigma \\circ ( \\lambda _ { 1 } x + \\sum _ { i = 2 } ^ { k } \\lambda _ { i } s _ { i } ) } \\end{array}$ , where $\\circ$ is coordinate-wise multiplication of vectors. The neural network is then trained on encoded images, which look like random pixel vectors to the human eye and yet lead to good classification accuracy $\\mathit { \\Theta } _ { \\left( < \\mathrm { 6 \\% } \\right. }$ accuracy loss on CIFAR-10, CIFAR-100, and ImageNet). ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Recently, [Carlini et al., 2020] gives an attack to recover private images of a small dataset, when the InstaHide encodings are revealed to the attacker (not in a Federated learning setting). Their first step is to train a neural network on a public dataset for similarity annotation, to infer whether a pair of InstaHide encodings contain the same private image. With the inferred similarities of all pairs of encodings, the attacker then runs a combinatorial algorithm (cubic time in size of private dataset) to cluster all encodings based on their original private images, and finally uses a regression algorithm (with the help of composite labels) to recover the private images. ",
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+    },
+    {
+        "type": "text",
+        "text": "Neither [Huang et al., 2020] or [Carlini et al., 2020] has evaluated their defense or attack in the Federated learning setting, where the attacker observes gradients of the encoded images instead of original encoded images. This necessitates the systematic evaluation in our next section. ",
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+        "type": "text",
+        "text": "5 Evaluation of defenses ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The main goal of our experiments is to understand the trade-offs between data utility (accuracy) and securely defending the state-of-the-art gradient inversion attack even in its strongest setting, without any relaxation of its implicit assumptions. Specifically, we grant the attacker the knowledge of 1) BatchNorm statistics of the private batch, and 2) labels of the private batch. ",
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+        "type": "text",
+        "text": "We vary key parameters for each defense, and evaluate their performance in terms of the test accuracy, computation overhead, and privacy risks (Section 5.2). We then investigate the feasibility of combining defenses (Section 5.3). We also estimate the computation cost of end-to-end recovery of a single image under evaluated defenses (Section 5.4). ",
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+    {
+        "type": "text",
+        "text": "As the feasibility of the state-of-the-art attack [Geiping et al., 2020] on a batch of high-resolution images remain elusive when its implicit assumptions no longer hold (see Figure 1), we focus on the evaluation with low-resolution in trying to understand whether current attacks can be mitigated. ",
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+    {
+        "type": "text",
+        "text": "5.1 Experimental setup ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Key parameters of defenses. We evaluate following defenses on CIFAR-10 dataset [Krizhevsky et al., 2009] with ResNet-18 architecture [He et al., 2016]. ",
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+        "type": "text",
+        "text": "• GradPrune (gradient pruning): gradient pruning set gradients of small magnitudes to zero. We vary the pruning ratio $p$ in $\\{ 0 . 5 , 0 . 7 , 0 . 9 , 0 . 9 5 , 0 . 9 9 , 0 . 9 9 9 \\}$ .   \n• MixUp: MixUp encodes a private image by linearly combining it with $k - 1$ other images from the training set. Following [Huang et al., 2020], we vary $k$ in $\\{ 4 , 6 \\}$ , and set the upper bound of a single coefficient to 0.65 (coefficients sum to 1).   \n• Intra-InstaHide: InstaHide [Huang et al., 2020] proposes two versions: Inter-InstaHide and Intra-InstaHide. The only difference is that at the mixup step, Inter-Instahide mixes up an image with images from a public dataset, whereas Intra-InstaHide only mixes with private images. Both versions apply a random sign flipping pattern on each mixed image. We evaluate Intra-InstaHide in our experiments, which is a weaker version of InstaHide. Similar to the evaluation of MixUp, we vary $k$ in $\\{ 4 , 6 \\}$ , and set the upper bound of a single coefficient to 0.65. Note that InstaHide flips signs of pixels in the image, which destroys the total variation prior. However, the absolute value of adjacent pixels should still be close. Therefore, for the InstaHide defense, we apply the total variation regularizer on $| x |$ , i.e. taking absolute value of each pixel in the reconstruction. We train the ResNet-18 architecture on CIFAR-10 using different defenses, and launch the attack.   \nWe provide more details of the experiments in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The attack. We use a subset of 50 CIFAR-10 images to evaluate the attack performance. Note that attacking MixUp and InstaHide involves another step to decode private images from the encoded images. We apply [Carlini et al., 2020]’s attack here as the decode step, where the attacker needs to eavesdrop $T$ epochs of training, instead of a single training step. We set $T = 2 0$ in our evaluation. We also grant the attacker the strongest power for the decode step to evaluate the upper bound of privacy leakage. Given a MixUp or Intra-InstaHide image which encodes $k$ private images, we assume the attacker knows: ",
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+    {
+        "type": "text",
+        "text": "1. The indices of $k$ images in the private dataset. In a realistic scenario, the attacker of [Carlini et al., 2020] would need to train a neural network to detect similarity of encodings, and run a combinatorial algorithm to solve an approximation of this mapping.   \n2. The mixing coefficients for each of the $k$ private image. In real Federated learning, this information is not available. ",
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+    },
+    {
+        "type": "text",
+        "text": "Hyper-parameters of the attack. The attack minimize the objective function given in Eq.3. We search $\\alpha _ { \\mathrm { T V } }$ in $\\{ 0 , 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 \\}$ for all defenses, and apply the best choice for each defense: 0.05 for GradPrune, 0.1 for MixUp, and 0.01 for Intra-InstaHide. We apply $\\alpha _ { \\mathrm { B N } } = 0 . 0 0 1$ for all defenses after searching it in $\\{ 0 , 0 . 0 0 0 5 , 0 . 0 0 1 , 0 . 0 1 , 0 . 0 5 , 0 . 0 1 \\}$ . We optimize the attack for 10, 000 iterations using Adam [Kingma and Ba, 2015], with initial learning rate 0.1. We decay the learning rate by a factor of 0.1 at $3 / \\bar { 8 } , 5 / 8 , 7 / 8$ of the optimization. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Batch size of the attack. [Zhu et al., 2019, Geiping et al., 2020] have shown that a small batch size is important for the success of the attack. We intentionally evaluate the attack with three small batch sizes to test the upper bound of privacy leakage, including the minimum (and unrealistic) batch size 1, and two small but realistic batch sizes, 16 and 32. ",
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+    },
+    {
+        "type": "text",
+        "text": "Metrics for reconstruction quality. We visualize reconstructions obtained under different defenses. Following [Yin et al., 2021], we also use the learned perceptual image patch similarity (LPIPS) score [Zhang et al., 2018b] to measure mismatch between reconstruction and original images: higher values suggest more mismatch (less privacy leakage). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/cbc88a741e67fa86b91e694e9db69bdd604257e78a1dc37ece158a4c167d3c24.jpg",
+        "image_caption": [
+            "Figure 3: Reconstruction results under different defenses with batch size being 1, 16 and 32. When batch size is 32, combining gradient pruning and Intra-InstaHide makes the reconstruction almost unrecognizable (the last column). See Figure 7 in Appendix A for the full version. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f536f4dc643bf33436d135e05a625be3731d6a94106d3fb21c4450727da1f518.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">None ，</td><td colspan=\"6\">GradPrune (p)</td><td colspan=\"2\">MixUp (k)</td><td colspan=\"2\">Intra-InstaHide (k)</td><td colspan=\"2\">GradPrune (p= 0.9) + MixUp| + Intra-InstaHide</td></tr><tr><td>0.5</td><td>0.7</td><td>0.9</td><td>0.95</td><td>0.99</td><td>|0.999 |</td><td>4</td><td>6</td><td>4 6</td><td>k=4</td><td>k=4</td></tr><tr><td>Parameter Test Acc.</td><td>93.37|</td><td colspan=\"2\">[93.19 |93.01 90.57|</td><td colspan=\"2\">|89.92 | 88.61</td><td colspan=\"3\">|83.58|92.31|9</td><td colspan=\"2\">90.04 88.20</td><td>91.37</td><td>86.10</td></tr><tr><td>Time (train)</td><td>1×</td><td colspan=\"2\"></td><td colspan=\"3\">1.04×</td><td colspan=\"2\">|90.41 1.06×</td><td colspan=\"2\">1.06×</td><td colspan=\"2\">1.10×</td></tr><tr><td></td><td colspan=\"14\">Attack batch size = 1</td></tr><tr><td rowspan=\"2\">Avg.LPIPS↓ BestLPIPS (LPIPS std.) 0.16</td><td rowspan=\"2\">0.19 0.19 0.02 0.02 0.05</td><td rowspan=\"2\">0.22 0.14 0.22</td><td rowspan=\"2\">0.35</td><td rowspan=\"2\">0.42 0.32</td><td rowspan=\"2\">0.52 0.52 0.36</td><td rowspan=\"2\"></td><td rowspan=\"2\">0.34 0.12</td><td rowspan=\"2\">0.46</td><td rowspan=\"2\"></td><td rowspan=\"2\">0.61</td><td rowspan=\"2\">0.41</td><td rowspan=\"2\">0.60 0.43</td></tr><tr><td>0.58 0.41 0.42 0.06</td></tr><tr><td colspan=\"10\">0.25 0.17 0.16 0.13 0.11 0.08 0.06 0.08 0.07</td><td>0.09</td><td>0.21 0.07</td><td></td><td>0.09</td></tr><tr><td>Avg.LPIPS ↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td>Attack batch size = 16</td><td></td><td></td><td></td><td>0.46</td><td></td><td></td></tr><tr><td rowspan=\"2\">Best LPIPS↓ (LPIPS std.)</td><td>0.45 0.18</td><td>0.46 0.19</td><td>0.47 0.19</td><td>0.51 0.31</td><td>0.55 0.43</td><td>0.58 0.61 0.47 0.51</td><td>0.34 0.11</td><td></td><td>0.31 0.62 0.13 0.41</td><td>0.63 0.44</td><td>0.22</td><td></td><td>0.68 0.54</td></tr><tr><td>0.12</td><td>0.12</td><td>0.11</td><td>0.07</td><td>0.05</td><td>0.04</td><td>0.03</td><td>0.09</td><td>0.09</td><td>0.08</td><td>0.08</td><td>0.10</td><td>0.07</td></tr><tr><td colspan=\"10\">Attack batch size = 32</td><td colspan=\"3\"></td><td></td><td></td></tr><tr><td>Avg.LPIPS↓ BestLPIPS↓</td><td>0.45 0.18</td><td>0.46 0.18</td><td>0.48 0.22</td><td>0.52 0.31</td><td>0.54 0.43</td><td>0.58</td><td>0.63</td><td>0.50</td><td>0.49 0.28</td><td>0.69 0.56</td><td>0.69 0.56</td><td>0.62</td><td>0.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>0.48</td><td>0.54</td><td>0.31</td><td></td><td></td><td></td><td>0.37</td><td>0.65</td></tr><tr><td>(LPIPS std.)</td><td>0.11</td><td>0.11</td><td></td><td>0.07</td><td>0.05</td><td>0.04</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.07</td><td>0.10</td><td></td></tr><tr><td></td><td></td><td></td><td>0.09</td><td></td><td></td><td></td><td></td><td>0.10</td><td></td><td>0.06</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.04</td><td></td><td>0.10</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/48913c3f139daf90cfae8821a03044d6c82c1975b98b52cda2675a66ace6a03b.jpg",
+        "image_caption": [
+            "Table 2: Utility-security trade-off of different defenses. We train the ResNet-18 model on the whole CIFAR-10 dataset, and report the averaged test accuracy and running time of 5 independent runs. We evaluate the attack on a subset of 50 CIFAR-10 images, and report the LPIPS score $\\downarrow \\colon$ lower values suggest more privacy leakage). We mark the least-leakage defense measured by the metric in green. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.2 Performance of defense methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We summarize the performance of each defense in Table 2, and visualize reconstructed images in Figure 3. We report the averaged and the best results for the metric of reconstruction quality, as a proxy for average-case and worst-case privacy leakage. ",
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+    {
+        "type": "text",
+        "text": "No defense. Without any defense, when batch size is 1, the attack can recover images well from the gradient. Increasing the batch size makes it difficult to recover well, but the recovered images are visually similar to the originals (see Figure 3). ",
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+    {
+        "type": "text",
+        "text": "Gradient pruning (GradPrune). Figure 3 shows that as the pruning ratio $p$ increases, there are more artifacts in the reconstructions. However, the reconstructions are still recognizable even when the pruning ratio $p = 0 . 9$ , thus the previous suggestion of using $p = 0 . 7$ by [Zhu et al., 2019] is no longer safe against the state-of-the-art attack. Our results suggest that, for CIFAR-10, defending the strongest attack with gradient pruning may require the pruning ratio $p \\geq 0 . 9 9 9$ . As a trade-off, such a high pruning ratio would introduce an accuracy loss of around $1 0 \\%$ (see Table 2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "MixUp. MixUp introduces a small computational overhead to training. MixUp with $k = 4$ only has a minor impact $( \\sim 2 \\% )$ on test accuracy, but it is not sufficient to defend the gradient inversion attack (see Figure 3). Increasing $k$ from 4 to 6 slightly reduces the leakage, however, the reconstruction is still highly recognizable. This suggests that MixUp alone may not be a practical defense against the state-of-the-art gradient inversion attack. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Intra-InstaHide. Intra-InstaHide with $k = 4$ incurs an extra ${ \\sim } 2 \\%$ accuracy loss compared with MixUp, but it achieves better defense performance: when batch size is 32, there are obvious artifacts and color shift in the reconstruction (see Figure 3). However, with batch size 32, Intra-InstaHide alone also cannot defend the state-of-the-art gradient inversion, as structures of private images are still vaguely identifiable in reconstructions. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Appendix A provides the whole reconstructed dataset under MixUp and Intra-InstaHide. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 Performance of combined defenses ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We notice that two types of defenses (i.e perturbing gradient and encoding inputs) are complementary to each other, which motivates an evaluation of combining gradient pruning with MixUp or IntraInstaHide. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Figure 3, when the batch size is 32, combining Intra-InstaHide $k = 4$ ) with gradient pruning $\\gamma = 0 . 9 ,$ ) makes the reconstruction almost unrecognizable. The combined defense yields a higher LPIPS score than using gradient pruning with $p = 0 . 9 9 9$ , but introduces a smaller accuracy loss ${ \\sim } 7 \\%$ compared with a no-defense pipeline). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that our evaluation uses the strongest attack and relatively small batch sizes. As shown in Appendix A, invalidating assumptions in Section 3 or increasing the batch size may hinder the attack with an even weaker defense (e.g. with a lower $p$ , or smaller $k$ ), which gives a better accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.4 Time estimate for end-to-end recovery of a single image ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 3 shows time estimates for the end-to-end recovery of a single image in a Federated learning setting with GradPrune or InstaHide defense. We do not estimate for MixUp since it has been shown to be a weak defense (see Section 5.2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our time estimates consider three fairly small dataset sizes. The largest size in our estimate is a small fraction of a dataset of ImageNet scale. We consider a client holds a dataset of $N$ private images and participates in Federated learning, which trains a ResNet-18 model with batch size $b = 1 2 8$ . Assumes that the resolution of the client’s data is $3 2 \\times 3 2 \\times 3$ . If the attacker uses a single NVIDIA GeForce RTX 2080 Ti GPU as his computation resource, and runs gradient inversion with 10,000 iterations of optimization, then $t$ , the running time for attacking a single batch is ${ \\sim } 0 . 2 5$ GPU hours (batch size $b$ has little impact on the attack’s running time, but a larger $b$ makes the attack less effective). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Non-defended and gradient pruning. Recovering a single image in a non-defended pipeline (or a pipeline that applies gradient pruning alone as the defense) only requires the attacker to invert gradient of a single step of training, which takes time $t$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "InstaHide. When InstaHide is applied, the current attack [Carlini et al., 2020] suggests that recovering a single image would involve recovering the whole dataset first. As discussed in Section 4, Carlini et al.’s attack consists of two steps: 1) recover InstaHide images from gradient of $T$ epochs. This would take $( N T / b ) \\times t$ GPU hours. 2) Run the decode attack [Carlini et al., 2020] on InstaHide images to recover the private dataset, which involves: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2a Train a neural network to detect similarity in recovered InstaHide images. Assume that training the network requires at least $n$ recovered InstaHide images, then collecting these images by running gradient inversion would take $( n / b ) \\times t$ GPU hours. The training takes $1 0 \\mathrm { G P U }$ hours according to [Carlini et al., 2020], so training the similarity network would take $( n / b ) \\times t + 1 0$ GPU hours in total. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8ad0c592653376326ec82c9030f2d6f5f70777630fa3c452e9d7909b4ff7c9bb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Size of client&#x27;s dataset (N)</td><td>No defense</td><td>GradPrune</td><td>InstaHide</td></tr><tr><td>5,000</td><td rowspan=\"3\">0.25</td><td rowspan=\"3\">0.25</td><td>934.48</td></tr><tr><td>50,000</td><td>46.579.01 (~ 5.5 GPU years)</td></tr><tr><td>500,000</td><td>4,215.524.32 (~ 493.4 GPU years)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Time estimates (NVIDIA GeForce RTX 2080 Ti GPU hours) of recovering a single image from the client’s dataset using the state-of-the-art gradient inversion attack [Geiping et al., 2020] under different defenses. We assume image resolution of the client’s data is $3 2 \\times 3 2 \\times 3$ . ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "2b Run the combinotorial algorithm to recover original images. Running time of this step has been shown to be at least quadratic in $m$ , the number of InstaHide encodings [Chen et al., 2021]. This step takes $1 / 6$ GPU hours with $m = 5 \\times 1 0 ^ { 3 }$ . Therefore for $m = N T$ , the running time is at least $\\begin{array} { r } { 1 / 6 \\times ( \\frac { N T } { 5 \\times 1 0 ^ { 3 } } ) ^ { 2 } } \\end{array}$ GPU hours. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In total, an attack on InstaHide in this real-world setting would take $( N T / b ) \\times t + ( n / b ) \\times t +$ $\\begin{array} { r } { 1 0 + 1 / 6 \\times ( \\frac { N T } { 5 \\times 1 0 ^ { 3 } } ) ^ { 2 } } \\end{array}$ GPU hours. We use $T = 5 0$ (used by [Carlini et al., 2020]), $n = 1 0 , 0 0 0$ and give estimate in Table 3. As shown, when InstaHide is applied on a small dataset $( N = 5 , 0 0 0 )$ ), the end-to-end recovery of a single image takes $> 3$ , $0 0 0 \\times$ longer than in a no-defense pipeline or GradPrune pipeline; when InstaHide is applied on a larger dataset $( N = 5 0 0 , 0 0 0 )$ , the computation cost for end-to-end recovery is enormous. ",
+        "bbox": [
+            173,
+            287,
+            826,
+            375
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            393,
+            307,
+            410
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper first points out that some state-of-the-art gradient inversion attacks have made strong assumptions about knowing BatchNorm statistics and private labels. Relaxing such assumptions can significantly weaken these attacks. ",
+        "bbox": [
+            174,
+            424,
+            825,
+            465
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The paper then reports the performance of a set of proposed defenses against gradient inversion attacks, and estimates the computation cost of an end-to-end recovery of a single image in different dataset sizes. Our evaluation shows that InstaHide without mixing with data from a public dataset combined with gradient pruning can defend the state-of-the-art attack, and the estimated time to recover a single image in a medium-size client dataset (e.g. of 500,000 images) is enormous. ",
+        "bbox": [
+            174,
+            472,
+            825,
+            542
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Based on our evaluation of the attack by [Geiping et al., 2020] and multiple defenses for plain-text gradients, we have the following observations: ",
+        "bbox": [
+            176,
+            547,
+            823,
+            577
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Using BatchNorm layers in your deep net but don’t share BatchNorm statistics of the private batch during Federated learning weakens the attack. We have demonstrated in Section 3 that exposing BatchNorm statistics to the attacker significantly improves the quality of gradient inversion. So a more secure configuration of Federated Learning would be to use BatchNorm layers, but do not share BatchNorm statistics in training, which has been shown feasible in [Andreux et al., 2020, Li et al., 2021].   \nUsing a large batch size weakens the attack; a batch size smaller than 32 is not safe. We have shown that a larger batch size hinders the attack by making it harder to guess the private labels (Section 3) and to recover the private images even with correct private labels (Section 5). Our experiments suggest that even with some weak defenses applied, a batch size smaller than 32 is not safe against the strongest gradient inversion attack.   \nCombining multiple defenses may achieve a better utility-privacy trade-off. In our experiment, for a batch size of 32, combining InstaHide $k = 4$ ) with gradient pruning $( p = 0 . 9 )$ ) achieves the best utility-privacy trade-off, by making the reconstruction almost unrecognizable at a cost of ${ \\sim } 7 \\%$ accuracy loss (using InstaHide also makes the end-to-end recovery of a single image more computationally expensive). Best parameters would vary for different deep learning tasks, but we strongly encourage Federated learning participants to explore the possibility of combining multiple defensive mechanisms, instead of only using one of them. ",
+        "bbox": [
+            217,
+            585,
+            825,
+            858
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We hope to extend our work by including evaluation of defenses for high-resolution images, the attack by [Yin et al., 2021] (when its implementation becomes available), and more defense mechanisms including those rely on adding noise to gradients. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 9
+    }
+]

parse/train/2rAiDBJgR_/2rAiDBJgR_.md

@@ -0,0 +1,471 @@
+# WeaveNet for Approximating Assignment Problems
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Assignment, a task to match a limited number of elements, is a fundamental   
+2 problem in informatics. Many assignment problems have no exact solvers due   
+3 to their NP-hardness or incomplete input, and their approximation algorithms   
+4 have been studied for a long time. However, individual practical applications   
+5 have various objective functions and prior assumptions, which usually differ from   
+6 academic studies. This gap hinders applying the algorithms to real problems   
+7 despite their theoretically ensured performance. In contrast, a learning-based   
+8 method can be a promising solution to fill the gap. To open a new vista for   
+9 real-world assignment problems, we propose a novel neural network architecture,   
+10 WeaveNet. Its core module, feature weaving layer, is stacked to model frequent   
+11 communication between elements in a parameter-efficient way for solving the   
+12 combinatorial problem of assignment. To evaluate the model, we approximated   
+13 one of the most popular non-linear assignment problems, stable matching with two   
+14 different strongly NP-hard settings. The experimental results showed its impressive   
+15 performance among the learning-based baselines. Furthermore, we achieved better   
+16 or comparative performance to the state-of-the-art algorithmic method, depending   
+17 on the size of problem instances.
+
+# 18 1 Introduction
+
+19 From multiple object tracking to job matching, assignment problems can represent a wide variety of   
+20 applications. An assignment problem is typically defined on a bipartite graph, a graph with two sets   
+21 of nodes $A$ and $B$ with edges $E = A \times B$ $\mathbf { \nabla } \cdot N = | A |$ , $M = | B |$ , $N \geq M$ ). On the graph, the task   
+22 is to find a matching $m \in \mathsf { \bar { \{ 0 , 1 \} } } ^ { A \times B }$ (a set of edges represented as a binary matrix) that satisfies   
+23 constraints and/or maximizes objectives. Depending on real-world scenes, there must be various   
+24 objectives and constraints for $m$ . A typical constraint is a one-to-one correspondence (i.e., every node   
+25 has at most one matched partner in $m$ ) and, for simplicity, we always assume it in this paper.   
+26 Matching stability is another example of such constraints. It is a non-linear constraint first introduced   
+7 for a hospital-student assignment problem (Gale and Shapley, $\textcircled { 1 9 6 2 }$ based on the preferences of   
+28 hospitals among students and vice versa. We say a matching $m$ is unstable when there exist $a \in A$   
+29 and $b \in B$ which are unmatched in $m$ ${ m } _ { a b } = 0$ ) but both prefer each other more than their partner   
+30 in $m$ . We can obtain a stable matching $m$ in $O ( N ^ { 2 } )$ by the Gale-Shapley (GS) algorithm $\underline { { \sqrt { \mathrm { G a l e } } } }$   
+and Shapley, 1962). However, when $m$ is expected to have the minimum difference in the total   
+32 satisfactions between sides $A$ and $B$ (known as sex-equal stable matching), the problem becomes   
+33 strongly NP-hard1 (Kato, 1993; McDermid and Irving, 2014)   
+34 In addition to the NP-hardness, we also face difficulties to obtain the best assignment when assignment   
+35 candidates may randomly disappear (e.g., multiple object tracking with occlusions (Emami et al.,   
+36 $\underline { 2 0 2 0 }$ or joint matching in multi-person pose estimation $( \overline { { \mathbb { C } \mathrm { a o } \ e t \ a l . } } , \overline { { 2 0 1 7 } } ) )$ . In such cases, we need   
+37 to compensate for the inputs of incomplete information by its stochastic properties. The traditional   
+38 methods often use sub-optimal approximations to avoid solving complex assignment problems.   
+39 A differential assignment model can be a future option that enables end-to-end training for such   
+40 applications.   
+41 Toward such future applications, this paper aims to propose an effective and promising differential   
+42 solver for assignment problems. The contribution of this paper is four-fold:
+
+1. We proposed WeaveNet, a novel neural network architecture for assignment problems and set-encoder, a novel local structure.   
+2. We proposed a novel technique, split batch normalization, to deal with a strong asymmetry in input distributions for sides $A$ and $B$ .   
+3. We focused on stable matching, a classical non-linear assignment problem actively studied even in recent years, and proposed a novel evaluation protoco $1 ^ { 2 }$ with pseudo costs, which enables us to compare learning-based solvers and algorithmic solvers directly.   
+4. We achieved a better performance with the state-of-the-art algorithmic baseline when $N = 2 0$ , and a comparative performance when $N = 3 0$ . We also outperformed any learning-based baselines with a large margin.
+
+# 53 2 Related work
+
+54 Despite the recent research interest in deep learning technology, we hardly have a fully differential   
+55 assignment solver. As long as authors know, there are two past attempts to solve assignment problems   
+56 by a fully differential model. $\underline { { \mathrm { L i } } } ( \underline { { \mathrm { 2 0 1 9 } } } )$ has tried to solve stable matching by multiple layer perceptrons   
+57 (MLP). Their contribution is in the proposed relaxation of the non-linear stability constraint to a   
+58 differential loss function. However, the MLP is too redundant to learn the assignment strategy without   
+59 overfitting. In addition, the proposed auxiliary loss to maintain the output to be one-to-one matching   
+60 (symmetric doubly stochastic function) overly constrains the solution search space. In this study, we   
+61 propose a parameter-efficient differential model and a weaker but sufficient constraint to output a   
+62 one-to-one matching.   
+63 The second attempt is made by $\boxed { \mathrm { G i b b o n s } ~ e t ~ a l . } \textcircled { 2 0 1 9 } $ , where Deep Bipartite Matching (DBM) is   
+64 proposed. They tested their model with the weapon-target assignment (WTA) problem. WTA is a   
+65 classical NP-hard problem whose state-of-the-art algorithm $( \overline { { \Delta \mathrm { h u j a } ~ e t ~ a l . } } , \overline { { 2 0 0 7 } } )$ could find optimal   
+66 solution when $N \leq 2 0$ in the experiment although there is no theoretical guarantee. In this sense,   
+67 we can consider WTA is empirically easier than sex-equal stable matching, for which we have no   
+68 such efficient solvers even for $N = 5$ . In addition, DBM is trained in a supervised manner or with   
+69 reinforcement learning, which is hard to apply to a larger $N$ . Furthermore, the implementation details   
+70 are not completely explained, and their dataset and source codes are not publicly available. Finally,   
+71 the architecture of DBM is still parameter-redundant, and their local structure is sub-optimal. In this   
+72 study, we propose a more parameter-efficient two-stream architecture, WeaveNet, with a novel local   
+73 structure, set-encoder, both of which have significant impacts on the performance.   
+74 In addition to the above methods, it is natural to consider using graph convolutional networks (GCNs).   
+75 However, there are no GCN methods for assignment problems due to the over-smoothing problem (Li   
+76 et al., 2018; Oono and Suzuki, 2020). Because any graph-convolutional layer summarizes the output   
+77 with neighboring nodes, its smoothing effect eliminates expressive power for node classification. To   
+78 avoid such elimination, GIN $\textcircled { | \textcircled { \times } { \textrm { v e t a l . } } \textcircled { 2 0 1 9 } }$ , the state-of-the-art GCN method, stacks only two   
+79 layers for a node classification task. Such elimination is critical for an assignment-problem solver   
+80 because it needs to identify any slight difference through frequent communication among nodes.   
+81 Unlike GIN, our model retains edge-wise features rather than node-wise summaries, which does not   
+82 cause the smoothing problem. Therefore, we can make the model very deep, which any traditional   
+83 graph convolutional networks cannot.
+
+# 84 3 Stable matching problem as a benchmark task
+
+To evaluate learning-based assignment solvers, we adopt two strongly NP-hard variants of stable matching. They have been actively studied for a long time (Kato, 1993; Iwama et al., 2010; Dworczak,
+
+87 2016; Gupta et al., 2019) and their state-of-the-art algorithm by $\boxed { \mathrm { T z i a v e l i s ~ } e t ~ a l . } ( \boxed { 2 0 1 9 } )$ must be a   
+88 strong baseline against learning-based methods. Hence, we set these two variants as the benchmark   
+89 task for learning-based assignment problems.   
+90 An instance $I$ of a stable matching problem consists of two sets of agents $A$ and $B$ on a bipartite   
+91 graph. Fig. 1 illustrates an example of $I$ . Each agent $a _ { i }$ in $A$ $( 0 < i \leq N )$ has a preference list $p _ { i } ^ { A }$ ,   
+92 which is an ordered set of elements in $B$ and $p _ { i j } ^ { A } = r a n k ( b _ { j } ; p _ { i } ^ { A } )$ is the index of $b _ { j }$ in the list $p _ { i } ^ { A }$ . $a _ { i }$   
+93 prefers $b _ { j }$ to $b _ { j ^ { \prime } }$ if $p _ { i j } ^ { A } < p _ { i j ^ { \prime } } ^ { A }$ . Similarly, each agent $b _ { j }$ in $B ( 0 < j \leq M )$ has a preference list $p _ { j } ^ { B }$ .
+
+$$
+\begin{array} { c c l } { { p _ { i } ^ { A } } } & { { \mathrm { a g e n t s } } } & { { \stackrel { \mathrm { m a t c h i n g ~ } m } { \longrightarrow } } } & { { \stackrel { \mathrm { ( b _ { j } ^ { A } ~ . ~ t h i n g ~ } m } { \longrightarrow } } } & { { p _ { j } ^ { B } } } \\ { { \stackrel {  } { \cong } } } & { { \{ b _ { 2 } , b _ { 1 } , b _ { 3 } \} } } & { { \stackrel { ( a _ { 1 } ) } { \Longleftrightarrow } } } & { { \stackrel { ( b _ { 1 } ) } { \longleftrightarrow } } } &  { ( b _ { 3 } , a _ { 2 } , a _ { 1 } \} } \\ { { \stackrel { \circleddash } { \geq } } } & { { \{ b _ { 3 } , b _ { 2 } , b _ { 1 } \} } } & { { \stackrel { \circleddash } { \Longleftrightarrow } } } & { { \stackrel { \circleddash } { \sum } } } & { { \{ a _ { 1 } , a _ { 2 } , a _ { 3 } \} } } \\ { { \stackrel { \circleddash } { \subseteq } } } & { { \{ b _ { 2 } , b _ { 3 } , b _ { 1 } \} } } & { { \stackrel { \circleddash } { \binom { a _ { 3 } } { 3 } } } } & { { \stackrel { \mathrm { m e t h i n g ~ p a r i n g ~ } m } { \uparrow \mathrm { t h e ~ b l o c k i n g ~ p a i r ~ } m } \stackrel { ( b _ { 3 } ) } { \{ a _ { 1 } , a _ { 3 } , a _ { 2 } \} } } } & { { \stackrel { \circleddash } { \sum } } } \end{array}
+$$
+
+Figure 1: An example of assignment, where $m$ (black edges) is not stable due to the blocking pair (the orange edge), while $m ^ { \prime }$ (green edges) is stable.
+
+94 For a matching $m$ , we say that an unmatched pair $\{ a _ { v } , b _ { w } \} ( m _ { v w } = 0 )$ blocks $m$ if $a _ { v }$ ’s partner   
+95 $b _ { j }$ $m _ { v j } = 1 )$ ) and $b _ { w }$ ’s partner $a _ { i }$ ( $m _ { i w } = 1 $ ) satisfy the conditions $p _ { v w } ^ { A } < p _ { v j } ^ { A }$ and $p _ { w v } ^ { B } < p _ { w i } ^ { B ^ { \ast } }$ . Here,   
+96 $\{ a _ { v } , b _ { w } \}$ is called a blocking pair (the orange edge blocks a matching of black edges in the figure).   
+97 A matching is stable if (and only if) it includes no blocking pair (the green edges in the figure). Note   
+98 that $I$ always has at least one stable matching, and the Gale-Shapley (GS) algorithm can find it in   
+99 $O ( N ^ { 2 } )$ . However, the GS algorithm has a biased nature, where one side is prioritized and the other   
+100 side only gets the least preferable result among all the possibilities of stable matching.   
+101 To compensate for the unfairness, we can introduce diverse objectives to maintain a stable matching   
+102 fair. Among them, the following two objectives make the stable matching problem strongly NP-hard.   
+103 The first one is Sex equality cost $( S E q )$ (Gusfield and Irving, 1989). It focuses on the unfairness   
+104 brought by the gap between the two sides’ satisfaction and defined by
+
+$$
+S E q ( m ; I ) = | P ( m ; A ) - P ( m ; B ) | , \quad P ( m ; A ) = \sum _ { \{ a _ { i } , b _ { j } \} \in m } p _ { i j } ^ { A } , \quad P ( m ; B ) = \sum _ { \{ a _ { i } , b _ { j } \} \in m } p _ { j i } ^ { B } .
+$$
+
+105 The other is Balance cost $( B a l ) ( \mathrm { \underline { { { F e d e r } } } \vert \mathrm { \underline { { { \vert 9 9 5 } } } ; \left[ \mathrm { { G u p t a \it { e t a l . } } \vert \mathrm { \underline { { { 2 0 1 9 } } } } } \right]}  } ,$ , which is a compromise between   
+106 side-equality and overall satisfaction. It is defined by
+
+$$
+B a l ( m ; I ) { = } \operatorname* { m a x } ( P ( m ; A ) , P ( m ; B ) ) .
+$$
+
+107 In the proposed evaluation protocol, we minimize either cost while maintaining stable one-to-one   
+108 matching.   
+109 Input and output data format for stable matching Learning-based approximation is realized by   
+110 a trainable function $F$ that outputs a matching $\hat { m } \in [ 0 , 1 ] ^ { N \times \widetilde { M } }$ , which is an $N \times M$ matrix. As   
+111 for the input, the value range of the preference rank depends on the problem size, which causes a   
+112 range shift of the input distribution. To avoid such shift, we linearly re-scal $ { \mathrm { e } } ^ { 3 }$ the rank of preference   
+113 $p _ { i j } ^ { * }$ $\mathbf { \Psi } ^ { \prime } * \in \{ A , B \} )$ from $[ 1 , N ]$ to a normalized score $s _ { i j } ^ { * }$ ranged in $( 0 , 1 ]$ to make it invariant to $N$ ,   
+114 where 1 for the highest rank. Then, we obtain the input as matrices $S ^ { A }$ and $S ^ { B }$ , where $s _ { i j } ^ { A }$ is the   
+115 $i j$ -element of $S ^ { A }$ .
+
+# 116 4 Deep-learning-based fair stable matching with WeaveNet
+
+# 4.1 WeaveNet
+
+18 One of the required properties of $F : ( S ^ { A } , \ S ^ { B } ) \to { \hat { m } }$ is to take all the agents’ preference into 19 account when determining the presence of each edge in the output $\hat { m }$ . Li (2019) implemented this by 3 The details of this linear re-scaling are based on Li $\bigoplus \iiiint \ M g \lVert \bigstar \rVert$ and described in A.1. Note that sections numbered with capital letters appear in the supplementary material.
+
+![](images/52beb3ce9d8cda69fe2150a714213608ef978ab0c72dceb0d8d4156dca5b9b6b.jpg)  
+Figure 2: WeaveNet architecture. $L$ feature weaving layers are stacked with shortcut paths to be a deep network. The encoded features are fed into Conv $( 1 \times 1 )$ layer to obtain logits $( { \hat { m } } ^ { \prime } { } ^ { \hat { A } } , { \hat { m } } ^ { \prime B } )$ . The output $\hat { m }$ will be binarized in prediction phase to represent a matching.
+
+120 MLPs, where $S ^ { A }$ and $S ^ { B }$ are destructured and concatenated into a single flat vector (with the length   
+121 of $2 N M )$ and fed to the MLP. Its output (a flat vector with the length of $N M$ ) is restructured into a   
+122 matrix $\hat { m }$ . The MLP model, however, would face difficulties due to the following four problems.   
+123 (a) Preference lists of multiple agents are encoded by independent parameters, though they share a   
+124 format so that we could efficiently process them in the same manner.   
+125 (b) MLP only supports a fixed-size input, so training different models for different cases of $N$   
+126 becomes mandatory.   
+127 (c) $F$ should be permutation invariant, which means the matching result should be unchanged even if   
+128 we shuffle the order of agents in $S ^ { A }$ and $S ^ { B }$ , but MLP does not satisfy.   
+129 (d) A shallow MLP model may be insufficient to approximate an exact solver for the NP-hard problem   
+130 when $N$ is large.   
+131 To address the above weaknesses of MLP, we propose the feature weaving network (WeaveNet) which   
+132 has the properties of (a) shared encoder, (b) variable-size input, (c) permutation invariance, and   
+133 (d) residual structure. The WeaveNet, as shown in Fig. $2 ,$ consists of $L$ feature weaving (FW)   
+134 layers. It has two streams of $A$ and $B$ . In a symmetric manner, each stream models the agent’s act of   
+135 selecting the one on the opposite side while sharing weights to enhance the parameter efficiency. The   
+136 shortcut paths at every two FW layers make them residual blocks, which allows the model to be as   
+137 deep as possible. We explain its details as follows.   
+138 Fig. $3$ illustrates the detail of a single FW layer, which is the core architecture of the proposed network.   
+139 FW layer is a two-stream layer whose inputs consist of a weftwise component $Z _ { \ell } ^ { \dot { A } }$ and a warpwise   
+140 component $Z _ { \ell } ^ { B }$ , which are the output of $( l - 1 )$ -th layer and $Z _ { 0 } ^ { A } = { \cal S } ^ { A }$ and $Z _ { 0 } ^ { B } = { \cal S } ^ { B }$ for the first   
+141 layer. The two components are symmetrically concatenated in each stream (cross-concatenation).   
+142 Then these concatenations are separated into agent-wise features, each of which is a set of outgoing  
+143 edge features of an agent (indicating the preference from that agent to every matching candidate).   
+144 These features are processed by the encoder $E _ { \ell }$ shared by every agent in both $A$ and $B$ . As for   
+145 an encoder that can embed variable-size input in a permutation invariant manner, we adopted the   
+146 structure inspired by DeepSet (Zaheer et al., 2017) and PointNet (Qi et al., 2017) (Fig. 4), which   
+47 consists of two convolutional layers with kernel size 1 and a set-wise max-pooling layer, followed by   
+48 batch-normalization and PReLU activation. We refer to this structure as set encoder.   
+149 Mathematical formulations $Z _ { \ell } ^ { A }$ in Fig. $\bigstar$ is a third-order tensor whose dimensions, in sequence,   
+150 corresponding to the agent, candidate, and feature dimension, with a size of $( N , M , D )$ . Similarly,   
+151 $Z _ { \ell } ^ { B }$ has a size of $( M , N , D )$ . The cross-concatenation is defined as
+
+![](images/9417b65197616b5b5286bdb81e8ecf41b452ea48e56713da6b584a9e3a5bf150.jpg)  
+Figure 3: Feature weaving layer orthogonally Figure 4: Illustration of the process in set enconcatenates the weftwise and warpwize com- coder $E _ { \ell }$ , where $z _ { \ell } ^ { a _ { i } }$ (colored in white) is once ponents $( Z _ { \ell } ^ { A }$ and $Z _ { \ell } ^ { B }$ ) in a symmetric way encoded to $D ^ { \prime }$ \` channel features (colored in pale (cross-concatenation). Then, the concatenated blue), then max-pooled to obtain statistics in the tensors are separated into $z _ { \ell } ^ { a _ { i } }$ (or $z _ { \ell } ^ { b _ { j } }$ ), which feature set (colored in blue). The statistics inrepresents a set of outgoing edges from agent formation is concatenated to each input feature $a _ { i }$ (or $b _ { j }$ ), and independently fed to $E _ { \ell }$ . and further encoded (color in a gradation).
+
+$$
+Z _ { \ell } ^ { \prime A } = c a t ( Z _ { \ell } ^ { A } , P _ { A  B } ( Z _ { \ell } ^ { B } ) ) ,
+$$
+
+152 where $P _ { A  B }$ swaps the first and second dimensions of the tensor, and $c a t ( \{ Z _ { 1 } , Z _ { 2 } , . . . \} )$ concatenates   
+153 the features of two tensors $Z _ { 1 }$ , $Z _ { 2 }$ . $Z _ { \ell } ^ { \prime A }$ is then sliced into agent-wise features $z _ { \ell } ^ { a _ { i } }$ and we obtain   
+154 $Z _ { \ell + 1 } ^ { A } = ( E _ { \ell } ( z _ { \ell } ^ { a _ { i } } ) | 0 < i \leq N )$ , which is also a third-order tensor (and fed to the next layer). We can   
+155 calculate $Z _ { \ell + 1 } ^ { B }$ in a symmetric manner (with the same encoder $E _ { \ell }$ ).   
+156 After the process of $L$ FW layers, $Z _ { L } ^ { A }$ and $Z _ { L } ^ { B }$ are further cross-concatenated and fed to the matching   
+157 estimator (in Fig. 2). It outputs a non-deterministic edge assignment $\hat { m }$ . In the training phase, $\hat { m }$   
+158 is input to an objective function, and the loss is minimized. In the prediction phase, the matching   
+159 is obtained by binarizing $\hat { m }$ . In this sense, matching estimation through a neural network can be   
+160 considered as an approximation by relaxing the binary assignment space $\{ \breve { 0 } , 1 \} ^ { N \times M }$ into a continuous   
+161 assignment space [0, 1] N⇥M .   
+162 Asymmetric variant with split batch normalization WeaveNet is designed to be fully symmetric   
+163 for $S ^ { A }$ and $S ^ { B }$ . Hence, it satisfies the equation $F ( S ^ { A } , S ^ { B } ) = F ( S ^ { B } , S ^ { A } ) ^ { \top }$ . This condition ensures   
+164 that the model architecture cannot distinguish the two sides $A$ and $B$ innately. This property is   
+165 beneficial when mathematically fair treatment between $A$ and $B$ is desirable. However, when inputs   
+166 from $A$ and $B$ are differently biased (e.g., the two sides have different trends of preference or the   
+167 objective is asymmetric for $A$ and $B$ ), this symmetric treatment degrades the performance. To   
+168 eliminate the bias difference without losing the parameter-efficiency, we further propose to a) apply   
+169 batch normalization independently for each stream (split batch normalization), and $\mathbf { b }$ ) adding a   
+170 side-identifiable code (e.g., 1 for $A$ and 0 for $B$ ) to $Z _ { 0 } ^ { A }$ and $Z _ { 0 } ^ { B }$ as a $( D { + } 1 )$ -th element of the feature.   
+171 We call this variant “asymmetric”.
+
+# 4.2 Relaxed continuous optimization for fair stable matching
+
+Generally, a combinatorial optimization problem has discrete objective functions and conditions, which are not differentiable. To optimize the model in an end-to-end manner without inaccessible ground truth, we optimize the model by relaxing such discrete loss functions into continuous ones.
+
+76 Assume we target to obtain a fair stable matching that has the minimum $S E q$ , for example. Then, we   
+77 have the following three loss functions.   
+178 ${ \mathcal { L } } _ { m }$ conditions the binarization of $\hat { m }$ to represent a matching.   
+179 $\mathcal { L } _ { s }$ conditions the matching to be stable.   
+180 $\mathcal { L } _ { f }$ minimizing the fairness cost $S E q$ of the matching
+
+181 The overall loss function is defined as
+
+$$
+\mathcal { L } _ { \mathrm { f s m } } ( \hat { m } ) = \lambda _ { m } \mathcal { L } _ { m } + \frac { 1 } { 2 } \sum _ { m \in \{ \hat { m } ^ { A } , \hat { m } ^ { B } \} } \big ( \lambda _ { s } \mathcal { L } _ { s } ( m ) + \lambda _ { f } \mathcal { L } _ { f } ( m ) \big ) ,
+$$
+
+where 182 $\hat { m } ^ { A } = \mathrm { s o f t m a x } ( \hat { m } )$ and $\hat { m } ^ { B } = \mathrm { s o f t m a x } ( \hat { m } ^ { \top } )$
+
+183 An important advantage of learning-based approximation is its flexibility. We can modify the above   
+184 loss functions to easily obtain other variants. For example, removing $\mathcal { L } _ { f }$ in Eq. $( 4 )$ leads to standard   
+185 stable matching, and replacing $\mathcal { L } _ { f }$ with $\mathcal { L } _ { b }$ (which minimizes $B a l$ ) leads to balanced stable matching,   
+186 as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { s m } } ( \hat { m } ) = \lambda _ { m } \mathcal { L } _ { m } + \displaystyle \frac { 1 } { 2 } \sum _ { m \in \{ \hat { m } ^ { A } , \hat { m } ^ { B } \} } \lambda _ { s } \mathcal { L } _ { s } ( m ) , } \\ & { \mathcal { L } _ { \mathrm { b s m } } ( \hat { m } ) = \lambda _ { m } \mathcal { L } _ { m } + \displaystyle \frac { 1 } { 2 } \sum _ { m \in \{ \hat { m } ^ { A } , \hat { m } ^ { B } \} } \big ( \lambda _ { s } \mathcal { L } _ { s } ( m ) + \lambda _ { b } \mathcal { L } _ { b } ( m ) \big ) . } \end{array}
+$$
+
+187 One-to-one matching constraint $\hat { m }$ can be safely converted into a binarized matching by column  
+188 wise or row-wise argmax operation when it is a symmetric doubly stochastic matrix $\check { ( \mathbb { L 1 } , \lfloor 2 0 1 9 \rangle ) }$ . To   
+189 satisfy this condition, we defined ${ \mathcal { L } } _ { m }$ with an average of the cosine distance as
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { m } ( \hat { m } ^ { A } , \hat { m } ^ { B } ) = 1 - \displaystyle \frac { 1 } { 2 } ( \mathbf { C } ( \hat { m } ^ { A } , \hat { m } ^ { B } ) + \mathbf { C } ( \hat { m } ^ { B } , \hat { m } ^ { A } ) ) , } \\ & { \mathbf { C } ( \hat { m } ^ { A } , \hat { m } ^ { B } ) = \displaystyle \frac { 1 } { N } \sum _ { i = 0 } ^ { N } \frac { \hat { m } _ { i * } ^ { A } \cdot \hat { m } _ { * i } ^ { B } } { \| \hat { m } _ { i * } ^ { A } \| _ { 2 } \| \hat { m } _ { * i } ^ { B } \| _ { 2 } } , } \end{array}
+$$
+
+190 where $\hat { m } _ { i * } ^ { A }$ means the $i$ -th row of $\hat { m } ^ { A }$ . This formulation binds $\hat { m }$ to be a symmetric4 doubly stochastic   
+191 ⇤ matrix when $\mathcal { L } _ { m } ( \hat { m } ^ { A } , \hat { m } ^ { B } ) = 0$ . The advantage of this implementation against the original one in Li   
+192 $\textcircled { 2 0 1 9 }$ is described in $\mathbf { B . l }$ with additional experimental results.
+
+193 Blocking pair suppression As for $L _ { s }$ , we used the function proposed in Li $\textcircled { 2 0 1 9 }$ , which is
+
+$$
+\begin{array} { c } { { \mathcal { L } _ { s } ( \hat { m } ; I ) = \displaystyle \sum _ { ( v , w ) \in A \times B } g ( a _ { v } ; b _ { w } , \hat { m } ) g ( b _ { w } ; a _ { v } , \hat { m } ) } } \\ { { g ( a _ { i } ; b _ { w } , \hat { m } ) = \displaystyle \sum _ { b _ { j } \ne b _ { w } } \hat { m } _ { i j } \cdot \operatorname * { m a x } ( S _ { i w } ^ { A } - S _ { i j } ^ { A } , 0 ) } } \\ { { g ( b _ { j } ; a _ { v } , \hat { m } ) = \displaystyle \sum _ { a _ { i } \ne a _ { v } } \hat { m } _ { j i } ^ { \top } \cdot \operatorname * { m a x } ( S _ { j v } ^ { B } - S _ { j i } ^ { B } , 0 ) , } } \end{array}
+$$
+
+194 where $g ( a _ { i } ; b _ { w } , \hat { m } )$ is a criterion known as ex-ante justified envy, which has a positive value when   
+195 $a _ { i }$ prefers $b _ { w }$ more than any $b _ { j }$ in $\{ b _ { j } | j \neq w , \hat { m } _ { i j } \stackrel { . } { > } 0 \}$ . This is the same for $g ( b _ { j } ; a _ { v } , \hat { m } )$ . Hence,   
+196 $\{ a _ { v } , b _ { w } \}$ becomes a (soft) blocking pair when both $g ( a _ { v } ; b _ { w } , \hat { m } )$ and $g ( b _ { w } ; a _ { v } , \hat { m } )$ are positive.
+
+197 Fairness measurements $\mathcal { L } _ { f } , \mathcal { L } _ { b }$ minimize $S E q ( m ; I ) , B a l ( m ; I )$ , respectively, and are defined as
+
+$$
+\mathcal { L } _ { f } ( \hat { m } ; I ) = \frac { 1 } { N } | S ( \hat { m } ; A ) - S ( \hat { m } ; B ) | \quad \mathcal { L } _ { b } ( \hat { m } ; I ) = - \frac { 1 } { N } \mathrm { m i n } ( S ( \hat { m } ; A ) , S ( \hat { m } ; B ) ) ,
+$$
+
+198 where
+
+$$
+S ( \hat { m } ; A ) = \sum _ { i = 1 } ^ { N } \sum _ { i = j } ^ { M } \hat { m } _ { i j } \cdot S _ { i j } ^ { A } , ~ S ( \hat { m } ; B ) = \sum _ { j = 1 } ^ { M } \sum _ { i = 1 } ^ { N } \hat { m } _ { i j } \cdot S _ { j i } ^ { B } .
+$$
+
+# 199 5 Experiments
+
+200 We evaluated WeaveNet with different sizes of $N$ . First, with test samples of $N < 1 0$ , we compared   
+201 its performance with learning-based baselines and optimal solutions obtained by a brute-force search.   
+202 Second, we compared WeaveNet with algorithmic baselines at $N = 2 0$ , 30, where neither existing   
+203 learning-based methods nor brute-force search work. We also demonstrated the generalization ability   
+204 of WeaveNet under the mismatched training/test dataset distributions. Third, we demonstrated the   
+205 performance of WeaveNet at $N = 1 0 0$ . Note that we always assume $M = N$ hereafter.
+
+Sample generation protocol In the experiments, we used the same method as Tziavelis et al.   
+$\underline { { \bar { ( 2 0 1 9 ) } } }$ to generate synthetic datasets that draw preference lists from the following distributions.
+
+Uniform (U) Each agent’s preference towards any matching candidate is totally random, defined by a uniform distribution $\mathcal { U } ( 0 , 1 )$ (larger value means prior in the preference list).
+
+Discrete $\mathbf { \eta } ^ { ( \mathbf { D } ) }$ Each agent has a preference of $\mathcal { U } ( 0 . 5 , 1 )$ towards a certain group of $\lfloor 0 . 4 N \rfloor$ popular candidates, while $\mathcal { U } ( 0 , 0 . 5 )$ towards the rest.
+
+Gauss (G) Each agent’s preference towards $i$ -th candidate is defined by a Gaussian distribution $\mathcal { N } ( i / N , 0 . 4 )$ .
+
+LibimSeTi (Lib) Simulate real rating activity on the online dating service LibimSeTi (Brozovsky and Petricek, 2007) based on the 2D distribution of frequency of each rating pair $( p _ { i j } ^ { A } , ~ p _ { j i } ^ { B } )$ .
+
+4 Here a (possibly non-square) matrix $\hat { m }$ $N \geq M )$ is symmetric if and only if $\hat { m } _ { i * } = \hat { m } _ { * i } , ( 0 < i \leq M )$
+
+216 Choosing the above preference distributions for group $A$ and $B$ respectively, we obtained five different   
+217 dataset settings, namely UU, DD, GG, UD, and Lib. We randomly generated 1,000 test samples and   
+218 1,000 validation samples for each of the five distribution settings.   
+219 Training protocol We trained any learning-based models $2 0 0 \mathrm { k }$ total iterations at $N \leq 3 0$ and $3 0 0 \mathrm { k }$   
+220 at $N = 1 0 0$ , with a batch size of 8. We randomly generated training samples at each iteration based   
+221 on the distribution of each dataset and used the Adam optimizer (Kingma and Ba, 2015). We set   
+222 learning rate 0.0001 and loss weights $\lambda _ { s } = 0 . 7$ , $\lambda _ { m } = 1 . 0$ , $\lambda _ { f } = \lambda _ { b } ^ { - } = 0 . 0 1$ based on a preliminary   
+223 experiment (see $\underline { { \overline { { | \mathbf { A . } 4 } } \mathbf { ) } } }$ .   
+224 Pseudo fairness costs for comparing learning-based results with algorithmic results Note that   
+225 for learning-based methods, there is a trade-off between fairness scores and stable matching rate.   
+226 Hence they may violate the constraints of stable one-to-one matching and yield an $S E q$ or $B a l$ even   
+227 lower than the ideal value. To compare the methods fairly with traditional algorithmic methods, we   
+228 evaluate our methods using pseudo $S E q \ ( p S E q )$ and pseudo Bal $( p B a l )$ cost, in which the cost of   
+229 violation cases is replaced by the worst result of the GS algorithm (prioritizing each side once and   
+230 adopting the worse one).
+
+# 5.1 Comparison with learning-based methods $( N = 3 , 5 , 7 , 9 )$
+
+Baselines and ablations In this experiment, we show results obtained by following baselines and WeaveNet variants. MLP is the model proposed in Li (2019). GIN is the state-of-the-art GCN model proposed in $\left| \mathrm { X u } \ e t \ a l . \right| \left( \mathbb { 2 0 } 1 9 \right)$ . We use each (normalized) preference list as a node feature and bipartite edges as the graph structure. After two graph-convolution calculations, as MLP, we destructed the node-wise embeddings and concatenated them into a single vector, which is fed to one Linear layer to output $\hat { m }$ . DBM is the model in Gibbons et al. (2019). SSWN is the single-stream WeaveNet, which is equivalent to a DBM adopting the set-encoder of WeaveNet. WN is the standard WeaveNet.
+
+![](images/1b2593afc1b1b2178d6e13c5b44d30776c7951a7ed534fe8969eaed4c4d4204b.jpg)  
+Figure 5: Change of the success rates of stable matching $( \uparrow )$ according to $N$
+
+![](images/3efed6a27f3e0ea5f76aedeec8ce97d7e56ffe986c8c73d26d110994f22f0fa9.jpg)  
+Figure 6: Change of $p S E q -$ ideal scores $( \downarrow )$ according to $N$ .
+
+![](images/23cb3aa0f4e1b9312c130ed61b539657cba37a7cc30dda2800a6d347c6961716.jpg)  
+Figure 7: Change of $p B a l -$ ideal scores $( \downarrow )$ according to $N$ .
+
+239 Fig. $\boldsymbol { \vert 5 \vert }$ shows the success rates of finding a stable matching, where we trained models to minimize   
+240 Eq. $( 5 )$ , considering only the stable matching constraints. Since MLP and GIN have size-dependency,   
+241 we trained the models independently for $N = 3$ , 5, 7, 9. The other models were trained with   
+242 $N = 1 0$ and tested on $N = 3$ , 5, 7, 9. We maintained models with $L = 6$ layers (the model names   
+243 are noted as XXX-6) to have a similar number of parameters with MLP for $N = 5$ (see $\underline { { \vert \bar { \mathbf { A . } } \bar { \mathbf { \xi } } ) } }$ , while   
+244 WN-18 is prepared to demonstrate the full performance (with the residual blocks).   
+245 MLP and GIN can hardly find stable matchings when $N \geq 5$ . Note that the number of total cases for   
+246 size $N$ instances is estimated by $N ! ^ { 2 ( N - 1 ) }$ . Hence, when $N = 3$ , there are only 1,296 cases at most,   
+247 and the test set will fully overlap with the training set. In contrast, when $N = 5$ , we have $4 . 3 \times 1 0 ^ { 1 6 }$   
+248 cases, and the overlap is negligible. Therefore, we can say that methods working only with $N = 3$   
+249 such as MLP and GIN, have little generalization ability.   
+250 DBM performs better than MLP but obviously worse than SSWN and WN. The performance gain of   
+251 SSWN-6 over DBM-6 represents the advantage of the set-encoder. Similarly, the improvement of   
+252 WN-6 over SSWN-6 shows the benefit of the two-stream architecture. Finally, that of WN-18 over   
+253 WN-6 demonstrates the impact of stacked layers on the performance. Fig. 15 of the appendix shows   
+54 some additional baselines, including a performance of our $L _ { m }$ against the original one proposed in Li   
+55 $\underline { { \left( 2 0 1 9 \right) } }$ .
+
+Figs. 6 and $\textcircled { 7 }$ show ${ p S E q }$ and $p B a l$ (their difference from the ideal values5 ), respectively. XXX-18f/b are trained to minimize Eqs. $\textcircled { \sharp }$ and $\textcircled{6}$ , respectivel ${ \boldsymbol { \imath } } ^ { 6 } .$ We omitted MLP and GIN due to their poor performance in Fig. 5. In the results, both SSWN and WN largely outperformed DBM, which again proved the advantage of the set-encoder. WN performed better than SSWN for larger $N$ , owing to the parameter efficiency of the two-stream architecture. Note that the performance gain of XXX-18f/b from XXX-18 proved the flexibility of general learning-based methods for customized objective functions.
+
+# 5.2 Comparison with algorithmic methods $N = 2 0$ , 30)
+
+As the algorithmic methods, we prepared four baselines. GS is the better result of applying the GS algorithm to prioritize each side once, which runs in $O ( N ^ { 2 } )$ . PolyMin minimizes some alternative fairness costs (the regret and egalitarian costs, which can be solved in $O ( N ^ { 2 } )$ and $\underline { { O } } ( N ^ { 3 } )$ , respectively (Gusfield, 1987; Irving et al., 1987; Feder, 1992)). DACC by $\boxed { \mathrm { D w o r c z a k } } ( \boxed { 2 0 1 6 } )$ is an approximate algorithm that runs in $\overline { { O ( N ^ { 4 } ) } }$ . PowerBalance is the state-of-the-art method that runs in $\mathcal { \hat { O } } ( N ^ { 2 } )$ .
+
+WN-60f/b(20/30) is WeaveNet with $L = 6 0$ layers trained with samples of $N = 2 0$ and $N = 3 0$ . Note that we used the asymmetric variant for UD and Lib. Moreover, we do not involve any traditional learning-based methods in this part since they scored clear performance drops with increasing $N$ (see Fig. $\textcircled { 5 }$ and the problem size of $N = 2 0$ , 30 is clearly beyond their capabilities, but an ablation with WeaveNet variants is reported in B.2.
+
+Table 1: Average $S E q$ (#) and success rate of stable matching ("). Bold and underlined scores shows the best and second best ones, respectively.   
+
+<table><tr><td rowspan="2">Agents (N × M) Datasets (Dist. Type)</td><td colspan="4">20 × 20</td><td rowspan="2"></td><td colspan="5">30 × 30</td></tr><tr><td>UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib</td></tr><tr><td>GS</td><td>41.89</td><td>18.81</td><td>19.52</td><td>70.97</td><td>19.66</td><td>94.03</td><td>43.46</td><td>36.56</td><td>163.77</td><td>39.78</td></tr><tr><td>PolyMin</td><td>19.93</td><td>11.83</td><td>20.57</td><td>87.08</td><td>18.47</td><td>35.52</td><td>21.21</td><td>37.37</td><td>209.62</td><td>31.85</td></tr><tr><td>DACC</td><td>24.34</td><td>20.13</td><td>23.07</td><td>101.75</td><td>20.40</td><td>40.87</td><td>34.35</td><td>40.59</td><td>240.48</td><td>33.88</td></tr><tr><td>Power Balance</td><td>16.28</td><td>8.93</td><td>17.07</td><td>71.09</td><td>15.40</td><td>18.45</td><td>11.05</td><td>27.22</td><td>163.90</td><td>21.57</td></tr><tr><td>WN-60f(20) (pSEq)</td><td>12.23</td><td>6.37</td><td>15.50</td><td>71.31</td><td>14.59</td><td>25.21</td><td>11.38</td><td>29.36</td><td>172.63</td><td>23.53</td></tr><tr><td>Stably Matched (%)</td><td>98.90</td><td>99.50</td><td>99.40</td><td>99.60</td><td>99.30</td><td>94.60</td><td>97.30</td><td>95.70</td><td>91.30</td><td>97.70</td></tr><tr><td>WN-60f(30) (p SEq)</td><td>12.16</td><td>6.53</td><td>15.56</td><td>71.34</td><td>14.53</td><td>18.30</td><td>10.52</td><td>27.39</td><td>170.35</td><td>22.17</td></tr><tr><td>Stably Matched (%)</td><td>99.10</td><td>99.40</td><td>99.40</td><td>99.50</td><td>99.80</td><td>98.10</td><td>99.00</td><td>98.00</td><td>93.90</td><td>98.60</td></tr></table>
+
+Table 2: Average Bal (#) and success rate of stable matching (").   
+
+<table><tr><td rowspan="2">Agents (N × M) Datasets (Dist. Type)</td><td colspan="4">20×20</td><td colspan="6"></td></tr><tr><td>UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib</td><td>UU</td><td>DD</td><td>30×30 GG</td><td>UD</td><td>Lib</td></tr><tr><td>GS</td><td>89.14</td><td>146.16</td><td>108.36</td><td>140.53</td><td>68.62</td><td>184.05</td><td>322.05</td><td>225.49</td><td>312.12</td><td>137.59</td></tr><tr><td>PolyMin</td><td>74.19</td><td>140.99</td><td>108.04</td><td>145.28</td><td>66.94</td><td>144.48</td><td>306.28</td><td>224.13</td><td>324.54</td><td>130.79</td></tr><tr><td>DACC</td><td>78.49</td><td>146.71</td><td>110.06</td><td>151.34</td><td>68.75</td><td>150.71</td><td>316.18</td><td>227.52</td><td>337.43</td><td>133.59</td></tr><tr><td>Power Balance</td><td>73.28</td><td>140.12</td><td>106.92</td><td>140.55</td><td>65.89</td><td>138.04</td><td>302.30</td><td>220.26</td><td>312.12</td><td>126.96</td></tr><tr><td>WN-60b(20) (pBal)</td><td>71.89</td><td>138.79</td><td>106.20</td><td>140.84</td><td>65.85</td><td>141.49</td><td>302.73</td><td>221.92</td><td>317.60</td><td>130.58</td></tr><tr><td>Stably Matched (%)</td><td>98.50</td><td>98.80</td><td>99.50</td><td>99.70</td><td>98.80</td><td>96.10</td><td>96.70</td><td>95.00</td><td>88.90</td><td>93.80</td></tr><tr><td>WN-60b(30) (pBal)</td><td>72.33</td><td>138.75</td><td>106.65</td><td>140.79</td><td>65.84</td><td>140.40</td><td>301.59</td><td>223.02</td><td>313.59</td><td>127.93</td></tr><tr><td>Stably Matched (%)</td><td>98.00</td><td>99.10</td><td>98.60</td><td>99.80</td><td>99.10</td><td>97.00</td><td>98.60</td><td>93.70</td><td>98.80</td><td>98.00</td></tr></table>
+
+274 We show the results in Tables 1 and 2. When $N = 2 0$ , except for UD, the proposed method constantly   
+275 performed better than any algorithmic methods for both $S E q$ and $B a l$ . When $N = 3 0$ , they are   
+276 comparative. For UD, GS performed even better than PowerBalance. That means that the ideal   
+277 solution constantly prioritizes one side (a kind of the strongest bias). Since we designed the WeaveNet   
+278 architecture to treat the sides evenly, this is the most challenging situation for WeaveNet. Nonetheless,   
+279 the proposed split batch normalization (with the side-identifiable code) achieved similar performance   
+280 to GS and PowerBalance. We show the performance drop with the fully symmetric version in $\boxed { \mathbf { B } . 2 }$   
+281 of the appendix, which is also interesting from the ethical viewpoint. It is noteworthy that the   
+282 model trained with $N = 2 0$ performs well even with $N = 3 0$ , which indicates that the method has   
+283 generalizability for size difference.   
+284 Generalization ability for different distributions A learning-based method should have a certain   
+285 generalizability for input distribution shifts. To test the ability, we evaluated the performance of   
+286 models trained with UU, DD, and GG on test sets of different distributions.
+
+Table 3: The generalizability of WeaveNet (trained/tested with $N = 3 0$ ).   
+
+<table><tr><td>train</td><td>WN-60f</td><td>UU</td><td>test DD</td><td>GG</td><td>Avg.</td></tr><tr><td>UU</td><td>pSEq Stably Matched (%)</td><td>18.30 98.10</td><td>25.81 94.90</td><td>29.09 93.60</td><td>21.10 95.53</td></tr><tr><td>DD</td><td>pSEq Stably Matched (%)</td><td>171.27 2.80</td><td>10.52 99.00</td><td>77.36 0.10</td><td>86.38 33.97</td></tr><tr><td>GG</td><td>pSEq Stably Matched (%)</td><td>21.38 97.30</td><td>12.85 98.10</td><td>27.39 98.00</td><td>20.54 97.80</td></tr></table>
+
+Table 4: Average SEq ( ) and Bal ( ) at $N = 1 0 0$ .   
+
+<table><tr><td>100 × 100,UU</td><td>SEq</td><td>Bal</td></tr><tr><td>GS</td><td>1259.39</td><td>1709.53</td></tr><tr><td>PolyMin</td><td>153.35</td><td>952.85</td></tr><tr><td>DACC</td><td>194.65</td><td>988.02</td></tr><tr><td>Power Balance</td><td>49.41</td><td>909.73</td></tr><tr><td>WN-80f/b+Hungarian</td><td></td><td></td></tr><tr><td>pSEqlpBal</td><td>257.99</td><td>1145.36</td></tr><tr><td>SEqiBal</td><td>68.36</td><td>919.75</td></tr><tr><td>Stably Matched (%)</td><td>89.4</td><td>80.8</td></tr></table>
+
+Table $3$ shows the results. Remarkably, there is a contrast between the model trained with DD and the others. The model with DD could hardly satisfy the one-to-one stable matching constraint when tested on UU/GG, and resulted in poor ${ p S E q }$ scores. In contrast, the model with GG achieved satisfying ${ p S E q }$ scores on UU/DD. Since GG generates preference lists based on a common preference score $( i / N$ for $i$ -th agent) with noise, agents in GG tend to have similar preference lists (i.e., hard to assign optimally). A model trained with such hard samples works well even for the test samples drawn from other distribution. UU has also performed well owing to its non-biased sampling strategy. On the other hand, DD worst performed due to its highly biased generation strategy. From these results, we confirmed that WeaveNet has certain robustness in the distribution shift as long as training samples are competitive enough.
+
+# 5.3 Demonstration with $N = 1 0 0$
+
+We further demonstrate the capability of WeaveNet under a larger size of problem instances, $N = 1 0 0$ In this case, we found that WN-80f and WN-80b failed to yield one-to-one matchings for $1 3 . 4 \%$ and $1 9 . 8 \%$ , respectively (see the Table 9 in B.2 for details). To compensate for this problem, we applied the Hungarian algorithm (Kuhn, 1955) to surely binarize $\hat { m }$ into a one-to-one matching. Table $^ 4$ shows WeaveNet’s relatively good $\overline { { S E q } }$ and $B a l$ scores. Even with the help of the Hungarian algorithm, they were strongly penalized in ${ p S E q }$ and $p B a l$ due to the poor stable matching rate. In other words, we can potentially fill the large gap by better constraining the output.
+
+Since this work is just a pilot study toward a practical differential assignment solver, there is still a lot of space for improvement. The proposed test protocol with stable matching will facilitate it since we can freely adjust the difficulty of the problem to develop and enhance the methods continuously.
+
+# 6 Conclusion
+
+This paper proposed a novel differential assignment solver, WeaveNet, and an evaluation protocol on two strongly NP-hard variants of stable matching. In the experiments, we demonstrated the advantage of set encoder and the two-stream architecture of Weavenet against the other learning-based methods. These techniques also achieved a better performance than the state-of-the-art algorithmic method when $N = 2 0$ and a comparative performance when $N = 3 0$ . Furthermore, the asymmetric variants, split batch normalization with the side-identifiable code, enabled the method to work even with the strongly biased dataset of UD. We also confirmed that the proposed method does not work at $N = 1 0 0$ , which will be an immediate task for this new field of differential assignment solver. We hope that this work becomes a starting point to open a new vista for real-world assignment problems.
+
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+0 Tomás Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45(2):233–284, 1992.   
+32 Tomás Feder. Stable networks and product graphs. American Mathematical Society, 1995.   
+33 David Gale and Lloyd S Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15, 1962.   
+5 Daniel Gibbons, Cheng-Chew Lim, and Peng Shi. Deep learning for bipartite assignment problems. In Proceedings of IEEE International Conference on Systems, Man and Cybernetics, pages 2318–   
+2325, 2019.   
+38 Sushmita Gupta, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. Balanced stable marriage: How close is close enough? In Workshop on Algorithms and Data Structures, pages 423–437, 2019. Dan Gusfield and Robert W Irving. The stable marriage problem: structure and algorithms. MIT press, 1989.   
+42 Dan Gusfield. Three fast algorithms for four problems in stable marriage. SIAM Journal on Computing, 16(1):111–128, 1987.   
+4 Robert W Irving, Paul Leather, and Dan Gusfield. An efficient algorithm for the “optimal” stable marriage. Journal of the ACM, 34(3):532–543, 1987.   
+46 Kazuo Iwama, Shuichi Miyazaki, and Hiroki Yanagisawa. Approximation algorithms for the sexequal stable marriage problem. ACM Transaction on Algorithms, 7(1), 2010.   
+48 Akiko Kato. Complexity of the sex-equal stable marriage problem. Japan Journal of Industrial and Applied Mathematics, 10(1):1, 1993.   
+0 Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of International Conference on Learning Representations, 2015.   
+52 Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83–97, 1955.   
+54 Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Proceedings of AAAI Conference on Artificial Intelligence, pages   
+3538–3545, 2018. Shira Li. Deep Learning for Two-Sided Matching Markets. Bachelor’s thesis, Harvard University,   
+2019.   
+59 Eric McDermid and Robert W Irving. Sex-equal stable matchings: Complexity and exact algorithms. Algorithmica, 68(3):545–570, 2014.   
+61 Kenta Oono and Taiji Suzuki. Graph neural networks exponentially lose expressive power for node classification. In Proceedings of International Conference on Learning Representations, 2020.   
+Charles R. Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, July 2017.   
+Nikolaos Tziavelis, Ioannis Giannakopoulos, Katerina Doka, Nectarios Koziris, and Panagiotis Karras. Equitable stable matchings in quadratic time. In Proceedings of Advances in Neural Information Processing Systems, pages 457–467, 2019.   
+Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In Proceedings of International Conference on Learning Representations, 2019.   
+Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. In Proceedings of Advances in Neural Information Processing Systems, volume 30, pages 3391–3401, 2017.
+
+# 374 Checklist
+
+3751. For all authors...
+
+376 (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contri  
+377 butions and scope? [Yes] The experimental results in Section 5 correspond to the main claims,   
+378 which are summarized as the contribution list in Section 1   
+379 (b) Did you describe the limitations of your work? [Yes] Our method works best among any   
+380 baselines when $N \leq 2 0$ , comparative to the state-of-the-art algorithmic baseline when $N = 3 0$ ,   
+381 but poorly when $N = 1 0 0$ . See Section $\underline { { \boldsymbol { \mathsf { F } . 3 } } }$   
+382 (c) Did you discuss any potential negative societal impacts of your work? [Yes] We briefly   
+383 discussed the fairness/unfairness achieved by our method in the paragraph “Asymmetric variant   
+384 with split batch normalization” in Section $4 . 1 .$ Namely, the asymmetric variant can better   
+385 optimize the objective than the symmetric one (which ensures mathematically equal treatment   
+386 for sides $A$ and $B$ ) but harms the equal treatment of the two sides.   
+387 (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?   
+388 [Yes]
+
+3892. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] We have no main theoretical results. The only theoretical discussion is for the computational cost of WeaveNet in A.2, whose assumption is the network shape explained in the paper.   
+(b) Did you include complete proofs of all theoretical results? [N/A] We have no main theoretical results. The only theoretical discussion is for the computational cost of WeaveNet in $\mathbf { A } . 2 ,$ where we provided enough detailed explanation as a proof.
+
+3963. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have included it in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We described the training details in Section 5 “Training protocol” and Section A.5.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We presented figures with the error bars in Section ${ \bf A . } 4$ of the appendix, which demonstrated the stable behavior of the proposed method against the random seed. For the other part, we have multiple settings, and we observed a stable trend in 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] We described it in the appendix, Section C.
+
+4094. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+410 (a) If your work uses existing assets, did you cite the creators? [Yes] See Section C in the appendix.   
+411 (b) Did you mention the license of the assets? [Yes] See Section C.   
+412 (c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We   
+413 provides a code to random-generate the problem instances, which contains no personal or any   
+414 other sensitive information, but only the distribution parameters of the LibimSeti dataset.   
+415 (d) Did you discuss whether and how consent was obtained from people whose data you’re   
+416 using/curating? [Yes] See Section C.   
+417 (e) Did you discuss whether the data you are using/curating contains personally identifiable   
+418 information or offensive content? [Yes] See Section C.
+
+4195. If you used crowdsourcing or conducted research with human subjects...
+
+420 (a) Did you include the full text of instructions given to participants and screenshots, if applicable?   
+421 [N/A] We used neither crowd-sourcing nor human subjects.   
+422 (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB)   
+423 approvals, if applicable? [N/A]   
+424 (c) Did you include the estimated hourly wage paid to participants and the total amount spent on   
+425 participant compensation? [N/A]

parse/train/2rAiDBJgR_/2rAiDBJgR__content_list.json

@@ -0,0 +1,1317 @@
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+    {
+        "type": "text",
+        "text": "WeaveNet for Approximating Assignment Problems ",
+        "text_level": 1,
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 Assignment, a task to match a limited number of elements, is a fundamental   \n2 problem in informatics. Many assignment problems have no exact solvers due   \n3 to their NP-hardness or incomplete input, and their approximation algorithms   \n4 have been studied for a long time. However, individual practical applications   \n5 have various objective functions and prior assumptions, which usually differ from   \n6 academic studies. This gap hinders applying the algorithms to real problems   \n7 despite their theoretically ensured performance. In contrast, a learning-based   \n8 method can be a promising solution to fill the gap. To open a new vista for   \n9 real-world assignment problems, we propose a novel neural network architecture,   \n10 WeaveNet. Its core module, feature weaving layer, is stacked to model frequent   \n11 communication between elements in a parameter-efficient way for solving the   \n12 combinatorial problem of assignment. To evaluate the model, we approximated   \n13 one of the most popular non-linear assignment problems, stable matching with two   \n14 different strongly NP-hard settings. The experimental results showed its impressive   \n15 performance among the learning-based baselines. Furthermore, we achieved better   \n16 or comparative performance to the state-of-the-art algorithmic method, depending   \n17 on the size of problem instances. ",
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+    {
+        "type": "text",
+        "text": "18 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "19 From multiple object tracking to job matching, assignment problems can represent a wide variety of   \n20 applications. An assignment problem is typically defined on a bipartite graph, a graph with two sets   \n21 of nodes $A$ and $B$ with edges $E = A \\times B$ $\\mathbf { \\nabla } \\cdot N = | A |$ , $M = | B |$ , $N \\geq M$ ). On the graph, the task   \n22 is to find a matching $m \\in \\mathsf { \\bar { \\{ 0 , 1 \\} } } ^ { A \\times B }$ (a set of edges represented as a binary matrix) that satisfies   \n23 constraints and/or maximizes objectives. Depending on real-world scenes, there must be various   \n24 objectives and constraints for $m$ . A typical constraint is a one-to-one correspondence (i.e., every node   \n25 has at most one matched partner in $m$ ) and, for simplicity, we always assume it in this paper.   \n26 Matching stability is another example of such constraints. It is a non-linear constraint first introduced   \n7 for a hospital-student assignment problem (Gale and Shapley, $\\textcircled { 1 9 6 2 }$ based on the preferences of   \n28 hospitals among students and vice versa. We say a matching $m$ is unstable when there exist $a \\in A$   \n29 and $b \\in B$ which are unmatched in $m$ ${ m } _ { a b } = 0$ ) but both prefer each other more than their partner   \n30 in $m$ . We can obtain a stable matching $m$ in $O ( N ^ { 2 } )$ by the Gale-Shapley (GS) algorithm $\\underline { { \\sqrt { \\mathrm { G a l e } } } }$   \nand Shapley, 1962). However, when $m$ is expected to have the minimum difference in the total   \n32 satisfactions between sides $A$ and $B$ (known as sex-equal stable matching), the problem becomes   \n33 strongly NP-hard1 (Kato, 1993; McDermid and Irving, 2014)   \n34 In addition to the NP-hardness, we also face difficulties to obtain the best assignment when assignment   \n35 candidates may randomly disappear (e.g., multiple object tracking with occlusions (Emami et al.,   \n36 $\\underline { 2 0 2 0 }$ or joint matching in multi-person pose estimation $( \\overline { { \\mathbb { C } \\mathrm { a o } \\ e t \\ a l . } } , \\overline { { 2 0 1 7 } } ) )$ . In such cases, we need   \n37 to compensate for the inputs of incomplete information by its stochastic properties. The traditional   \n38 methods often use sub-optimal approximations to avoid solving complex assignment problems.   \n39 A differential assignment model can be a future option that enables end-to-end training for such   \n40 applications.   \n41 Toward such future applications, this paper aims to propose an effective and promising differential   \n42 solver for assignment problems. The contribution of this paper is four-fold: ",
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+    {
+        "type": "text",
+        "text": "1. We proposed WeaveNet, a novel neural network architecture for assignment problems and set-encoder, a novel local structure.   \n2. We proposed a novel technique, split batch normalization, to deal with a strong asymmetry in input distributions for sides $A$ and $B$ .   \n3. We focused on stable matching, a classical non-linear assignment problem actively studied even in recent years, and proposed a novel evaluation protoco $1 ^ { 2 }$ with pseudo costs, which enables us to compare learning-based solvers and algorithmic solvers directly.   \n4. We achieved a better performance with the state-of-the-art algorithmic baseline when $N = 2 0$ , and a comparative performance when $N = 3 0$ . We also outperformed any learning-based baselines with a large margin. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "53 2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            352,
+            316,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "54 Despite the recent research interest in deep learning technology, we hardly have a fully differential   \n55 assignment solver. As long as authors know, there are two past attempts to solve assignment problems   \n56 by a fully differential model. $\\underline { { \\mathrm { L i } } } ( \\underline { { \\mathrm { 2 0 1 9 } } } )$ has tried to solve stable matching by multiple layer perceptrons   \n57 (MLP). Their contribution is in the proposed relaxation of the non-linear stability constraint to a   \n58 differential loss function. However, the MLP is too redundant to learn the assignment strategy without   \n59 overfitting. In addition, the proposed auxiliary loss to maintain the output to be one-to-one matching   \n60 (symmetric doubly stochastic function) overly constrains the solution search space. In this study, we   \n61 propose a parameter-efficient differential model and a weaker but sufficient constraint to output a   \n62 one-to-one matching.   \n63 The second attempt is made by $\\boxed { \\mathrm { G i b b o n s } ~ e t ~ a l . } \\textcircled { 2 0 1 9 } $ , where Deep Bipartite Matching (DBM) is   \n64 proposed. They tested their model with the weapon-target assignment (WTA) problem. WTA is a   \n65 classical NP-hard problem whose state-of-the-art algorithm $( \\overline { { \\Delta \\mathrm { h u j a } ~ e t ~ a l . } } , \\overline { { 2 0 0 7 } } )$ could find optimal   \n66 solution when $N \\leq 2 0$ in the experiment although there is no theoretical guarantee. In this sense,   \n67 we can consider WTA is empirically easier than sex-equal stable matching, for which we have no   \n68 such efficient solvers even for $N = 5$ . In addition, DBM is trained in a supervised manner or with   \n69 reinforcement learning, which is hard to apply to a larger $N$ . Furthermore, the implementation details   \n70 are not completely explained, and their dataset and source codes are not publicly available. Finally,   \n71 the architecture of DBM is still parameter-redundant, and their local structure is sub-optimal. In this   \n72 study, we propose a more parameter-efficient two-stream architecture, WeaveNet, with a novel local   \n73 structure, set-encoder, both of which have significant impacts on the performance.   \n74 In addition to the above methods, it is natural to consider using graph convolutional networks (GCNs).   \n75 However, there are no GCN methods for assignment problems due to the over-smoothing problem (Li   \n76 et al., 2018; Oono and Suzuki, 2020). Because any graph-convolutional layer summarizes the output   \n77 with neighboring nodes, its smoothing effect eliminates expressive power for node classification. To   \n78 avoid such elimination, GIN $\\textcircled { | \\textcircled { \\times } { \\textrm { v e t a l . } } \\textcircled { 2 0 1 9 } }$ , the state-of-the-art GCN method, stacks only two   \n79 layers for a node classification task. Such elimination is critical for an assignment-problem solver   \n80 because it needs to identify any slight difference through frequent communication among nodes.   \n81 Unlike GIN, our model retains edge-wise features rather than node-wise summaries, which does not   \n82 cause the smoothing problem. Therefore, we can make the model very deep, which any traditional   \n83 graph convolutional networks cannot. ",
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+        "type": "text",
+        "text": "84 3 Stable matching problem as a benchmark task ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To evaluate learning-based assignment solvers, we adopt two strongly NP-hard variants of stable matching. They have been actively studied for a long time (Kato, 1993; Iwama et al., 2010; Dworczak, ",
+        "bbox": [
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+        "type": "text",
+        "text": "87 2016; Gupta et al., 2019) and their state-of-the-art algorithm by $\\boxed { \\mathrm { T z i a v e l i s ~ } e t ~ a l . } ( \\boxed { 2 0 1 9 } )$ must be a   \n88 strong baseline against learning-based methods. Hence, we set these two variants as the benchmark   \n89 task for learning-based assignment problems.   \n90 An instance $I$ of a stable matching problem consists of two sets of agents $A$ and $B$ on a bipartite   \n91 graph. Fig. 1 illustrates an example of $I$ . Each agent $a _ { i }$ in $A$ $( 0 < i \\leq N )$ has a preference list $p _ { i } ^ { A }$ ,   \n92 which is an ordered set of elements in $B$ and $p _ { i j } ^ { A } = r a n k ( b _ { j } ; p _ { i } ^ { A } )$ is the index of $b _ { j }$ in the list $p _ { i } ^ { A }$ . $a _ { i }$   \n93 prefers $b _ { j }$ to $b _ { j ^ { \\prime } }$ if $p _ { i j } ^ { A } < p _ { i j ^ { \\prime } } ^ { A }$ . Similarly, each agent $b _ { j }$ in $B ( 0 < j \\leq M )$ has a preference list $p _ { j } ^ { B }$ . ",
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+        "img_path": "images/61a559b69c64397abd5957420ddfd0e787f09873d0fe34175a15454ae91cb9f9.jpg",
+        "text": "$$\n\\begin{array} { c c l } { { p _ { i } ^ { A } } } & { { \\mathrm { a g e n t s } } } & { { \\stackrel { \\mathrm { m a t c h i n g ~ } m } { \\longrightarrow } } } & { { \\stackrel { \\mathrm { ( b _ { j } ^ { A } ~ . ~ t h i n g ~ } m } { \\longrightarrow } } } & { { p _ { j } ^ { B } } } \\\\ { { \\stackrel {  } { \\cong } } } & { { \\{ b _ { 2 } , b _ { 1 } , b _ { 3 } \\} } } & { { \\stackrel { ( a _ { 1 } ) } { \\Longleftrightarrow } } } & { { \\stackrel { ( b _ { 1 } ) } { \\longleftrightarrow } } } &  { ( b _ { 3 } , a _ { 2 } , a _ { 1 } \\} } \\\\ { { \\stackrel { \\circleddash } { \\geq } } } & { { \\{ b _ { 3 } , b _ { 2 } , b _ { 1 } \\} } } & { { \\stackrel { \\circleddash } { \\Longleftrightarrow } } } & { { \\stackrel { \\circleddash } { \\sum } } } & { { \\{ a _ { 1 } , a _ { 2 } , a _ { 3 } \\} } } \\\\ { { \\stackrel { \\circleddash } { \\subseteq } } } & { { \\{ b _ { 2 } , b _ { 3 } , b _ { 1 } \\} } } & { { \\stackrel { \\circleddash } { \\binom { a _ { 3 } } { 3 } } } } & { { \\stackrel { \\mathrm { m e t h i n g ~ p a r i n g ~ } m } { \\uparrow \\mathrm { t h e ~ b l o c k i n g ~ p a i r ~ } m } \\stackrel { ( b _ { 3 } ) } { \\{ a _ { 1 } , a _ { 3 } , a _ { 2 } \\} } } } & { { \\stackrel { \\circleddash } { \\sum } } } \\end{array}\n$$",
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+        "image_caption": [
+            "Figure 1: An example of assignment, where $m$ (black edges) is not stable due to the blocking pair (the orange edge), while $m ^ { \\prime }$ (green edges) is stable. "
+        ],
+        "image_footnote": [],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "94 For a matching $m$ , we say that an unmatched pair $\\{ a _ { v } , b _ { w } \\} ( m _ { v w } = 0 )$ blocks $m$ if $a _ { v }$ ’s partner   \n95 $b _ { j }$ $m _ { v j } = 1 )$ ) and $b _ { w }$ ’s partner $a _ { i }$ ( $m _ { i w } = 1 $ ) satisfy the conditions $p _ { v w } ^ { A } < p _ { v j } ^ { A }$ and $p _ { w v } ^ { B } < p _ { w i } ^ { B ^ { \\ast } }$ . Here,   \n96 $\\{ a _ { v } , b _ { w } \\}$ is called a blocking pair (the orange edge blocks a matching of black edges in the figure).   \n97 A matching is stable if (and only if) it includes no blocking pair (the green edges in the figure). Note   \n98 that $I$ always has at least one stable matching, and the Gale-Shapley (GS) algorithm can find it in   \n99 $O ( N ^ { 2 } )$ . However, the GS algorithm has a biased nature, where one side is prioritized and the other   \n100 side only gets the least preferable result among all the possibilities of stable matching.   \n101 To compensate for the unfairness, we can introduce diverse objectives to maintain a stable matching   \n102 fair. Among them, the following two objectives make the stable matching problem strongly NP-hard.   \n103 The first one is Sex equality cost $( S E q )$ (Gusfield and Irving, 1989). It focuses on the unfairness   \n104 brought by the gap between the two sides’ satisfaction and defined by ",
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+        "text": "$$\nS E q ( m ; I ) = | P ( m ; A ) - P ( m ; B ) | , \\quad P ( m ; A ) = \\sum _ { \\{ a _ { i } , b _ { j } \\} \\in m } p _ { i j } ^ { A } , \\quad P ( m ; B ) = \\sum _ { \\{ a _ { i } , b _ { j } \\} \\in m } p _ { j i } ^ { B } .\n$$",
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+        "text": "105 The other is Balance cost $( B a l ) ( \\mathrm { \\underline { { { F e d e r } } } \\vert \\mathrm { \\underline { { { \\vert 9 9 5 } } } ; \\left[ \\mathrm { { G u p t a \\it { e t a l . } } \\vert \\mathrm { \\underline { { { 2 0 1 9 } } } } } \\right]}  } ,$ , which is a compromise between   \n106 side-equality and overall satisfaction. It is defined by ",
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+        "text": "107 In the proposed evaluation protocol, we minimize either cost while maintaining stable one-to-one   \n108 matching.   \n109 Input and output data format for stable matching Learning-based approximation is realized by   \n110 a trainable function $F$ that outputs a matching $\\hat { m } \\in [ 0 , 1 ] ^ { N \\times \\widetilde { M } }$ , which is an $N \\times M$ matrix. As   \n111 for the input, the value range of the preference rank depends on the problem size, which causes a   \n112 range shift of the input distribution. To avoid such shift, we linearly re-scal $ { \\mathrm { e } } ^ { 3 }$ the rank of preference   \n113 $p _ { i j } ^ { * }$ $\\mathbf { \\Psi } ^ { \\prime } * \\in \\{ A , B \\} )$ from $[ 1 , N ]$ to a normalized score $s _ { i j } ^ { * }$ ranged in $( 0 , 1 ]$ to make it invariant to $N$ ,   \n114 where 1 for the highest rank. Then, we obtain the input as matrices $S ^ { A }$ and $S ^ { B }$ , where $s _ { i j } ^ { A }$ is the   \n115 $i j$ -element of $S ^ { A }$ . ",
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+        "text": "116 4 Deep-learning-based fair stable matching with WeaveNet ",
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+        "text": "4.1 WeaveNet ",
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+        "text": "18 One of the required properties of $F : ( S ^ { A } , \\ S ^ { B } ) \\to { \\hat { m } }$ is to take all the agents’ preference into 19 account when determining the presence of each edge in the output $\\hat { m }$ . Li (2019) implemented this by 3 The details of this linear re-scaling are based on Li $\\bigoplus \\iiiint \\ M g \\lVert \\bigstar \\rVert$ and described in A.1. Note that sections numbered with capital letters appear in the supplementary material. ",
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+        "img_path": "images/52beb3ce9d8cda69fe2150a714213608ef978ab0c72dceb0d8d4156dca5b9b6b.jpg",
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+            "Figure 2: WeaveNet architecture. $L$ feature weaving layers are stacked with shortcut paths to be a deep network. The encoded features are fed into Conv $( 1 \\times 1 )$ layer to obtain logits $( { \\hat { m } } ^ { \\prime } { } ^ { \\hat { A } } , { \\hat { m } } ^ { \\prime B } )$ . The output $\\hat { m }$ will be binarized in prediction phase to represent a matching. "
+        ],
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+        "type": "text",
+        "text": "120 MLPs, where $S ^ { A }$ and $S ^ { B }$ are destructured and concatenated into a single flat vector (with the length   \n121 of $2 N M )$ and fed to the MLP. Its output (a flat vector with the length of $N M$ ) is restructured into a   \n122 matrix $\\hat { m }$ . The MLP model, however, would face difficulties due to the following four problems.   \n123 (a) Preference lists of multiple agents are encoded by independent parameters, though they share a   \n124 format so that we could efficiently process them in the same manner.   \n125 (b) MLP only supports a fixed-size input, so training different models for different cases of $N$   \n126 becomes mandatory.   \n127 (c) $F$ should be permutation invariant, which means the matching result should be unchanged even if   \n128 we shuffle the order of agents in $S ^ { A }$ and $S ^ { B }$ , but MLP does not satisfy.   \n129 (d) A shallow MLP model may be insufficient to approximate an exact solver for the NP-hard problem   \n130 when $N$ is large.   \n131 To address the above weaknesses of MLP, we propose the feature weaving network (WeaveNet) which   \n132 has the properties of (a) shared encoder, (b) variable-size input, (c) permutation invariance, and   \n133 (d) residual structure. The WeaveNet, as shown in Fig. $2 ,$ consists of $L$ feature weaving (FW)   \n134 layers. It has two streams of $A$ and $B$ . In a symmetric manner, each stream models the agent’s act of   \n135 selecting the one on the opposite side while sharing weights to enhance the parameter efficiency. The   \n136 shortcut paths at every two FW layers make them residual blocks, which allows the model to be as   \n137 deep as possible. We explain its details as follows.   \n138 Fig. $3$ illustrates the detail of a single FW layer, which is the core architecture of the proposed network.   \n139 FW layer is a two-stream layer whose inputs consist of a weftwise component $Z _ { \\ell } ^ { \\dot { A } }$ and a warpwise   \n140 component $Z _ { \\ell } ^ { B }$ , which are the output of $( l - 1 )$ -th layer and $Z _ { 0 } ^ { A } = { \\cal S } ^ { A }$ and $Z _ { 0 } ^ { B } = { \\cal S } ^ { B }$ for the first   \n141 layer. The two components are symmetrically concatenated in each stream (cross-concatenation).   \n142 Then these concatenations are separated into agent-wise features, each of which is a set of outgoing  \n143 edge features of an agent (indicating the preference from that agent to every matching candidate).   \n144 These features are processed by the encoder $E _ { \\ell }$ shared by every agent in both $A$ and $B$ . As for   \n145 an encoder that can embed variable-size input in a permutation invariant manner, we adopted the   \n146 structure inspired by DeepSet (Zaheer et al., 2017) and PointNet (Qi et al., 2017) (Fig. 4), which   \n47 consists of two convolutional layers with kernel size 1 and a set-wise max-pooling layer, followed by   \n48 batch-normalization and PReLU activation. We refer to this structure as set encoder.   \n149 Mathematical formulations $Z _ { \\ell } ^ { A }$ in Fig. $\\bigstar$ is a third-order tensor whose dimensions, in sequence,   \n150 corresponding to the agent, candidate, and feature dimension, with a size of $( N , M , D )$ . Similarly,   \n151 $Z _ { \\ell } ^ { B }$ has a size of $( M , N , D )$ . The cross-concatenation is defined as ",
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+            "Figure 3: Feature weaving layer orthogonally Figure 4: Illustration of the process in set enconcatenates the weftwise and warpwize com- coder $E _ { \\ell }$ , where $z _ { \\ell } ^ { a _ { i } }$ (colored in white) is once ponents $( Z _ { \\ell } ^ { A }$ and $Z _ { \\ell } ^ { B }$ ) in a symmetric way encoded to $D ^ { \\prime }$ \\` channel features (colored in pale (cross-concatenation). Then, the concatenated blue), then max-pooled to obtain statistics in the tensors are separated into $z _ { \\ell } ^ { a _ { i } }$ (or $z _ { \\ell } ^ { b _ { j } }$ ), which feature set (colored in blue). The statistics inrepresents a set of outgoing edges from agent formation is concatenated to each input feature $a _ { i }$ (or $b _ { j }$ ), and independently fed to $E _ { \\ell }$ . and further encoded (color in a gradation). "
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+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/47472ca27630d6aef80907cb05ec6cf38890ff6ec6aa48b40d314204e2b5a554.jpg",
+        "text": "$$\nZ _ { \\ell } ^ { \\prime A } = c a t ( Z _ { \\ell } ^ { A } , P _ { A  B } ( Z _ { \\ell } ^ { B } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "152 where $P _ { A  B }$ swaps the first and second dimensions of the tensor, and $c a t ( \\{ Z _ { 1 } , Z _ { 2 } , . . . \\} )$ concatenates   \n153 the features of two tensors $Z _ { 1 }$ , $Z _ { 2 }$ . $Z _ { \\ell } ^ { \\prime A }$ is then sliced into agent-wise features $z _ { \\ell } ^ { a _ { i } }$ and we obtain   \n154 $Z _ { \\ell + 1 } ^ { A } = ( E _ { \\ell } ( z _ { \\ell } ^ { a _ { i } } ) | 0 < i \\leq N )$ , which is also a third-order tensor (and fed to the next layer). We can   \n155 calculate $Z _ { \\ell + 1 } ^ { B }$ in a symmetric manner (with the same encoder $E _ { \\ell }$ ).   \n156 After the process of $L$ FW layers, $Z _ { L } ^ { A }$ and $Z _ { L } ^ { B }$ are further cross-concatenated and fed to the matching   \n157 estimator (in Fig. 2). It outputs a non-deterministic edge assignment $\\hat { m }$ . In the training phase, $\\hat { m }$   \n158 is input to an objective function, and the loss is minimized. In the prediction phase, the matching   \n159 is obtained by binarizing $\\hat { m }$ . In this sense, matching estimation through a neural network can be   \n160 considered as an approximation by relaxing the binary assignment space $\\{ \\breve { 0 } , 1 \\} ^ { N \\times M }$ into a continuous   \n161 assignment space [0, 1] N⇥M .   \n162 Asymmetric variant with split batch normalization WeaveNet is designed to be fully symmetric   \n163 for $S ^ { A }$ and $S ^ { B }$ . Hence, it satisfies the equation $F ( S ^ { A } , S ^ { B } ) = F ( S ^ { B } , S ^ { A } ) ^ { \\top }$ . This condition ensures   \n164 that the model architecture cannot distinguish the two sides $A$ and $B$ innately. This property is   \n165 beneficial when mathematically fair treatment between $A$ and $B$ is desirable. However, when inputs   \n166 from $A$ and $B$ are differently biased (e.g., the two sides have different trends of preference or the   \n167 objective is asymmetric for $A$ and $B$ ), this symmetric treatment degrades the performance. To   \n168 eliminate the bias difference without losing the parameter-efficiency, we further propose to a) apply   \n169 batch normalization independently for each stream (split batch normalization), and $\\mathbf { b }$ ) adding a   \n170 side-identifiable code (e.g., 1 for $A$ and 0 for $B$ ) to $Z _ { 0 } ^ { A }$ and $Z _ { 0 } ^ { B }$ as a $( D { + } 1 )$ -th element of the feature.   \n171 We call this variant “asymmetric”. ",
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+        "type": "text",
+        "text": "4.2 Relaxed continuous optimization for fair stable matching ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Generally, a combinatorial optimization problem has discrete objective functions and conditions, which are not differentiable. To optimize the model in an end-to-end manner without inaccessible ground truth, we optimize the model by relaxing such discrete loss functions into continuous ones. ",
+        "bbox": [
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+        "text": "76 Assume we target to obtain a fair stable matching that has the minimum $S E q$ , for example. Then, we   \n77 have the following three loss functions.   \n178 ${ \\mathcal { L } } _ { m }$ conditions the binarization of $\\hat { m }$ to represent a matching.   \n179 $\\mathcal { L } _ { s }$ conditions the matching to be stable.   \n180 $\\mathcal { L } _ { f }$ minimizing the fairness cost $S E q$ of the matching ",
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+        "type": "text",
+        "text": "181 The overall loss function is defined as ",
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+        "type": "equation",
+        "img_path": "images/c1f8d26616dc870d474e951d973a2b99388e3cfb117a1302eb28fc0c01706f4d.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { f s m } } ( \\hat { m } ) = \\lambda _ { m } \\mathcal { L } _ { m } + \\frac { 1 } { 2 } \\sum _ { m \\in \\{ \\hat { m } ^ { A } , \\hat { m } ^ { B } \\} } \\big ( \\lambda _ { s } \\mathcal { L } _ { s } ( m ) + \\lambda _ { f } \\mathcal { L } _ { f } ( m ) \\big ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 182 $\\hat { m } ^ { A } = \\mathrm { s o f t m a x } ( \\hat { m } )$ and $\\hat { m } ^ { B } = \\mathrm { s o f t m a x } ( \\hat { m } ^ { \\top } )$ ",
+        "bbox": [
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+        "text": "183 An important advantage of learning-based approximation is its flexibility. We can modify the above   \n184 loss functions to easily obtain other variants. For example, removing $\\mathcal { L } _ { f }$ in Eq. $( 4 )$ leads to standard   \n185 stable matching, and replacing $\\mathcal { L } _ { f }$ with $\\mathcal { L } _ { b }$ (which minimizes $B a l$ ) leads to balanced stable matching,   \n186 as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f700b0ebd18b8dca3cefc37c266c51637cf9f36a987bf79145804f7e96c96a04.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { s m } } ( \\hat { m } ) = \\lambda _ { m } \\mathcal { L } _ { m } + \\displaystyle \\frac { 1 } { 2 } \\sum _ { m \\in \\{ \\hat { m } ^ { A } , \\hat { m } ^ { B } \\} } \\lambda _ { s } \\mathcal { L } _ { s } ( m ) , } \\\\ & { \\mathcal { L } _ { \\mathrm { b s m } } ( \\hat { m } ) = \\lambda _ { m } \\mathcal { L } _ { m } + \\displaystyle \\frac { 1 } { 2 } \\sum _ { m \\in \\{ \\hat { m } ^ { A } , \\hat { m } ^ { B } \\} } \\big ( \\lambda _ { s } \\mathcal { L } _ { s } ( m ) + \\lambda _ { b } \\mathcal { L } _ { b } ( m ) \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "187 One-to-one matching constraint $\\hat { m }$ can be safely converted into a binarized matching by column  \n188 wise or row-wise argmax operation when it is a symmetric doubly stochastic matrix $\\check { ( \\mathbb { L 1 } , \\lfloor 2 0 1 9 \\rangle ) }$ . To   \n189 satisfy this condition, we defined ${ \\mathcal { L } } _ { m }$ with an average of the cosine distance as ",
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+    {
+        "type": "equation",
+        "img_path": "images/06630cf0ac9131b456878e945ae7f778eff62d2a9997e436019bc516b31d38d0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { m } ( \\hat { m } ^ { A } , \\hat { m } ^ { B } ) = 1 - \\displaystyle \\frac { 1 } { 2 } ( \\mathbf { C } ( \\hat { m } ^ { A } , \\hat { m } ^ { B } ) + \\mathbf { C } ( \\hat { m } ^ { B } , \\hat { m } ^ { A } ) ) , } \\\\ & { \\mathbf { C } ( \\hat { m } ^ { A } , \\hat { m } ^ { B } ) = \\displaystyle \\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } \\frac { \\hat { m } _ { i * } ^ { A } \\cdot \\hat { m } _ { * i } ^ { B } } { \\| \\hat { m } _ { i * } ^ { A } \\| _ { 2 } \\| \\hat { m } _ { * i } ^ { B } \\| _ { 2 } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "190 where $\\hat { m } _ { i * } ^ { A }$ means the $i$ -th row of $\\hat { m } ^ { A }$ . This formulation binds $\\hat { m }$ to be a symmetric4 doubly stochastic   \n191 ⇤ matrix when $\\mathcal { L } _ { m } ( \\hat { m } ^ { A } , \\hat { m } ^ { B } ) = 0$ . The advantage of this implementation against the original one in Li   \n192 $\\textcircled { 2 0 1 9 }$ is described in $\\mathbf { B . l }$ with additional experimental results. ",
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+        "type": "text",
+        "text": "193 Blocking pair suppression As for $L _ { s }$ , we used the function proposed in Li $\\textcircled { 2 0 1 9 }$ , which is ",
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+    {
+        "type": "equation",
+        "img_path": "images/ac864e04edf63a148f4c1c2465c7b76dd20d098f802d586f5e4dcc625d096a12.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\mathcal { L } _ { s } ( \\hat { m } ; I ) = \\displaystyle \\sum _ { ( v , w ) \\in A \\times B } g ( a _ { v } ; b _ { w } , \\hat { m } ) g ( b _ { w } ; a _ { v } , \\hat { m } ) } } \\\\ { { g ( a _ { i } ; b _ { w } , \\hat { m } ) = \\displaystyle \\sum _ { b _ { j } \\ne b _ { w } } \\hat { m } _ { i j } \\cdot \\operatorname * { m a x } ( S _ { i w } ^ { A } - S _ { i j } ^ { A } , 0 ) } } \\\\ { { g ( b _ { j } ; a _ { v } , \\hat { m } ) = \\displaystyle \\sum _ { a _ { i } \\ne a _ { v } } \\hat { m } _ { j i } ^ { \\top } \\cdot \\operatorname * { m a x } ( S _ { j v } ^ { B } - S _ { j i } ^ { B } , 0 ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "194 where $g ( a _ { i } ; b _ { w } , \\hat { m } )$ is a criterion known as ex-ante justified envy, which has a positive value when   \n195 $a _ { i }$ prefers $b _ { w }$ more than any $b _ { j }$ in $\\{ b _ { j } | j \\neq w , \\hat { m } _ { i j } \\stackrel { . } { > } 0 \\}$ . This is the same for $g ( b _ { j } ; a _ { v } , \\hat { m } )$ . Hence,   \n196 $\\{ a _ { v } , b _ { w } \\}$ becomes a (soft) blocking pair when both $g ( a _ { v } ; b _ { w } , \\hat { m } )$ and $g ( b _ { w } ; a _ { v } , \\hat { m } )$ are positive. ",
+        "bbox": [
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+        "type": "text",
+        "text": "197 Fairness measurements $\\mathcal { L } _ { f } , \\mathcal { L } _ { b }$ minimize $S E q ( m ; I ) , B a l ( m ; I )$ , respectively, and are defined as ",
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+        "img_path": "images/8eca96ddb07b9e1c08d51f58c59537a7598f70459573c3587b66abae79cdee54.jpg",
+        "text": "$$\n\\mathcal { L } _ { f } ( \\hat { m } ; I ) = \\frac { 1 } { N } | S ( \\hat { m } ; A ) - S ( \\hat { m } ; B ) | \\quad \\mathcal { L } _ { b } ( \\hat { m } ; I ) = - \\frac { 1 } { N } \\mathrm { m i n } ( S ( \\hat { m } ; A ) , S ( \\hat { m } ; B ) ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "198 where ",
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+        "img_path": "images/b426b95bb61c692e92ef74ee27d017a077186feadc9fa25836ee5dd11a3c0413.jpg",
+        "text": "$$\nS ( \\hat { m } ; A ) = \\sum _ { i = 1 } ^ { N } \\sum _ { i = j } ^ { M } \\hat { m } _ { i j } \\cdot S _ { i j } ^ { A } , ~ S ( \\hat { m } ; B ) = \\sum _ { j = 1 } ^ { M } \\sum _ { i = 1 } ^ { N } \\hat { m } _ { i j } \\cdot S _ { j i } ^ { B } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "199 5 Experiments ",
+        "text_level": 1,
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+        "text": "200 We evaluated WeaveNet with different sizes of $N$ . First, with test samples of $N < 1 0$ , we compared   \n201 its performance with learning-based baselines and optimal solutions obtained by a brute-force search.   \n202 Second, we compared WeaveNet with algorithmic baselines at $N = 2 0$ , 30, where neither existing   \n203 learning-based methods nor brute-force search work. We also demonstrated the generalization ability   \n204 of WeaveNet under the mismatched training/test dataset distributions. Third, we demonstrated the   \n205 performance of WeaveNet at $N = 1 0 0$ . Note that we always assume $M = N$ hereafter. ",
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+    },
+    {
+        "type": "text",
+        "text": "Sample generation protocol In the experiments, we used the same method as Tziavelis et al.   \n$\\underline { { \\bar { ( 2 0 1 9 ) } } }$ to generate synthetic datasets that draw preference lists from the following distributions. ",
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+        "type": "text",
+        "text": "Uniform (U) Each agent’s preference towards any matching candidate is totally random, defined by a uniform distribution $\\mathcal { U } ( 0 , 1 )$ (larger value means prior in the preference list). ",
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+        "type": "text",
+        "text": "Discrete $\\mathbf { \\eta } ^ { ( \\mathbf { D } ) }$ Each agent has a preference of $\\mathcal { U } ( 0 . 5 , 1 )$ towards a certain group of $\\lfloor 0 . 4 N \\rfloor$ popular candidates, while $\\mathcal { U } ( 0 , 0 . 5 )$ towards the rest. ",
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+    {
+        "type": "text",
+        "text": "Gauss (G) Each agent’s preference towards $i$ -th candidate is defined by a Gaussian distribution $\\mathcal { N } ( i / N , 0 . 4 )$ . ",
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+        "type": "text",
+        "text": "LibimSeTi (Lib) Simulate real rating activity on the online dating service LibimSeTi (Brozovsky and Petricek, 2007) based on the 2D distribution of frequency of each rating pair $( p _ { i j } ^ { A } , ~ p _ { j i } ^ { B } )$ . ",
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+        "type": "text",
+        "text": "4 Here a (possibly non-square) matrix $\\hat { m }$ $N \\geq M )$ is symmetric if and only if $\\hat { m } _ { i * } = \\hat { m } _ { * i } , ( 0 < i \\leq M )$ ",
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+        "type": "text",
+        "text": "216 Choosing the above preference distributions for group $A$ and $B$ respectively, we obtained five different   \n217 dataset settings, namely UU, DD, GG, UD, and Lib. We randomly generated 1,000 test samples and   \n218 1,000 validation samples for each of the five distribution settings.   \n219 Training protocol We trained any learning-based models $2 0 0 \\mathrm { k }$ total iterations at $N \\leq 3 0$ and $3 0 0 \\mathrm { k }$   \n220 at $N = 1 0 0$ , with a batch size of 8. We randomly generated training samples at each iteration based   \n221 on the distribution of each dataset and used the Adam optimizer (Kingma and Ba, 2015). We set   \n222 learning rate 0.0001 and loss weights $\\lambda _ { s } = 0 . 7$ , $\\lambda _ { m } = 1 . 0$ , $\\lambda _ { f } = \\lambda _ { b } ^ { - } = 0 . 0 1$ based on a preliminary   \n223 experiment (see $\\underline { { \\overline { { | \\mathbf { A . } 4 } } \\mathbf { ) } } }$ .   \n224 Pseudo fairness costs for comparing learning-based results with algorithmic results Note that   \n225 for learning-based methods, there is a trade-off between fairness scores and stable matching rate.   \n226 Hence they may violate the constraints of stable one-to-one matching and yield an $S E q$ or $B a l$ even   \n227 lower than the ideal value. To compare the methods fairly with traditional algorithmic methods, we   \n228 evaluate our methods using pseudo $S E q \\ ( p S E q )$ and pseudo Bal $( p B a l )$ cost, in which the cost of   \n229 violation cases is replaced by the worst result of the GS algorithm (prioritizing each side once and   \n230 adopting the worse one). ",
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+    {
+        "type": "text",
+        "text": "5.1 Comparison with learning-based methods $( N = 3 , 5 , 7 , 9 )$ ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines and ablations In this experiment, we show results obtained by following baselines and WeaveNet variants. MLP is the model proposed in Li (2019). GIN is the state-of-the-art GCN model proposed in $\\left| \\mathrm { X u } \\ e t \\ a l . \\right| \\left( \\mathbb { 2 0 } 1 9 \\right)$ . We use each (normalized) preference list as a node feature and bipartite edges as the graph structure. After two graph-convolution calculations, as MLP, we destructed the node-wise embeddings and concatenated them into a single vector, which is fed to one Linear layer to output $\\hat { m }$ . DBM is the model in Gibbons et al. (2019). SSWN is the single-stream WeaveNet, which is equivalent to a DBM adopting the set-encoder of WeaveNet. WN is the standard WeaveNet. ",
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+    {
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+        "img_path": "images/1b2593afc1b1b2178d6e13c5b44d30776c7951a7ed534fe8969eaed4c4d4204b.jpg",
+        "image_caption": [
+            "Figure 5: Change of the success rates of stable matching $( \\uparrow )$ according to $N$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/3efed6a27f3e0ea5f76aedeec8ce97d7e56ffe986c8c73d26d110994f22f0fa9.jpg",
+        "image_caption": [
+            "Figure 6: Change of $p S E q -$ ideal scores $( \\downarrow )$ according to $N$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/23cb3aa0f4e1b9312c130ed61b539657cba37a7cc30dda2800a6d347c6961716.jpg",
+        "image_caption": [
+            "Figure 7: Change of $p B a l -$ ideal scores $( \\downarrow )$ according to $N$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "239 Fig. $\\boldsymbol { \\vert 5 \\vert }$ shows the success rates of finding a stable matching, where we trained models to minimize   \n240 Eq. $( 5 )$ , considering only the stable matching constraints. Since MLP and GIN have size-dependency,   \n241 we trained the models independently for $N = 3$ , 5, 7, 9. The other models were trained with   \n242 $N = 1 0$ and tested on $N = 3$ , 5, 7, 9. We maintained models with $L = 6$ layers (the model names   \n243 are noted as XXX-6) to have a similar number of parameters with MLP for $N = 5$ (see $\\underline { { \\vert \\bar { \\mathbf { A . } } \\bar { \\mathbf { \\xi } } ) } }$ , while   \n244 WN-18 is prepared to demonstrate the full performance (with the residual blocks).   \n245 MLP and GIN can hardly find stable matchings when $N \\geq 5$ . Note that the number of total cases for   \n246 size $N$ instances is estimated by $N ! ^ { 2 ( N - 1 ) }$ . Hence, when $N = 3$ , there are only 1,296 cases at most,   \n247 and the test set will fully overlap with the training set. In contrast, when $N = 5$ , we have $4 . 3 \\times 1 0 ^ { 1 6 }$   \n248 cases, and the overlap is negligible. Therefore, we can say that methods working only with $N = 3$   \n249 such as MLP and GIN, have little generalization ability.   \n250 DBM performs better than MLP but obviously worse than SSWN and WN. The performance gain of   \n251 SSWN-6 over DBM-6 represents the advantage of the set-encoder. Similarly, the improvement of   \n252 WN-6 over SSWN-6 shows the benefit of the two-stream architecture. Finally, that of WN-18 over   \n253 WN-6 demonstrates the impact of stacked layers on the performance. Fig. 15 of the appendix shows   \n54 some additional baselines, including a performance of our $L _ { m }$ against the original one proposed in Li   \n55 $\\underline { { \\left( 2 0 1 9 \\right) } }$ . ",
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+    {
+        "type": "text",
+        "text": "Figs. 6 and $\\textcircled { 7 }$ show ${ p S E q }$ and $p B a l$ (their difference from the ideal values5 ), respectively. XXX-18f/b are trained to minimize Eqs. $\\textcircled { \\sharp }$ and $\\textcircled{6}$ , respectivel ${ \\boldsymbol { \\imath } } ^ { 6 } .$ We omitted MLP and GIN due to their poor performance in Fig. 5. In the results, both SSWN and WN largely outperformed DBM, which again proved the advantage of the set-encoder. WN performed better than SSWN for larger $N$ , owing to the parameter efficiency of the two-stream architecture. Note that the performance gain of XXX-18f/b from XXX-18 proved the flexibility of general learning-based methods for customized objective functions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 Comparison with algorithmic methods $N = 2 0$ , 30) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "As the algorithmic methods, we prepared four baselines. GS is the better result of applying the GS algorithm to prioritize each side once, which runs in $O ( N ^ { 2 } )$ . PolyMin minimizes some alternative fairness costs (the regret and egalitarian costs, which can be solved in $O ( N ^ { 2 } )$ and $\\underline { { O } } ( N ^ { 3 } )$ , respectively (Gusfield, 1987; Irving et al., 1987; Feder, 1992)). DACC by $\\boxed { \\mathrm { D w o r c z a k } } ( \\boxed { 2 0 1 6 } )$ is an approximate algorithm that runs in $\\overline { { O ( N ^ { 4 } ) } }$ . PowerBalance is the state-of-the-art method that runs in $\\mathcal { \\hat { O } } ( N ^ { 2 } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "WN-60f/b(20/30) is WeaveNet with $L = 6 0$ layers trained with samples of $N = 2 0$ and $N = 3 0$ . Note that we used the asymmetric variant for UD and Lib. Moreover, we do not involve any traditional learning-based methods in this part since they scored clear performance drops with increasing $N$ (see Fig. $\\textcircled { 5 }$ and the problem size of $N = 2 0$ , 30 is clearly beyond their capabilities, but an ablation with WeaveNet variants is reported in B.2. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/afe97a4ad627ce4cefafcd881d496107e756f722c66c95c5b966b0356e1bef1c.jpg",
+        "table_caption": [
+            "Table 1: Average $S E q$ (#) and success rate of stable matching (\"). Bold and underlined scores shows the best and second best ones, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Agents (N × M) Datasets (Dist. Type)</td><td colspan=\"4\">20 × 20</td><td rowspan=\"2\"></td><td colspan=\"5\">30 × 30</td></tr><tr><td>UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib</td></tr><tr><td>GS</td><td>41.89</td><td>18.81</td><td>19.52</td><td>70.97</td><td>19.66</td><td>94.03</td><td>43.46</td><td>36.56</td><td>163.77</td><td>39.78</td></tr><tr><td>PolyMin</td><td>19.93</td><td>11.83</td><td>20.57</td><td>87.08</td><td>18.47</td><td>35.52</td><td>21.21</td><td>37.37</td><td>209.62</td><td>31.85</td></tr><tr><td>DACC</td><td>24.34</td><td>20.13</td><td>23.07</td><td>101.75</td><td>20.40</td><td>40.87</td><td>34.35</td><td>40.59</td><td>240.48</td><td>33.88</td></tr><tr><td>Power Balance</td><td>16.28</td><td>8.93</td><td>17.07</td><td>71.09</td><td>15.40</td><td>18.45</td><td>11.05</td><td>27.22</td><td>163.90</td><td>21.57</td></tr><tr><td>WN-60f(20) (pSEq)</td><td>12.23</td><td>6.37</td><td>15.50</td><td>71.31</td><td>14.59</td><td>25.21</td><td>11.38</td><td>29.36</td><td>172.63</td><td>23.53</td></tr><tr><td>Stably Matched (%)</td><td>98.90</td><td>99.50</td><td>99.40</td><td>99.60</td><td>99.30</td><td>94.60</td><td>97.30</td><td>95.70</td><td>91.30</td><td>97.70</td></tr><tr><td>WN-60f(30) (p SEq)</td><td>12.16</td><td>6.53</td><td>15.56</td><td>71.34</td><td>14.53</td><td>18.30</td><td>10.52</td><td>27.39</td><td>170.35</td><td>22.17</td></tr><tr><td>Stably Matched (%)</td><td>99.10</td><td>99.40</td><td>99.40</td><td>99.50</td><td>99.80</td><td>98.10</td><td>99.00</td><td>98.00</td><td>93.90</td><td>98.60</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/8990148bd09c23c87742796e9bfc1ee3657edfdfb54139c707065188255ee551.jpg",
+        "table_caption": [
+            "Table 2: Average Bal (#) and success rate of stable matching (\"). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Agents (N × M) Datasets (Dist. Type)</td><td colspan=\"4\">20×20</td><td colspan=\"6\"></td></tr><tr><td>UU</td><td>DD</td><td>GG</td><td>UD</td><td>Lib</td><td>UU</td><td>DD</td><td>30×30 GG</td><td>UD</td><td>Lib</td></tr><tr><td>GS</td><td>89.14</td><td>146.16</td><td>108.36</td><td>140.53</td><td>68.62</td><td>184.05</td><td>322.05</td><td>225.49</td><td>312.12</td><td>137.59</td></tr><tr><td>PolyMin</td><td>74.19</td><td>140.99</td><td>108.04</td><td>145.28</td><td>66.94</td><td>144.48</td><td>306.28</td><td>224.13</td><td>324.54</td><td>130.79</td></tr><tr><td>DACC</td><td>78.49</td><td>146.71</td><td>110.06</td><td>151.34</td><td>68.75</td><td>150.71</td><td>316.18</td><td>227.52</td><td>337.43</td><td>133.59</td></tr><tr><td>Power Balance</td><td>73.28</td><td>140.12</td><td>106.92</td><td>140.55</td><td>65.89</td><td>138.04</td><td>302.30</td><td>220.26</td><td>312.12</td><td>126.96</td></tr><tr><td>WN-60b(20) (pBal)</td><td>71.89</td><td>138.79</td><td>106.20</td><td>140.84</td><td>65.85</td><td>141.49</td><td>302.73</td><td>221.92</td><td>317.60</td><td>130.58</td></tr><tr><td>Stably Matched (%)</td><td>98.50</td><td>98.80</td><td>99.50</td><td>99.70</td><td>98.80</td><td>96.10</td><td>96.70</td><td>95.00</td><td>88.90</td><td>93.80</td></tr><tr><td>WN-60b(30) (pBal)</td><td>72.33</td><td>138.75</td><td>106.65</td><td>140.79</td><td>65.84</td><td>140.40</td><td>301.59</td><td>223.02</td><td>313.59</td><td>127.93</td></tr><tr><td>Stably Matched (%)</td><td>98.00</td><td>99.10</td><td>98.60</td><td>99.80</td><td>99.10</td><td>97.00</td><td>98.60</td><td>93.70</td><td>98.80</td><td>98.00</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "274 We show the results in Tables 1 and 2. When $N = 2 0$ , except for UD, the proposed method constantly   \n275 performed better than any algorithmic methods for both $S E q$ and $B a l$ . When $N = 3 0$ , they are   \n276 comparative. For UD, GS performed even better than PowerBalance. That means that the ideal   \n277 solution constantly prioritizes one side (a kind of the strongest bias). Since we designed the WeaveNet   \n278 architecture to treat the sides evenly, this is the most challenging situation for WeaveNet. Nonetheless,   \n279 the proposed split batch normalization (with the side-identifiable code) achieved similar performance   \n280 to GS and PowerBalance. We show the performance drop with the fully symmetric version in $\\boxed { \\mathbf { B } . 2 }$   \n281 of the appendix, which is also interesting from the ethical viewpoint. It is noteworthy that the   \n282 model trained with $N = 2 0$ performs well even with $N = 3 0$ , which indicates that the method has   \n283 generalizability for size difference.   \n284 Generalization ability for different distributions A learning-based method should have a certain   \n285 generalizability for input distribution shifts. To test the ability, we evaluated the performance of   \n286 models trained with UU, DD, and GG on test sets of different distributions. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
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+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/40d4345f4af067c99de280b456edbc3dfa462157dd305cdc76a50515330548bd.jpg",
+        "table_caption": [
+            "Table 3: The generalizability of WeaveNet (trained/tested with $N = 3 0$ ). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>train</td><td>WN-60f</td><td>UU</td><td>test DD</td><td>GG</td><td>Avg.</td></tr><tr><td>UU</td><td>pSEq Stably Matched (%)</td><td>18.30 98.10</td><td>25.81 94.90</td><td>29.09 93.60</td><td>21.10 95.53</td></tr><tr><td>DD</td><td>pSEq Stably Matched (%)</td><td>171.27 2.80</td><td>10.52 99.00</td><td>77.36 0.10</td><td>86.38 33.97</td></tr><tr><td>GG</td><td>pSEq Stably Matched (%)</td><td>21.38 97.30</td><td>12.85 98.10</td><td>27.39 98.00</td><td>20.54 97.80</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8b33aff7b40fdbcc95d4c935f79a4e236250bd5f44bcff9039298f8f0189485b.jpg",
+        "table_caption": [
+            "Table 4: Average SEq ( ) and Bal ( ) at $N = 1 0 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>100 × 100,UU</td><td>SEq</td><td>Bal</td></tr><tr><td>GS</td><td>1259.39</td><td>1709.53</td></tr><tr><td>PolyMin</td><td>153.35</td><td>952.85</td></tr><tr><td>DACC</td><td>194.65</td><td>988.02</td></tr><tr><td>Power Balance</td><td>49.41</td><td>909.73</td></tr><tr><td>WN-80f/b+Hungarian</td><td></td><td></td></tr><tr><td>pSEqlpBal</td><td>257.99</td><td>1145.36</td></tr><tr><td>SEqiBal</td><td>68.36</td><td>919.75</td></tr><tr><td>Stably Matched (%)</td><td>89.4</td><td>80.8</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table $3$ shows the results. Remarkably, there is a contrast between the model trained with DD and the others. The model with DD could hardly satisfy the one-to-one stable matching constraint when tested on UU/GG, and resulted in poor ${ p S E q }$ scores. In contrast, the model with GG achieved satisfying ${ p S E q }$ scores on UU/DD. Since GG generates preference lists based on a common preference score $( i / N$ for $i$ -th agent) with noise, agents in GG tend to have similar preference lists (i.e., hard to assign optimally). A model trained with such hard samples works well even for the test samples drawn from other distribution. UU has also performed well owing to its non-biased sampling strategy. On the other hand, DD worst performed due to its highly biased generation strategy. From these results, we confirmed that WeaveNet has certain robustness in the distribution shift as long as training samples are competitive enough. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 Demonstration with $N = 1 0 0$ ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We further demonstrate the capability of WeaveNet under a larger size of problem instances, $N = 1 0 0$ In this case, we found that WN-80f and WN-80b failed to yield one-to-one matchings for $1 3 . 4 \\%$ and $1 9 . 8 \\%$ , respectively (see the Table 9 in B.2 for details). To compensate for this problem, we applied the Hungarian algorithm (Kuhn, 1955) to surely binarize $\\hat { m }$ into a one-to-one matching. Table $^ 4$ shows WeaveNet’s relatively good $\\overline { { S E q } }$ and $B a l$ scores. Even with the help of the Hungarian algorithm, they were strongly penalized in ${ p S E q }$ and $p B a l$ due to the poor stable matching rate. In other words, we can potentially fill the large gap by better constraining the output. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Since this work is just a pilot study toward a practical differential assignment solver, there is still a lot of space for improvement. The proposed test protocol with stable matching will facilitate it since we can freely adjust the difficulty of the problem to develop and enhance the methods continuously. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper proposed a novel differential assignment solver, WeaveNet, and an evaluation protocol on two strongly NP-hard variants of stable matching. In the experiments, we demonstrated the advantage of set encoder and the two-stream architecture of Weavenet against the other learning-based methods. These techniques also achieved a better performance than the state-of-the-art algorithmic method when $N = 2 0$ and a comparative performance when $N = 3 0$ . Furthermore, the asymmetric variants, split batch normalization with the side-identifiable code, enabled the method to work even with the strongly biased dataset of UD. We also confirmed that the proposed method does not work at $N = 1 0 0$ , which will be an immediate task for this new field of differential assignment solver. We hope that this work becomes a starting point to open a new vista for real-world assignment problems. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "18 References   \n19 Ravindra K Ahuja, Arvind Kumar, Krishna C Jha, and James B Orlin. Exact and heuristic algorithms for the weapon-target assignment problem. Operations research, 55(6):1136–1146, 2007.   \n21 Lukas Brozovsky and Vaclav Petricek. Recommender system for online dating service. In Proceedings of Conference Znalosti 2007, Ostrava, 2007.   \n23 Zhe Cao, Tomas Simon, Shih-En Wei, and Yaser Sheikh. Realtime multi-person 2d pose estimation using part affinity fields. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.   \n6 Piotr Dworczak. Deferred acceptance with compensation chains. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 65–66, 2016.   \n28 Patrick Emami, Panos M. Pardalos, Lily Elefteriadou, and Sanjay Ranka. Machine learning methods for data association in multi-object tracking. ACM Computing Surveys, 53(4), 2020.   \n0 Tomás Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45(2):233–284, 1992.   \n32 Tomás Feder. Stable networks and product graphs. American Mathematical Society, 1995.   \n33 David Gale and Lloyd S Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15, 1962.   \n5 Daniel Gibbons, Cheng-Chew Lim, and Peng Shi. Deep learning for bipartite assignment problems. In Proceedings of IEEE International Conference on Systems, Man and Cybernetics, pages 2318–   \n2325, 2019.   \n38 Sushmita Gupta, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. Balanced stable marriage: How close is close enough? In Workshop on Algorithms and Data Structures, pages 423–437, 2019. Dan Gusfield and Robert W Irving. The stable marriage problem: structure and algorithms. MIT press, 1989.   \n42 Dan Gusfield. Three fast algorithms for four problems in stable marriage. SIAM Journal on Computing, 16(1):111–128, 1987.   \n4 Robert W Irving, Paul Leather, and Dan Gusfield. An efficient algorithm for the “optimal” stable marriage. Journal of the ACM, 34(3):532–543, 1987.   \n46 Kazuo Iwama, Shuichi Miyazaki, and Hiroki Yanagisawa. Approximation algorithms for the sexequal stable marriage problem. ACM Transaction on Algorithms, 7(1), 2010.   \n48 Akiko Kato. Complexity of the sex-equal stable marriage problem. Japan Journal of Industrial and Applied Mathematics, 10(1):1, 1993.   \n0 Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of International Conference on Learning Representations, 2015.   \n52 Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83–97, 1955.   \n54 Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Proceedings of AAAI Conference on Artificial Intelligence, pages   \n3538–3545, 2018. Shira Li. Deep Learning for Two-Sided Matching Markets. Bachelor’s thesis, Harvard University,   \n2019.   \n59 Eric McDermid and Robert W Irving. Sex-equal stable matchings: Complexity and exact algorithms. Algorithmica, 68(3):545–570, 2014.   \n61 Kenta Oono and Taiji Suzuki. Graph neural networks exponentially lose expressive power for node classification. In Proceedings of International Conference on Learning Representations, 2020.   \nCharles R. Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, July 2017.   \nNikolaos Tziavelis, Ioannis Giannakopoulos, Katerina Doka, Nectarios Koziris, and Panagiotis Karras. Equitable stable matchings in quadratic time. In Proceedings of Advances in Neural Information Processing Systems, pages 457–467, 2019.   \nKeyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In Proceedings of International Conference on Learning Representations, 2019.   \nManzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. In Proceedings of Advances in Neural Information Processing Systems, volume 30, pages 3391–3401, 2017. ",
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+    {
+        "type": "text",
+        "text": "374 Checklist ",
+        "text_level": 1,
+        "bbox": [
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+            106
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3751. For all authors... ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "376 (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contri  \n377 butions and scope? [Yes] The experimental results in Section 5 correspond to the main claims,   \n378 which are summarized as the contribution list in Section 1   \n379 (b) Did you describe the limitations of your work? [Yes] Our method works best among any   \n380 baselines when $N \\leq 2 0$ , comparative to the state-of-the-art algorithmic baseline when $N = 3 0$ ,   \n381 but poorly when $N = 1 0 0$ . See Section $\\underline { { \\boldsymbol { \\mathsf { F } . 3 } } }$   \n382 (c) Did you discuss any potential negative societal impacts of your work? [Yes] We briefly   \n383 discussed the fairness/unfairness achieved by our method in the paragraph “Asymmetric variant   \n384 with split batch normalization” in Section $4 . 1 .$ Namely, the asymmetric variant can better   \n385 optimize the objective than the symmetric one (which ensures mathematically equal treatment   \n386 for sides $A$ and $B$ ) but harms the equal treatment of the two sides.   \n387 (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?   \n388 [Yes] ",
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+        "type": "text",
+        "text": "3892. If you are including theoretical results... ",
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+    {
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] We have no main theoretical results. The only theoretical discussion is for the computational cost of WeaveNet in A.2, whose assumption is the network shape explained in the paper.   \n(b) Did you include complete proofs of all theoretical results? [N/A] We have no main theoretical results. The only theoretical discussion is for the computational cost of WeaveNet in $\\mathbf { A } . 2 ,$ where we provided enough detailed explanation as a proof. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have included it in the supplemental material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We described the training details in Section 5 “Training protocol” and Section A.5.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We presented figures with the error bars in Section ${ \\bf A . } 4$ of the appendix, which demonstrated the stable behavior of the proposed method against the random seed. For the other part, we have multiple settings, and we observed a stable trend in the results.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We described it in the appendix, Section C. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4094. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "410 (a) If your work uses existing assets, did you cite the creators? [Yes] See Section C in the appendix.   \n411 (b) Did you mention the license of the assets? [Yes] See Section C.   \n412 (c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We   \n413 provides a code to random-generate the problem instances, which contains no personal or any   \n414 other sensitive information, but only the distribution parameters of the LibimSeti dataset.   \n415 (d) Did you discuss whether and how consent was obtained from people whose data you’re   \n416 using/curating? [Yes] See Section C.   \n417 (e) Did you discuss whether the data you are using/curating contains personally identifiable   \n418 information or offensive content? [Yes] See Section C. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4195. If you used crowdsourcing or conducted research with human subjects... ",
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+        "text": "420 (a) Did you include the full text of instructions given to participants and screenshots, if applicable?   \n421 [N/A] We used neither crowd-sourcing nor human subjects.   \n422 (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB)   \n423 approvals, if applicable? [N/A]   \n424 (c) Did you include the estimated hourly wage paid to participants and the total amount spent on   \n425 participant compensation? [N/A] ",
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parse/train/ByQpn1ZA-/ByQpn1ZA-.md

@@ -0,0 +1,358 @@
+# MANY PATHS TO EQUILIBRIUM: GANS DO NOT NEED TO DECREASE A DIVERGENCE AT EVERY STEP
+
+William Fedus∗ 1, Mihaela Rosca∗ 2, Balaji Lakshminarayanan2, Andrew M. Dai1, Shakir Mohamed2 and Ian Goodfellow1
+
+1Google Brain   
+2DeepMind
+
+# ABSTRACT
+
+Generative adversarial networks (GANs) are a family of generative models that do not minimize a single training criterion. Unlike other generative models, the data distribution is learned via a game between a generator (the generative model) and a discriminator (a teacher providing training signal) that each minimize their own cost. GANs are designed to reach a Nash equilibrium at which each player cannot reduce their cost without changing the other players’ parameters. One useful approach for the theory of GANs is to show that a divergence between the training distribution and the model distribution obtains its minimum value at equilibrium. Several recent research directions have been motivated by the idea that this divergence is the primary guide for the learning process and that every step of learning should decrease the divergence. We show that this view is overly restrictive. During GAN training, the discriminator provides learning signal in situations where the gradients of the divergences between distributions would not be useful. We provide empirical counterexamples to the view of GAN training as divergence minimization. Specifically, we demonstrate that GANs are able to learn distributions in situations where the divergence minimization point of view predicts they would fail. We also show that gradient penalties motivated from the divergence minimization perspective are equally helpful when applied in other contexts in which the divergence minimization perspective does not predict they would be helpful. This contributes to a growing body of evidence that GAN training may be more usefully viewed as approaching Nash equilibria via trajectories that do not necessarily minimize a specific divergence at each step.
+
+# 1 INTRODUCTION
+
+Generative adversarial networks (GANs) (Goodfellow et al., 2014) are generative models based on a competition between a generator network $G$ and a discriminator network $D$ . The generator network $G$ represents a probability distribution $p _ { \mathrm { m o d e l } } ( { \pmb x } )$ . To obtain a sample from this distribution, we apply the generator network to a noise vector $_ { z }$ sampled from $p _ { z }$ , that is $\pmb { x } = G ( \pmb { z } )$ . Typically, $_ { z }$ is drawn from a Gaussian or uniform distribution, but any distribution with sufficient diversity is possible. The discriminator $D ( \pmb { x } )$ attempts to distinguish whether an input value $_ { \textbf { \em x } }$ is real (came from the training data) or fake (came from the generator).
+
+The goal of the training process is to recover the true distribution $p _ { \mathrm { d a t a } }$ that generated the data. Several variants of the GAN training process have been proposed. Different variants of GANs have been interpreted as approximately minimizing different divergences or distances between $p _ { \mathrm { d a t a } }$ and $p _ { \mathrm { m o d e l } }$ . However, it has been difficult to understand whether the improvements are caused by a change in the underlying divergence or the learning dynamics.
+
+We conduct several experiments to assess whether the improvements associated with new GAN methods are due to the reasons cited in their design motivation. We perform a comprehensive study of GANs on simplified, synthetic tasks for which the true $p _ { \mathrm { d a t a } }$ is known and the relevant distances are straightforward to calculate, to assess the performance of proposed models against baseline methods. We also evaluate GANs using several independent evaluation measures on real data to better understand new approaches. Our contributions are:
+
+• We aim to clarify terminology used in recent papers, where the terms “standard GAN,” “regular GAN,” or “traditional GAN” are used without definition (e.g., (Arjovsky et al., 2017; Denton et al., 2015; Salimans et al., 2016; Donahue et al., 2016)). The original GAN paper described two different losses: the “minimax” loss and the “non-saturating” loss, equations (10) and (13) of Goodfellow (2016), respectively. Recently, it has become important to clarify this terminology, because many of the criticisms of “standard GANs”, e.g. Arjovsky et al. (2017), are applicable only to the minimax GAN, while the non-saturating GAN is the standard for GAN implementations. The non-saturating GAN was recommended for use in practice and implemented in the original paper of Goodfellow et al. (2014), and is the default in subsequent papers (Radford et al., 2015; Salimans et al., 2016; Donahue et al., 2016; Nowozin et al., 2016)1. To avoid confusion we will always indicate whether we mean minimax GAN (M-GAN) or non-saturating GAN (NS-GAN). We demonstrate that gradient penalties designed in the divergence minimization framework— to improve Wasserstein GANs (Gulrajani et al., 2017) or justified from a game theory perspective to improve minimax GANs (Kodali et al., 2017)—also improve the non-saturating GAN on both synthetic and real data. We observe improved sample quality and diversity. We find that non-saturating GANs are able to fit problems that cannot be fit by JensenShannon divergence minimization. Specifically, Figure 1 shows a GAN using the loss from the original non-saturating GAN succeeding on a task where the Jensen-Shannon divergence provides no useful gradient. Figure 2 shows that the non-saturating GAN does not suffer from vanishing gradients when applied to two widely separated Gaussian distributions.
+
+# 2 VARIANTS OF GENERATIVE ADVERSARIAL NETWORKS
+
+# 2.1 NON-SATURATING AND MINIMAX GANS
+
+In the original GAN formulation (Goodfellow et al., 2014), the output of the discriminator is a probability and the cost function for the discriminator is given by the negative log-likelihood of the binary discrimination task of classifying samples as real or fake:
+
+$$
+\begin{array} { r } { J ^ { ( D ) } ( D , G ) = - \underset { x \sim p _ { \mathrm { d a t a } } } { \mathbb { E } } \left[ \log D ( \pmb { x } ) \right] - \underset { z \sim p _ { z } } { \mathbb { E } } \left[ \log ( 1 - D ( G ( z ) ) ) \right] . } \end{array}
+$$
+
+The theoretical analysis in (Goodfellow et al., 2014) is based on a zero-sum game in which the generator maximizes $J ^ { ( D ) }$ , a situation that we refer to here as “minimax GANs”. In minimax GANs the generator attempts to generate samples that have low probability of being fake, by minimizing the objective (2). However, in practice, Goodfellow et al. (2014) recommend implementing an alternative cost function that instead ensures that generated samples have high probability of being real, and the generator instead minimizes an alternative objective (3).
+
+$$
+\begin{array} { r l } { \mathbf { M i n i m a x } } & { \boldsymbol { J } ^ { ( G ) } ( G ) = \underset { \boldsymbol { z } \sim p _ { \boldsymbol { z } } } { \mathbb { E } } \log [ 1 - D ( G ( \boldsymbol { z } ) ) ] . } \\ { \mathbf { N o n - s a t u r a t i n g } } & { \boldsymbol { J } ^ { ( G ) } ( G ) = \underset { \boldsymbol { z } \sim p _ { \boldsymbol { z } } } { - } \log \boldsymbol { D } ( G ( \boldsymbol { z } ) ) . } \end{array}
+$$
+
+We refer to the alternative objective as non-saturating, due to the non-saturating behavior of the gradient (see figure 2), and was the implementation used in the code of the original paper. We use the non-saturating objective (3) in all our experiments
+
+As shown in (Goodfellow et al., 2014), whenever $D$ successfully minimizes $J ^ { ( D ) }$ optimally, maximizing $J ^ { ( D ) }$ with respect to the generator is equivalent to minimizing the Jensen-Shannon divergence. Goodfellow et al. (2014) use this observation to establish that there is a unique Nash equilibrium in function space corresponding to $p _ { \mathrm { d a t a } } = p _ { \mathrm { m o d e l } }$ .
+
+# 2.2 WASSERSTEIN GAN
+
+Wasserstein GANs (Arjovsky et al., 2017) modify the discriminator to emit an unconstrained real number rather than a probability (analogous to emitting the logits rather than the probabilities used in the original GAN paper). The cost function for the WGAN then omits the log-sigmoid functions used in the original GAN paper. The cost function for the discriminator is now:
+
+$$
+\begin{array} { r } { W ^ { ( D ) } ( D , G ) = \underset { \pmb { x } \sim p _ { \mathrm { d a t a } } } { \mathbb { E } } \left[ D ( \pmb { x } ) \right] - \underset { \pmb { z } \sim p _ { \pmb { z } } } { \mathbb { E } } \left[ D ( G ( \pmb { z } ) ) \right] . } \end{array}
+$$
+
+The cost function for the generator is simply $W ^ { ( G ) } = - W ^ { ( D ) } ( D , G )$ . When the discriminator is Lipschitz smooth, this approach approximately minimizes the earth mover’s distance between pdata and $p _ { \mathrm { m o d e l } }$ . To enforce Lipschitz smoothness, the weights of $D$ are clipped to lie within $( - c , c )$ where $c$ is some small real number.
+
+# 2.3 GRADIENT PENALTIES FOR GENERATIVE ADVERSARIAL NETWORKS
+
+Multiple formulations of gradient penalties have been proposed for GANs. As introduced in WGANGP (Gulrajani et al., 2017), the gradient penalty is justified from the perspective of the Wasserstein distance, by imposing properties which hold for an optimal critic as an additional training criterion. In this approach, the gradient penalty is typically a penalty on the gradient norm, and is applied on a linear interpolation between data points and samples, thus smoothing out the space between the two distributions.
+
+Kodali et al. (2017) introduce DRAGAN with a gradient penalty from the perspective of regret minimization, by setting the regularization function to be a gradient penalty on points around the data manifold, as in Follow The Regularized Leader (Cesa-Bianchi & Lugosi, 2006), a standard no-regret algorithm. This encourages the discriminator to be close to linear around the data manifold, thus bringing the set of possible discriminators closer to a convex set, the set of linear functions. We also note that they used the minimax version of the game to define the loss, in which the generator maximizes $J ^ { ( D ) }$ rather than minimizing $J ^ { ( G ) }$ .
+
+To formalize the above, both proposed gradient penalties of the form:
+
+$$
+\underset { \hat { x } \sim p _ { \hat { x } } } { \mathbb { E } } \left[ ( \| \nabla _ { \hat { x } } D ( \hat { x } ) \| _ { 2 } - 1 ) ^ { 2 } \right] ,
+$$
+
+where $p _ { \hat { x } }$ is defined as the distribution defined by the sampling process:
+
+$$
+x \sim p _ { \mathrm { d a t a } } ; \qquad x _ { \mathrm { m o d e l } } \sim p _ { \mathrm { m o d e l } } ; \qquad x _ { \mathrm { n o i s e } } \sim p _ { \mathrm { n o i s e } }
+$$
+
+$$
+\begin{array} { c l c r } { { \alpha \sim U ( 0 , 1 ) } } \\ { { \hat { x } = \alpha x + ( 1 - \alpha ) \tilde { x } . } } \end{array}
+$$
+
+As we will note in our experimental section, Kodali et al. (2017) also reported that mode-collapse is reduced using their version of the gradient penalty.
+
+# 2.3.1 NON-SATURATING GAN WITH GRADIENT PENALTY
+
+We consider the non-saturating GAN objective (3) supplemented by two gradient penalties: the penalty proposed by Gulrajani et al. (2017), which we refer to as “GAN-GP”; the gradient penalty proposed by DRAGAN (Kodali et al., 2017), which we refer to as DRAGAN-NS, to emphasize that we use the non-saturating generator loss function. In both cases, the gradient penalty applies only to the discriminator, with the generator loss remaining unchanged (as defined in Equation 3). In this setting, the loss of the discriminator becomes:
+
+![](images/c345cd545745234de5c42fbdca6bc4c7cc41d9e5674043d0e483861441be6be1.jpg)  
+Figure 1: Visualization of experiment 1 training dynamics in two dimensions, demonstrated specifically in the case where the model is initialized so that it it represents a linear manifold parallel to the linear manifold of the training data. Here the GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). Specifically, this is an illustration of the parallel line thought experiment from (Arjovsky et al., 2017). When run in practice with a non-saturating GAN, the GAN succeeds. In the same setting, minimization of Jensen-Shannon divergence would fail. This indicates that while Jensen-Shannon divergence is useful for characterizing GAN equilibrium, it does not necessarily tell us much about non-equilibrium learning dynamics.
+
+$$
+\tilde { J } ^ { ( D ) } ( D , G ) = - \underset { x \sim p _ { \mathrm { d a t a } } } { \mathbb { E } } \left[ \log D ( x ) \right] - \underset { z \sim p _ { z } } { \mathbb { E } } \left[ \log ( 1 - D ( G ( z ) ) ) \right] + \lambda \underset { \hat { x } \sim p _ { \hat { \alpha } } } { \mathbb { E } } \left[ ( \lVert \nabla _ { \hat { x } } D ( \hat { x } ) \rVert _ { 2 } - 1 ) ^ { 2 } \right]
+$$
+
+We consider these GAN variants because:
+
+• We want to assess whether gradient penalties are effective outside their original defining scope. Namely, we perform experiments to determine whether the benefit obtained by applying the gradient penalty for Wasserstein GANs is obtained from properties of the earth mover’s distance, or from the penalty itself. Similarly, we evaluate whether the DRAGAN gradient penalty is beneficial outside the minimax GAN setting.   
+• We want to assess whether the exact form of the gradient penalty matters.   
+We compare three models, to control over different aspects of training: same gradient penalty but different underlying adversarial losses (GAN-GP versus WGAN-GP), as well as the same underlying adversarial loss, but different gradient penalties (GAN-GP versus DRAGAN-NS).
+
+We note that we do not compare with the original DRAGAN formulation, which uses the minimax GAN formulation, since in this work we focus on non-saturating GAN variants.
+
+# 3 MANY PATHS TO EQUILIBRIUM
+
+The original GAN paper (Goodfellow et al., 2014) used the correspondence between $J ^ { ( D ) } ( D ^ { * } , G )$ and the Jensen-Shannon divergence to characterize the Nash equilibrium of minimax GANs. It is important to keep in mind that there are many ways for the learning process to approach this equilibrium point, and the majority of them do not correspond to gradually reducing the JensenShannon divergence at each step. Divergence minimization is useful for understanding the outcome of training, but GAN training is not the same thing as running gradient descent on a divergence and GAN training may not encounter the same problems as gradient descent applied to a divergence.
+
+Arjovsky et al. (2017) describe the learning process of GANs from the perspective of divergence minimization and show that the Jensen-Shannon divergence is unable to provide a gradient that will bring $p _ { \mathrm { d a t a } }$ and $p _ { \mathrm { m o d e l } }$ together if both are sharp manifolds that do not overlap early in the learning process. Following this line of reasoning, they suggest that when applied to probability distributions that are supported only on low dimensional manifolds, the Kullback Leibler (KL), Jensen Shannon (JS) and Total Variation (TV) divergences do not provide a useful gradient for learning algorithms based on gradient descent, the “traditional GANs” is inappropriate for fitting such low dimensional manifolds (“traditional GAN” seems to refer to the minimax version of GANs used for theoretical analysis in the original paper, and there is no explicit statement about whether the argument is intended to apply to the non-saturating GAN implemented in the code accompanying the original GAN paper). In Section 4 we show that non-saturating GANs are able to learn on tasks where the data distribution lies on the low dimensional manifold.
+
+![](images/e40fa277017297d4a93812edd2dc820808d2dc596cb8741b91762c1344aef944.jpg)  
+Figure 2: (Left) A recreation of Figure 2 of Arjovsky et al. (2017). This figure is used by Arjovsky et al. (2017) to show that a model they call the “traditional GAN” suffers from vanishing gradients in the areas where $D ( x )$ is flat. This plot is correct if “traditional GAN” is used to refer to the minimax GAN, but it does not apply to the non-saturating GAN. (Right) A plot of both generator losses from the original GAN paper, as a function of the generator output. Even when the model distribution is highly separated from the data distribution, non-saturating GANs are able to bring the model distribution closer to the data distribution because the loss function has strong gradient when the generator samples are far from the data samples, even when the discriminator itself has nearly zero gradient. While it is true that the $\begin{array} { r } { \frac { 1 } { 2 } \log ( 1 - D ( x ) ) } \end{array}$ loss has a vanishing gradient on the right half of the plot, the original GAN paper instead recommends implementing $- { \frac { 1 } { 2 } } \log D ( x )$ . This latter, recommended loss function has a vanishing gradient only on the left side of the plot. It makes sense for the gradient to vanish on the left because generator samples in that area have already reached the area where data samples lie.
+
+We show that non-saturating GANs do not suffer from vanishing gradients for two widely separated Gaussians in Figure 2. The fact that the gradient of the recommended loss does not actually vanish explains why GANs with the non-saturating objective (3), are able to bring together two widely separated Gaussian distributions. Note that the gradient for this loss does not vanish even when the discriminator is optimal. The discriminator has vanishing gradients but the generator loss amplifies small differences in discriminator outputs to recover strong gradients. This means it is possible to train the GAN by changing the loss rather than the discriminator.
+
+For the parallel lines thought experiment (Arjovsky et al., 2017) (see Figure 1), the main problem with the Jensen-Shannon divergence is that it is parameterized in terms of the density function, and the two density functions have no support in common. Most GANs, and many other models, can solve this problem by parameterizing their loss functions in terms of samples from the two distributions rather than in terms of their density functions.
+
+# 4 SYNTHETIC EXPERIMENTS
+
+To assess the learning process of GANs we empirically examine GAN training on pathological tasks where the data is constructed to lie on a low dimensional manifold, and show the model is able to learn the data distribution in cases where using the underlying divergence obtained at optimality would not provide useful gradients. We then evaluate convergence properties of common GAN variants on this task where the parameters generating the distribution are known.
+
+# 4.1 EXPERIMENT I: 1-D DATA MANIFOLD AND 1-D GENERATOR
+
+In our first experiment, we generate synthetic training data that lies along a one-dimensional line and design a one-dimensional generative model, however, we embed the problem in a higher $d .$ - dimensional space where $d \gg 1$ . This experiment is essentially an implementation of a thought experiment from Arjovsky et al. (2017).
+
+Specifically, in a $d$ -dimensional space, we define $p _ { \mathrm { d a t a } }$ by randomly generating parameters defining the distribution once at the beginning of the experiment. We generate a random $b _ { r } \in \mathbb { R } ^ { d }$ and random $W _ { r } \in \mathbb { R } ^ { 1 \times d }$ . Our latent $z _ { r } \in \mathbb { R } \sim N ( 0 , \sigma )$ where $\sigma$ is the standard deviation of the normal distribution. The synthetic training data of $m$ examples is then given by
+
+$$
+\{ x ^ { ( i ) } \} _ { i = 1 } ^ { m } = \{ z _ { r } ^ { ( i ) } \} _ { i = 1 } ^ { m } W _ { r } + b _ { r }
+$$
+
+The real synthetic data is therefore Gaussian distributed on a 1-D surface within the space, where the position is determined by $b _ { r }$ and the orientation is determined by $W _ { r }$ .
+
+The generator also assumes the same functional form, that is, it is also intrinsically one dimensional,
+
+$$
+G _ { \theta } ( z ) = z W _ { \theta } + b _ { \theta }
+$$
+
+where $b _ { \theta } \in \mathbb { R } ^ { d }$ and $W _ { \theta } \in \mathbb { R } ^ { 1 \times d }$ . The discriminator is a single hidden layer ReLU network, which is of higher complexity than the generator so that it may learn non-linear boundaries in the space.
+
+This experiment captures the idea of sharp, non-overlapping manifolds that motivate alternative GAN losses. Further, because we know the true generating parameters of the training data, we may explicitly test convergence properties of the various methodologies.
+
+# 4.2 EXPERIMENT II: 1-D DATA MANIFOLD AND OVERCOMPLETE GENERATOR
+
+In our second experiment, the synthetic training data is still the same (lying on a 1-D line) and given by Eq. 11 but now the generator is overcomplete for this task, and has a higher latent dimension $g$ , where $1 < g \le d$ .
+
+$$
+G ( z ) = z W _ { \theta } + b _ { \theta }
+$$
+
+where matrix $W _ { \theta } \in \mathbb { R } ^ { g \times d }$ and vector $b _ { \theta } \in \mathbb { R } ^ { d }$ , so that the generator is able to represent a manifold with too high of a dimensionality. The generator parameterizes a multivariate Gaussian $N ( x ; \mu , \Sigma )$ with $\mu = b$ . The covariance matrix elements $\dot { \Sigma _ { i j } } = E [ \sigma ^ { 2 } ( X _ { i } - \mu _ { i } ) ( X _ { j } - \mu _ { j } ) ] = \sigma ^ { 2 } \dot { E } [ ( X _ { i } -$ $\mu _ { i } ) ( X _ { j } - \mu _ { j } ) ]$ . In vector notation, $\Sigma = \sigma ^ { 2 } W ^ { T } W$ .
+
+# 4.3 RESULTS
+
+To evaluate the convergence of an experimental trial, we report the square Fréchet distance (Fréchet (1957)) between the true data Gaussian distribution and the fitted Gaussian parameters. In our notation, where the $r$ subscript denotes real data, the $\theta$ subscript denotes the generator Gaussian parameters and $\| x \| ^ { 2 }$ is the squared $l _ { 2 }$ norm of $x$ , the Fréchet distance is defined as (Dowson $\&$ Landau (1982)):
+
+$$
+d ^ { 2 } ( \mu _ { r } , \mu _ { \theta } , \Sigma _ { r } , \Sigma _ { \theta } ) = \| \mu _ { \theta } - \mu _ { r } \| ^ { 2 } + \mathrm { T r } \left( \Sigma _ { r } + \Sigma _ { \theta } - 2 ( \Sigma _ { r } \Sigma _ { \theta } ) ^ { - 1 / 2 } \right)
+$$
+
+Every GAN variant was trained for 200000 steps. For each step, the generator is updated once and the discriminator is updated 5 times. Throughout the paper, the number of steps will correspond to the number of generator updates.
+
+The main conclusions from our synthetic data experiments are:
+
+• Gradient penalties (both applied near the data manifold, DRAGAN-NS, and at an interpolation between data and samples, GAN-GP) stabilize training and improve convergence (Figures 3, 9, 10).   
+Despite the inability of Jensen-Shannon divergence minimization to solve this problem, we find that the non-saturating GAN succeeds in converging to the 1D data manifold (Figure 3). However, in higher dimensions the resulting fit is not as strong as the other methods: Figure 10 shows that increasing the number of dimensions while keeping the learning rate fixed can decrease the performance of the non-saturating GAN model.   
+Non-saturating GANs are able to learn data distributions which are disjoint from the training sample distribution at initialization (or another point in training), as demonstrated in Figure 1.   
+Updating the discriminator 5 times per generator update does not result in vanishing gradients when using the non-saturating cost. However, when scaling the number of discriminator updates to 100 per generator update, non-saturating GANs perform worse than when using a smaller number of updates (1, 5, 10). Gradient penalties help here too: GAN-GP scales better with the number of discriminator updates. The results are detailed in Appendix Section A.2.   
+• An over-capacity generator with the ability to have more directions of high variance than the underlying data is able to capture the data distribution using non-saturating GAN training (Figure 4).
+
+# 4.4 HYPERPARAMETER SENSITIVITY
+
+We assess the robustness of the considered models by looking at results across hyperparameters for both experiment 1 and experiment 2. In one setting, we keep the input dimension fixed while varying the learning rate (Figure 9); in another setting, we keep the learning rate fixed, while varying the input dimension (Figure 10). In both cases, the results are averaged out over 1000 runs per setting, each starting from a different random seed. We notice that:
+
+• The non-saturating GAN model (with no gradient penalty) is most sensitive to hyperparameters.   
+• Gradient penalties make the non-saturating GAN model more robust.   
+• Both Wasserstein GAN formulations are quite robust to hyperparameter changes.   
+• For certain hyperparameter settings, there is no performance difference between the two gradient penalties for the non-saturating GAN, when averaging across random seeds. This is especially visible in Experiment 1, when the number of latent variables is 1. This could be due to the fact that the data sits on a low dimensional manifold, and because the discriminator is a small, shallow network.
+
+# 5 REAL DATA EXPERIMENTS
+
+To assess the effectiveness of the gradient penalty on standard datasets for the non-saturating GAN formulation, we train a non-saturating GAN, a non-saturating GAN with the gradient penalty introduced by (Gulrajani et al., 2017) (denoted by GAN-GP), a non-saturating GAN with the gradient penalty introduced by (Kodali et al., 2017) (denoted by DRAGAN-NS), and a Wasserstein GAN with gradient penalty (WGAN-GP) on three datasets: Color MNIST (Metz et al., 2016) - data dimensionality (28, 28, 3), CelebA (Liu et al., 2015) - data dimensionality (64, 64, 3) and CIFAR-10 (Krizhevsky, 2009) - data dimensionality (32, 32, 3), as seen in Figure 6.
+
+For all our experiments we used $\lambda = 1 0$ as the gradient penalty coefficient and used batch normalization (Ioffe & Szegedy, 2015); Kodali et al. (2017) suggests that batch normalization is not neeeded for DRAGAN, but we found that it also improved our DRAGAN-NS results. We used the Adam optimizer (Kingma & Ba, 2014) with $\beta _ { 1 } = 0 . 5$ and $\beta _ { 2 } = 0 . 9$ and a batch size of 64. The input data was scaled to be between -1 and 1. We did not add any noise to the discriminator inputs or activations, as that regularization technique can be interpreted as having the same goal as gradient penalties, and we wanted to avoid a confounding factor. We trained all Color MNIST models for 100000 steps, and CelebA and CIFAR-10 models for 200000 steps. We note that the experimental results on real data for the non-saturating GAN and for the Improved Wasserstein GAN (WGAN-GP) are quoted with permission from an earlier publication by Rosca et al. (2017).
+
+![](images/73c9856186ab835d1a3a36f708d693d9fd5a76f7a8243f278a7290805ca7a30a.jpg)  
+Figure 3: Visualization of experiment 1 training dynamics in two dimensions. Here the GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). We note that this is a visual illustration, and the results have not been averaged out over multiple seeds. Exact plots may vary on different runs. However, a single example of success is sufficient to refute claims that this this task is impossible for this model.
+
+We note that the WGAN-GP model was the only model for which we did 5 discriminator updates in real data experiments. All other models (DCGAN, DRAGAN-NS, GAN-GP) used one discriminator update for generator update.
+
+For all reported results, we sweep over two hyperparameters:
+
+• Learning rates for the discriminator and generator. Following Radford et al. (2015), we tried learning rates of 0.0001, 0.0002, 0.0003 for both the discriminator and the generator. We note that this is consistent with WGAN-GP, where the authors use 0.0002 for CIFAR-10 experiments.   
+• Number of latents. For CelebA and CIFAR-10 we try latent sizes 100, 128 and 150, while for Color MNIST we try 10, 50, 75.
+
+![](images/547232301dbc7de185f58c14971f802086c3adeab0c20de259ecfbeebc49be2b.jpg)  
+Figure 4: Visualization of experiment 2 training dynamics in two dimensions - where the GAN model has 3 latent variables. Here the rank one GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). We observe how for poor initialization the non-saturating GAN suffers from mode collapse. However, adding a gradient penalty stabilizes training. We note that this is a visual illustration, and the results have not been averaged out over multiple seeds. Exact plots may vary on different runs.
+
+# 5.1 EVALUATION
+
+Unlike the synthetic case, here we are unable to evaluate the performance of our models relative to the true solution, since that is unknown. Moreover, there is no single metric that can evaluate the performance of GANs. We thus complement visual inspection with three metrics, each measuring a different criteria related to model performance. We use the Inception Score (Salimans et al., 2016) to measure how visually appealing CIFAR-10 samples are, the MS-SSIM metric (Wang et al., 2003; Odena et al., 2016) to check sample diversity, and an Improved Wasserstein independent critic to assess overfitting, as well as sample quality (Danihelka et al., 2017). For a more detailed explanation of these metrics, we refer to Rosca et al. (2017). In all our experiments, we control over discriminator and generator architectures, using the ones used by DCGAN (Radford et al., 2015) and the original WGAN paper (Arjovsky et al., 2017)2. We note that the WGAN-GP paper used a different architecture when reporting the Inception Score on CIFAR10, and thus their results are not directly comparable.
+
+For all the metrics, we report both the hyperparameter sensitivity of the model (by showing quartile statistics), as well as the 10 best results according to the metric. The sample diversity measure needs to be seen in context with the value reported on the test set: too high diversity can mean failure to capture the data distribution. For all other metrics, higher is better.
+
+![](images/e194aa4cd2fbbf76bd867f710668fca21af89252d739a0c91ca2ce2faff9625d.jpg)  
+Figure 5: The square Fréchet distance between the learned Gaussian and the true Gaussian distribution. For reference, we also plot the distance obtained by a randomly initialized generator with the same architecture as the trained generators. Results are averaged over 1000 runs. Lower values are better.
+
+![](images/90a54d4069466c009ace763dbd8f1d757d7480e357df414e3d53e2c58276309b.jpg)  
+Figure 6: Examples from the three datasets explored in this paper: Color MNIST (left), CIFAR-10 (middle) and CelebA (right).
+
+# 5.2 VISUAL SAMPLE INSPECTION
+
+By visually inspecting the results of our models, we noticed that applying gradient penalties to the non-saturating GAN results in more stable training across the board. When training the non-saturating GAN with no gradient penalty, we did observe cases of severe mode collapse (see Figure 17). Gradient penalties improves upon that, but we can still observe mode collapse. Each non-saturating GAN variant with gradient penalty (DRAGAN-NS and GAN-GP) only produced mode collapse on one dataset, see Figure 21). We also noticed that for certain learning rates, WGAN-GPs fail to learn the data distribution (Figure 22). For the GAN-GP and DRAGAN-NS models, most hyperparameters produced samples of equal quality - the models are quite robust. We show samples from the GAN-GP, DRAGAN-NS and WGAN-GP models in Figures 18, 19 and 20.
+
+# 5.3 METRICS
+
+We show that gradient penalties make non-saturating GANs more robust to hyperparameter changes. For this, we report not only the best obtained results, but rather a box plot of the obtained results showing the quartiles obtained by each sweep, along with the top 10 best results explicitly shown in the graph (note that for each model we tried 27 different hyperparameter settings, corresponding to 3 discriminator learning rates, 3 generator learning rates and 3 generator input sizes). We report two Inception Score metrics for CIFAR-10, one using the standard Inception network used when the metric was introduced (Salimans et al., 2016), trained on the Imagenet dataset, as well as a VGG style network trained on CIFAR-10 (for details on the architecture, we refer the reader to Rosca et al. (2017)). We report the former to be compatible with existing literature, and the latter to obtain a more meaningful metric, since the network doing the evaluation was trained on the same dataset as the one we evaluate, hence the learned features will be more relevant for the task at hand. When reporting sample diversity, we subtract the average pairwise image similarity (as reported by MS-SSIM) computed as the mean of the similarity of every pair of images from 5 batches from the test set. Note that we can only apply this measure to CelebA, since for datasets such as CIFAR-10 different classes are represented by very different images, making this metric meaningless across class borders. Since our models are completely unsupervised, we do not compute the similarity across samples of the same class as in (Odena et al., 2016). The Inception Score and sample diversity metric results can be seen in Figure 8. The results obtained using the Independent Wasserstein critic on all datasets can be found in Figure 7.
+
+# 5.4 KEY TAKEAWAYS FROM REAL DATA EXPERIMENTS
+
+When analyzing the results obtained by training non-saturating GANs using gradient penalties (GAN-GP and DRAGAN-NS), we notice that:
+
+• Both gradient penalties help when training non-saturating GANs, by making the models more robust to hyperparameters.   
+• On CelebA, for various hyperparameter settings WGAN-GP fails to learn the data distribution and produces samples that do not look like faces (Figure 22). This results in a higher sample diversity than the reference diversity obtained on the test set, as reported by our diversity metric - see Figure 8(a) which compares sample diversity for the considered models across hyperparameters. The same figure shows that for most hyperparameter values, the WGAN-GP model produces higher diversity than the one obtained on the test set (indicating failure to capture the data distribution), while for most hyperparameters non-saturating GAN variants produce samples with lower diversity than that of the test set (indicating mode collapse). However, WGAN-GP is closer to the reference value for more hyperparameters, compared to the non-saturating GAN variants.   
+Even if we are only interested in the best results (without looking across the hyperparameter sweep), we see that the gradient penalties tend to improve results for non-saturating GANs.
+
+![](images/1bc94b416f45cc9e9848171fd1c2a5d27a8d505e6d2de9bb5188957904e36d39.jpg)  
+Figure 7: Negative Wasserstein distance estimated using an independent Wasserstein critic on the three datasets we evaluate on. The metric captures overfitting to the training data and low quality samples. Higher is better; the 10 black dots represent the results obtained with the 10 best hyperparameter settings.
+
+![](images/293c7484f28ca209e7e830b20f89bbd97a442331074e43ceb8dad2dc506c17fd.jpg)  
+Figure 8: Left plot shows sample diversity results on CelebA. It is important to look at this measure relative to the measure on the test set: too much diversity can mean failure to capture the data distribution, too little is indicative of mode collapse. To illustrate this, we report the diversity obtained when adding normal noise with zero mean and 0.1 standard deviation to the test set: this results in more diversity than the original data. The black dots report the results closest to the reference values obtained on the test set by each model. Middle plot: Inception Score results on CIFAR-10. Right most plot shows Inception Score computed using a VGG style network trained on CIFAR-10. As a reference benchmark, we also compute these scores using samples from test data split; diversity: 0.621, Inception Score: 11.25, Inception Score (VGG net trained on CIFAR-10): 9.18.
+
+• The non-saturating GAN trained with gradient penalties produces better samples which give better Inception Scores, both when looking at the results obtained from the best set of hyperparameters and when looking at the entire sweep.
+
+• While the non-saturating GAN variants are much faster to train than the WGAN-GP model (since we do only one discriminator update per generator update), they perform similarly to the WGAN-GP model. Thus, non-saturating GANs with penalties offer a better computation versus performance tradeoff. When we trained WGAN-GP models in which we update the discriminator only once per generator update, we noticed a decrease in sample quality for all datasets, reflected by our reported metrics, as seen in Figure 15.
+
+When looking at the independent Wasserstein critic results, we see that the WGAN-GP models perform best on Color MNIST and CIFAR-10. However, on CelebA the Independent Wasserstein Critic can distinguish between validation data examples and samples from the model (see Figure 7(b)). This is consistent with what we have seen by examining samples: the hyperparameters which result in samples of reduced quality are the same with a reduced negative Wasserstein distance.
+
+• The sample diversity metric and the Independent Wasserstein critic detect mode collapse. When DRAGAN-NS collapses for two hyperparameter settings, the negative Wasserstein distance reported by the critic for these jobs is low, showing that the critic captures the difference in distributions, and the sample diversity reported for those settings is greatly reduced (Figure 16).
+
+# 6 DISCUSSION
+
+We have shown that viewing the training dynamics of GANs through the lens of the underlying divergence at optimality can be misleading. On low-dimensional synthetic problems, we showed that non-saturating GANs are able to learn the true data distribution where Jensen-Shannon divergence minimization would fail. We also showed that gradient penalty regularizers help improve the training dynamics and robustness of non-saturating GANs. It is worth noting that one of the gradient penalty regularizers was originally proposed for Wasserstein GANs, motivated by properties of the Wasserstein distance; evaluating non-saturating GANs with similar gradient penalty regularizers helps disentangle the improvements arising from optimizing a different divergence (or distance) and the improvements from better training dynamics.
+
+Comparison between explored gradient penalties: As described in Section 2.3, we have evaluated two gradient penalties on non-saturating GANs. We now turn our attention to the distinction between the two gradient penalties. We have already noted that for a few hyperparameter settings, DRAGAN-NS produced samples with mode collapse, while the GAN-GP model did not. By looking at the resulting metrics, we note that there is no clear winner between the two types of gradient penalties. To assess whether the two penalties have a different regularization effect, we also tried applying both (with a gradient penalty coefficient of 10 for both, or of 5 for both), but that did not result in better models. This could be because the two penalties have a very similar effect, or due to optimization considerations (they might conflict with each other).
+
+Other gradient penalties: Besides the gradient penalties explored in this work, several other regularizers have been proposed for stabilizing GAN training. Roth et al. (2017) proposed a gradient penalty aiming to smooth the discriminator of $f$ -GANs (including the minimax GAN), which we refer to as $f$ -GAN-GP, inspired by Sønderby et al. (2016) and Arjovsky & Bottou (2017). Their gradient penalty is different from the ones explored here; specifically, their gradient penalty is weighted by the square of the discriminator’s probability of real for each data instance and the penalty is applied to data and samples (no noise is added). In Fisher-GAN (Mroueh & Sercu, 2017), an equality constraint that is added on the magnitude of the output of the discriminator on data as well as samples is directly penalized, as opposed to the magnitude of the discriminator gradients, as in WGAN-GP. Similar to WGAN-GP, the penalty was introduced in the framework of integral probability metrics, but it can be directly applied to other approaches to GAN training. Unlike WGAN-GP, Fisher GAN uses augmented Lagrangians to impose the equality constraint, instead of a penalty method. To the best of our knowledge, this has not been tried yet and we leave it for future work.
+
+The regularizers assessed in this work (the penalties proposed by DRAGAN and WGAN-GP), as well as others (such as $f$ -GAN-GP and Fisher-GAN) are similar in spirit, but have been proposed from distinct theoretical considerations. Future study of GAN regularizers will determine how these regularizers interact, and help us understand the mechanism by which they stabilize GAN training and motivate new approaches.
+
+# ACKNOWLEDGEMENTS
+
+We thank Ivo Danihelka and Jascha Sohl-Dickstein for helpful feedback and discussions.
+
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+
+# A RESULTS
+
+# A.1 SYNTHETIC EXPERIMENTS
+
+We present here more detailed results for our synthetic experiments.
+
+![](images/9feedfb917ce896ce0f638dbd99b9672543ce68a6206bb3d178708dd2963f782.jpg)  
+Figure 9: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different GAN variants, when varying the learning rate while keeping the input dimension fixed. Results averaged over 1000 runs. Lower values are better.
+
+![](images/a79722e76322fede6a0b090597f2bd6d06c54c2b00ae1e13db54394d9d4c8e7b.jpg)  
+Figure 10: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different GAN variants, when varying the learning rate while keeping the input dimension fixed. Results averaged over 1000 runs. Lower values are better.
+
+# A.2 THE EFFECT OF THE NUMBER OF DISCRIMINATOR UPDATES ON GAN AND GAN-GP
+
+In this section we assess the affects of varying the discriminator update count per generator update. We notice that using 100 discriminator updates per generator update results in a bad distribution fit for the non saturating GAN. GAN-GP scales better with the number of discriminator updates but
+
+increasing the number of discriminator updates does not always result in a closer match to the true distribution for this model either.
+
+![](images/40e37fd6f78f3b3bd7ea80633fe27f82ebcbd3898650fe397289f90ef59d17e3.jpg)  
+Figure 11: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training non saturating GANs, with varying the learning rates. Results averaged over 1000 runs. Lower values are better.
+
+![](images/a570050acc62544be4a40a0deb4d60634217977a5ff1800771fbba8de038adf1.jpg)  
+Figure 12: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training non saturating GANs, with varying the learning rates. Results averaged over 1000 runs. Lower values are better.
+
+# A.3 REAL DATA EXPERIMENTS
+
+We present here generated samples and other evaluation metrics on real data.
+
+![](images/f9426148f90cb290256f8d8ec1016dbcfa04dc9cb616d4b886269b0df175884e.jpg)  
+Square Fréchet distance across learning rate,input dimensior $_ { 1 = 1 0 0 0 }$ model:GAN-GP   
+Figure 13: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training GAN-GP, with varying the learning rates. Results averaged over 1000 runs. Lower values are better.
+
+![](images/c47257ce4fe1589913dec915d601418cce9349ae7bdc7f6a57232ac074d2ec67.jpg)  
+Square Fréchet distance across learning rate,input dimension $= 1 0 0 0$ model:GAN-GP   
+Figure 14: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training GAN-GP, with varying the learning rates. Results averaged over 1000 runs. Lower values are better.
+
+![](images/800af0f570aeda60dde80dd85d1d44525837355e7ca2da761f1585cf31d6a30e.jpg)  
+Figure 15: Comparison across models when doing one update for the discriminator in Wasserstein GAN (WGAN-GP-1). The reduced performance in consistent with the observed decrease in sample quality when examining results. Inception Score results obtained on the test set: with Imagenet trained classifier: 11.25, With CIFAR-10 trained classifier: 9.18. Higher is better; the 10 black dots represent the results obtained with the 10 best hyperparameter settings.
+
+![](images/0322fbe267a01d347845c8fd7017c65c6d62ada52b7f3de8d21106224ce5502a.jpg)
+
+![](images/42788a9608370983211d53dbcac4d8b91c18f908dc4c6b8a17743a9b0b82a723.jpg)  
+Figure 16: The metrics employed are able to capture mode collapse. Looking at the 5 worst values (the black dots) in a hyperparameter sweep according to sample diversity and negative Wasserstein distance as estimated by an Independent Wasserstein critic, we see that these metrics are able to capture the two examples of model collapse that we have seen when training DRGAN-NS on CelebA, as shown in Figure 21. For sample diversity, the worst results are computed by the biggest absolute difference to the reference point (test set diversity), while for negative Wasserstein distance the worst results are computed by choosing the lowest value.   
+Figure 17: Examples of mode collapse obtained for some hyperparameter settings with non-saturating GAN.
+
+![](images/7d088b54336a8dc9a8891a3a1bcd072d0972ca5b7ae8b5d2d79aab08ae20c336.jpg)  
+Figure 18: CIFAR-10 samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models.
+
+![](images/a711eff18b0844b96d3908cad4f96cc40761338f88c2dc1a13507fe63736414a.jpg)  
+Figure 19: CelebA samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models.
+
+![](images/7dfbe1e13c92d0ab8c6fb51b7f41f563ee7ecd555205833bb2d40bf0b514eb54.jpg)  
+Figure 20: CMNIST samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models.
+
+![](images/8a70e568414599f0a2bb97fe4e429b779085eff465cb4e90605236243b1eaa99.jpg)  
+Figure 21: Mode collapse when adding gradient penalties to non-saturating GANs. GAN-GP only had two instances of mode collapse, namely color mode collapse on Color-MNIST (left), while DRAGAN-NS only had two instances of mode collapse, which ocurred when trained on CelebA (right and middle).
+
+![](images/605fce6c9b6f16262e6b6d48270ec4a6a546091b135d989204bd18eea580180f.jpg)  
+Figure 22: Examples of failure to capture the data distribution with WGAN-GP. The model puts too much mass around the data distribution when trained on the CelebA dataset.

parse/train/ByQpn1ZA-/ByQpn1ZA-_content_list.json

@@ -0,0 +1,1844 @@
+[
+    {
+        "type": "text",
+        "text": "MANY PATHS TO EQUILIBRIUM: GANS DO NOT NEED TO DECREASE A DIVERGENCE AT EVERY STEP ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "William Fedus∗ 1, Mihaela Rosca∗ 2, Balaji Lakshminarayanan2, Andrew M. Dai1, Shakir Mohamed2 and Ian Goodfellow1 ",
+        "bbox": [
+            287,
+            175,
+            710,
+            204
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Google Brain   \n2DeepMind ",
+        "bbox": [
+            450,
+            217,
+            545,
+            244
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            286,
+            544,
+            301
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative adversarial networks (GANs) are a family of generative models that do not minimize a single training criterion. Unlike other generative models, the data distribution is learned via a game between a generator (the generative model) and a discriminator (a teacher providing training signal) that each minimize their own cost. GANs are designed to reach a Nash equilibrium at which each player cannot reduce their cost without changing the other players’ parameters. One useful approach for the theory of GANs is to show that a divergence between the training distribution and the model distribution obtains its minimum value at equilibrium. Several recent research directions have been motivated by the idea that this divergence is the primary guide for the learning process and that every step of learning should decrease the divergence. We show that this view is overly restrictive. During GAN training, the discriminator provides learning signal in situations where the gradients of the divergences between distributions would not be useful. We provide empirical counterexamples to the view of GAN training as divergence minimization. Specifically, we demonstrate that GANs are able to learn distributions in situations where the divergence minimization point of view predicts they would fail. We also show that gradient penalties motivated from the divergence minimization perspective are equally helpful when applied in other contexts in which the divergence minimization perspective does not predict they would be helpful. This contributes to a growing body of evidence that GAN training may be more usefully viewed as approaching Nash equilibria via trajectories that do not necessarily minimize a specific divergence at each step. ",
+        "bbox": [
+            233,
+            318,
+            764,
+            623
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            643,
+            336,
+            660
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative adversarial networks (GANs) (Goodfellow et al., 2014) are generative models based on a competition between a generator network $G$ and a discriminator network $D$ . The generator network $G$ represents a probability distribution $p _ { \\mathrm { m o d e l } } ( { \\pmb x } )$ . To obtain a sample from this distribution, we apply the generator network to a noise vector $_ { z }$ sampled from $p _ { z }$ , that is $\\pmb { x } = G ( \\pmb { z } )$ . Typically, $_ { z }$ is drawn from a Gaussian or uniform distribution, but any distribution with sufficient diversity is possible. The discriminator $D ( \\pmb { x } )$ attempts to distinguish whether an input value $_ { \\textbf { \\em x } }$ is real (came from the training data) or fake (came from the generator). ",
+        "bbox": [
+            174,
+            675,
+            825,
+            773
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The goal of the training process is to recover the true distribution $p _ { \\mathrm { d a t a } }$ that generated the data. Several variants of the GAN training process have been proposed. Different variants of GANs have been interpreted as approximately minimizing different divergences or distances between $p _ { \\mathrm { d a t a } }$ and $p _ { \\mathrm { m o d e l } }$ . However, it has been difficult to understand whether the improvements are caused by a change in the underlying divergence or the learning dynamics. ",
+        "bbox": [
+            174,
+            780,
+            825,
+            851
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We conduct several experiments to assess whether the improvements associated with new GAN methods are due to the reasons cited in their design motivation. We perform a comprehensive study of GANs on simplified, synthetic tasks for which the true $p _ { \\mathrm { d a t a } }$ is known and the relevant distances are straightforward to calculate, to assess the performance of proposed models against baseline methods. We also evaluate GANs using several independent evaluation measures on real data to better understand new approaches. Our contributions are: ",
+        "bbox": [
+            176,
+            857,
+            823,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            145
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We aim to clarify terminology used in recent papers, where the terms “standard GAN,” “regular GAN,” or “traditional GAN” are used without definition (e.g., (Arjovsky et al., 2017; Denton et al., 2015; Salimans et al., 2016; Donahue et al., 2016)). The original GAN paper described two different losses: the “minimax” loss and the “non-saturating” loss, equations (10) and (13) of Goodfellow (2016), respectively. Recently, it has become important to clarify this terminology, because many of the criticisms of “standard GANs”, e.g. Arjovsky et al. (2017), are applicable only to the minimax GAN, while the non-saturating GAN is the standard for GAN implementations. The non-saturating GAN was recommended for use in practice and implemented in the original paper of Goodfellow et al. (2014), and is the default in subsequent papers (Radford et al., 2015; Salimans et al., 2016; Donahue et al., 2016; Nowozin et al., 2016)1. To avoid confusion we will always indicate whether we mean minimax GAN (M-GAN) or non-saturating GAN (NS-GAN). We demonstrate that gradient penalties designed in the divergence minimization framework— to improve Wasserstein GANs (Gulrajani et al., 2017) or justified from a game theory perspective to improve minimax GANs (Kodali et al., 2017)—also improve the non-saturating GAN on both synthetic and real data. We observe improved sample quality and diversity. We find that non-saturating GANs are able to fit problems that cannot be fit by JensenShannon divergence minimization. Specifically, Figure 1 shows a GAN using the loss from the original non-saturating GAN succeeding on a task where the Jensen-Shannon divergence provides no useful gradient. Figure 2 shows that the non-saturating GAN does not suffer from vanishing gradients when applied to two widely separated Gaussian distributions. ",
+        "bbox": [
+            214,
+            156,
+            828,
+            457
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 VARIANTS OF GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            476,
+            665,
+            492
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NON-SATURATING AND MINIMAX GANS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            507,
+            501,
+            521
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the original GAN formulation (Goodfellow et al., 2014), the output of the discriminator is a probability and the cost function for the discriminator is given by the negative log-likelihood of the binary discrimination task of classifying samples as real or fake: ",
+        "bbox": [
+            174,
+            532,
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+            575
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/63683057d7a10abbae58d8c8f32dd9de072fdf842d1fd68d8eee2b9a67252f23.jpg",
+        "text": "$$\n\\begin{array} { r } { J ^ { ( D ) } ( D , G ) = - \\underset { x \\sim p _ { \\mathrm { d a t a } } } { \\mathbb { E } } \\left[ \\log D ( \\pmb { x } ) \\right] - \\underset { z \\sim p _ { z } } { \\mathbb { E } } \\left[ \\log ( 1 - D ( G ( z ) ) ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            593,
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+            618
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The theoretical analysis in (Goodfellow et al., 2014) is based on a zero-sum game in which the generator maximizes $J ^ { ( D ) }$ , a situation that we refer to here as “minimax GANs”. In minimax GANs the generator attempts to generate samples that have low probability of being fake, by minimizing the objective (2). However, in practice, Goodfellow et al. (2014) recommend implementing an alternative cost function that instead ensures that generated samples have high probability of being real, and the generator instead minimizes an alternative objective (3). ",
+        "bbox": [
+            174,
+            626,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e2b1da23d2b1ce9df67d5dcf5233e45f588cf8a547b5f5c7604a66942f83676.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathbf { M i n i m a x } } & { \\boldsymbol { J } ^ { ( G ) } ( G ) = \\underset { \\boldsymbol { z } \\sim p _ { \\boldsymbol { z } } } { \\mathbb { E } } \\log [ 1 - D ( G ( \\boldsymbol { z } ) ) ] . } \\\\ { \\mathbf { N o n - s a t u r a t i n g } } & { \\boldsymbol { J } ^ { ( G ) } ( G ) = \\underset { \\boldsymbol { z } \\sim p _ { \\boldsymbol { z } } } { - } \\log \\boldsymbol { D } ( G ( \\boldsymbol { z } ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            715,
+            687,
+            768
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We refer to the alternative objective as non-saturating, due to the non-saturating behavior of the gradient (see figure 2), and was the implementation used in the code of the original paper. We use the non-saturating objective (3) in all our experiments ",
+        "bbox": [
+            176,
+            771,
+            825,
+            814
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As shown in (Goodfellow et al., 2014), whenever $D$ successfully minimizes $J ^ { ( D ) }$ optimally, maximizing $J ^ { ( D ) }$ with respect to the generator is equivalent to minimizing the Jensen-Shannon divergence. Goodfellow et al. (2014) use this observation to establish that there is a unique Nash equilibrium in function space corresponding to $p _ { \\mathrm { d a t a } } = p _ { \\mathrm { m o d e l } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 WASSERSTEIN GAN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            178,
+            356,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Wasserstein GANs (Arjovsky et al., 2017) modify the discriminator to emit an unconstrained real number rather than a probability (analogous to emitting the logits rather than the probabilities used in the original GAN paper). The cost function for the WGAN then omits the log-sigmoid functions used in the original GAN paper. The cost function for the discriminator is now: ",
+        "bbox": [
+            173,
+            204,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d5e3a89f9f9eb1cbe44e95f4a0f90c95259a7fc414cfb1d4edd5929605e2b0c.jpg",
+        "text": "$$\n\\begin{array} { r } { W ^ { ( D ) } ( D , G ) = \\underset { \\pmb { x } \\sim p _ { \\mathrm { d a t a } } } { \\mathbb { E } } \\left[ D ( \\pmb { x } ) \\right] - \\underset { \\pmb { z } \\sim p _ { \\pmb { z } } } { \\mathbb { E } } \\left[ D ( G ( \\pmb { z } ) ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            266,
+            669,
+            292
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The cost function for the generator is simply $W ^ { ( G ) } = - W ^ { ( D ) } ( D , G )$ . When the discriminator is Lipschitz smooth, this approach approximately minimizes the earth mover’s distance between pdata and $p _ { \\mathrm { m o d e l } }$ . To enforce Lipschitz smoothness, the weights of $D$ are clipped to lie within $( - c , c )$ where $c$ is some small real number. ",
+        "bbox": [
+            173,
+            299,
+            825,
+            357
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 GRADIENT PENALTIES FOR GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            373,
+            691,
+            388
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multiple formulations of gradient penalties have been proposed for GANs. As introduced in WGANGP (Gulrajani et al., 2017), the gradient penalty is justified from the perspective of the Wasserstein distance, by imposing properties which hold for an optimal critic as an additional training criterion. In this approach, the gradient penalty is typically a penalty on the gradient norm, and is applied on a linear interpolation between data points and samples, thus smoothing out the space between the two distributions. ",
+        "bbox": [
+            173,
+            398,
+            826,
+            483
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Kodali et al. (2017) introduce DRAGAN with a gradient penalty from the perspective of regret minimization, by setting the regularization function to be a gradient penalty on points around the data manifold, as in Follow The Regularized Leader (Cesa-Bianchi & Lugosi, 2006), a standard no-regret algorithm. This encourages the discriminator to be close to linear around the data manifold, thus bringing the set of possible discriminators closer to a convex set, the set of linear functions. We also note that they used the minimax version of the game to define the loss, in which the generator maximizes $J ^ { ( D ) }$ rather than minimizing $J ^ { ( G ) }$ . ",
+        "bbox": [
+            173,
+            489,
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+            590
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To formalize the above, both proposed gradient penalties of the form: ",
+        "bbox": [
+            173,
+            595,
+            627,
+            611
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e5fa869ae3541d02b19cd0ddac7dd88e156ab7c05f67ecddc758c43be474b67.jpg",
+        "text": "$$\n\\underset { \\hat { x } \\sim p _ { \\hat { x } } } { \\mathbb { E } } \\left[ ( \\| \\nabla _ { \\hat { x } } D ( \\hat { x } ) \\| _ { 2 } - 1 ) ^ { 2 } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            628,
+            593,
+            654
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { \\hat { x } }$ is defined as the distribution defined by the sampling process: ",
+        "bbox": [
+            174,
+            665,
+            640,
+            680
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5e3349c6a9a41ee539b400018ae615a49099d18314280a248f2e59ad76335dca.jpg",
+        "text": "$$\nx \\sim p _ { \\mathrm { d a t a } } ; \\qquad x _ { \\mathrm { m o d e l } } \\sim p _ { \\mathrm { m o d e l } } ; \\qquad x _ { \\mathrm { n o i s e } } \\sim p _ { \\mathrm { n o i s e } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            671,
+            703
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58a9d7f978b07cdbf920d0cd288638928a0e3a99abd895ca3d842a3c488a9df2.jpg",
+        "text": "$$\n\\begin{array} { c l c r } { { \\alpha \\sim U ( 0 , 1 ) } } \\\\ { { \\hat { x } = \\alpha x + ( 1 - \\alpha ) \\tilde { x } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            777,
+            568,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As we will note in our experimental section, Kodali et al. (2017) also reported that mode-collapse is reduced using their version of the gradient penalty. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3.1 NON-SATURATING GAN WITH GRADIENT PENALTY ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider the non-saturating GAN objective (3) supplemented by two gradient penalties: the penalty proposed by Gulrajani et al. (2017), which we refer to as “GAN-GP”; the gradient penalty proposed by DRAGAN (Kodali et al., 2017), which we refer to as DRAGAN-NS, to emphasize that we use the non-saturating generator loss function. In both cases, the gradient penalty applies only to the discriminator, with the generator loss remaining unchanged (as defined in Equation 3). In this setting, the loss of the discriminator becomes: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c345cd545745234de5c42fbdca6bc4c7cc41d9e5674043d0e483861441be6be1.jpg",
+        "image_caption": [
+            "Figure 1: Visualization of experiment 1 training dynamics in two dimensions, demonstrated specifically in the case where the model is initialized so that it it represents a linear manifold parallel to the linear manifold of the training data. Here the GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). Specifically, this is an illustration of the parallel line thought experiment from (Arjovsky et al., 2017). When run in practice with a non-saturating GAN, the GAN succeeds. In the same setting, minimization of Jensen-Shannon divergence would fail. This indicates that while Jensen-Shannon divergence is useful for characterizing GAN equilibrium, it does not necessarily tell us much about non-equilibrium learning dynamics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a54a9d6aeeec6f2c2a4a1eae3069e47015d9759f25025cbabef9bb200f1cc8b.jpg",
+        "text": "$$\n\\tilde { J } ^ { ( D ) } ( D , G ) = - \\underset { x \\sim p _ { \\mathrm { d a t a } } } { \\mathbb { E } } \\left[ \\log D ( x ) \\right] - \\underset { z \\sim p _ { z } } { \\mathbb { E } } \\left[ \\log ( 1 - D ( G ( z ) ) ) \\right] + \\lambda \\underset { \\hat { x } \\sim p _ { \\hat { \\alpha } } } { \\mathbb { E } } \\left[ ( \\lVert \\nabla _ { \\hat { x } } D ( \\hat { x } ) \\rVert _ { 2 } - 1 ) ^ { 2 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider these GAN variants because: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• We want to assess whether gradient penalties are effective outside their original defining scope. Namely, we perform experiments to determine whether the benefit obtained by applying the gradient penalty for Wasserstein GANs is obtained from properties of the earth mover’s distance, or from the penalty itself. Similarly, we evaluate whether the DRAGAN gradient penalty is beneficial outside the minimax GAN setting.   \n• We want to assess whether the exact form of the gradient penalty matters.   \nWe compare three models, to control over different aspects of training: same gradient penalty but different underlying adversarial losses (GAN-GP versus WGAN-GP), as well as the same underlying adversarial loss, but different gradient penalties (GAN-GP versus DRAGAN-NS). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We note that we do not compare with the original DRAGAN formulation, which uses the minimax GAN formulation, since in this work we focus on non-saturating GAN variants. ",
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+    {
+        "type": "text",
+        "text": "3 MANY PATHS TO EQUILIBRIUM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The original GAN paper (Goodfellow et al., 2014) used the correspondence between $J ^ { ( D ) } ( D ^ { * } , G )$ and the Jensen-Shannon divergence to characterize the Nash equilibrium of minimax GANs. It is important to keep in mind that there are many ways for the learning process to approach this equilibrium point, and the majority of them do not correspond to gradually reducing the JensenShannon divergence at each step. Divergence minimization is useful for understanding the outcome of training, but GAN training is not the same thing as running gradient descent on a divergence and GAN training may not encounter the same problems as gradient descent applied to a divergence. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Arjovsky et al. (2017) describe the learning process of GANs from the perspective of divergence minimization and show that the Jensen-Shannon divergence is unable to provide a gradient that will bring $p _ { \\mathrm { d a t a } }$ and $p _ { \\mathrm { m o d e l } }$ together if both are sharp manifolds that do not overlap early in the learning process. Following this line of reasoning, they suggest that when applied to probability distributions that are supported only on low dimensional manifolds, the Kullback Leibler (KL), Jensen Shannon (JS) and Total Variation (TV) divergences do not provide a useful gradient for learning algorithms based on gradient descent, the “traditional GANs” is inappropriate for fitting such low dimensional manifolds (“traditional GAN” seems to refer to the minimax version of GANs used for theoretical analysis in the original paper, and there is no explicit statement about whether the argument is intended to apply to the non-saturating GAN implemented in the code accompanying the original GAN paper). In Section 4 we show that non-saturating GANs are able to learn on tasks where the data distribution lies on the low dimensional manifold. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e40fa277017297d4a93812edd2dc820808d2dc596cb8741b91762c1344aef944.jpg",
+        "image_caption": [
+            "Figure 2: (Left) A recreation of Figure 2 of Arjovsky et al. (2017). This figure is used by Arjovsky et al. (2017) to show that a model they call the “traditional GAN” suffers from vanishing gradients in the areas where $D ( x )$ is flat. This plot is correct if “traditional GAN” is used to refer to the minimax GAN, but it does not apply to the non-saturating GAN. (Right) A plot of both generator losses from the original GAN paper, as a function of the generator output. Even when the model distribution is highly separated from the data distribution, non-saturating GANs are able to bring the model distribution closer to the data distribution because the loss function has strong gradient when the generator samples are far from the data samples, even when the discriminator itself has nearly zero gradient. While it is true that the $\\begin{array} { r } { \\frac { 1 } { 2 } \\log ( 1 - D ( x ) ) } \\end{array}$ loss has a vanishing gradient on the right half of the plot, the original GAN paper instead recommends implementing $- { \\frac { 1 } { 2 } } \\log D ( x )$ . This latter, recommended loss function has a vanishing gradient only on the left side of the plot. It makes sense for the gradient to vanish on the left because generator samples in that area have already reached the area where data samples lie. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We show that non-saturating GANs do not suffer from vanishing gradients for two widely separated Gaussians in Figure 2. The fact that the gradient of the recommended loss does not actually vanish explains why GANs with the non-saturating objective (3), are able to bring together two widely separated Gaussian distributions. Note that the gradient for this loss does not vanish even when the discriminator is optimal. The discriminator has vanishing gradients but the generator loss amplifies small differences in discriminator outputs to recover strong gradients. This means it is possible to train the GAN by changing the loss rather than the discriminator. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the parallel lines thought experiment (Arjovsky et al., 2017) (see Figure 1), the main problem with the Jensen-Shannon divergence is that it is parameterized in terms of the density function, and the two density functions have no support in common. Most GANs, and many other models, can solve this problem by parameterizing their loss functions in terms of samples from the two distributions rather than in terms of their density functions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 SYNTHETIC EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To assess the learning process of GANs we empirically examine GAN training on pathological tasks where the data is constructed to lie on a low dimensional manifold, and show the model is able to learn the data distribution in cases where using the underlying divergence obtained at optimality would not provide useful gradients. We then evaluate convergence properties of common GAN variants on this task where the parameters generating the distribution are known. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT I: 1-D DATA MANIFOLD AND 1-D GENERATOR ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our first experiment, we generate synthetic training data that lies along a one-dimensional line and design a one-dimensional generative model, however, we embed the problem in a higher $d .$ - dimensional space where $d \\gg 1$ . This experiment is essentially an implementation of a thought experiment from Arjovsky et al. (2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically, in a $d$ -dimensional space, we define $p _ { \\mathrm { d a t a } }$ by randomly generating parameters defining the distribution once at the beginning of the experiment. We generate a random $b _ { r } \\in \\mathbb { R } ^ { d }$ and random $W _ { r } \\in \\mathbb { R } ^ { 1 \\times d }$ . Our latent $z _ { r } \\in \\mathbb { R } \\sim N ( 0 , \\sigma )$ where $\\sigma$ is the standard deviation of the normal distribution. The synthetic training data of $m$ examples is then given by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/553385b778976bb190f3672cff611cf586aad876c5691dabdde3e727cdd127b0.jpg",
+        "text": "$$\n\\{ x ^ { ( i ) } \\} _ { i = 1 } ^ { m } = \\{ z _ { r } ^ { ( i ) } \\} _ { i = 1 } ^ { m } W _ { r } + b _ { r }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The real synthetic data is therefore Gaussian distributed on a 1-D surface within the space, where the position is determined by $b _ { r }$ and the orientation is determined by $W _ { r }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The generator also assumes the same functional form, that is, it is also intrinsically one dimensional, ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9dbfa00f4c01196c486dcf0fa01d86893b11745f64b474fd2fcb05e6eece7787.jpg",
+        "text": "$$\nG _ { \\theta } ( z ) = z W _ { \\theta } + b _ { \\theta }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            479,
+            565,
+            497
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $b _ { \\theta } \\in \\mathbb { R } ^ { d }$ and $W _ { \\theta } \\in \\mathbb { R } ^ { 1 \\times d }$ . The discriminator is a single hidden layer ReLU network, which is of higher complexity than the generator so that it may learn non-linear boundaries in the space. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This experiment captures the idea of sharp, non-overlapping manifolds that motivate alternative GAN losses. Further, because we know the true generating parameters of the training data, we may explicitly test convergence properties of the various methodologies. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENT II: 1-D DATA MANIFOLD AND OVERCOMPLETE GENERATOR ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our second experiment, the synthetic training data is still the same (lying on a 1-D line) and given by Eq. 11 but now the generator is overcomplete for this task, and has a higher latent dimension $g$ , where $1 < g \\le d$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3bd0b7d24010b2a796e40ed2ea5504e2bf76d7a39204aaf70cbce9579a82cc3b.jpg",
+        "text": "$$\nG ( z ) = z W _ { \\theta } + b _ { \\theta }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            688,
+            562,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where matrix $W _ { \\theta } \\in \\mathbb { R } ^ { g \\times d }$ and vector $b _ { \\theta } \\in \\mathbb { R } ^ { d }$ , so that the generator is able to represent a manifold with too high of a dimensionality. The generator parameterizes a multivariate Gaussian $N ( x ; \\mu , \\Sigma )$ with $\\mu = b$ . The covariance matrix elements $\\dot { \\Sigma _ { i j } } = E [ \\sigma ^ { 2 } ( X _ { i } - \\mu _ { i } ) ( X _ { j } - \\mu _ { j } ) ] = \\sigma ^ { 2 } \\dot { E } [ ( X _ { i } -$ $\\mu _ { i } ) ( X _ { j } - \\mu _ { j } ) ]$ . In vector notation, $\\Sigma = \\sigma ^ { 2 } W ^ { T } W$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the convergence of an experimental trial, we report the square Fréchet distance (Fréchet (1957)) between the true data Gaussian distribution and the fitted Gaussian parameters. In our notation, where the $r$ subscript denotes real data, the $\\theta$ subscript denotes the generator Gaussian parameters and $\\| x \\| ^ { 2 }$ is the squared $l _ { 2 }$ norm of $x$ , the Fréchet distance is defined as (Dowson $\\&$ Landau (1982)): ",
+        "bbox": [
+            174,
+            814,
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+            886
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11e1f0ab6e5e7111471afb83d368c4535eb1472ba63dd108d3c68479fd8a4b6f.jpg",
+        "text": "$$\nd ^ { 2 } ( \\mu _ { r } , \\mu _ { \\theta } , \\Sigma _ { r } , \\Sigma _ { \\theta } ) = \\| \\mu _ { \\theta } - \\mu _ { r } \\| ^ { 2 } + \\mathrm { T r } \\left( \\Sigma _ { r } + \\Sigma _ { \\theta } - 2 ( \\Sigma _ { r } \\Sigma _ { \\theta } ) ^ { - 1 / 2 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Every GAN variant was trained for 200000 steps. For each step, the generator is updated once and the discriminator is updated 5 times. Throughout the paper, the number of steps will correspond to the number of generator updates. ",
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The main conclusions from our synthetic data experiments are: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Gradient penalties (both applied near the data manifold, DRAGAN-NS, and at an interpolation between data and samples, GAN-GP) stabilize training and improve convergence (Figures 3, 9, 10).   \nDespite the inability of Jensen-Shannon divergence minimization to solve this problem, we find that the non-saturating GAN succeeds in converging to the 1D data manifold (Figure 3). However, in higher dimensions the resulting fit is not as strong as the other methods: Figure 10 shows that increasing the number of dimensions while keeping the learning rate fixed can decrease the performance of the non-saturating GAN model.   \nNon-saturating GANs are able to learn data distributions which are disjoint from the training sample distribution at initialization (or another point in training), as demonstrated in Figure 1.   \nUpdating the discriminator 5 times per generator update does not result in vanishing gradients when using the non-saturating cost. However, when scaling the number of discriminator updates to 100 per generator update, non-saturating GANs perform worse than when using a smaller number of updates (1, 5, 10). Gradient penalties help here too: GAN-GP scales better with the number of discriminator updates. The results are detailed in Appendix Section A.2.   \n• An over-capacity generator with the ability to have more directions of high variance than the underlying data is able to capture the data distribution using non-saturating GAN training (Figure 4). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 HYPERPARAMETER SENSITIVITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            486,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We assess the robustness of the considered models by looking at results across hyperparameters for both experiment 1 and experiment 2. In one setting, we keep the input dimension fixed while varying the learning rate (Figure 9); in another setting, we keep the learning rate fixed, while varying the input dimension (Figure 10). In both cases, the results are averaged out over 1000 runs per setting, each starting from a different random seed. We notice that: ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• The non-saturating GAN model (with no gradient penalty) is most sensitive to hyperparameters.   \n• Gradient penalties make the non-saturating GAN model more robust.   \n• Both Wasserstein GAN formulations are quite robust to hyperparameter changes.   \n• For certain hyperparameter settings, there is no performance difference between the two gradient penalties for the non-saturating GAN, when averaging across random seeds. This is especially visible in Experiment 1, when the number of latent variables is 1. This could be due to the fact that the data sits on a low dimensional manifold, and because the discriminator is a small, shallow network. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 REAL DATA EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            760,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To assess the effectiveness of the gradient penalty on standard datasets for the non-saturating GAN formulation, we train a non-saturating GAN, a non-saturating GAN with the gradient penalty introduced by (Gulrajani et al., 2017) (denoted by GAN-GP), a non-saturating GAN with the gradient penalty introduced by (Kodali et al., 2017) (denoted by DRAGAN-NS), and a Wasserstein GAN with gradient penalty (WGAN-GP) on three datasets: Color MNIST (Metz et al., 2016) - data dimensionality (28, 28, 3), CelebA (Liu et al., 2015) - data dimensionality (64, 64, 3) and CIFAR-10 (Krizhevsky, 2009) - data dimensionality (32, 32, 3), as seen in Figure 6. ",
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For all our experiments we used $\\lambda = 1 0$ as the gradient penalty coefficient and used batch normalization (Ioffe & Szegedy, 2015); Kodali et al. (2017) suggests that batch normalization is not neeeded for DRAGAN, but we found that it also improved our DRAGAN-NS results. We used the Adam optimizer (Kingma & Ba, 2014) with $\\beta _ { 1 } = 0 . 5$ and $\\beta _ { 2 } = 0 . 9$ and a batch size of 64. The input data was scaled to be between -1 and 1. We did not add any noise to the discriminator inputs or activations, as that regularization technique can be interpreted as having the same goal as gradient penalties, and we wanted to avoid a confounding factor. We trained all Color MNIST models for 100000 steps, and CelebA and CIFAR-10 models for 200000 steps. We note that the experimental results on real data for the non-saturating GAN and for the Improved Wasserstein GAN (WGAN-GP) are quoted with permission from an earlier publication by Rosca et al. (2017). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/73c9856186ab835d1a3a36f708d693d9fd5a76f7a8243f278a7290805ca7a30a.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of experiment 1 training dynamics in two dimensions. Here the GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). We note that this is a visual illustration, and the results have not been averaged out over multiple seeds. Exact plots may vary on different runs. However, a single example of success is sufficient to refute claims that this this task is impossible for this model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We note that the WGAN-GP model was the only model for which we did 5 discriminator updates in real data experiments. All other models (DCGAN, DRAGAN-NS, GAN-GP) used one discriminator update for generator update. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For all reported results, we sweep over two hyperparameters: ",
+        "bbox": [
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+            804,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Learning rates for the discriminator and generator. Following Radford et al. (2015), we tried learning rates of 0.0001, 0.0002, 0.0003 for both the discriminator and the generator. We note that this is consistent with WGAN-GP, where the authors use 0.0002 for CIFAR-10 experiments.   \n• Number of latents. For CelebA and CIFAR-10 we try latent sizes 100, 128 and 150, while for Color MNIST we try 10, 50, 75. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/547232301dbc7de185f58c14971f802086c3adeab0c20de259ecfbeebc49be2b.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of experiment 2 training dynamics in two dimensions - where the GAN model has 3 latent variables. Here the rank one GAN model (red points) converges upon the one dimensional synthetic data distribution (blue points). We observe how for poor initialization the non-saturating GAN suffers from mode collapse. However, adding a gradient penalty stabilizes training. We note that this is a visual illustration, and the results have not been averaged out over multiple seeds. Exact plots may vary on different runs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.1 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Unlike the synthetic case, here we are unable to evaluate the performance of our models relative to the true solution, since that is unknown. Moreover, there is no single metric that can evaluate the performance of GANs. We thus complement visual inspection with three metrics, each measuring a different criteria related to model performance. We use the Inception Score (Salimans et al., 2016) to measure how visually appealing CIFAR-10 samples are, the MS-SSIM metric (Wang et al., 2003; Odena et al., 2016) to check sample diversity, and an Improved Wasserstein independent critic to assess overfitting, as well as sample quality (Danihelka et al., 2017). For a more detailed explanation of these metrics, we refer to Rosca et al. (2017). In all our experiments, we control over discriminator and generator architectures, using the ones used by DCGAN (Radford et al., 2015) and the original WGAN paper (Arjovsky et al., 2017)2. We note that the WGAN-GP paper used a different architecture when reporting the Inception Score on CIFAR10, and thus their results are not directly comparable. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For all the metrics, we report both the hyperparameter sensitivity of the model (by showing quartile statistics), as well as the 10 best results according to the metric. The sample diversity measure needs to be seen in context with the value reported on the test set: too high diversity can mean failure to capture the data distribution. For all other metrics, higher is better. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/e194aa4cd2fbbf76bd867f710668fca21af89252d739a0c91ca2ce2faff9625d.jpg",
+        "image_caption": [
+            "Figure 5: The square Fréchet distance between the learned Gaussian and the true Gaussian distribution. For reference, we also plot the distance obtained by a randomly initialized generator with the same architecture as the trained generators. Results are averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/90a54d4069466c009ace763dbd8f1d757d7480e357df414e3d53e2c58276309b.jpg",
+        "image_caption": [
+            "Figure 6: Examples from the three datasets explored in this paper: Color MNIST (left), CIFAR-10 (middle) and CelebA (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 VISUAL SAMPLE INSPECTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By visually inspecting the results of our models, we noticed that applying gradient penalties to the non-saturating GAN results in more stable training across the board. When training the non-saturating GAN with no gradient penalty, we did observe cases of severe mode collapse (see Figure 17). Gradient penalties improves upon that, but we can still observe mode collapse. Each non-saturating GAN variant with gradient penalty (DRAGAN-NS and GAN-GP) only produced mode collapse on one dataset, see Figure 21). We also noticed that for certain learning rates, WGAN-GPs fail to learn the data distribution (Figure 22). For the GAN-GP and DRAGAN-NS models, most hyperparameters produced samples of equal quality - the models are quite robust. We show samples from the GAN-GP, DRAGAN-NS and WGAN-GP models in Figures 18, 19 and 20. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 METRICS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We show that gradient penalties make non-saturating GANs more robust to hyperparameter changes. For this, we report not only the best obtained results, but rather a box plot of the obtained results showing the quartiles obtained by each sweep, along with the top 10 best results explicitly shown in the graph (note that for each model we tried 27 different hyperparameter settings, corresponding to 3 discriminator learning rates, 3 generator learning rates and 3 generator input sizes). We report two Inception Score metrics for CIFAR-10, one using the standard Inception network used when the metric was introduced (Salimans et al., 2016), trained on the Imagenet dataset, as well as a VGG style network trained on CIFAR-10 (for details on the architecture, we refer the reader to Rosca et al. (2017)). We report the former to be compatible with existing literature, and the latter to obtain a more meaningful metric, since the network doing the evaluation was trained on the same dataset as the one we evaluate, hence the learned features will be more relevant for the task at hand. When reporting sample diversity, we subtract the average pairwise image similarity (as reported by MS-SSIM) computed as the mean of the similarity of every pair of images from 5 batches from the test set. Note that we can only apply this measure to CelebA, since for datasets such as CIFAR-10 different classes are represented by very different images, making this metric meaningless across class borders. Since our models are completely unsupervised, we do not compute the similarity across samples of the same class as in (Odena et al., 2016). The Inception Score and sample diversity metric results can be seen in Figure 8. The results obtained using the Independent Wasserstein critic on all datasets can be found in Figure 7. ",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.4 KEY TAKEAWAYS FROM REAL DATA EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When analyzing the results obtained by training non-saturating GANs using gradient penalties (GAN-GP and DRAGAN-NS), we notice that: ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• Both gradient penalties help when training non-saturating GANs, by making the models more robust to hyperparameters.   \n• On CelebA, for various hyperparameter settings WGAN-GP fails to learn the data distribution and produces samples that do not look like faces (Figure 22). This results in a higher sample diversity than the reference diversity obtained on the test set, as reported by our diversity metric - see Figure 8(a) which compares sample diversity for the considered models across hyperparameters. The same figure shows that for most hyperparameter values, the WGAN-GP model produces higher diversity than the one obtained on the test set (indicating failure to capture the data distribution), while for most hyperparameters non-saturating GAN variants produce samples with lower diversity than that of the test set (indicating mode collapse). However, WGAN-GP is closer to the reference value for more hyperparameters, compared to the non-saturating GAN variants.   \nEven if we are only interested in the best results (without looking across the hyperparameter sweep), we see that the gradient penalties tend to improve results for non-saturating GANs. ",
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1bc94b416f45cc9e9848171fd1c2a5d27a8d505e6d2de9bb5188957904e36d39.jpg",
+        "image_caption": [
+            "Figure 7: Negative Wasserstein distance estimated using an independent Wasserstein critic on the three datasets we evaluate on. The metric captures overfitting to the training data and low quality samples. Higher is better; the 10 black dots represent the results obtained with the 10 best hyperparameter settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/293c7484f28ca209e7e830b20f89bbd97a442331074e43ceb8dad2dc506c17fd.jpg",
+        "image_caption": [
+            "Figure 8: Left plot shows sample diversity results on CelebA. It is important to look at this measure relative to the measure on the test set: too much diversity can mean failure to capture the data distribution, too little is indicative of mode collapse. To illustrate this, we report the diversity obtained when adding normal noise with zero mean and 0.1 standard deviation to the test set: this results in more diversity than the original data. The black dots report the results closest to the reference values obtained on the test set by each model. Middle plot: Inception Score results on CIFAR-10. Right most plot shows Inception Score computed using a VGG style network trained on CIFAR-10. As a reference benchmark, we also compute these scores using samples from test data split; diversity: 0.621, Inception Score: 11.25, Inception Score (VGG net trained on CIFAR-10): 9.18. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• The non-saturating GAN trained with gradient penalties produces better samples which give better Inception Scores, both when looking at the results obtained from the best set of hyperparameters and when looking at the entire sweep. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• While the non-saturating GAN variants are much faster to train than the WGAN-GP model (since we do only one discriminator update per generator update), they perform similarly to the WGAN-GP model. Thus, non-saturating GANs with penalties offer a better computation versus performance tradeoff. When we trained WGAN-GP models in which we update the discriminator only once per generator update, we noticed a decrease in sample quality for all datasets, reflected by our reported metrics, as seen in Figure 15. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "When looking at the independent Wasserstein critic results, we see that the WGAN-GP models perform best on Color MNIST and CIFAR-10. However, on CelebA the Independent Wasserstein Critic can distinguish between validation data examples and samples from the model (see Figure 7(b)). This is consistent with what we have seen by examining samples: the hyperparameters which result in samples of reduced quality are the same with a reduced negative Wasserstein distance. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• The sample diversity metric and the Independent Wasserstein critic detect mode collapse. When DRAGAN-NS collapses for two hyperparameter settings, the negative Wasserstein distance reported by the critic for these jobs is low, showing that the critic captures the difference in distributions, and the sample diversity reported for those settings is greatly reduced (Figure 16). ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We have shown that viewing the training dynamics of GANs through the lens of the underlying divergence at optimality can be misleading. On low-dimensional synthetic problems, we showed that non-saturating GANs are able to learn the true data distribution where Jensen-Shannon divergence minimization would fail. We also showed that gradient penalty regularizers help improve the training dynamics and robustness of non-saturating GANs. It is worth noting that one of the gradient penalty regularizers was originally proposed for Wasserstein GANs, motivated by properties of the Wasserstein distance; evaluating non-saturating GANs with similar gradient penalty regularizers helps disentangle the improvements arising from optimizing a different divergence (or distance) and the improvements from better training dynamics. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Comparison between explored gradient penalties: As described in Section 2.3, we have evaluated two gradient penalties on non-saturating GANs. We now turn our attention to the distinction between the two gradient penalties. We have already noted that for a few hyperparameter settings, DRAGAN-NS produced samples with mode collapse, while the GAN-GP model did not. By looking at the resulting metrics, we note that there is no clear winner between the two types of gradient penalties. To assess whether the two penalties have a different regularization effect, we also tried applying both (with a gradient penalty coefficient of 10 for both, or of 5 for both), but that did not result in better models. This could be because the two penalties have a very similar effect, or due to optimization considerations (they might conflict with each other). ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Other gradient penalties: Besides the gradient penalties explored in this work, several other regularizers have been proposed for stabilizing GAN training. Roth et al. (2017) proposed a gradient penalty aiming to smooth the discriminator of $f$ -GANs (including the minimax GAN), which we refer to as $f$ -GAN-GP, inspired by Sønderby et al. (2016) and Arjovsky & Bottou (2017). Their gradient penalty is different from the ones explored here; specifically, their gradient penalty is weighted by the square of the discriminator’s probability of real for each data instance and the penalty is applied to data and samples (no noise is added). In Fisher-GAN (Mroueh & Sercu, 2017), an equality constraint that is added on the magnitude of the output of the discriminator on data as well as samples is directly penalized, as opposed to the magnitude of the discriminator gradients, as in WGAN-GP. Similar to WGAN-GP, the penalty was introduced in the framework of integral probability metrics, but it can be directly applied to other approaches to GAN training. Unlike WGAN-GP, Fisher GAN uses augmented Lagrangians to impose the equality constraint, instead of a penalty method. To the best of our knowledge, this has not been tried yet and we leave it for future work. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The regularizers assessed in this work (the penalties proposed by DRAGAN and WGAN-GP), as well as others (such as $f$ -GAN-GP and Fisher-GAN) are similar in spirit, but have been proposed from distinct theoretical considerations. Future study of GAN regularizers will determine how these regularizers interact, and help us understand the mechanism by which they stabilize GAN training and motivate new approaches. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            728,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We thank Ivo Danihelka and Jascha Sohl-Dickstein for helpful feedback and discussions. ",
+        "bbox": [
+            176,
+            752,
+            754,
+            767
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+            102,
+            287,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.1 SYNTHETIC EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We present here more detailed results for our synthetic experiments. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/9feedfb917ce896ce0f638dbd99b9672543ce68a6206bb3d178708dd2963f782.jpg",
+        "image_caption": [
+            "Figure 9: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different GAN variants, when varying the learning rate while keeping the input dimension fixed. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a79722e76322fede6a0b090597f2bd6d06c54c2b00ae1e13db54394d9d4c8e7b.jpg",
+        "image_caption": [
+            "Figure 10: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different GAN variants, when varying the learning rate while keeping the input dimension fixed. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A.2 THE EFFECT OF THE NUMBER OF DISCRIMINATOR UPDATES ON GAN AND GAN-GP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we assess the affects of varying the discriminator update count per generator update. We notice that using 100 discriminator updates per generator update results in a bad distribution fit for the non saturating GAN. GAN-GP scales better with the number of discriminator updates but ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "increasing the number of discriminator updates does not always result in a closer match to the true distribution for this model either. ",
+        "bbox": [
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+        "page_idx": 16
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+    {
+        "type": "image",
+        "img_path": "images/40e37fd6f78f3b3bd7ea80633fe27f82ebcbd3898650fe397289f90ef59d17e3.jpg",
+        "image_caption": [
+            "Figure 11: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training non saturating GANs, with varying the learning rates. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 16
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+    {
+        "type": "image",
+        "img_path": "images/a570050acc62544be4a40a0deb4d60634217977a5ff1800771fbba8de038adf1.jpg",
+        "image_caption": [
+            "Figure 12: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training non saturating GANs, with varying the learning rates. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.3 REAL DATA EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            853,
+            405,
+            868
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We present here generated samples and other evaluation metrics on real data. ",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/f9426148f90cb290256f8d8ec1016dbcfa04dc9cb616d4b886269b0df175884e.jpg",
+        "image_caption": [
+            "Square Fréchet distance across learning rate,input dimensior $_ { 1 = 1 0 0 0 }$ model:GAN-GP ",
+            "Figure 13: Synthetic Experiment 1. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training GAN-GP, with varying the learning rates. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/c47257ce4fe1589913dec915d601418cce9349ae7bdc7f6a57232ac074d2ec67.jpg",
+        "image_caption": [
+            "Square Fréchet distance across learning rate,input dimension $= 1 0 0 0$ model:GAN-GP ",
+            "Figure 14: Synthetic Experiment 2. The square Fréchet distance between the generated Gaussian parameters and true Gaussian parameters for different number of discriminator updates when training GAN-GP, with varying the learning rates. Results averaged over 1000 runs. Lower values are better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/800af0f570aeda60dde80dd85d1d44525837355e7ca2da761f1585cf31d6a30e.jpg",
+        "image_caption": [
+            "Figure 15: Comparison across models when doing one update for the discriminator in Wasserstein GAN (WGAN-GP-1). The reduced performance in consistent with the observed decrease in sample quality when examining results. Inception Score results obtained on the test set: with Imagenet trained classifier: 11.25, With CIFAR-10 trained classifier: 9.18. Higher is better; the 10 black dots represent the results obtained with the 10 best hyperparameter settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/0322fbe267a01d347845c8fd7017c65c6d62ada52b7f3de8d21106224ce5502a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/42788a9608370983211d53dbcac4d8b91c18f908dc4c6b8a17743a9b0b82a723.jpg",
+        "image_caption": [
+            "Figure 16: The metrics employed are able to capture mode collapse. Looking at the 5 worst values (the black dots) in a hyperparameter sweep according to sample diversity and negative Wasserstein distance as estimated by an Independent Wasserstein critic, we see that these metrics are able to capture the two examples of model collapse that we have seen when training DRGAN-NS on CelebA, as shown in Figure 21. For sample diversity, the worst results are computed by the biggest absolute difference to the reference point (test set diversity), while for negative Wasserstein distance the worst results are computed by choosing the lowest value. ",
+            "Figure 17: Examples of mode collapse obtained for some hyperparameter settings with non-saturating GAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            416,
+            583,
+            576,
+            705
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d088b54336a8dc9a8891a3a1bcd072d0972ca5b7ae8b5d2d79aab08ae20c336.jpg",
+        "image_caption": [
+            "Figure 18: CIFAR-10 samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            256,
+            762,
+            738,
+            886
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/a711eff18b0844b96d3908cad4f96cc40761338f88c2dc1a13507fe63736414a.jpg",
+        "image_caption": [
+            "Figure 19: CelebA samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            135,
+            815,
+            299
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/7dfbe1e13c92d0ab8c6fb51b7f41f563ee7ecd555205833bb2d40bf0b514eb54.jpg",
+        "image_caption": [
+            "Figure 20: CMNIST samples obtained from the GAN-GP, DRAGAN-NS, and WGAN-GP models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            256,
+            411,
+            736,
+            534
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/8a70e568414599f0a2bb97fe4e429b779085eff465cb4e90605236243b1eaa99.jpg",
+        "image_caption": [
+            "Figure 21: Mode collapse when adding gradient penalties to non-saturating GANs. GAN-GP only had two instances of mode collapse, namely color mode collapse on Color-MNIST (left), while DRAGAN-NS only had two instances of mode collapse, which ocurred when trained on CelebA (right and middle). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            647,
+            815,
+            809
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/605fce6c9b6f16262e6b6d48270ec4a6a546091b135d989204bd18eea580180f.jpg",
+        "image_caption": [
+            "Figure 22: Examples of failure to capture the data distribution with WGAN-GP. The model puts too much mass around the data distribution when trained on the CelebA dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            406,
+            815,
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+        ],
+        "page_idx": 20
+    }
+]

parse/train/GFV8IVyMM4n/GFV8IVyMM4n.md

@@ -0,0 +1,339 @@
+# Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis
+
+Tianchang Shen 1,2,3 Jun Gao1,2,3 Kangxue Yin 1
+
+Ming-Yu Liu 1 Sanja Fidler1,2,3
+
+NVIDIA1 University of Toronto2 Vector Institute3
+
+{frshen, jung, kangxuey, mingyul, sfidler}@nvidia.com
+
+# Abstract
+
+We introduce DMTET, a deep 3D conditional generative model that can synthesize high-resolution 3D shapes using simple user guides such as coarse voxels. It marries the merits of implicit and explicit 3D representations by leveraging a novel hybrid 3D representation. Compared to the current implicit approaches, which are trained to regress the signed distance values, DMTET directly optimizes for the reconstructed surface, which enables us to synthesize finer geometric details with fewer artifacts. Unlike deep 3D generative models that directly generate explicit representations such as meshes, our model can synthesize shapes with arbitrary topology. The core of DMTET includes a deformable tetrahedral grid that encodes a discretized signed distance function and a differentiable marching tetrahedra layer that converts the implicit signed distance representation to the explicit surface mesh representation. This combination allows joint optimization of the surface geometry and topology as well as generation of the hierarchy of subdivisions using reconstruction and adversarial losses defined explicitly on the surface mesh. Our approach significantly outperforms existing work on conditional shape synthesis from coarse voxel inputs, trained on a dataset of complex 3D animal shapes. Project page: https://nv-tlabs.github.io/DMTet/.
+
+# 1 Introduction
+
+Fields such as simulation, architecture, gaming, and film rely on high-quality 3D content with rich geometric details and complex topology. However, creating such content requires tremendous expert human effort. It takes a significant amount of development time to create each individual 3D asset. In contrast, creating rough 3D shapes with simple building blocks like voxels has been widely adopted. For example, Minecraft has been used by hundreds of millions of users for creating 3D content. Most of them are non-experts. Developing A.I. tools that enable regular people to upscale coarse, voxelized objects into high resolution, beautiful 3D shapes would bring us one step closer to democratizing high-quality 3D content creation. Similar tools can be envisioned for turning 3D scans of objects recorded by modern phones into high-quality forms. Our work aspires to create such capabilities.
+
+A powerful 3D representation is a critical component of a learning-based 3D content creation framework. A good 3D representation for high-quality reconstruction and synthesis should capture local geometric details and represent objects with arbitrary topology while also being memory and computationally efficient for fast inference in interactive applications.
+
+Recently, neural implicit representations [8, 39, 42, 51], which use a neural network to implicitly represent a shape via a signed distance field (SDF) or an occupancy field (OF), have emerged as an effective 3D representation. Neural implicits have the benefit of representing complex geometry and topology, not limited to a predefined resolution. The success of these methods has been shown in shape compression [49, 13, 51], single-image shape generation [47, 60, 48], and point cloud reconstruction [57]. However, most of the current implicit approaches are trained by regressing to SDF or OF values and cannot utilize an explicit supervision on the target surface, which imposes useful constraints for training. To mitigate this issue, several works [45, 31] proposed to utilize iso-surfacing techniques such as the Marching Cubes (MC) algorithm to extract a surface mesh from the implicit representation, which, however, is computationally expensive.
+
+In this work, we introduce DMTET, a deep 3D conditional generative model for high-resolution 3D shape synthesis from user guides in the form of coarse voxels. In the heart of DMTET is a new differentiable shape representation that marries implicit and explicit 3D representations. In contrast to deep implicit approaches optimized for predicting sign distance (or occupancy) values, our model employs additional supervision on the surface, which empirically renders higher quality shapes with finer geometric details. Compared to methods that learn to directly generate explicit representations, such as meshes [54], by committing to a preset topology, our DMTET can produce shapes with arbitrary topology. Specifically, DMTET predicts the underlying surface parameterized by an implicit function encoded via a deformable tetrahedral grid. The underlying surface is converted into an explicit mesh with a Marching Tetrahedra (MT) algorithm, which we show is differentiable and more performant than the Marching Cubes. DMTET maintains efficiency by learning to adapt the grid resolution by deforming and selectively subdividing tetrahedra. This has the effect of spending computation only on the relevant regions in space. We achieve further gains in the overall quality of the output shape with learned surface subdivision. Our DMTET is end-to-end differentiable, allowing the network to jointly optimize the geometry and topology of the surface, as well as the hierarchy of subdivisions using a loss function defined explicitly on the surface mesh.
+
+We demonstrate our DMTET on two challenging tasks: 3D shape synthesis from coarse voxel inputs and point cloud 3D reconstruction. We outperform existing state-of-the-art methods by a significant margin while being 10 times faster than alternative implicit representation-based methods at inference time. In summary, we make the following technical contributions:
+
+1. We show that using Marching Tetrahedra (MT) as a differentiable iso-surfacing layer allows topological change for the underlying shape represented by a implicit field, in contrast to the analysis in prior works [31, 45].   
+2. We incorporate MT in a DL framework and introduce DMTET, a hybrid representation that combines implicit and explicit surface representations. We demonstrate that the additional supervision (e.g. chamfer distance, adversarial loss) defined directly on the extracted surface from implicit field improves the shape synthesis quality.   
+3. We introduce a coarse-to-fine optimization strategy that scales DMTET to high resolution during training. We thus achieves better reconstruction quality than state-of-the-art methods on challenging 3D shape synthesis tasks, while requiring a lower computation cost.
+
+# 2 Related Work
+
+We review the related work on learning-based 3D synthesis methods based on their 3D representations.
+
+Voxel-based Methods Early work [59, 10, 38] represented 3D shapes as voxels, which store the coarse occupancy (inside/outside) values on a regular grid, which makes powerful convolutional neural networks native and renders impressive results on 3D reconstruction and synthesis [12, 11, 58, 2]. For high-resolution shape synthesis, DECOR-GAN [6] transfers geometric details from a high-resolution shape represented in voxel to a low-resolution shape by utilizing a discriminator defined on 3D patches of the voxel grid. However, the computational and memory costs grow cubically as the resolution increases, prohibiting the reconstruction of fine geometric details and smooth curves. One common way to address this limitation is building hierarchical structures such as octrees [46, 52, 55, 56, 24, 52], which adapt the grid resolution locally based on the underlying shape. In this paper, we adopt a hierarchical deformable tetrahedral grid to utilize the resolution better. Unlike octree-based shape synthesis, our network learns grid deformation and subdivision jointly to better represent the surface without relying on explicit supervision from a pre-computed hierarchy.
+
+Deep Implicit Fields (DIFs) represent a 3D shape as a zero level set of a continuous function parameterized by a neural network [39, 44, 17, 40]. This formulation can represent arbitrary typology and has infinite resolution. DIF-based shape synthesis approaches have demonstrated strong performance in many applications, including single view 3D reconstruction [60, 30, 47, 48], shape manipulation, and synthesis [26, 21, 28, 14, 1, 9]. However, as these approaches are trained by minimizing the reconstruction loss of function values at a set of sampled 3D locations (a rough proxy of the surface), they tend to render artifacts when synthesizing fine details. Furthermore, if one desires a mesh to be extracted from a DIF, an expensive iso-surfacing step based on Marching Cubes [36] or Marching Tetrahedra [15] is required. Due to the computational burden, iso-surfacing is often done on a smaller resolution, hence prone to quantization errors. Lei et al. [29] proposes an analytic meshing solution to reduce the error, but is only applicable to DIFs parametrized by MLPs with ReLU activation. Our representation scales to high resolution and does not require additional modification to the backward pass for training end-to-end. DMTET can represent arbitrary typology, and is trained via direct supervision on the generated surface. Recent works [1, 9] learn to regress unsigned distance to triangle soup or point cloud. However, their iso-surfacing formulation is not differentiable in contrast to DMTET.
+
+![](images/0d5058ec53aafbd0e0995b8e5ac5432565d0802e926641001c8ac4ebef8fb2be.jpg)  
+Figure 1: DMTET reconstructs the shape implicitly in a coarse-to-fine manner by predicting the SDF defined on a deformable tetrahedral grid. It then converts the SDF to a surface mesh by a differentiable Marching Tetrahedra layer. DMTET is trained by optimizing the objective function defined on the final surface.
+
+Surface-based Methods directly predict triangular meshes and have achieved impressive results for reconstructing and synthesizing simpler shapes [54, 23, 5, 41, 7]. Typically, they predefined the topology of the shape, e.g. equivalent to a sphere [54, 5, 25], or a union of primitives [43, 53, 19] or a set of segmented parts [61, 62, 50]. As a result, they can not model a distribution of shapes with complex topology variations. Recently, DefTet [18] represents a mesh with a deformable tetrahedral grid where the grid vertex coordinates and the occupancy values are learned. However, similar to voxel-based methods, the computational costf increases cubically with the grid resolution. Furthermore, as the occupancy loss for supervising topology learning and the surface loss for supervising geometry learning do not support joint training, it tends to generate suboptimal results. In contrast, our method is able to synthesize high-resolution 3D shapes, not shown in previous work.
+
+# 3 Deep Marching Tetrahedra
+
+We now introduce our DMTET for synthesizing high-quality 3D objects. The schematic illustration is provided in Fig. 1. Our model relies on a new, hybrid 3D representation specifically designed for high-resolution reconstruction and synthesis, which we describe in Sec. 3.1. In Sec. 3.2, we describe the neural network architecture of DMTET that predicts the shape representation from inputs such as coarse voxels. We provide the training objectives in Sec. 3.3.
+
+# 3.1 3D Representation
+
+We represent a shape using a sign distance field (SDF) encoded with a deformable tetrahedral grid, adopted from DefTet [18, 20]. The grid fully tetrahedralizes a unit cube, where each cell in the volume is a tetahedron with 4 vertices and faces. The key aspect of this representation is that the grid vertices can deform to represent the geometry of the shape more efficiently. While the original DefTet encoded occupancy defined on each tetrahedron, we here encode signed distance values defined on the vertices of the grid and represent the underlying surface implicitly (Sec. 3.1.1). The use of signed distance values, instead of occupancy values, provides more flexibility in representing the underlying surface. For greater representation power while keeping memory and computation manageable, we further selectively subdivide the tetrahedra around the predicted surface (Sec. 3.1.2). We convert the signed distance-based implicit representation into a triangular mesh using a marching tetrahedra layer, which we discuss in Sec. 3.1.3. The final mesh is further converted into a parameterized surface with a differentiable surface subdivision module, described in Sec. 3.1.4.
+
+# 3.1.1 Deformable Tetrahedral Mesh as an Approximation of an Implicit Function
+
+We adopt and extend the deformable tetrahedral grid introduced in Gao et al. [18], which we denote with $( V _ { T } , T )$ , where $V _ { T }$ are the vertices in the tetrahedral grid $T$ . Following the notation in [18], each tetrahedron $T _ { k } \in T$ is represented with four vertices $\left\{ v _ { a _ { k } } , v _ { b _ { k } } , v _ { c _ { k } } , v _ { d _ { k } } \right\}$ , with $k \in \{ 1 , . . . . , K \}$ , where $K$ is the total number of tetrahedra and $v _ { i _ { k } } \in V _ { T }$ .
+
+We represent the sign distance field by interpolating SDF values defined on the vertices of the grid. Specifically, we denote the SDF value in vertex $v _ { i } \in V _ { T }$ as $s ( v _ { i } )$ . SDF values for the points that lie inside the tetrahedron follow a barycentric interpolation of the SDF values of the four vertices that encapsulates the point.
+
+# 3.1.2 Volume Subdivision
+
+We represent shape in a coarse to fine manner for efficiency. We determine the surface tetrahedra $T _ { s u r f }$ by checking whether a tetrahedron has vertices with different SDF signs – indicating that it intersects the surface encoded by the SDF. We subdivide $T _ { s u r f }$ as well as their immediate neighbors and increase resolution by adding the mid point to each edge. We compute SDF values of the new vertices by averaging the SDF values on the edge (Fig. 2).
+
+![](images/690ebadbdb5fb11a0781110c51057baf1fbf00e37c739b19a6285e80b9e34a71.jpg)  
+Figure 2: Volume Subdivision: Each surface tet.(blue) is divided into 8 tet.(red) by adding midpoints.
+
+# 3.1.3 Marching Tetrahedra for converting between an Implicit and Explicit Representation
+
+![](images/66197fc00f7d532542743103dce49d9e142b692a641aa4f468f4386db848bf60.jpg)  
+Figure 3: Three unique surface configurations in MT. Vertex color indicates the sign of signed distance value. Notice that flipping the signs of all vertices will result in the same surface configuration. Position of the vertex is linearly interpolated along the edges with sign change.
+
+We use the Marching Tetrahedra [15] algorithm to convert the encoded SDF into an explicit triangular mesh. Given the SDF values $\mathbf { \bar { \{ } }  s ( v _ { a } ) , s ( \mathbf { \bar { { v } } } _ { b } ) , s ( v _ { c } ) , s ( v _ { d } ) \}$ of the vertices of a tetrahedron, MT determines the surface typology inside the tetrahedron based on the signs of $s ( v )$ , which is illustrated in Fig. 3. The total number of configurations is $2 ^ { 4 } = { \bar { 1 } } 6$ , which falls into 3 unique cases after considering rotation symmetry. Once the surface typology inside the tetrahedron is identified, the vertex location of the iso-surface is computed at the zero crossings of the linear interpolation along the tetrahedron’s edges, as shown in Fig. 3.
+
+Prior works [45, 31] argue that the singularity in this formulation, i.e. when $s ( v _ { a } ) = s ( v _ { b } )$ , prevents the change of surface typology (sign change of $s ( v _ { a } ) )$ ) during training. However, we find that, in practise, the equation is only evaluated when $\mathrm { s i g n } ( s ( v _ { a } ) ) \neq \mathrm { s i g n } ( s ( v _ { b } ) )$ . Thus, during training, the singularity never happens and the gradient from a loss defined on the extracted iso-surface (Sec. 3.3), can be back-propagated to both vertex positions and SDF values via the chain rule. A more detailed analysis is in the Appendix.
+
+# 3.1.4 Surface Subdivision
+
+Having a surface mesh as output allows us to further increase the representation power and the visual quality of the shapes with a differentiable surface subdivision module. We follow the scheme of the Loop Subdivision method [35], but instead of using a fixed set of parameters for subdivision, we make these parameters learnable in DMTET. Specifically, learnable parameters include the positions of each mesh vertex $\boldsymbol { v } _ { i } ^ { \prime }$ , as well as $\alpha _ { i }$ which controls the generated surface via weighting the smoothness of neighbouring vertices. Note that different from Liu et al. [33], we only predict the per-vertex parameter at the beginning and carry it over to subsequent subdivision iterations to attain a lower computational cost. We provide more details in Appendix.
+
+# 3.2 DMTET: 3D Deep Conditional Generative Model
+
+Our DMTET is a neural network that utilizes our proposed 3D representation and aims to output a high resolution 3D mesh $M$ from input $x$ (a point cloud or a coarse voxelized shape). We describe the architecture (Fig. 4) of the generator for each module of our 3D representation in Sec. 3.2.1, with the architecture of the discriminator presented in Sec. 3.2.2. Further details are in Appendix.
+
+![](images/fa49995de6dd2e0e3e4ada8be106cd973661cac4b1241747a493f2249be01cff.jpg)  
+Figure 4: Our generator and discriminator architectures. The generator is composed of two parts—one utilizes MLP to generate the initial predictions for all grid vertices and the other uses GCN to refine the surface.
+
+# 3.2.1 3D Generator
+
+Input Encoder We use PVCNN [34] as an input encoder to extract a 3D feature volume $F _ { v o l } ( x )$ from a point cloud. When the input is a coarse voxelized shape, we sample points on its surface. We compute a feature vector $F _ { v o l } ( v , x )$ for a grid vertex $v \in \mathbb { R } ^ { 3 }$ via trilinear interpolation.
+
+Initial Prediction of SDF We predict the SDF value for each vertex in the initial deformable tetrahedral grid using a fully-connected network $s ( v ) = M L P ( F _ { v o l } ( v , x ) , v )$ . The fully-connected network additionally outputs a feature vector $f ( v )$ , which is used for the surface refinement in the volume subdivision stage.
+
+Surface Refinement with Volume Subdivision After obtaining the initial SDF, we iteratively refine the surface and subdivide the tetrahedral grid. We first identify surface tetrahedra $T _ { s u r f }$ based on the current $s ( v )$ value. We then build a graph $G = ( V _ { s u r f } , E _ { s u r f } )$ , where $V _ { s u r f } , E _ { s u r f }$ correspond to the vertices and edges in $T _ { s u r f }$ . We then predict the position offsets $\Delta v _ { i }$ and SDF residual values $\Delta s ( v _ { i } )$ for each vertex $i$ in $V _ { s u r f }$ using a Graph Convolutional Network [32] (GCN):
+
+$$
+\begin{array} { r c l } { f _ { v _ { i } } ^ { \prime } } & { = } & { \mathsf { c o n c a t } ( v _ { i } , s ( v _ { i } ) , F _ { v o l } ( v _ { i } , x ) , f ( v _ { i } ) ) , } \\ { ( \Delta v _ { i } , \Delta s ( v _ { i } ) , \overline { { f ( v _ { i } ) } } ) _ { i = 1 , \cdots N _ { s u r f } } } & { = } & { \mathsf { G C N } \big ( ( f _ { v _ { i } } ^ { \prime } ) _ { i = 1 , \cdots N _ { s u r f } } , G \big ) , } \end{array}
+$$
+
+where $N _ { s u r f }$ is the total number of vertices in $V _ { s u r f }$ and $\overline { { f ( v _ { i } ) } }$ is the updated per-vertex feature. The vertex position and the SDF value for vertex $v _ { i }$ are updated as $v _ { i } ^ { \prime } = v _ { i } + \Delta v _ { i }$ and $s ( v _ { i } ^ { \prime } ) =$ $s ( v _ { i } ) + \Delta s ( v _ { i } )$ . This refinement step can potentially flip the sign of the SDF values to refine the local typology, and also move the vertices thus improving the local geometry.
+
+After surface refinement, we perform the volume subdivision step followed by an additional surface refinement step. In particular, we re-identify $T _ { s u r f }$ and subdivide $T _ { s u r f }$ and their immediate neighbors. We drop the unsubdivided tetrahedra from the full tetrahedral grid in both steps, which saves memory and computation, as the size of the $T _ { s u r f }$ is proportional to the surface area of the object, and scales up quadratically rather than cubically as the grid resolution increases.
+
+Note that the SDF values and positions of the vertices are inherited from the level before subdivision, thus, the loss computed at the final surface can back-propagate to all vertices from all levels. Therefore, our DMTET automatically learns to subdivide the tetrahedra and does not need an additional loss term in the intermediate steps to supervise the learning of the octree hierarchy as in the prior work [52].
+
+Learnable Surface Subdivision After extracting the surface mesh using MT, we can further apply learnable surface subdivision. Specifically, we build a new graph on the extracted mesh, and use GCN to predict the updated position of each vertex $\boldsymbol { v } _ { i } ^ { \prime }$ , and $\alpha _ { i }$ for Loop Subvidision. This step removes the quantization errors and mitigates the approximation errors from the classic Loop Subdivision by adjusting $\alpha _ { i }$ , which are fixed in the classic method.
+
+# 3.2.2 3D Discriminator
+
+We apply a 3D discriminator $D$ on the final surface predicted from the generator. We empirically find that using a 3D CNN from DECOR-GAN [6] as the discriminator on the signed distance field that is computed from the predicted mesh is effective to capture the local details. Specifically, we first randomly select a high-curvature vertex $v$ from the target mesh and compute the ground truth signed distance field $S _ { r e a l } \in \mathbb { R } ^ { N \times N \times N }$ at a voxelized region around $v$ . Similarly, we compute the signed distance field of the predicted surface mesh $M$ at the same location to obtain $S _ { p r e d } \in \mathbb { R } ^ { N \times \tilde { N } \times N }$ . Note that $S _ { p r e d }$ is an analytical function of the mesh $M$ , and thus the gradient to $S _ { p r e d }$ can backpropagate to the vertex positions in $M$ . We feed $S _ { r e a l }$ or $S _ { p r e d }$ into the discriminator, along with the feature vector $F _ { v o l } ( v , x )$ in position $v$ . The discriminator then predicts the probability indicating whether the input comes from the real or generated shapes.
+
+# 3.3 Loss Function
+
+DMTET is end-to-end trainable. We supervise all modules to minimize the error defined on the final predicted mesh $M$ . Our loss function contains three different terms: a surface alignment loss to encourage the alignment with ground truth surface, an adversarial loss to improve realism of the generated shape, and regularizations to regularize the behavior of SDF and vertex deformations.
+
+Surface Alignment loss We sample a set of points $P _ { g t }$ from the surface of the ground truth mesh $M _ { g t }$ . Similarly, we also sample a set of points from $M _ { p r e d }$ to obtain $P _ { p r e d }$ , and minimize the L2 Chamfer Distance and the normal consistency loss between $P _ { g t }$ and $P _ { p r e d }$ :
+
+$$
+L _ { \mathrm { c d } } = \sum _ { p \in P _ { p r e d } } \operatorname* { m i n } _ { q \in P _ { g t } } | | p - q | | _ { 2 } + \sum _ { q \in P _ { g t } } \operatorname* { m i n } _ { p \in P _ { p r e d } } | | q - p | | _ { 2 } , L _ { \mathrm { n o m a l } } = \sum _ { p \in P _ { p r e d } } ( 1 - | \Vec { \mathbf { n } } _ { p } \cdot \Vec { \mathbf { n } } _ { \Vec { q } } | ) ,
+$$
+
+where $\hat { q }$ is the point that corresponds to $p$ when computing the Chamfer Distance, and $\vec { \bf n } _ { p } , \vec { \bf n } _ { \hat { q } }$ denotes the normal direction at point $p , \hat { q }$ .
+
+Adversarial Loss We use the adversarial loss proposed in LSGAN [37]:
+
+$$
+L _ { \mathrm { D } } = \frac { 1 } { 2 } [ ( D ( M _ { g t } ) - 1 ) ^ { 2 } + D ( M _ { p r e d } ) ^ { 2 } ] , L _ { \mathrm { G } } = \frac { 1 } { 2 } [ ( D ( M _ { p r e d } ) - 1 ) ^ { 2 } ] .
+$$
+
+Regularizations The above loss functions operate on the extracted surface, thus, only the vertices that are close to the iso-surface in the tetrahedral grid receive gradients, while the other vertices do not. Moreover, the surface losses do not provide information about what is inside/outside, since flipping the SDF sign of all vertices in a tetrahedron would result in the same surface being extracted by MT. This may lead to disconnected components during training. To alleviate this issue, we add a SDF loss to regularize SDF values:
+
+$$
+L _ { \mathrm { S D F } } = \sum _ { v _ { i } \in V _ { T } } | s ( v _ { i } ) - S D F ( v _ { i } , M _ { g t } ) | ^ { 2 } ,
+$$
+
+where $S D F ( v _ { i } , M _ { g t } )$ denotes the SDF value of point $v _ { i }$ to the mesh $M _ { g t }$ . In addition, we apply the $L _ { 2 }$ regularization loss on the predicted vertex deformations to avoid artifacts: $\begin{array} { r } { L _ { \mathrm { d e f } } = \sum _ { v _ { i } \in V _ { T } } | | \Delta v _ { i } | | _ { 2 } } \end{array}$ .
+
+The final loss is a weighted sum of all five loss terms:
+
+$$
+L = \lambda _ { \mathrm { c d } } L _ { \mathrm { c d } } + \lambda _ { \mathrm { n o r m a l } } L _ { \mathrm { n o r m a l } } + \lambda _ { \mathrm { G } } L _ { \mathrm { G } } + \lambda _ { \mathrm { S D F } } L _ { \mathrm { S D F } } + \lambda _ { \mathrm { d e f } } L _ { \mathrm { d e f } } ,
+$$
+
+where $\lambda _ { \mathrm { c d } } , \lambda _ { \mathrm { n o r m a l } } , \lambda _ { \mathrm { G } } , \lambda _ { \mathrm { S D F } } , \lambda _ { \mathrm { d e f } }$ are hyperparameters (provided in the Supplement).
+
+# 4 Experiments
+
+We first evaluate DMTET in the challenging application of generating high-quality animal shapes from coarse voxels. We further evaluate DMTET in reconstructing 3D shapes from noisy point clouds on ShapeNet by comparing to existing state-of-the-art methods.
+
+# 4.1 3D Shape Synthesis from Coarse Voxels
+
+Experimental Settings We collected 1562 animal models from the TurboSquid website1. These models have a wide range of diversity, ranging from cats, dogs, bears, giraffes, to rhinoceros, goats, etc. We provide visualizations in Supplement. Among 1562 shapes, we randomly select 1120 shapes for training, and the remaining 442 shapes for testing. We follow the pipeline in Kaolin [27] to convert shapes to watertight meshes. To prepare the input to the network, we first voxelize the mesh into the resolution of $1 6 ^ { \overleftarrow { 3 } }$ , and then sample 3000 points from the surface after applying marching cubes to the $1 6 ^ { 3 }$ voxel grid. Note that this preprocessing is agnostic to the representation of the input coarse shape, allowing us to evaluate on different resolution voxels, or even meshes.
+
+We compare our model with the official implementation of ConvOnet [44], which achieved SOTA performance on voxel upsampling. We also compare to DECOR-GAN [6], which obtained impressive results on transferring styles from a high-resolution voxel shape to a low-resolution voxel. Note that the original setting of DECOR-GAN is different from ours. For a fair comparison, we use all 1120 training shapes as the high-resolution style shapes during training, and retrieve the closet training shape to the test shape as the style shape during inference, which we refer as DECOR-Retv. We also compare against a randomly selected style shape as reference, denoted as DECOR-Rand.
+
+1https://www.turbosquid.com, we obtain consent via an agreement with TurboSquid, and following license at https://blog.turbosquid.com/turbosquid-3d-model-license/
+
+![](images/4b1bbd56197269ad4335baabf87fb00fe595064f3e4ec463efdf1d0b619d8092.jpg)  
+Figure 5: Qualitative results on 3D shapes Synthesis from Coarse Voxels. Comparing with all baselines, our method reconstructs shapes with much higher quality. Adding GAN further improves the realism of the generated shape. We also show the retrieved shapes from the training set in the second last column.
+
+Metrics We evaluate L2 and L1 Chamfer Distance, as well as normal consistency score to assess how well the methods reconstruct the corresponding high-resolution shape following [44]. We also report Light Field Distance [4] (LFD) which measures the visual similarity in 2D rendered views. In addition, we evaluate Cls score following [6]. Specifically, we render the predicted 3D shapes and train a patch-based image classifier to distinguish whether images are from the renderings of real or generated shapes. The mean classification accuracy of the trained classifier is reported as Cls (lower is better). More details are in the Supplement.
+
+Experimental Results We provide quantitative results in Table 1 with qualitative examples in Fig. 5. Our DMTET achieves significant improvements over all baselines in terms of all metrics. Compared to both ConvOnet [44] and DECORGAN [6], our DMTET reconstructs shapes with better quality when training without adversarial loss (5th column in Fig. 5). Further geometric details, including nails, ears, eyes, mouths, etc, are captured when trained with the adversarial loss (6th column in Fig. 5), significantly improving the realism and visual quality of the generated shape. To demonstrate the generalization ability of our
+
+![](images/51a010aed7513fe7dfaf318abbe714fe80699ba1341c5a33b907b24eb8d658e0.jpg)  
+Figure 6: Qualitative Results of synthesizing highresolution shapes from coarse voxels collected online.
+
+DMTET, we collect human-created low-resolution voxels from Turbosquid (shapes unseen in training). We provide qualitative results in Fig. 6. Despite the fact that these human-created shapes have noticeable differences with our coarse voxels used in training, e.g., different ratios of body parts compared with our training shapes (larger head, thinner legs, longer necks), our model faithfully generates high-quality 3D details conditioned on each coarse voxel – an exciting result.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>L2 Chamfer↓</td><td rowspan=1 colspan=1>L1 Chamfer↓</td><td rowspan=1 colspan=1>Norm. Cons.↑</td><td rowspan=1 colspan=1>LFD↓</td><td rowspan=1 colspan=1>Cls</td></tr><tr><td rowspan=1 colspan=1>ConvOnet [44]</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>2.41</td><td rowspan=1 colspan=1>0.901</td><td rowspan=1 colspan=1>3220</td><td rowspan=1 colspan=1>0.63</td></tr><tr><td rowspan=1 colspan=1>DECOR [6]-Retv.</td><td rowspan=1 colspan=1>1.32</td><td rowspan=1 colspan=1>3.81</td><td rowspan=1 colspan=1>0.876</td><td rowspan=1 colspan=1>3689</td><td rowspan=1 colspan=1>0.66</td></tr><tr><td rowspan=1 colspan=1>DECOR [6]-Rand.</td><td rowspan=1 colspan=1>2.38</td><td rowspan=1 colspan=1>6.85</td><td rowspan=1 colspan=1>0.797</td><td rowspan=1 colspan=1>5338</td><td rowspan=1 colspan=1>0.67</td></tr><tr><td rowspan=1 colspan=1>DMTET wo Adv.</td><td rowspan=1 colspan=1>0.76</td><td rowspan=1 colspan=1>2.20</td><td rowspan=1 colspan=1>0.916</td><td rowspan=1 colspan=1>2846</td><td rowspan=1 colspan=1>0.58</td></tr><tr><td rowspan=1 colspan=1>DMTET</td><td rowspan=1 colspan=1>0.75</td><td rowspan=1 colspan=1>2.19</td><td rowspan=1 colspan=1>0.918</td><td rowspan=1 colspan=1>2823</td><td rowspan=1 colspan=1>0.54</td></tr></table>
+
+Table 1: Super Resolution of Animal Shapes: DMTET significantly outperforms all baselines in all metrics.
+
+User Studies We conduct user studies via Amazon Machanical Turk (AMT) to further evaluate the performance of all methods. In particular, we present two shapes that are predicted from two different models to the AMT workers and ask them to evaluate which one is a better looking shape and which one features more realistic details. Detailed experimental settings are provided in the Supplement. We compare DMTET against ConvONet [44], DECOR [6]-Retv, as well as DMTET without adversarial loss (w.o. Adv.). Quantitative results are reported in Table 2. Human judges agree that the shapes generated from our model have better details, compared to all baselines, in a vast majority of the cases. Ablations on using adversarial loss demonstrate the effectiveness of generating higher quality geometry using a discriminator during training.
+
+Ablation Studies To evaluate the effectiveness of our volume subdivision and surface subdivision modules, we ablate by sequentially introducing them to the base model (we refer as $\mathrm { D M T E T } _ { B }$ ) which we train on 100-resolution uniform tetrahedral grid without both volume and surface subdivision modules and adversarial
+
+Table 2: User Study on 3D Shape Synthesis from Coarse voxels. In each cell, we report percentages of shapes for which the users agree are better looking (left) or have better details (right).   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ConvONet[44]</td><td rowspan=1 colspan=1>DECOR[6]-Retv.</td><td rowspan=1 colspan=1>DMTETWoAdv</td></tr><tr><td rowspan=1 colspan=1>Baselinewins</td><td rowspan=1 colspan=1>5% 15%</td><td rowspan=1 colspan=1>26%/17%</td><td rowspan=1 colspan=1>29% /25%</td></tr><tr><td rowspan=1 colspan=1>DMTETwins</td><td rowspan=1 colspan=1>95%/95%</td><td rowspan=1 colspan=1>74% /83%</td><td rowspan=1 colspan=1>71% 175%</td></tr></table>
+
+loss. We conduct user studies to evaluate the improvement after each step using the protocol described in the above paragraph. We first reduce the initial resolution to 70 and employ volume subdivision to support higher output resolution (we refer this model as $\mathbf { D M T E T } _ { V }$ ) and compare with $\mathbf { D M T E T } _ { B }$ Predictions by $\mathrm { D M T E T } _ { V }$ wins $78 \%$ of cases over $\mathrm { D M T E T } _ { B }$ for better looking, and $61 \%$ of cases for realistic details, showing that the volume subdivision module is effective in synthesizing shape details. We then add surface subdivision on top of the $\mathbf { D M T E T } _ { V }$ and compare with it. The new model wins $62 \%$ of cases over $\mathrm { D M T E T } _ { V }$ for better looking, and $62 \%$ of cases for realistic details as well, demonstrating the effect of surface subdivision module in enhancing the shape details.
+
+# 4.2 Point Cloud 3D Reconstruction
+
+Experimental Settings We follow the setting from DefTet [18], and use all 13 categories in ShapeNet [3] core data2, which we pre-process using Kaolin [27] to watertight meshes. We sample 5000 points for each shape and add Gaussian noise with zero mean of standard deviation 0.005. For quantitative evaluation, we report the L1 Chamfer Distance in the main paper, and refer readers to the Supplement for results in other metrics (3D IoU, L2 Chamfer Distance and F1 score). We additionally report average inference time on the same Nvidia V100 GPU.
+
+We compare DMTET against state-of-the-art 3D reconstruction approaches using different representations: voxels [10], deforming a mesh with a fixed template [54], deforming a mesh generated from a volumetric representation [22], DefTet [18], and implicit functions [44]. For a fair comparison, we use the same point cloud encoder for all the methods, and adopt the decoders in the original papers to generate shapes in different representations. We also remove the adversarial loss in this application, since baselines also do not have it. We further compare with oracle performance of MC/MT where the ground truth SDF is utilized to extract iso-surface using MC/MT.
+
+Experimental Results Quantitative results are summarized in Table 3, with a few qualitative examples shown in Fig. 7. Compared to DMC [31], which also predicts the SDF values and supervises with a surface loss, DMTET achieves much better reconstruction quality since training using the marching tetrahedra layer is more efficient than calculating an expectation over all possible configurations within one grid cell as done in DMC [31]. Compared to a method that deforms a fixed template (sphere) [54], we reconstruct shapes with different topologies, achieving more faithful results compared to the ground truth shape. When compared with other explicit surface representations that also support different topology [18, 22], our method achieves higher quality results for local geometry, benefiting from the fact that the typology is jointly optimized with the geometry, whereas it is separately supervised by an occupancy loss in [18, 22]. Compared to a neural implicit method [44], we generate higher quality shapes with less artifacts, while running significantly faster at inference. Finally, compared to a voxel-based method [10] at the same resolution, our method recovers more geometric details, benefiting from the predicted vertex deformations as well as the surface loss.
+
+![](images/7bd33ba212cc763b44caca6f6748c49136fb2f72c0e054ac16d83ee028728622.jpg)  
+Figure 7: Qualitative results on 3D Reconstruction from Point Clouds: Our model reconstructs shapes with more geometric details compared to baselines.
+
+Table 3: Quantitative Results on Point Cloud Reconstruction (Chamfer L1). Note that all the networks in the baselines are not designed for this task, and thus we use the same encoder and their decoder for a fair comparison. We also ablate ourselves by operating on fixed grid (DMTET wo (Def, Vol., Surf.)), removing volume subdivision (DMTET wo Vol.), or surface subdivision (DMTET wo Surf.), or the both (DMTET wo (Vol., Surf.)).   
+
+<table><tr><td>Category</td><td>Airplane</td><td>Bench</td><td>Dresser</td><td>Car</td><td>Chair</td><td>Display</td><td>Lamp</td><td>Speaker</td><td>Rifle</td><td>Sofa</td><td>Table</td><td>Phone</td><td>Vessel</td><td>Mean↓</td><td>Time(ms)</td></tr><tr><td>3D-R2N2[10]</td><td>1.48</td><td>1.59</td><td>1.64</td><td>1.62</td><td>1.70</td><td>1.66</td><td>1.74</td><td>1.74</td><td>1.37</td><td>1.60</td><td>1.78</td><td>1.55</td><td>1.51</td><td>1.61</td><td>174</td></tr><tr><td>DMC [31]</td><td>1.57</td><td>1.47</td><td>1.29</td><td>1.67</td><td>1.44</td><td>1.25</td><td>2.15</td><td>1.49</td><td>1.45</td><td>1.19</td><td>1.33</td><td>0.88</td><td>1.70</td><td>1.45</td><td>349</td></tr><tr><td>Pixel2mesh [54]</td><td>0.98</td><td>1.28</td><td>1.44</td><td>1.19</td><td>1.91</td><td>1.25</td><td>2.07</td><td>1.61</td><td>0.91</td><td>1.15</td><td>1.82</td><td>0.83</td><td>1.12</td><td>1.35</td><td>30</td></tr><tr><td>ConvOnet [44]</td><td>0.82</td><td>0.95</td><td>0.96</td><td>1.12</td><td>1.03</td><td>0.93</td><td>1.22</td><td>1.12</td><td>0.79</td><td>0.91</td><td>0.94</td><td>0.67</td><td>0.99</td><td>0.95</td><td>866</td></tr><tr><td>MeshRCNN [22]</td><td>0.88</td><td>1.01</td><td>1.05</td><td>1.14</td><td>1.10</td><td>0.99</td><td>1.20</td><td>1.21</td><td>0.83</td><td>0.96</td><td>1.00</td><td>0.71</td><td>1.03</td><td>1.01</td><td>228</td></tr><tr><td>DEFTET[18]</td><td>0.85</td><td>0.94</td><td>0.97</td><td>1.13</td><td>1.04</td><td>0.92</td><td>1.28</td><td>1.17</td><td>0.85</td><td>0.90</td><td>0.93</td><td>0.65</td><td>0.99</td><td>0.97</td><td>61</td></tr><tr><td>DMTET wo (Def, Vol., Surf.)]</td><td>0.82</td><td>0.96</td><td>0.94</td><td>0.98</td><td>0.99</td><td>0.90</td><td>1.04</td><td>1.03</td><td>0.80</td><td>0.86</td><td>0.93</td><td>0.65</td><td>0.89</td><td>0.91</td><td></td></tr><tr><td>DMTET wo (Vol.,Surf.)</td><td>0.69</td><td>0.82</td><td>0.88</td><td>0.92</td><td>0.92</td><td>0.82</td><td>0.89</td><td>0.97</td><td>0.65</td><td>0.81</td><td>0.84</td><td>0.61</td><td>0.80</td><td>0.81</td><td>52 52</td></tr><tr><td>DMTET wo Vol.</td><td>0.65</td><td>0.78</td><td>0.84</td><td>0.89</td><td>0.89</td><td>0.79</td><td>0.86</td><td>0.95</td><td>0.61</td><td>0.78</td><td>0.79</td><td>0.60</td><td>0.78</td><td>0.79</td><td>67</td></tr><tr><td>DMTET wo Surf.</td><td>0.63</td><td>0.77</td><td>0.84</td><td>0.88</td><td>0.88</td><td>0.79</td><td>0.84</td><td>0.94</td><td>0.60</td><td>0.78</td><td>0.79</td><td>0.59</td><td>0.76</td><td>0.78</td><td>108</td></tr><tr><td>DMTET</td><td>0.62</td><td>0.76</td><td>0.83</td><td>0.87</td><td>0.88</td><td>0.78</td><td>0.84</td><td>0.94</td><td>0.59</td><td>0.77</td><td>0.78</td><td>0.57</td><td>0.76</td><td>0.77</td><td>129</td></tr></table>
+
+# 4.2.1 Analysis
+
+We investigate how each component in our representation affects the performance and reconstruction quality.
+
+Comparisons with Oracle Performance of MC/MT We first demonstrate the effect of learning on explicit surface via MT. We compare with the oracle performance of extracting the iso-surface with MT/MC from the ground truth signed distance fields on the Chair test set in ShapeNet, which contains diverse high-quality details. Specifically, for MC/MT, we first compute the discretized SDF at different grid resolutions, and compare the extracted surface to the ground truth surface.
+
+![](images/ca6df925a51dd1d96de02ca23469bf92f0ad16b5ce533ad7239d3a0faf8c0a73.jpg)  
+Figure 8: Comparing our DMTET with oracle performance of MC and MT.
+
+As shown in Fig. 8, MT consistently outperforms MC when querying the same number of points. We found the staggered grids pattern in tetrahedral grid [16, 18] better captures thin structures at a limited resolution (Fig. 9). This makes MT a better choice for efficiency reasons. The usage of tetrahedral mesh in DMTET follows this motivation. Without deforming the grid, DMTET outperforms the oracle performance of MT by a large margin when querying the same number of points, although DMTET predicts the surface from noisy point cloud. This demonstrates that directly optimizing the reconstructed surface can mitigate the discretization errors imposed by MT to a large extent.
+
+![](images/430b26fbbcba7dc56f2f3f84c2d486be08926dbae7665f553626b9f093f6b9b3.jpg)  
+Figure 9: We compare trained DMTET to oracle performance of MT and MC. Number in bracket indicates number of SDF points queried.
+
+Ablation Studies We further provide ablation studies on the entire ShapeNet test set, which is summarized in Tab. 3. We first compare the version where we only predict SDF values without learning to deform the vertices and volume/surface subdivision with the version that predicts both SDF and the deformation. Predicting deformation along with SDF is significantly more performant, since vertex movements allow for a better reconstruction of the underlying surface. This is especially true for categories with thin structures (e.g. lamp) where the grid vertices are desired to align with them. We further ablate the use of volume subdivision and surface subdivision. We show that each component provides an improvement. In particular, volume subdivision has a significant
+
+improvement for object categories with fine-grained structural details, such as airplane and lamp, which require higher grid resolutions to model the occupancy change. Surface subdivision generates shapes with a parametric surface, avoiding the quantization errors in the planar faces and produces more visually pleasing results.
+
+# 5 Conclusion
+
+In this paper, we introduced a deep 3D conditional generative model that can synthesize highresolution 3D shapes using simple user guides such as coarse voxels. Our DMTET features a novel 3D representation that marries implicit and explicit representations by leveraging the advantages of both. We experimentally show that our approach synthesizes significantly higher quality shapes with better geometric details than existing methods, confirmed by quantitative metrics and an extensive user study. By showcasing the ability to upscale coarse voxels such as Minecraft shapes, we hope that we take one step closer to democratizing 3D content creation.
+
+# 6 Broad Impact
+
+Many fields such as AR/VR, robotics, architecture, gaming and film rely on high-quality 3D content. Creating such content, however, requires human experts, i.e., experienced artists, and a significant amount of development time. In contrast, platforms like Minecraft enable millions of users around the world to carve out coarse shapes with simple blocks. Our work aims at creating A.I. tools that would enable even novice users to upscale simple, low-resolution shapes into high resolution, beautiful 3D content. Our method currently focuses on 3D animal shapes. We are not currently aware of and do not foresee nefarious use cases of our method.
+
+# 7 Disclosure of Funding
+
+This work was funded by NVIDIA. Tianchang Shen and Jun Gao acknowledge additional revenue in the form of student scholarships from University of Toronto and the Vector Institute, which are not in direct support of this work.
+
+# References
+
+[1] Matan Atzmon and Yaron Lipman. Sal: Sign agnostic learning of shapes from raw data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2565–2574, 2020.   
+[2] Andrew Brock, Theodore Lim, James M Ritchie, and Nick Weston. Generative and discriminative voxel modeling with convolutional neural networks. arXiv preprint arXiv:1608.04236, 2016. [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 arXiv:1512.03012, 2015.   
+[4] Ding-Yun Chen, Xiao-Pei Tian, Yu-Te Shen, and Ming Ouhyoung. On visual similarity based 3d model retrieval. In Computer graphics forum, volume 22, pages 223–232. Wiley Online Library, 2003.   
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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] We provide extensive experiments in Sec. 4.   
+(b) Did you describe the limitations of your work? [Yes] We provide the discussion on limitations an failure cases in Supplement.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] We provide the discussion in the Board Impact section with further discussions in Supplement.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [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)? [No] The code is currently quite uncleaned and requires many dependencies. We are planning to release the code after cleaning.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We provide training details in both Sec. 4 in the main paper and Supplement.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Training existing 3D models, including ours, on large-scale 3D datasets is too computation costly to repeat multiple times.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We provide in the Supplement.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We used ShapeNet [3] core dataset in Sec. 4.2. We also used official code to reproduce baselines with citations. In particular, ConvONet [44] and DECOR-GAN [6].   
+(b) Did you mention the license of the assets? [Yes] We provided the license of ShapeNet and Turbosquid.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] The Turbosquid data we are using contains proprietary information.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We discussed in the Sec. 4 and provide further details in the Supplementary Materials.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We provide discussion on this in Supplement.
+
+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? [Yes] We provide details in the paper, with full text and screenshot in Supplement.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] We did not anticipate the potential participant risks, as we only conduct human studies on generated animals.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [Yes] We provide details in Supplement

parse/train/GFV8IVyMM4n/GFV8IVyMM4n_content_list.json

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+        "text": "Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis ",
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+        "text": "Tianchang Shen 1,2,3 Jun Gao1,2,3 Kangxue Yin 1 ",
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+        "text": "NVIDIA1 University of Toronto2 Vector Institute3 ",
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+        "type": "text",
+        "text": "{frshen, jung, kangxuey, mingyul, sfidler}@nvidia.com ",
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+        "text": "Abstract ",
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+        "text": "We introduce DMTET, a deep 3D conditional generative model that can synthesize high-resolution 3D shapes using simple user guides such as coarse voxels. It marries the merits of implicit and explicit 3D representations by leveraging a novel hybrid 3D representation. Compared to the current implicit approaches, which are trained to regress the signed distance values, DMTET directly optimizes for the reconstructed surface, which enables us to synthesize finer geometric details with fewer artifacts. Unlike deep 3D generative models that directly generate explicit representations such as meshes, our model can synthesize shapes with arbitrary topology. The core of DMTET includes a deformable tetrahedral grid that encodes a discretized signed distance function and a differentiable marching tetrahedra layer that converts the implicit signed distance representation to the explicit surface mesh representation. This combination allows joint optimization of the surface geometry and topology as well as generation of the hierarchy of subdivisions using reconstruction and adversarial losses defined explicitly on the surface mesh. Our approach significantly outperforms existing work on conditional shape synthesis from coarse voxel inputs, trained on a dataset of complex 3D animal shapes. Project page: https://nv-tlabs.github.io/DMTet/. ",
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+        "text": "1 Introduction ",
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+        "text": "Fields such as simulation, architecture, gaming, and film rely on high-quality 3D content with rich geometric details and complex topology. However, creating such content requires tremendous expert human effort. It takes a significant amount of development time to create each individual 3D asset. In contrast, creating rough 3D shapes with simple building blocks like voxels has been widely adopted. For example, Minecraft has been used by hundreds of millions of users for creating 3D content. Most of them are non-experts. Developing A.I. tools that enable regular people to upscale coarse, voxelized objects into high resolution, beautiful 3D shapes would bring us one step closer to democratizing high-quality 3D content creation. Similar tools can be envisioned for turning 3D scans of objects recorded by modern phones into high-quality forms. Our work aspires to create such capabilities. ",
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+        "text": "A powerful 3D representation is a critical component of a learning-based 3D content creation framework. A good 3D representation for high-quality reconstruction and synthesis should capture local geometric details and represent objects with arbitrary topology while also being memory and computationally efficient for fast inference in interactive applications. ",
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+        "text": "Recently, neural implicit representations [8, 39, 42, 51], which use a neural network to implicitly represent a shape via a signed distance field (SDF) or an occupancy field (OF), have emerged as an effective 3D representation. Neural implicits have the benefit of representing complex geometry and topology, not limited to a predefined resolution. The success of these methods has been shown in shape compression [49, 13, 51], single-image shape generation [47, 60, 48], and point cloud reconstruction [57]. However, most of the current implicit approaches are trained by regressing to SDF or OF values and cannot utilize an explicit supervision on the target surface, which imposes useful constraints for training. To mitigate this issue, several works [45, 31] proposed to utilize iso-surfacing techniques such as the Marching Cubes (MC) algorithm to extract a surface mesh from the implicit representation, which, however, is computationally expensive. ",
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+    {
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+        "text": "In this work, we introduce DMTET, a deep 3D conditional generative model for high-resolution 3D shape synthesis from user guides in the form of coarse voxels. In the heart of DMTET is a new differentiable shape representation that marries implicit and explicit 3D representations. In contrast to deep implicit approaches optimized for predicting sign distance (or occupancy) values, our model employs additional supervision on the surface, which empirically renders higher quality shapes with finer geometric details. Compared to methods that learn to directly generate explicit representations, such as meshes [54], by committing to a preset topology, our DMTET can produce shapes with arbitrary topology. Specifically, DMTET predicts the underlying surface parameterized by an implicit function encoded via a deformable tetrahedral grid. The underlying surface is converted into an explicit mesh with a Marching Tetrahedra (MT) algorithm, which we show is differentiable and more performant than the Marching Cubes. DMTET maintains efficiency by learning to adapt the grid resolution by deforming and selectively subdividing tetrahedra. This has the effect of spending computation only on the relevant regions in space. We achieve further gains in the overall quality of the output shape with learned surface subdivision. Our DMTET is end-to-end differentiable, allowing the network to jointly optimize the geometry and topology of the surface, as well as the hierarchy of subdivisions using a loss function defined explicitly on the surface mesh. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We demonstrate our DMTET on two challenging tasks: 3D shape synthesis from coarse voxel inputs and point cloud 3D reconstruction. We outperform existing state-of-the-art methods by a significant margin while being 10 times faster than alternative implicit representation-based methods at inference time. In summary, we make the following technical contributions: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We show that using Marching Tetrahedra (MT) as a differentiable iso-surfacing layer allows topological change for the underlying shape represented by a implicit field, in contrast to the analysis in prior works [31, 45].   \n2. We incorporate MT in a DL framework and introduce DMTET, a hybrid representation that combines implicit and explicit surface representations. We demonstrate that the additional supervision (e.g. chamfer distance, adversarial loss) defined directly on the extracted surface from implicit field improves the shape synthesis quality.   \n3. We introduce a coarse-to-fine optimization strategy that scales DMTET to high resolution during training. We thus achieves better reconstruction quality than state-of-the-art methods on challenging 3D shape synthesis tasks, while requiring a lower computation cost. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We review the related work on learning-based 3D synthesis methods based on their 3D representations. ",
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+    {
+        "type": "text",
+        "text": "Voxel-based Methods Early work [59, 10, 38] represented 3D shapes as voxels, which store the coarse occupancy (inside/outside) values on a regular grid, which makes powerful convolutional neural networks native and renders impressive results on 3D reconstruction and synthesis [12, 11, 58, 2]. For high-resolution shape synthesis, DECOR-GAN [6] transfers geometric details from a high-resolution shape represented in voxel to a low-resolution shape by utilizing a discriminator defined on 3D patches of the voxel grid. However, the computational and memory costs grow cubically as the resolution increases, prohibiting the reconstruction of fine geometric details and smooth curves. One common way to address this limitation is building hierarchical structures such as octrees [46, 52, 55, 56, 24, 52], which adapt the grid resolution locally based on the underlying shape. In this paper, we adopt a hierarchical deformable tetrahedral grid to utilize the resolution better. Unlike octree-based shape synthesis, our network learns grid deformation and subdivision jointly to better represent the surface without relying on explicit supervision from a pre-computed hierarchy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Deep Implicit Fields (DIFs) represent a 3D shape as a zero level set of a continuous function parameterized by a neural network [39, 44, 17, 40]. This formulation can represent arbitrary typology and has infinite resolution. DIF-based shape synthesis approaches have demonstrated strong performance in many applications, including single view 3D reconstruction [60, 30, 47, 48], shape manipulation, and synthesis [26, 21, 28, 14, 1, 9]. However, as these approaches are trained by minimizing the reconstruction loss of function values at a set of sampled 3D locations (a rough proxy of the surface), they tend to render artifacts when synthesizing fine details. Furthermore, if one desires a mesh to be extracted from a DIF, an expensive iso-surfacing step based on Marching Cubes [36] or Marching Tetrahedra [15] is required. Due to the computational burden, iso-surfacing is often done on a smaller resolution, hence prone to quantization errors. Lei et al. [29] proposes an analytic meshing solution to reduce the error, but is only applicable to DIFs parametrized by MLPs with ReLU activation. Our representation scales to high resolution and does not require additional modification to the backward pass for training end-to-end. DMTET can represent arbitrary typology, and is trained via direct supervision on the generated surface. Recent works [1, 9] learn to regress unsigned distance to triangle soup or point cloud. However, their iso-surfacing formulation is not differentiable in contrast to DMTET. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/0d5058ec53aafbd0e0995b8e5ac5432565d0802e926641001c8ac4ebef8fb2be.jpg",
+        "image_caption": [
+            "Figure 1: DMTET reconstructs the shape implicitly in a coarse-to-fine manner by predicting the SDF defined on a deformable tetrahedral grid. It then converts the SDF to a surface mesh by a differentiable Marching Tetrahedra layer. DMTET is trained by optimizing the objective function defined on the final surface. "
+        ],
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+        "type": "text",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surface-based Methods directly predict triangular meshes and have achieved impressive results for reconstructing and synthesizing simpler shapes [54, 23, 5, 41, 7]. Typically, they predefined the topology of the shape, e.g. equivalent to a sphere [54, 5, 25], or a union of primitives [43, 53, 19] or a set of segmented parts [61, 62, 50]. As a result, they can not model a distribution of shapes with complex topology variations. Recently, DefTet [18] represents a mesh with a deformable tetrahedral grid where the grid vertex coordinates and the occupancy values are learned. However, similar to voxel-based methods, the computational costf increases cubically with the grid resolution. Furthermore, as the occupancy loss for supervising topology learning and the surface loss for supervising geometry learning do not support joint training, it tends to generate suboptimal results. In contrast, our method is able to synthesize high-resolution 3D shapes, not shown in previous work. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3 Deep Marching Tetrahedra ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now introduce our DMTET for synthesizing high-quality 3D objects. The schematic illustration is provided in Fig. 1. Our model relies on a new, hybrid 3D representation specifically designed for high-resolution reconstruction and synthesis, which we describe in Sec. 3.1. In Sec. 3.2, we describe the neural network architecture of DMTET that predicts the shape representation from inputs such as coarse voxels. We provide the training objectives in Sec. 3.3. ",
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+    {
+        "type": "text",
+        "text": "3.1 3D Representation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We represent a shape using a sign distance field (SDF) encoded with a deformable tetrahedral grid, adopted from DefTet [18, 20]. The grid fully tetrahedralizes a unit cube, where each cell in the volume is a tetahedron with 4 vertices and faces. The key aspect of this representation is that the grid vertices can deform to represent the geometry of the shape more efficiently. While the original DefTet encoded occupancy defined on each tetrahedron, we here encode signed distance values defined on the vertices of the grid and represent the underlying surface implicitly (Sec. 3.1.1). The use of signed distance values, instead of occupancy values, provides more flexibility in representing the underlying surface. For greater representation power while keeping memory and computation manageable, we further selectively subdivide the tetrahedra around the predicted surface (Sec. 3.1.2). We convert the signed distance-based implicit representation into a triangular mesh using a marching tetrahedra layer, which we discuss in Sec. 3.1.3. The final mesh is further converted into a parameterized surface with a differentiable surface subdivision module, described in Sec. 3.1.4. ",
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+    {
+        "type": "text",
+        "text": "3.1.1 Deformable Tetrahedral Mesh as an Approximation of an Implicit Function ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We adopt and extend the deformable tetrahedral grid introduced in Gao et al. [18], which we denote with $( V _ { T } , T )$ , where $V _ { T }$ are the vertices in the tetrahedral grid $T$ . Following the notation in [18], each tetrahedron $T _ { k } \\in T$ is represented with four vertices $\\left\\{ v _ { a _ { k } } , v _ { b _ { k } } , v _ { c _ { k } } , v _ { d _ { k } } \\right\\}$ , with $k \\in \\{ 1 , . . . . , K \\}$ , where $K$ is the total number of tetrahedra and $v _ { i _ { k } } \\in V _ { T }$ . ",
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+        "type": "text",
+        "text": "We represent the sign distance field by interpolating SDF values defined on the vertices of the grid. Specifically, we denote the SDF value in vertex $v _ { i } \\in V _ { T }$ as $s ( v _ { i } )$ . SDF values for the points that lie inside the tetrahedron follow a barycentric interpolation of the SDF values of the four vertices that encapsulates the point. ",
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+    {
+        "type": "text",
+        "text": "3.1.2 Volume Subdivision ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We represent shape in a coarse to fine manner for efficiency. We determine the surface tetrahedra $T _ { s u r f }$ by checking whether a tetrahedron has vertices with different SDF signs – indicating that it intersects the surface encoded by the SDF. We subdivide $T _ { s u r f }$ as well as their immediate neighbors and increase resolution by adding the mid point to each edge. We compute SDF values of the new vertices by averaging the SDF values on the edge (Fig. 2). ",
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+    {
+        "type": "image",
+        "img_path": "images/690ebadbdb5fb11a0781110c51057baf1fbf00e37c739b19a6285e80b9e34a71.jpg",
+        "image_caption": [
+            "Figure 2: Volume Subdivision: Each surface tet.(blue) is divided into 8 tet.(red) by adding midpoints. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.1.3 Marching Tetrahedra for converting between an Implicit and Explicit Representation ",
+        "text_level": 1,
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+        "type": "image",
+        "img_path": "images/66197fc00f7d532542743103dce49d9e142b692a641aa4f468f4386db848bf60.jpg",
+        "image_caption": [
+            "Figure 3: Three unique surface configurations in MT. Vertex color indicates the sign of signed distance value. Notice that flipping the signs of all vertices will result in the same surface configuration. Position of the vertex is linearly interpolated along the edges with sign change. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use the Marching Tetrahedra [15] algorithm to convert the encoded SDF into an explicit triangular mesh. Given the SDF values $\\mathbf { \\bar { \\{ } }  s ( v _ { a } ) , s ( \\mathbf { \\bar { { v } } } _ { b } ) , s ( v _ { c } ) , s ( v _ { d } ) \\}$ of the vertices of a tetrahedron, MT determines the surface typology inside the tetrahedron based on the signs of $s ( v )$ , which is illustrated in Fig. 3. The total number of configurations is $2 ^ { 4 } = { \\bar { 1 } } 6$ , which falls into 3 unique cases after considering rotation symmetry. Once the surface typology inside the tetrahedron is identified, the vertex location of the iso-surface is computed at the zero crossings of the linear interpolation along the tetrahedron’s edges, as shown in Fig. 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prior works [45, 31] argue that the singularity in this formulation, i.e. when $s ( v _ { a } ) = s ( v _ { b } )$ , prevents the change of surface typology (sign change of $s ( v _ { a } ) )$ ) during training. However, we find that, in practise, the equation is only evaluated when $\\mathrm { s i g n } ( s ( v _ { a } ) ) \\neq \\mathrm { s i g n } ( s ( v _ { b } ) )$ . Thus, during training, the singularity never happens and the gradient from a loss defined on the extracted iso-surface (Sec. 3.3), can be back-propagated to both vertex positions and SDF values via the chain rule. A more detailed analysis is in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "3.1.4 Surface Subdivision ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Having a surface mesh as output allows us to further increase the representation power and the visual quality of the shapes with a differentiable surface subdivision module. We follow the scheme of the Loop Subdivision method [35], but instead of using a fixed set of parameters for subdivision, we make these parameters learnable in DMTET. Specifically, learnable parameters include the positions of each mesh vertex $\\boldsymbol { v } _ { i } ^ { \\prime }$ , as well as $\\alpha _ { i }$ which controls the generated surface via weighting the smoothness of neighbouring vertices. Note that different from Liu et al. [33], we only predict the per-vertex parameter at the beginning and carry it over to subsequent subdivision iterations to attain a lower computational cost. We provide more details in Appendix. ",
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+    {
+        "type": "text",
+        "text": "3.2 DMTET: 3D Deep Conditional Generative Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our DMTET is a neural network that utilizes our proposed 3D representation and aims to output a high resolution 3D mesh $M$ from input $x$ (a point cloud or a coarse voxelized shape). We describe the architecture (Fig. 4) of the generator for each module of our 3D representation in Sec. 3.2.1, with the architecture of the discriminator presented in Sec. 3.2.2. Further details are in Appendix. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fa49995de6dd2e0e3e4ada8be106cd973661cac4b1241747a493f2249be01cff.jpg",
+        "image_caption": [
+            "Figure 4: Our generator and discriminator architectures. The generator is composed of two parts—one utilizes MLP to generate the initial predictions for all grid vertices and the other uses GCN to refine the surface. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2.1 3D Generator ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Input Encoder We use PVCNN [34] as an input encoder to extract a 3D feature volume $F _ { v o l } ( x )$ from a point cloud. When the input is a coarse voxelized shape, we sample points on its surface. We compute a feature vector $F _ { v o l } ( v , x )$ for a grid vertex $v \\in \\mathbb { R } ^ { 3 }$ via trilinear interpolation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Initial Prediction of SDF We predict the SDF value for each vertex in the initial deformable tetrahedral grid using a fully-connected network $s ( v ) = M L P ( F _ { v o l } ( v , x ) , v )$ . The fully-connected network additionally outputs a feature vector $f ( v )$ , which is used for the surface refinement in the volume subdivision stage. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surface Refinement with Volume Subdivision After obtaining the initial SDF, we iteratively refine the surface and subdivide the tetrahedral grid. We first identify surface tetrahedra $T _ { s u r f }$ based on the current $s ( v )$ value. We then build a graph $G = ( V _ { s u r f } , E _ { s u r f } )$ , where $V _ { s u r f } , E _ { s u r f }$ correspond to the vertices and edges in $T _ { s u r f }$ . We then predict the position offsets $\\Delta v _ { i }$ and SDF residual values $\\Delta s ( v _ { i } )$ for each vertex $i$ in $V _ { s u r f }$ using a Graph Convolutional Network [32] (GCN): ",
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+    {
+        "type": "equation",
+        "img_path": "images/c23b00128dc037edafe2fde1487ee41eb4ff642d329832475e1941d503f3144b.jpg",
+        "text": "$$\n\\begin{array} { r c l } { f _ { v _ { i } } ^ { \\prime } } & { = } & { \\mathsf { c o n c a t } ( v _ { i } , s ( v _ { i } ) , F _ { v o l } ( v _ { i } , x ) , f ( v _ { i } ) ) , } \\\\ { ( \\Delta v _ { i } , \\Delta s ( v _ { i } ) , \\overline { { f ( v _ { i } ) } } ) _ { i = 1 , \\cdots N _ { s u r f } } } & { = } & { \\mathsf { G C N } \\big ( ( f _ { v _ { i } } ^ { \\prime } ) _ { i = 1 , \\cdots N _ { s u r f } } , G \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $N _ { s u r f }$ is the total number of vertices in $V _ { s u r f }$ and $\\overline { { f ( v _ { i } ) } }$ is the updated per-vertex feature. The vertex position and the SDF value for vertex $v _ { i }$ are updated as $v _ { i } ^ { \\prime } = v _ { i } + \\Delta v _ { i }$ and $s ( v _ { i } ^ { \\prime } ) =$ $s ( v _ { i } ) + \\Delta s ( v _ { i } )$ . This refinement step can potentially flip the sign of the SDF values to refine the local typology, and also move the vertices thus improving the local geometry. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After surface refinement, we perform the volume subdivision step followed by an additional surface refinement step. In particular, we re-identify $T _ { s u r f }$ and subdivide $T _ { s u r f }$ and their immediate neighbors. We drop the unsubdivided tetrahedra from the full tetrahedral grid in both steps, which saves memory and computation, as the size of the $T _ { s u r f }$ is proportional to the surface area of the object, and scales up quadratically rather than cubically as the grid resolution increases. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the SDF values and positions of the vertices are inherited from the level before subdivision, thus, the loss computed at the final surface can back-propagate to all vertices from all levels. Therefore, our DMTET automatically learns to subdivide the tetrahedra and does not need an additional loss term in the intermediate steps to supervise the learning of the octree hierarchy as in the prior work [52]. ",
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+    {
+        "type": "text",
+        "text": "Learnable Surface Subdivision After extracting the surface mesh using MT, we can further apply learnable surface subdivision. Specifically, we build a new graph on the extracted mesh, and use GCN to predict the updated position of each vertex $\\boldsymbol { v } _ { i } ^ { \\prime }$ , and $\\alpha _ { i }$ for Loop Subvidision. This step removes the quantization errors and mitigates the approximation errors from the classic Loop Subdivision by adjusting $\\alpha _ { i }$ , which are fixed in the classic method. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2.2 3D Discriminator ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We apply a 3D discriminator $D$ on the final surface predicted from the generator. We empirically find that using a 3D CNN from DECOR-GAN [6] as the discriminator on the signed distance field that is computed from the predicted mesh is effective to capture the local details. Specifically, we first randomly select a high-curvature vertex $v$ from the target mesh and compute the ground truth signed distance field $S _ { r e a l } \\in \\mathbb { R } ^ { N \\times N \\times N }$ at a voxelized region around $v$ . Similarly, we compute the signed distance field of the predicted surface mesh $M$ at the same location to obtain $S _ { p r e d } \\in \\mathbb { R } ^ { N \\times \\tilde { N } \\times N }$ . Note that $S _ { p r e d }$ is an analytical function of the mesh $M$ , and thus the gradient to $S _ { p r e d }$ can backpropagate to the vertex positions in $M$ . We feed $S _ { r e a l }$ or $S _ { p r e d }$ into the discriminator, along with the feature vector $F _ { v o l } ( v , x )$ in position $v$ . The discriminator then predicts the probability indicating whether the input comes from the real or generated shapes. ",
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+    {
+        "type": "text",
+        "text": "3.3 Loss Function ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "DMTET is end-to-end trainable. We supervise all modules to minimize the error defined on the final predicted mesh $M$ . Our loss function contains three different terms: a surface alignment loss to encourage the alignment with ground truth surface, an adversarial loss to improve realism of the generated shape, and regularizations to regularize the behavior of SDF and vertex deformations. ",
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+        "type": "text",
+        "text": "Surface Alignment loss We sample a set of points $P _ { g t }$ from the surface of the ground truth mesh $M _ { g t }$ . Similarly, we also sample a set of points from $M _ { p r e d }$ to obtain $P _ { p r e d }$ , and minimize the L2 Chamfer Distance and the normal consistency loss between $P _ { g t }$ and $P _ { p r e d }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/e64ab678ee6659ce63b55c98d5f2f734f874712d27e15225187c2b8e7093ea13.jpg",
+        "text": "$$\nL _ { \\mathrm { c d } } = \\sum _ { p \\in P _ { p r e d } } \\operatorname* { m i n } _ { q \\in P _ { g t } } | | p - q | | _ { 2 } + \\sum _ { q \\in P _ { g t } } \\operatorname* { m i n } _ { p \\in P _ { p r e d } } | | q - p | | _ { 2 } , L _ { \\mathrm { n o m a l } } = \\sum _ { p \\in P _ { p r e d } } ( 1 - | \\Vec { \\mathbf { n } } _ { p } \\cdot \\Vec { \\mathbf { n } } _ { \\Vec { q } } | ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { q }$ is the point that corresponds to $p$ when computing the Chamfer Distance, and $\\vec { \\bf n } _ { p } , \\vec { \\bf n } _ { \\hat { q } }$ denotes the normal direction at point $p , \\hat { q }$ . ",
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+    {
+        "type": "text",
+        "text": "Adversarial Loss We use the adversarial loss proposed in LSGAN [37]: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/17f344150cf5ac968e7b3ff206f3a406b00d72d8b6b004e8975ea72d1606bac8.jpg",
+        "text": "$$\nL _ { \\mathrm { D } } = \\frac { 1 } { 2 } [ ( D ( M _ { g t } ) - 1 ) ^ { 2 } + D ( M _ { p r e d } ) ^ { 2 } ] , L _ { \\mathrm { G } } = \\frac { 1 } { 2 } [ ( D ( M _ { p r e d } ) - 1 ) ^ { 2 } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Regularizations The above loss functions operate on the extracted surface, thus, only the vertices that are close to the iso-surface in the tetrahedral grid receive gradients, while the other vertices do not. Moreover, the surface losses do not provide information about what is inside/outside, since flipping the SDF sign of all vertices in a tetrahedron would result in the same surface being extracted by MT. This may lead to disconnected components during training. To alleviate this issue, we add a SDF loss to regularize SDF values: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/776fd095945e78462a0165ae55b342a5542feb18a1f752dc0ac69338d741825d.jpg",
+        "text": "$$\nL _ { \\mathrm { S D F } } = \\sum _ { v _ { i } \\in V _ { T } } | s ( v _ { i } ) - S D F ( v _ { i } , M _ { g t } ) | ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $S D F ( v _ { i } , M _ { g t } )$ denotes the SDF value of point $v _ { i }$ to the mesh $M _ { g t }$ . In addition, we apply the $L _ { 2 }$ regularization loss on the predicted vertex deformations to avoid artifacts: $\\begin{array} { r } { L _ { \\mathrm { d e f } } = \\sum _ { v _ { i } \\in V _ { T } } | | \\Delta v _ { i } | | _ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The final loss is a weighted sum of all five loss terms: ",
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/dda99802456be2f83ede0a92476294d583b92d4aa933604c884ab3838aa4b2aa.jpg",
+        "text": "$$\nL = \\lambda _ { \\mathrm { c d } } L _ { \\mathrm { c d } } + \\lambda _ { \\mathrm { n o r m a l } } L _ { \\mathrm { n o r m a l } } + \\lambda _ { \\mathrm { G } } L _ { \\mathrm { G } } + \\lambda _ { \\mathrm { S D F } } L _ { \\mathrm { S D F } } + \\lambda _ { \\mathrm { d e f } } L _ { \\mathrm { d e f } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda _ { \\mathrm { c d } } , \\lambda _ { \\mathrm { n o r m a l } } , \\lambda _ { \\mathrm { G } } , \\lambda _ { \\mathrm { S D F } } , \\lambda _ { \\mathrm { d e f } }$ are hyperparameters (provided in the Supplement). ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We first evaluate DMTET in the challenging application of generating high-quality animal shapes from coarse voxels. We further evaluate DMTET in reconstructing 3D shapes from noisy point clouds on ShapeNet by comparing to existing state-of-the-art methods. ",
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+    {
+        "type": "text",
+        "text": "4.1 3D Shape Synthesis from Coarse Voxels ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Experimental Settings We collected 1562 animal models from the TurboSquid website1. These models have a wide range of diversity, ranging from cats, dogs, bears, giraffes, to rhinoceros, goats, etc. We provide visualizations in Supplement. Among 1562 shapes, we randomly select 1120 shapes for training, and the remaining 442 shapes for testing. We follow the pipeline in Kaolin [27] to convert shapes to watertight meshes. To prepare the input to the network, we first voxelize the mesh into the resolution of $1 6 ^ { \\overleftarrow { 3 } }$ , and then sample 3000 points from the surface after applying marching cubes to the $1 6 ^ { 3 }$ voxel grid. Note that this preprocessing is agnostic to the representation of the input coarse shape, allowing us to evaluate on different resolution voxels, or even meshes. ",
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+    {
+        "type": "text",
+        "text": "We compare our model with the official implementation of ConvOnet [44], which achieved SOTA performance on voxel upsampling. We also compare to DECOR-GAN [6], which obtained impressive results on transferring styles from a high-resolution voxel shape to a low-resolution voxel. Note that the original setting of DECOR-GAN is different from ours. For a fair comparison, we use all 1120 training shapes as the high-resolution style shapes during training, and retrieve the closet training shape to the test shape as the style shape during inference, which we refer as DECOR-Retv. We also compare against a randomly selected style shape as reference, denoted as DECOR-Rand. ",
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+    {
+        "type": "text",
+        "text": "1https://www.turbosquid.com, we obtain consent via an agreement with TurboSquid, and following license at https://blog.turbosquid.com/turbosquid-3d-model-license/ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/4b1bbd56197269ad4335baabf87fb00fe595064f3e4ec463efdf1d0b619d8092.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results on 3D shapes Synthesis from Coarse Voxels. Comparing with all baselines, our method reconstructs shapes with much higher quality. Adding GAN further improves the realism of the generated shape. We also show the retrieved shapes from the training set in the second last column. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Metrics We evaluate L2 and L1 Chamfer Distance, as well as normal consistency score to assess how well the methods reconstruct the corresponding high-resolution shape following [44]. We also report Light Field Distance [4] (LFD) which measures the visual similarity in 2D rendered views. In addition, we evaluate Cls score following [6]. Specifically, we render the predicted 3D shapes and train a patch-based image classifier to distinguish whether images are from the renderings of real or generated shapes. The mean classification accuracy of the trained classifier is reported as Cls (lower is better). More details are in the Supplement. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experimental Results We provide quantitative results in Table 1 with qualitative examples in Fig. 5. Our DMTET achieves significant improvements over all baselines in terms of all metrics. Compared to both ConvOnet [44] and DECORGAN [6], our DMTET reconstructs shapes with better quality when training without adversarial loss (5th column in Fig. 5). Further geometric details, including nails, ears, eyes, mouths, etc, are captured when trained with the adversarial loss (6th column in Fig. 5), significantly improving the realism and visual quality of the generated shape. To demonstrate the generalization ability of our ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/51a010aed7513fe7dfaf318abbe714fe80699ba1341c5a33b907b24eb8d658e0.jpg",
+        "image_caption": [
+            "Figure 6: Qualitative Results of synthesizing highresolution shapes from coarse voxels collected online. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DMTET, we collect human-created low-resolution voxels from Turbosquid (shapes unseen in training). We provide qualitative results in Fig. 6. Despite the fact that these human-created shapes have noticeable differences with our coarse voxels used in training, e.g., different ratios of body parts compared with our training shapes (larger head, thinner legs, longer necks), our model faithfully generates high-quality 3D details conditioned on each coarse voxel – an exciting result. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/df1e7551c170f87070b67404a130987968feb5b96f31ec1b8553941510bf4ad7.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Super Resolution of Animal Shapes: DMTET significantly outperforms all baselines in all metrics. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>L2 Chamfer↓</td><td rowspan=1 colspan=1>L1 Chamfer↓</td><td rowspan=1 colspan=1>Norm. Cons.↑</td><td rowspan=1 colspan=1>LFD↓</td><td rowspan=1 colspan=1>Cls</td></tr><tr><td rowspan=1 colspan=1>ConvOnet [44]</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>2.41</td><td rowspan=1 colspan=1>0.901</td><td rowspan=1 colspan=1>3220</td><td rowspan=1 colspan=1>0.63</td></tr><tr><td rowspan=1 colspan=1>DECOR [6]-Retv.</td><td rowspan=1 colspan=1>1.32</td><td rowspan=1 colspan=1>3.81</td><td rowspan=1 colspan=1>0.876</td><td rowspan=1 colspan=1>3689</td><td rowspan=1 colspan=1>0.66</td></tr><tr><td rowspan=1 colspan=1>DECOR [6]-Rand.</td><td rowspan=1 colspan=1>2.38</td><td rowspan=1 colspan=1>6.85</td><td rowspan=1 colspan=1>0.797</td><td rowspan=1 colspan=1>5338</td><td rowspan=1 colspan=1>0.67</td></tr><tr><td rowspan=1 colspan=1>DMTET wo Adv.</td><td rowspan=1 colspan=1>0.76</td><td rowspan=1 colspan=1>2.20</td><td rowspan=1 colspan=1>0.916</td><td rowspan=1 colspan=1>2846</td><td rowspan=1 colspan=1>0.58</td></tr><tr><td rowspan=1 colspan=1>DMTET</td><td rowspan=1 colspan=1>0.75</td><td rowspan=1 colspan=1>2.19</td><td rowspan=1 colspan=1>0.918</td><td rowspan=1 colspan=1>2823</td><td rowspan=1 colspan=1>0.54</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "User Studies We conduct user studies via Amazon Machanical Turk (AMT) to further evaluate the performance of all methods. In particular, we present two shapes that are predicted from two different models to the AMT workers and ask them to evaluate which one is a better looking shape and which one features more realistic details. Detailed experimental settings are provided in the Supplement. We compare DMTET against ConvONet [44], DECOR [6]-Retv, as well as DMTET without adversarial loss (w.o. Adv.). Quantitative results are reported in Table 2. Human judges agree that the shapes generated from our model have better details, compared to all baselines, in a vast majority of the cases. Ablations on using adversarial loss demonstrate the effectiveness of generating higher quality geometry using a discriminator during training. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Ablation Studies To evaluate the effectiveness of our volume subdivision and surface subdivision modules, we ablate by sequentially introducing them to the base model (we refer as $\\mathrm { D M T E T } _ { B }$ ) which we train on 100-resolution uniform tetrahedral grid without both volume and surface subdivision modules and adversarial ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a97fa21e21618d409e2083e585facfbf1958aaa1268147ae1b4384ddd53a54e6.jpg",
+        "table_caption": [
+            "Table 2: User Study on 3D Shape Synthesis from Coarse voxels. In each cell, we report percentages of shapes for which the users agree are better looking (left) or have better details (right). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ConvONet[44]</td><td rowspan=1 colspan=1>DECOR[6]-Retv.</td><td rowspan=1 colspan=1>DMTETWoAdv</td></tr><tr><td rowspan=1 colspan=1>Baselinewins</td><td rowspan=1 colspan=1>5% 15%</td><td rowspan=1 colspan=1>26%/17%</td><td rowspan=1 colspan=1>29% /25%</td></tr><tr><td rowspan=1 colspan=1>DMTETwins</td><td rowspan=1 colspan=1>95%/95%</td><td rowspan=1 colspan=1>74% /83%</td><td rowspan=1 colspan=1>71% 175%</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "loss. We conduct user studies to evaluate the improvement after each step using the protocol described in the above paragraph. We first reduce the initial resolution to 70 and employ volume subdivision to support higher output resolution (we refer this model as $\\mathbf { D M T E T } _ { V }$ ) and compare with $\\mathbf { D M T E T } _ { B }$ Predictions by $\\mathrm { D M T E T } _ { V }$ wins $78 \\%$ of cases over $\\mathrm { D M T E T } _ { B }$ for better looking, and $61 \\%$ of cases for realistic details, showing that the volume subdivision module is effective in synthesizing shape details. We then add surface subdivision on top of the $\\mathbf { D M T E T } _ { V }$ and compare with it. The new model wins $62 \\%$ of cases over $\\mathrm { D M T E T } _ { V }$ for better looking, and $62 \\%$ of cases for realistic details as well, demonstrating the effect of surface subdivision module in enhancing the shape details. ",
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+    {
+        "type": "text",
+        "text": "4.2 Point Cloud 3D Reconstruction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Experimental Settings We follow the setting from DefTet [18], and use all 13 categories in ShapeNet [3] core data2, which we pre-process using Kaolin [27] to watertight meshes. We sample 5000 points for each shape and add Gaussian noise with zero mean of standard deviation 0.005. For quantitative evaluation, we report the L1 Chamfer Distance in the main paper, and refer readers to the Supplement for results in other metrics (3D IoU, L2 Chamfer Distance and F1 score). We additionally report average inference time on the same Nvidia V100 GPU. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare DMTET against state-of-the-art 3D reconstruction approaches using different representations: voxels [10], deforming a mesh with a fixed template [54], deforming a mesh generated from a volumetric representation [22], DefTet [18], and implicit functions [44]. For a fair comparison, we use the same point cloud encoder for all the methods, and adopt the decoders in the original papers to generate shapes in different representations. We also remove the adversarial loss in this application, since baselines also do not have it. We further compare with oracle performance of MC/MT where the ground truth SDF is utilized to extract iso-surface using MC/MT. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental Results Quantitative results are summarized in Table 3, with a few qualitative examples shown in Fig. 7. Compared to DMC [31], which also predicts the SDF values and supervises with a surface loss, DMTET achieves much better reconstruction quality since training using the marching tetrahedra layer is more efficient than calculating an expectation over all possible configurations within one grid cell as done in DMC [31]. Compared to a method that deforms a fixed template (sphere) [54], we reconstruct shapes with different topologies, achieving more faithful results compared to the ground truth shape. When compared with other explicit surface representations that also support different topology [18, 22], our method achieves higher quality results for local geometry, benefiting from the fact that the typology is jointly optimized with the geometry, whereas it is separately supervised by an occupancy loss in [18, 22]. Compared to a neural implicit method [44], we generate higher quality shapes with less artifacts, while running significantly faster at inference. Finally, compared to a voxel-based method [10] at the same resolution, our method recovers more geometric details, benefiting from the predicted vertex deformations as well as the surface loss. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/7bd33ba212cc763b44caca6f6748c49136fb2f72c0e054ac16d83ee028728622.jpg",
+        "image_caption": [
+            "Figure 7: Qualitative results on 3D Reconstruction from Point Clouds: Our model reconstructs shapes with more geometric details compared to baselines. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/1585e0bf24ed2ee50f6489bda6d09d78fef7cc3f9d898b9448c222e335133ad1.jpg",
+        "table_caption": [
+            "Table 3: Quantitative Results on Point Cloud Reconstruction (Chamfer L1). Note that all the networks in the baselines are not designed for this task, and thus we use the same encoder and their decoder for a fair comparison. We also ablate ourselves by operating on fixed grid (DMTET wo (Def, Vol., Surf.)), removing volume subdivision (DMTET wo Vol.), or surface subdivision (DMTET wo Surf.), or the both (DMTET wo (Vol., Surf.)). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Category</td><td>Airplane</td><td>Bench</td><td>Dresser</td><td>Car</td><td>Chair</td><td>Display</td><td>Lamp</td><td>Speaker</td><td>Rifle</td><td>Sofa</td><td>Table</td><td>Phone</td><td>Vessel</td><td>Mean↓</td><td>Time(ms)</td></tr><tr><td>3D-R2N2[10]</td><td>1.48</td><td>1.59</td><td>1.64</td><td>1.62</td><td>1.70</td><td>1.66</td><td>1.74</td><td>1.74</td><td>1.37</td><td>1.60</td><td>1.78</td><td>1.55</td><td>1.51</td><td>1.61</td><td>174</td></tr><tr><td>DMC [31]</td><td>1.57</td><td>1.47</td><td>1.29</td><td>1.67</td><td>1.44</td><td>1.25</td><td>2.15</td><td>1.49</td><td>1.45</td><td>1.19</td><td>1.33</td><td>0.88</td><td>1.70</td><td>1.45</td><td>349</td></tr><tr><td>Pixel2mesh [54]</td><td>0.98</td><td>1.28</td><td>1.44</td><td>1.19</td><td>1.91</td><td>1.25</td><td>2.07</td><td>1.61</td><td>0.91</td><td>1.15</td><td>1.82</td><td>0.83</td><td>1.12</td><td>1.35</td><td>30</td></tr><tr><td>ConvOnet [44]</td><td>0.82</td><td>0.95</td><td>0.96</td><td>1.12</td><td>1.03</td><td>0.93</td><td>1.22</td><td>1.12</td><td>0.79</td><td>0.91</td><td>0.94</td><td>0.67</td><td>0.99</td><td>0.95</td><td>866</td></tr><tr><td>MeshRCNN [22]</td><td>0.88</td><td>1.01</td><td>1.05</td><td>1.14</td><td>1.10</td><td>0.99</td><td>1.20</td><td>1.21</td><td>0.83</td><td>0.96</td><td>1.00</td><td>0.71</td><td>1.03</td><td>1.01</td><td>228</td></tr><tr><td>DEFTET[18]</td><td>0.85</td><td>0.94</td><td>0.97</td><td>1.13</td><td>1.04</td><td>0.92</td><td>1.28</td><td>1.17</td><td>0.85</td><td>0.90</td><td>0.93</td><td>0.65</td><td>0.99</td><td>0.97</td><td>61</td></tr><tr><td>DMTET wo (Def, Vol., Surf.)]</td><td>0.82</td><td>0.96</td><td>0.94</td><td>0.98</td><td>0.99</td><td>0.90</td><td>1.04</td><td>1.03</td><td>0.80</td><td>0.86</td><td>0.93</td><td>0.65</td><td>0.89</td><td>0.91</td><td></td></tr><tr><td>DMTET wo (Vol.,Surf.)</td><td>0.69</td><td>0.82</td><td>0.88</td><td>0.92</td><td>0.92</td><td>0.82</td><td>0.89</td><td>0.97</td><td>0.65</td><td>0.81</td><td>0.84</td><td>0.61</td><td>0.80</td><td>0.81</td><td>52 52</td></tr><tr><td>DMTET wo Vol.</td><td>0.65</td><td>0.78</td><td>0.84</td><td>0.89</td><td>0.89</td><td>0.79</td><td>0.86</td><td>0.95</td><td>0.61</td><td>0.78</td><td>0.79</td><td>0.60</td><td>0.78</td><td>0.79</td><td>67</td></tr><tr><td>DMTET wo Surf.</td><td>0.63</td><td>0.77</td><td>0.84</td><td>0.88</td><td>0.88</td><td>0.79</td><td>0.84</td><td>0.94</td><td>0.60</td><td>0.78</td><td>0.79</td><td>0.59</td><td>0.76</td><td>0.78</td><td>108</td></tr><tr><td>DMTET</td><td>0.62</td><td>0.76</td><td>0.83</td><td>0.87</td><td>0.88</td><td>0.78</td><td>0.84</td><td>0.94</td><td>0.59</td><td>0.77</td><td>0.78</td><td>0.57</td><td>0.76</td><td>0.77</td><td>129</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2.1 Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            710,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We investigate how each component in our representation affects the performance and reconstruction quality. ",
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+    },
+    {
+        "type": "text",
+        "text": "Comparisons with Oracle Performance of MC/MT We first demonstrate the effect of learning on explicit surface via MT. We compare with the oracle performance of extracting the iso-surface with MT/MC from the ground truth signed distance fields on the Chair test set in ShapeNet, which contains diverse high-quality details. Specifically, for MC/MT, we first compute the discretized SDF at different grid resolutions, and compare the extracted surface to the ground truth surface. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca6df925a51dd1d96de02ca23469bf92f0ad16b5ce533ad7239d3a0faf8c0a73.jpg",
+        "image_caption": [
+            "Figure 8: Comparing our DMTET with oracle performance of MC and MT. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            593,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 8, MT consistently outperforms MC when querying the same number of points. We found the staggered grids pattern in tetrahedral grid [16, 18] better captures thin structures at a limited resolution (Fig. 9). This makes MT a better choice for efficiency reasons. The usage of tetrahedral mesh in DMTET follows this motivation. Without deforming the grid, DMTET outperforms the oracle performance of MT by a large margin when querying the same number of points, although DMTET predicts the surface from noisy point cloud. This demonstrates that directly optimizing the reconstructed surface can mitigate the discretization errors imposed by MT to a large extent. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
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+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/430b26fbbcba7dc56f2f3f84c2d486be08926dbae7665f553626b9f093f6b9b3.jpg",
+        "image_caption": [
+            "Figure 9: We compare trained DMTET to oracle performance of MT and MC. Number in bracket indicates number of SDF points queried. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            167,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ablation Studies We further provide ablation studies on the entire ShapeNet test set, which is summarized in Tab. 3. We first compare the version where we only predict SDF values without learning to deform the vertices and volume/surface subdivision with the version that predicts both SDF and the deformation. Predicting deformation along with SDF is significantly more performant, since vertex movements allow for a better reconstruction of the underlying surface. This is especially true for categories with thin structures (e.g. lamp) where the grid vertices are desired to align with them. We further ablate the use of volume subdivision and surface subdivision. We show that each component provides an improvement. In particular, volume subdivision has a significant ",
+        "bbox": [
+            501,
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+            392
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "improvement for object categories with fine-grained structural details, such as airplane and lamp, which require higher grid resolutions to model the occupancy change. Surface subdivision generates shapes with a parametric surface, avoiding the quantization errors in the planar faces and produces more visually pleasing results. ",
+        "bbox": [
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+        "text": "5 Conclusion ",
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+        "text": "In this paper, we introduced a deep 3D conditional generative model that can synthesize highresolution 3D shapes using simple user guides such as coarse voxels. Our DMTET features a novel 3D representation that marries implicit and explicit representations by leveraging the advantages of both. We experimentally show that our approach synthesizes significantly higher quality shapes with better geometric details than existing methods, confirmed by quantitative metrics and an extensive user study. By showcasing the ability to upscale coarse voxels such as Minecraft shapes, we hope that we take one step closer to democratizing 3D content creation. ",
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+        "text": "6 Broad Impact ",
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+        "text": "Many fields such as AR/VR, robotics, architecture, gaming and film rely on high-quality 3D content. Creating such content, however, requires human experts, i.e., experienced artists, and a significant amount of development time. In contrast, platforms like Minecraft enable millions of users around the world to carve out coarse shapes with simple blocks. Our work aims at creating A.I. tools that would enable even novice users to upscale simple, low-resolution shapes into high resolution, beautiful 3D content. Our method currently focuses on 3D animal shapes. We are not currently aware of and do not foresee nefarious use cases of our method. ",
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+        "text": "7 Disclosure of Funding ",
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+        "text": "This work was funded by NVIDIA. Tianchang Shen and Jun Gao acknowledge additional revenue in the form of student scholarships from University of Toronto and the Vector Institute, which are not in direct support of this work. ",
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parse/train/H1sUHgb0Z/H1sUHgb0Z.md

@@ -0,0 +1,392 @@
+# LEARNING FROM NOISY SINGLY-LABELED DATA
+
+Ashish Khetan   
+University of Illinois at Urbana-Champaign   
+Urbana, IL 61801   
+khetan2@illinois.edu
+
+Zachary C. Lipton Amazon Web Services Seattle, WA 98101 liptoz@amazon.com
+
+Animashree Anandkumar Amazon Web Services Seattle, WA 98101 anima@amazon.com
+
+# ABSTRACT
+
+Supervised learning depends on annotated examples, which are taken to be the ground truth. But these labels often come from noisy crowdsourcing platforms, like Amazon Mechanical Turk. Practitioners typically collect multiple labels per example and aggregate the results to mitigate noise (the classic crowdsourcing problem). Given a fixed annotation budget and unlimited unlabeled data, redundant annotation comes at the expense of fewer labeled examples. This raises two fundamental questions: (1) How can we best learn from noisy workers? (2) How should we allocate our labeling budget to maximize the performance of a classifier? We propose a new algorithm for jointly modeling labels and worker quality from noisy crowd-sourced data. The alternating minimization proceeds in rounds, estimating worker quality from disagreement with the current model and then updating the model by optimizing a loss function that accounts for the current estimate of worker quality. Unlike previous approaches, even with only one annotation per example, our algorithm can estimate worker quality. We establish a generalization error bound for models learned with our algorithm and establish theoretically that it’s better to label many examples once (vs less multiply) when worker quality exceeds a threshold. Experiments conducted on both ImageNet (with simulated noisy workers) and MS-COCO (using the real crowdsourced labels) confirm our algorithm’s benefits.
+
+# 1 INTRODUCTION
+
+Recent advances in supervised learning owe, in part, to the availability of large annotated datasets. For instance, the performance of modern image classifiers saturates only with millions of labeled examples. This poses an economic problem: Assembling such datasets typically requires the labor of human annotators. If we confined the labor pool to experts, this work might be prohibitively expensive. Therefore, most practitioners turn to crowdsourcing platforms such as Amazon Mechanical Turk (AMT), which connect employers with low-skilled workers who perform simple tasks, such as classifying images, at low cost.
+
+Compared to experts, crowd-workers provide noisier annotations, possibly owing to high variation in worker skill; and a per-answer compensation structure that encourages rapid answers, even at the expense of accuracy. To address variation in worker skill, practitioners typically collect multiple independent labels for each training example from different workers. In practice, these labels are often aggregated by applying a simple majority vote. Academics have proposed many efficient algorithms for estimating the ground truth from noisy annotations. Research addressing the crowd-sourcing problem goes back to the early 1970s. Dawid & Skene (1979) proposed a probabilistic model to jointly estimate worker skills and ground truth labels and used expectation maximization (EM) to estimate the parameters. Whitehill et al. (2009); Welinder et al. (2010); Zhou et al. (2015) proposed generalizations of the Dawid-Skene model, e.g. by estimating the difficulty of each example.
+
+Although the downstream goal of many crowdsourcing projects is to train supervised learning models, research in the two disciplines tends to proceed in isolation. Crowdsourcing research seldom accounts for the downstream utility of the produced annotations as training data in machine learning (ML) algorithms. And ML research seldom exploits the noisy labels collected from multiple human workers. A few recent papers use the original noisy labels and the corresponding worker identities together with the predictions of a supervised learning model trained on those same labels, to estimate the ground truth (Branson et al., 2017; Guan et al., 2017; Welinder et al., 2010). However, these papers do not realize the full potential of combining modeling and crowd-sourcing. In particular, they are unable to estimate worker qualities when there is only one label per training example.
+
+This paper presents a new supervised learning algorithm that alternately models the labels and worker quality. The EM algorithm bootstraps itself in the following way: Given a trained model, the algorithm estimates worker qualities using the disagreement between workers and the current predictions of the learning algorithm. Given estimated worker qualities, our algorithm optimizes a suitably modified loss function. We show that accurate estimates of worker quality can be obtained even when only collecting one label per example provided that each worker labels sufficiently many examples. An accurate estimate of the worker qualities leads to learning a better model. This addresses a shortcoming of the prior work and overcomes a significant hurdle to achieving practical crowdsourcing without redundancy.
+
+We give theoretical guarantees on the performance of our algorithm. We analyze the two alternating steps: (a) estimating worker qualities from disagreement with the model, (b) learning a model by optimizing the modified loss function. We obtain a bound on the accuracy of the estimated worker qualities and the generalization error of the model. Through the generalization error bound, we establish that it is better to label many examples once than to label less examples multiply when worker quality is above a threshold. Empirically, we verify our approach on several multi-class classification datasets: ImageNet and CIFAR10 (with simulated noisy workers), and MS-COCO (using the real noisy annotator labels). Our experiments validate that when the cost of obtaining unlabeled examples is negligible and the total annotation budget is fixed, it is best to collect a single label per training example for as many examples as possible. We emphasize that although this paper applies our approach to classification problems, the main ideas of the algorithm can be extended to other tasks in supervised learning.
+
+# 2 RELATED WORK
+
+The traditional crowdsourcing problem addresses the challenge of aggregating multiple noisy labels. A naive approach is to aggregate the labels based on majority voting. More sophisticated agreementbased algorithms jointly model worker skills and ground truth labels, estimating both using EM or similar techniques (Dawid & Skene, 1979; Jin & Ghahramani, 2003; Whitehill et al., 2009; Welinder et al., 2010; Zhou et al., 2012; Liu et al., 2012; Dalvi et al., 2013; Liu et al., 2012). Zhang et al. (2014) shows that the EM algorithm with spectral initialization achieves minimax optimal performance under the Dawid-Skene model. Karger et al. (2014) introduces a message-passing algorithm for estimating binary labels under the Dawid-Skene model, showing that it performs strictly better than majority voting when the number of labels per example exceeds some threshold. Similar observations are made by (Bragg et al., 2016). A primary criticism of EM-based approaches is that in practice, it’s rare to collect more than 3 to 5 labels per example; and with so little redundancy, the small gains achieved by EM over majority voting are not compelling to practitioners. In contrast, our algorithm performs well in the low-redundancy setting. Even with just one label per example, we can accurately estimate worker quality.
+
+Several prior crowdsourcing papers incorporate the predictions of a supervised learning model, together with the noisy labels, to estimate the ground truth labels. Welinder et al. (2010) consider binary classification and frames the problem as a generative Bayesian model on the features of the examples and the labels. Branson et al. (2017) consider a generalization of the Dawid-Skene model and estimate its parameters using supervised learning in the loop. In particular, they consider a joint probability over observed image features, ground truth labels, and the worker labels and compute the maximum likelihood estimate of the true labels using alternating minimization. We also consider a joint probability model but it is significantly different from theirs as we assume that the optimal labeling function gives the ground truth labels. We maximize the joint likelihood using a variation of expectation maximization to learn the optimal labeling function and the true labels. Further, they train the supervised learning model using the intermediate predictions of the labels whereas we train the model by minimizing a weighted loss function where the weights are the intermediate posterior probability distribution of the labels. Moreover, with only one label per example, their algorithm fails and estimates all the workers to be equally good. They only consider binary classification, whereas we verify our algorithm on multi-class (ten classes) classification problem.
+
+A rich body of work addresses human-in-loop annotation for computer vision tasks. However, these works assume that humans are experts, i.e., that they give noiseless annotations (Branson et al., 2010; Deng et al., 2013; Wah et al., 2011). We assume workers are unreliable and have varying skills. A recent work by Ratner et al. (2016) also proposes to use predictions of a supervised learning model to estimate the ground truth. However, their algorithm is significantly different than ours as it does not use iterative estimation technique, and their approach of incorporating worker quality parameters in the supervised learning model is different. Their theoretical results are limited to the linear classifiers.
+
+Another line of work employs active learning, iteratively filtering out examples for which aggregated labels have high confidence and collect additional labels for the remaining examples (Whitehill et al., 2009; Welinder & Perona, 2010; Khetan & Oh, 2016). The underlying modeling assumption in these papers is that the questions have varying levels of difficulty. At each iteration, these approaches employ an EM-based algorithm to estimate the ground truth label of the remaining unclassified examples. For simplicity, our paper does not address example difficulties, but we could easily extend our model and algorithm to accommodate this complexity.
+
+Several papers analyze whether repeated labeling is useful. Sheng et al. (2008) analyzed the effect of repeated labeling and showed that it depends upon the relative cost of getting an unlabeled example and the cost of labeling. Ipeirotis et al. (2014) shows that if worker quality is below a threshold then repeated labeling is useful, otherwise not. Lin et al. (2014a; 2016) argues that it also depends upon expressiveness of the classifier in addition to the factors considered by others. However, these works do not exploit predictions of the supervised learning algorithm to estimate the ground truth labels, and hence their findings do not extend to our methodology.
+
+Another body of work that is relevant to our problem is learning with noisy labels where usual assumption is that all the labels are generated through the same noisy rate given their ground truth label. Recently Natarajan et al. (2013) proposed a generic unbiased loss function for binary classification with noisy labels. They employed a modified loss function that can be expressed as a weighted sum of the original loss function, and gave theoretical bounds on the performance. However, their weights become unstably large when the noise rate is large, and hence the weights need to be tuned. Sukhbaatar et al. (2014); Jindal et al. (2016) learns noise rate as parameters of the model. A recent work by Guan et al. (2017) trains an individual softmax layer for each expert and then predicts their weighted sum where weights are also learned by the model. It is not scalable to crowdsourcing scenario where there are thousands of workers. There are works that aim to create noise-robust models (Joulin et al., 2016; Krause et al., 2016), but they are not relevant to our work.
+
+# 3 PROBLEM FORMULATION
+
+Let $\mathcal { D }$ be the underlying true distribution generating pairs $( X , Y ) \in { \mathcal { X } } \times { \mathcal { K } }$ from which $n$ i.i.d. samples $( X _ { 1 } , Y _ { 1 } ) , ( X _ { 2 } , Y _ { 2 } ) , \cdot \cdot \cdot , ( X _ { n } , Y _ { n } )$ are drawn, where $\kappa$ denotes the set of possible labels ${ \mathcal { K } } : = \{ 1 , 2 , \cdots , K \}$ , and $\mathcal { X } \subseteq \mathbb { R } ^ { d }$ denotes the set of euclidean features. We denote the marginal distribution of $Y$ by $\{ q _ { 1 } , q _ { 2 } , \cdots , q _ { K } \}$ , which is unknown to us. Consider a pool of $m$ workers indexed by $1 , 2 , \cdots , m$ . We use $[ m ]$ to denote the set $\{ 1 , 2 , \cdots , m \}$ . For each $i$ -th sample $X _ { i }$ , $r$ workers $\{ w _ { i j } \} _ { j \in [ r ] } \in [ m ] ^ { r }$ are selected randomly, independent of the sample $X _ { i }$ . Each selected worker provides a noisy label $Z _ { i j }$ for the sample $X _ { i }$ , where the distribution of $Z _ { i j }$ depends on the selected worker and the true label $Y _ { i }$ . We call $r$ the redundancy and, for simplicity, assume it to be the same for each sample. However, our algorithm can also be applied when redundancy varies across the samples. We use $Z _ { i } ^ { ( r ) }$ to denote $\{ Z _ { i j } \} _ { j \in [ r ] }$ , the set of $r$ labels collected on the $i$ -th example, and $w _ { i } ^ { ( r ) }$ to denote $\{ w _ { i j } \} _ { j \in [ r ] }$ .
+
+Following Dawid & Skene (1979), we assume the probability that the $a$ -th worker labels an item in class $k \in \mathcal { K }$ as class $s \in \kappa$ is independent of any particular chosen item, that is, it is a constant over $i \in [ n ]$ . Let us denote this constant by $\pi _ { k s }$ ; by definition, $\textstyle \sum _ { s \in { \mathcal { K } } } \pi _ { k s } = 1$ for all $k \in \mathcal { K }$ , and we call $\pi ^ { ( a ) } \in \left[ 0 , 1 \right] ^ { K \times K }$ the confusion matrix of the $a$ -th worker. In particular, the distribution of $Z$ is:
+
+$$
+\mathbb { P } \left[ Z _ { i j } = s \mid Y _ { i } = k , w _ { i j } = a \right] = \pi _ { k s } ^ { ( a ) } .
+$$
+
+The diagonal entries of the confusion matrix correspond to the probabilities of correctly labeling an example. The off-diagonal entries represent the probability of mislabeling. We use $\pi$ to denote the collection of confusion matrices $\{ \pi ^ { ( \bar { a } ) } \} _ { a \in [ m ] }$ .
+
+We assume nr workers $w _ { 1 , 1 } , w _ { 1 , 2 } , \cdots , w _ { n , r }$ are selected uniformly at random from a pool of $m$ workers with replacement and a batch of $r$ workers are assigned to each of the examples $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$ . The corrupted labels along with the worker information $( X _ { 1 } , Z _ { 1 } ^ { ( r ) } , w _ { 1 } ^ { ( r ) } ) , \cdot \cdot \cdot , ( X _ { n } , Z _ { n } ^ { ( r ) } , w _ { n } ^ { ( r ) } )$ are what the learning algorithm sees.
+
+Let $\mathcal { F }$ be the hypothesis class, and $f \in \mathcal { F } , f : \mathcal { X } \to \mathbb { R } ^ { K }$ , denote a vector valued predictor function. Let $\ell ( f ( X ) , Y )$ denote a loss function. For a predictor $f$ , its $\ell$ -risk under $\mathcal { D }$ is defined as
+
+$$
+\begin{array} { r l r } { R _ { \ell , { \mathcal D } } ( f ) } & { : = } & { { \mathbb E } _ { ( X , Y ) \sim { \mathcal D } } \left[ \ell ( f ( X ) , Y ) \right] . } \end{array}
+$$
+
+Given the observed samples $( X _ { 1 } , Z _ { 1 } ^ { ( r ) } , w _ { 1 } ^ { ( r ) } ) , \cdot \cdot \cdot , ( X _ { n } , Z _ { n } ^ { ( r ) } , w _ { n } ^ { ( r ) } )$ , we want to learn a good predictor function ${ \widehat { f } } \in { \mathcal { F } }$ such that its risk under the true distribution $\mathcal { D }$ , $R _ { \ell , \mathcal { D } } ( \widehat { f } )$ is minimal. Having access to only noisy labels $Z ^ { ( r ) }$ by workers $w ^ { ( r ) }$ , we compute $\widehat { f }$ as the one which minimizes a suitably modified loss function $\ell _ { \widehat { \pi } , \widehat { q } } ( f ( X ) , Z ^ { ( r ) } , w ^ { ( r ) } )$ . Where $\widehat { \pi }$ denote an estimate of confusion matrix $\pi$ , and $\widehat { q }$ an estimate of $q$ b b, the prior distribution on $Y$ b. We define $\ell _ { \widehat { \pi } , \widehat { q } }$ in the following section.
+
+# 4 ALGORITHM
+
+Assume that there exists a function $f ^ { * } \in { \mathcal { F } }$ such that $f ^ { * } ( X _ { i } ) = Y _ { i }$ for all $i \in [ n ]$ . Under the Dawid-Skene model (described in previous section), the joint likelihood of true labeling function $f ^ { * } ( X _ { i } )$ and observed labels $\{ Z _ { i j } \} _ { i \in [ n ] , j \in [ r ] }$ as a function of confusion matrices of workers $\pi$ can be written as
+
+$$
+\begin{array} { l } { L \left( \pi ; f ^ { * } , \{ X _ { i } \} _ { i \in [ n ] } , \{ Z _ { i j } \} _ { i \in [ n ] , j \in [ r ] } \right) : = } \\ { \displaystyle \prod _ { i = 1 } ^ { n } \left( \sum _ { k \in { \cal K } } q _ { k } \mathbb { I } [ f ^ { * } ( X _ { i } ) = k ] \left( \prod _ { j = 1 } ^ { r } \left( \sum _ { s \in { \cal K } } \mathbb { I } [ Z _ { i j } = s ] \pi _ { k s } ^ { ( w _ { i j } ) } \right) \right) \right) \ : . } \end{array}
+$$
+
+$q _ { k }$ ’s are the marginal distribution of the true labels $Y _ { i }$ ’s. We estimate the worker confusion matrices $\pi$ and the true labeling function $f ^ { * }$ by maximizing the likelihood function $L ( \pi ; f ^ { * } ( X ) , Z )$ . Observe that the likelihood function $L ( \pi ; f ^ { * } ( X ) , Z )$ is different than the standard likelihood function of Dawid-Skene model in that we replace each true hidden labels $Y _ { i }$ by $f ^ { * } ( X _ { i } )$ . Like the EM algorithm introduced in (Dawid & Skene, 1979), we propose ‘Model Bootstrapped EM’ (MBEM) to estimate confusion matrices $\pi$ and the true labeling function $f ^ { * }$ . EM converges to the true confusion matrices and the true labels given an appropriate spectral initialization of worker confusion matrices (Zhang et al., 2014). We show in Section 4.4 that MBEM converges under mild conditions when the worker quality is above a threshold and the number of training examples is sufficiently large. In the following two subsections, we motivate and explain our iterative algorithm to estimate the true labeling function $f ^ { * }$ given a good estimate of worker confusion matrices $\pi$ and vice-versa.
+
+# 4.1 LEARNING WITH NOISY LABELS
+
+To begin, we ask, what is the optimal approach to learn the predictor function $\widehat { f }$ when for each worker we have $\widehat { \pi }$ , a good estimation of the true confusion matrix $\pi$ , and $\widehat { q } ,$ , an estimate of the prior? b bA recent paper, Natarajan et al. (2013) proposes minimizing an unbiased loss function specifically, a weighted sum of the original loss over each possible ground truth label. They provide weights for binary classification where each example is labeled by only one worker. Consider a worker with confusion matrix $\pi$ , where $\pi _ { y } > 1 / 2$ and $\pi _ { - y } > 1 / 2$ represent her probability of correctly labeling the examples belonging to class $y$ and $- y$ respectively. Then their weights are $\pi _ { - y } / ( \pi _ { y } + \bar { \pi } _ { - y } - 1 \bar { ) }$ for class $y$ and $- ( 1 - \pi _ { y } ) / ( \pi _ { y } + \pi _ { - y } - 1 )$ for class $- y$ . It is evident that their weights become unstably large when the probabilities of correct classification $\pi _ { y }$ and $\pi _ { - y }$ are close to $1 / 2$ , limiting the method’s usefulness in practice. As explained below, for the same scenario, our weights would be $\pi _ { y } / ( 1 + \pi _ { y } - \pi _ { - y } )$ for class $y$ and $( 1 - \bar { \pi } _ { - y } ) / ( 1 + \pi _ { y } - \pi _ { - y } )$ for class $- y$ . Inspired by their idea, we propose weighing the loss function according to the posterior distribution of the true label given the $Z ^ { ( r ) }$ observed labels and an estimate of the confusion matrices of the worker who provided those labels. In particular, we define $\ell _ { \widehat { \pi } , \widehat { q } }$ to be
+
+$$
+\begin{array} { r c l } { \ell _ { \widehat { \pi } , \widehat { q } } ( f ( X ) , Z ^ { ( r ) } , w ^ { ( r ) } ) } & { : = } & { \displaystyle \sum _ { k \in K } \mathbb { P } _ { \widehat { \pi } , \widehat { q } } [ Y = k \mid Z ^ { ( r ) } ; w ^ { ( r ) } ] \ell ( f ( X ) , Y = k ) . } \end{array}
+$$
+
+If the observed label is uniformly random, then all weights are equal and the loss is identical for all predictor functions $f$ . Absent noise, we recover the original loss function. Under the Dawid-Skene model, given the observed noisy labels $Z ^ { ( r ) }$ , an estimate of confusion matrices $\widehat { \pi }$ , and an estimate of prior $\widehat { q }$ b, the posterior distribution of the true labels can be computed as follows:
+
+$$
+\begin{array} { r l r } { \mathbb { P } _ { \widehat { \pi } , \widehat { q } } [ Y _ { i } = k \mid Z _ { i } ^ { ( r ) } ; { w } _ { i } ^ { ( r ) } ] } & { = } & { \frac { \widehat { q } _ { k } \prod _ { j = 1 } ^ { r } \Big ( \sum _ { s \in { \mathcal K } } \mathbb { I } [ Z _ { i j } = s ] \widehat { \pi } _ { k s } ^ { ( w _ { i j } ) } \Big ) } { \sum _ { k ^ { \prime } \in { \mathcal K } } \Big ( \widehat { q } _ { k ^ { \prime } } \prod _ { j = 1 } ^ { r } \Big ( \sum _ { s \in { \mathcal K } } \mathbb { I } [ Z _ { i j } = s ] \widehat { \pi } _ { k ^ { \prime } s } ^ { ( w _ { i j } ) } \Big ) \Big ) } , } \end{array}
+$$
+
+where $\mathbb { I } [ . ]$ is the indicator function which takes value one if the identity inside it is true, otherwise zero. We give guarantees on the performance of the proposed loss function in Theorem 4.1. In practice, it is robust to noise level and significantly outperforms the unbiased loss function. Given $\ell _ { \widehat { \pi } , \widehat { q } } .$ , we learn the predictor function $\hat { f }$ by minimizing the empirical risk
+
+$$
+\widehat { f }  \arg \operatorname* { m i n } _ { f \in \mathcal { F } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell _ { \widehat { \pi } , \widehat { q } } ( f ( X _ { i } ) , Z _ { i } ^ { ( r ) } , w _ { i } ^ { ( r ) } ) .
+$$
+
+# 4.2 ESTIMATING ANNOTATOR NOISE
+
+The next question is: how do we get a good estimate $\widehat { \pi }$ of the true confusion matrix $\pi$ for each worker. If redundancy $r$ bis sufficiently large, we can employ the EM algorithm. However, in practical applications, redundancy is typically three or five. With so little redundancy, the standard applications of EM are of limited use. In this paper we look to transcend this problem, posing the question: Can we estimate confusion matrices of workers even when there is only one label per example? While this isn’t possible in the standard approach, we can overcome this obstacle by incorporating a supervised learning model into the process of assessing worker quality.
+
+Under the Dawid-Skene model, the EM algorithm estimates the ground truth labels and the confusion matrices in the following way: It alternately fixes the ground truth labels and the confusion matrices by their estimates and and updates its estimate of the other by maximizing the likelihood of the observed labels. The alternating maximization begins by initializing the ground truth labels with a majority vote. With only 1 label per example, EM estimates that all the workers are perfect.
+
+We propose using model predictions as estimates of the ground truth labels. Our model is initially trained on the majority vote of the labels. In particular, if the model prediction is $\{ t _ { i } \} _ { i \in [ n ] }$ , where $t _ { i } \in \mathcal { K }$ , then the maximum likelihood estimate of confusion matrices and the prior distribution is given below. For the $a$ -th worker, ${ \widehat \pi } _ { k s } ^ { ( a ) }$ for $k , s \in \mathcal { K }$ , and $\widehat { q _ { k } }$ for $k \in \mathcal { K }$ , we have,
+
+$$
+\widehat { \pi } _ { k s } ^ { ( a ) } = \frac { \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { r } \mathbb { I } [ w _ { i j } = a ] \mathbb { I } [ t _ { i } = k ] \mathbb { I } [ Z _ { i j } = s ] } { \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { r } \mathbb { I } [ w _ { i j } = a ] \mathbb { I } [ t _ { i } = k ] } , \qquad \widehat { q } _ { k } = ( 1 / n ) \sum _ { i = 1 } ^ { n } \mathbb { I } [ t _ { i } = k ]
+$$
+
+The estimate is effective when the hypothesis class $\mathcal { F }$ is expressive enough and the learner is robust to noise. Thus the model should, in general, have small training error on correctly labeled examples and large training error on wrongly labeled examples. Consider the case when there is only one label per example. The model will be trained on the raw noisy labels given by the workers. For simplicity, assume that each worker is either a hammer (always correct) or a spammer (chooses labels uniformly random). By comparing model predictions with the training labels, we can identify which workers are hammers and which are spammers, as long as each worker labels sufficiently many examples. We expect a hammer to agree with the model more often than a spammer.
+
+# 4.3 ITERATIVE ALGORITHM
+
+Building upon the previous two ideas, we present ‘Model Bootstrapped EM’, an iterative algorithm for efficient learning from noisy labels with small redundancy. MBEM takes data, noisy labels, and the corresponding worker IDs, and returns the best predictor function $\widehat { f }$ in the hypothesis class $\mathcal { F }$ . In the first round, we compute the weights of the modified loss function $\ell _ { \widehat { \pi } , \widehat { q } }$ by using the weighted b bmajority vote. Then we obtain an estimate of the worker confusion matrices $\widehat { \pi }$ using the maximum blikelihood estimator by taking the model predictions as the ground truth labels. In the second round, weights of the loss function are computed as the posterior probability distribution of the ground truth labels conditioned on the noisy labels and the estimate of the confusion matrices obtained in the previous round. In our experiments, only two rounds are required to achieve substantial improvements over baselines.
+
+# Algorithm 1 Model Bootstrapped EM (MBEM)
+
+Input: $\{ ( X _ { i } , Z _ { i } ^ { ( r ) } , w _ { i } ^ { ( r ) } ) \} _ { i \in [ n ] }$ , $T$ : number of iterations   
+Output: $\widehat { f }$ : predictor function   
+Initialize posterior distribution using weighted majority vote   
+$\mathbb P _ { \widehat { \pi } , \widehat { q } } [ Y _ { i } = k \mid Z _ { i } ^ { ( r ) } ; w _ { i } ^ { ( r ) } ] \gets ( 1 / r ) \bar { \sum _ { j = 1 } ^ { r } } \mathbb I [ Z _ { i j } = k ]$ , for $k \in \mathcal { K } , i \in [ n ]$   
+b bRepeat $T$ times: learn predictor function $\widehat { f }$ $\begin{array} { r } { \widehat { f } \longleftarrow \arg \operatorname* { m i n } _ { f \in \mathcal { F } _ { \ast } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sum _ { k \in K } \mathbb { P } _ { \widehat { \pi } , \widehat { q } } [ Y _ { i } = k \mid Z _ { i } ^ { ( r ) } ; w _ { i } ^ { ( r ) } ] \ell ( f ( X _ { i } ) , Y _ { i } = k ) } \end{array}$ predict on training examples $t _ { i } \gets \arg \operatorname* { m a x } _ { k \in \mathcal { K } } \widehat { f } ( X _ { i } ) _ { k }$ , for $i \in [ n ]$ bestimate confusion matrices $\widehat { \pi }$ and prior class distribution $\widehat { q }$ given $\{ t _ { i } \} _ { i \in [ n ] }$ ${ \widehat { \pi } } ^ { ( a ) } \gets$ Equation (7), for $a \in [ m ]$ ; $\widehat { q } \gets$ Equation (7) b bestimate label posterior distribution given ${ \widehat { \pi } } , { \widehat { q } }$ $\mathbb { P } _ { \widehat { \pi } , \widehat { q } } [ Y _ { i } = k \mid \bar { Z _ { i } ^ { ( r ) } } ; w _ { i } ^ { ( r ) } ] , \gets \mathrm { I }$ bquation (5), for $k \in \mathcal { K } , i \in [ n ]$   
+b bReturn f
+
+# 4.4 PERFORMANCE GUARANTEES
+
+The following result gives guarantee on the excess risk for the learned predictor function $\widehat { f }$ in terms of the VC dimension of the hypothesis class $\mathcal { F }$ . Recall that risk of a function $f$ w.r.t. loss function $\ell$ is defined to be $R _ { \ell , \mathcal { D } } ( f ) : = \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } \left[ \ell ( f ( X ) , Y ) \right]$ , Equation (2). We assume that the classification problem is binary, and the distribution $q$ , prior on ground truth labels $Y$ , is uniform and is known to us. We give guarantees on the excess risk of the predictor function $\widehat { f }$ , and accuracy of $\widehat { \pi }$ estimated bin the second round. For the purpose of analysis, we assume that fresh samples are used in each round for computing function $\widehat { f }$ and estimating $\widehat { \pi }$ . In other words, we assume that $\widehat { f }$ and $\widehat { \pi }$ are each computed using $n / 4$ bfresh samples in the first two rounds. We define $\alpha$ and $\beta _ { \epsilon }$ bto capture the average worker quality. Here, we give their concise bound for a special case when all the workers are identical, and their confusion matrix is represented by a single parameter, $0 \leq \rho < 1 / 2$ . Where $\pi _ { k k } = 1 - \rho$ $\pi _ { k s } ~ = ~ \rho$ $k \neq s$ . re t. e with probability for this special cas $\rho$ .  i $\beta _ { \epsilon } \leq$ $\begin{array} { r } { ( \rho + \epsilon ) ^ { r } \sum _ { u = 0 } ^ { r } \binom { r } { u } ( \tau ^ { u } + \tau ^ { r - u } ) ^ { - 1 } } \end{array}$ $\tau : = ( \rho + \epsilon ) / ( 1 - \rho - \epsilon )$ $\alpha$ $\rho$ general definition of $\alpha$ and $\beta _ { \epsilon }$ for any confusion matrices $\pi$ is provided in the Appendix.
+
+Theorem 4.1. Define $N : = n r$ to be the number of total annotations collected on n training examples with redundancy $r$ . Suppose $\operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f ) \leq 1 / 4$ . For any hypothesis class $\mathcal { F }$ with a finite VC dimension $V$ , and any $\delta < 1$ , there exists a universal constant $C$ such that if $N$ is large enough and satisfies
+
+$$
+N \ge \operatorname* { m a x } \left\{ C r \big ( \big ( \sqrt V + \sqrt { \log ( 1 / \delta ) } \big ) / ( 1 - 2 \alpha ) \big ) ^ { 2 } , 2 ^ { 1 2 } m \log ( 2 ^ { 6 } m / \delta ) \right\} ,
+$$
+
+then for binary classification with 0-1 loss function $\ell$ , $\hat { f }$ and $\widehat { \pi }$ returned by Algorithm $I$ after $T = 2$ iterations satisfies
+
+$$
+R _ { \ell , \mathcal { D } } ( \widehat { f } ) - \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f ) \leq \frac { C \sqrt { r } } { 1 - 2 \beta _ { \epsilon } } \left( \sqrt { \frac { V } { N } } + \sqrt { \frac { \log ( 1 / \delta ) } { N } } \right) ,
+$$
+
+and $\| \widehat { \pi } ^ { ( a ) } - \pi ^ { ( a ) } \| _ { \infty } \leq \epsilon _ { 1 }$ for all $a \in [ m ]$ , with probability at least √ $1 - \delta$ . Where $\epsilon : = 2 ^ { 4 } \gamma +$ $2 ^ { 8 } \sqrt { m \log ( 2 ^ { 6 } m \delta ) / N }$ , and $\begin{array} { r } { \gamma : = \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f ) + C ( \sqrt { V } + \sqrt { \log ( 1 / \delta ) } ) / ( ( 1 - 2 \alpha ) \sqrt { N / r } ) } \end{array}$ . $\epsilon _ { 1 }$ is defined to be $\epsilon$ with $\alpha$ in it replaced by $\beta _ { \epsilon }$ .
+
+The price we pay in generalization error bound on $\widehat { f }$ is $( 1 - 2 \beta _ { \epsilon } )$ . Note that, when $n$ is large, $\epsilon$ goes to zero, and $\beta _ { \epsilon } \leq 2 \rho ( 1 - \rho )$ , for $r = 1$ .
+
+If $\mathrm { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f )$ is sufficiently small, VC dimension is finite, and $\rho$ is bounded away from $1 / 2$ then for $n = { \cal O } ( m \log ( m ) / r )$ , we get $\epsilon _ { 1 }$ to be sufficiently small. Therefore, for any redundancy $r$ , error in confusion matrix estimation is small when the number of training examples is sufficiently large. Hence, for $N$ large enough, using Equation (9) and the bound on $\beta _ { \epsilon }$ , we get that for fixed total annotation budget, the optimal choice of redundancy $r$ is 1 when the worker quality $( 1 - \rho )$ is above a threshold. In particular, if $( 1 - \rho ) \ge 0 . 8 2 5$ then label once is the optimal strategy. However, in experiments we observe that with our algorithm the choice of $r = 1$ is optimal even for much smaller values of worker quality.
+
+# 5 EXPERIMENTS
+
+We experimentally investigate our algorithm, MBEM, on multiple large datasets. On CIFAR-10 (Krizhevsky & Hinton, 2009) and ImageNet (Deng et al., 2009), we draw noisy labels from synthetic worker models. We confirm our results on multiple worker models. On the MS-COCO dataset (Lin et al., 2014b), we accessed the real raw data that was used to produce this annotation. We compare MBEM against the following baselines:
+
+• MV: First aggregate labels by performing a majority vote, then train the model. • weighted-MV: Model learned using weighted loss function with weights set by majority vote. • EM: First aggregate labels using EM. Then train model in the standard fashion. (Dawid & Skene, 1979) • weighted-EM: Model learned using weighted loss function with weights set by standard EM. • oracle weighted EM: This model is learned by minimizing $\ell _ { \pi }$ , using the true confusion matrices. • oracle correctly labeled: This baseline is trained using the standard loss function $\ell$ but only using those training examples for which at least one of the $r$ workers has given the true label.
+
+Note that oracle models cannot be deployed in practice. We show them to build understanding only. In the plots, the dashed lines correspond to MV and EM algorithm. The black dashed-dotted line shows generalization error if the model is trained using ground truth labels on all the training examples. For experiments with synthetic noisy workers, we consider two models of worker skill:
+
+• hammer-spammer: Each worker is either a hammer (always correct) with probability $\gamma$ or a spammer (chooses labels uniformly at random).   
+• class-wise hammer-spammer: Each worker can be a hammer for some subset of classes and a spammer for the others. The confusion matrix in this case has two types of rows: (a) hammer class: row with all off-diagonal elements being 0. (b) spammer class: row with all elements being $1 / | \mathcal { K } |$ . A worker is a hammer for any class $k \in \mathcal { K }$ with probability $\gamma$ .
+
+We sample $m$ confusion matrices $\{ \pi ^ { ( a ) } \} _ { a \in [ m ] }$ according to the given worker skill distribution for a given $\gamma$ . We assign $r$ workers uniformly at random to each example. Given the ground truth labels, we generate noisy labels according to the probabilities given in a worker’s confusion matrix, using Equation (1). While our synthetic workers are sampled from these specific worker skill models, our algorithms do not use this information to estimate the confusion matrices. A Python implementation of the MBEM algorithm is available for download at https://github.com/khetan2/MBEM.
+
+CIFAR-10 This dataset has a total of $6 0 \mathrm { K }$ images belonging to 10 different classes where each class is represented by an equal number of images. We use 50K images for training the model and 10K images for testing. We use the ground truth labels to generate noisy labels from synthetic workers. We choose $m = 1 0 0$ , and for each worker, sample confusion matrix of size $1 0 \times 1 0$ according to the worker skill distribution. All our experiments are carried out with a 20-layer ResNet which achieves an accuracy of $9 1 . 5 \%$ . With the larger ResNet-200, we can obtain a higher accuracy of $9 3 . 5 \%$ but to save training time we restrict our attention to ResNet-20. We run MBEM 1 for $T = 2$ rounds. We assume that the prior distribution $\widehat { q }$ is uniform. We report mean accuracy of 5 runs and its standard error for all the experiments.
+
+![](images/6c945ae9fff5521cc6e41d06a130396dbff3f15c56211f1937aa4caf459564d6.jpg)  
+Figure 1: Plots for CIFAR-10. Line colors- black: oracle correctly labeled, red: oracle weighted EM, blue: MBEM, green: weighted EM, yellow: weighted MV.
+
+Figure 1 shows plots for CIFAR-10 dataset under various settings. The three plots in the first row correspond to “hammer-spammer” worker skill distribution and the plots in the second row correspond to “class-wise hammer-spammer” distribution. In the first plot, we fix redundancy $r = 1$ , and plot generalization error of the model for varying hammer probability $\gamma$ . MBEM significantly outperforms all baselines and closely matches the Oracle weighted EM. This implies MBEM recovers worker confusion matrices accurately even when we have only one label per example. When there is only one label per example, MV, weighted-MV, EM, and weighted-EM all reduce learning with the standard loss function $\ell$ .
+
+In the second plot, we fix hammer probability $\gamma = 0 . 2$ , and vary redundancy $r$ . This plot shows that weighted-MV and weighted-EM perform significantly better than MV and EM and confirms that our approach of weighing the loss function with posterior probability is effective. MBEM performs much better than weighted-EM at small redundancy, demonstrating the effect of our bootstrapping idea. However, when redundancy is large, EM works as good as MBEM.
+
+In the third plot, we show that when the total annotation budget is fixed, it is optimal to collect one label per example for as many examples as possible. We fixed hammer probability $\gamma = 0 . 2$ . Here, when redundancy is increased from 1 to 2, the number of of available training examples is reduced by $50 \%$ , and so on. Performance of weighted-EM improves when redundancy is increased from 1 to 5, showing that with the standard EM algorithm it might be better to collect redundant annotations for fewer example (as it leads to better estimation of worker qualities) than to singly annotate more examples. However, MBEM always performs better than the standard EM algorithm, achieving lowest generalization error with many singly annotated examples. Unlike standard EM, MBEM can estimate worker qualities even with singly annotated examples by comparing them with model predictions. This corroborates our theoretical result that label-once is the optimal strategy when worker quality is above a threshold. The plots corresponding to class-wise hammer-spammer workers follow the same trend. Estimation of confusion matrices in this setting is difficult and hence the gap between MBEM and the baselines is less pronounced.
+
+ImageNet The ImageNet-1K dataset contains 1.2M training examples and 50K validation examples. We divide test set in two parts: 10K for validation and 40K for test. Each example belongs to one of the possible 1000 classes. We implement our algorithms using a ResNet-18 that achieves top
+
+![](images/30f356ecd82f96c6537d9c31616006d1638d277695f800b639ed35818b630b05.jpg)  
+Figure 2: Plots for ImageNet. Solid lines represent top-5 error, dashed-lines represent top-1 error. Line colors- blue: MBEM, green: weighted majority vote, yellow: majority vote
+
+<table><tr><td rowspan=1 colspan=1>Approach</td><td rowspan=1 colspan=1>F1 score</td></tr><tr><td rowspan=1 colspan=1>majority vote</td><td rowspan=1 colspan=1>0.433</td></tr><tr><td rowspan=1 colspan=1>EM</td><td rowspan=1 colspan=1>0.447</td></tr><tr><td rowspan=1 colspan=1>MBEM</td><td rowspan=1 colspan=1>0.451</td></tr><tr><td rowspan=1 colspan=1>ground truth labels</td><td rowspan=1 colspan=1>0.512</td></tr></table>
+
+![](images/052f3a805394aed41772f861c3670f86759ecf43e7f19897d4beb4aeb74f5e6c.jpg)  
+Figure 3: Results on raw MS-COCO annotations.
+
+1 accuracy of $6 9 . 5 \%$ and top-5 accuracy of $8 9 \%$ on ground truth labels. We use $m = 1 0 0 0$ simulated workers. Although in general, a worker can mislabel an example to one of the 1000 possible classes, our simulated workers mislabel an example to only one of the 10 possible classes. This captures the intuition that even with a larger number of classes, perhaps only a small number are easily confused for each other. Therefore, each workers’ confusion matrix is of size $1 0 \times 1 0$ . Note that without this assumption, there is little hope of estimating a $1 0 0 0 \times 1 0 0 0$ confusion matrix for each worker by collecting only approximately 1200 noisy labels from a worker. The rest of the settings are the same as in our CIFAR-10 experiments. In Figure 2, we fix total annotation budget to be 1.2M and vary redundancy from 1 to 9. When redundancy is 9, we have only $( 1 . 2 / 9 ) \mathbf { M }$ training examples, each labeled by 9 workers. MBEM outperforms baselines in each of the plots, achieving the minimum generalization error with many singly annotated training examples.
+
+MS-COCO These experiments use the real raw annotations collected when MS-COCO was crowdsourced. Each image in the dataset has multiple objects (approximately 3 on average). For validation set images (out of 40K), labels were collected from 9 workers on average. Each worker marks which out of the 80 possible objects are present. However, on many examples workers disagree. These annotations were collected to label bounding boxes but we ask a different question: what is the best way to learn a model to perform multi-object classification, using these noisy annotations. We use 35K images for training the model and 1K for validation and 4K for testing. We use raw noisy annotations for training the model and the final MS-COCO annotations as the ground truth for the validation and test set. We use ResNet-98 deep learning model and train independent binary classifier for each of the 80 object classes. Table in Figure 3 shows generalization F1 score of four different algorithms: majority vote, EM, MBEM using all 9 noisy annotations on each of the training examples, and a model trained using the ground truth labels. MBEM performs significantly better than the standard majority vote and slightly improves over EM. In the plot, we fix the total annotation budget to 35K. We vary redundancy from 1 to 7, and accordingly reduce the number of training examples to keep the total number of annotations fixed. When redundancy is $r \ < \ 9$ we select uniformly at random $r$ of the original 9 noisy annotations. Again, we find it best to singly annotate as many examples as possible when the total annotation budget is fixed. MBEM significantly outperforms majority voting and EM at small redundancy.
+
+# 6 CONCLUSION
+
+We introduced a new algorithm for learning from noisy crowd workers. We also presented a new theoretical and empirical demonstration of the insight that when examples are cheap and annotations expensive, it’s better to label many examples once than to label few multiply when worker quality is above a threshold. Many avenues seem ripe for future work. We are especially keen to incorporate our approach into active query schemes, choosing not only which examples to annotate, but which annotator to route them to based on our models current knowledge of both the data and the worker confusion matrices.
+
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+
+# APPENDIX
+
+# A PROOF OF THEOREM 4.1
+
+Assuming the prior on $Y$ , distribution $q$ , to be uniform, we change the notation for the modified loss function $\ell _ { \widehat { \pi } , \widehat { q } }$ to $\ell _ { \widehat { \pi } }$ . Observe that for binary classification, $Z ^ { ( r ) } \bar { \in } \{ \pm 1 \} ^ { r }$ . Let $\rho$ denote the posterior b bdistribution of $Y$ b, Equation (5), when $q$ is uniform. Let $\tau$ denote the probability of observing an instance of $Z ^ { ( r ) }$ as a function of the latent true confusion matrices $\pi$ , conditioned on the ground truth label $Y = y$ .
+
+$$
+\rho _ { \pi } ( y , Z ^ { ( r ) } , w ^ { ( r ) } ) : = \mathbb { P } _ { \widehat { \pi } } [ Y = y \mid Z ^ { ( r ) } ; w ^ { ( r ) } ] , \qquad \tau _ { \pi } ( y , Z ^ { ( r ) } , w ^ { ( r ) } ) : = \mathbb { P } _ { \pi } [ Z ^ { ( r ) } \mid Y = y ; w ^ { ( r ) } ] .
+$$
+
+Let $W$ denote the uniform distribution over a pool of $m$ workers, from which nr workers are selected i.i.d. with replacement, and a batch of $r$ workers are assigned to each example $X _ { i }$ . We define the following quantities which play an important role in our analysis.
+
+$$
+\begin{array} { r c l } { \displaystyle \beta _ { \widetilde \pi } ( y ) } & { : = } & { \displaystyle \mathbb { E } _ { w \sim W } \left[ \sum _ { Z ^ { ( r ) } \in \{ \pm 1 \} ^ { r } } \rho _ { \widetilde \pi } ( - y , Z ^ { ( r ) } , w ^ { ( r ) } ) \tau _ { \pi } ( y , Z ^ { ( r ) } , w ^ { ( r ) } ) \right] . } \\ { \displaystyle \beta _ { \widetilde \pi } } & { : = } & { \displaystyle \mathbb { E } _ { w \sim W } \left[ \operatorname* { m a x } _ { y \in \{ \pm 1 \} } \left\{ \sum _ { Z ^ { ( r ) } \in \{ \pm 1 \} ^ { r } } \rho _ { \widetilde \pi } ( - y , Z ^ { ( r ) } , w ^ { ( r ) } ) \tau _ { \pi } ( y , Z ^ { ( r ) } , w ^ { ( r ) } ) \right\} \right] . } \\ { \displaystyle \alpha ( y ) } & { : = } & { \displaystyle \mathbb { E } _ { w \sim W } \left[ \mathbb { P } _ { \pi } [ Z = - y \mid Y = y ; w ] \right] . } \\ { \displaystyle \alpha } & { : = } & { \displaystyle \mathbb { E } _ { w \sim W } \left[ \operatorname* { m a x } _ { y \in \{ \pm 1 \} } \mathbb { P } _ { \pi } [ Z = - y \mid Y = y ; w ] \right] . } \end{array}
+$$
+
+For any give $\widehat { \pi }$ with $| \widehat { \pi } _ { k s } ^ { ( a ) } - \pi _ { k s } ^ { ( a ) } | \le \epsilon$ , for all $a \in [ m ] , k , s \in \mathcal { K }$ , we can compute $\beta _ { \epsilon }$ from the $\beta _ { \widehat { \pi } }$ b such that $\beta _ { \widehat { \pi } } ~ \leq ~ \beta _ { \epsilon }$ . For the special case described in Section 4.4, we have the bfollowing bound on $\beta _ { \epsilon }$ .
+
+$$
+\begin{array} { r c l } { \beta _ { \epsilon } } & { \le } & { \displaystyle \sum _ { u = 0 } ^ { r } \frac { ( \rho + \epsilon ) ^ { ( r - u ) } ( 1 - \rho - \epsilon ) ^ { u } } { ( \rho + \epsilon ) ^ { u } ( 1 - \rho - \epsilon ) ^ { ( r - u ) } + ( \rho + \epsilon ) ^ { ( r - u ) } ( 1 - \rho - \epsilon ) ^ { u } } \binom { r } { u } ( 1 - \rho ) ^ { r - u } \rho ^ { u } } \\ & { = } & { \displaystyle ( \rho + \epsilon ) ^ { r } \sum _ { u = 0 } ^ { r } \binom { r } { u } \left( \left( \frac { \rho + \epsilon } { 1 - \rho - \epsilon } \right) ^ { u } + \left( \frac { \rho + \epsilon } { 1 - \rho - \epsilon } \right) ^ { r - u } \right) ^ { - 1 } . } \end{array}
+$$
+
+It can easily be checked that $\begin{array} { r } { \beta _ { \epsilon } \leq ( \rho + \epsilon ) ^ { r } \sum _ { u = 0 } ^ { \lceil r / 2 \rceil } \binom { r } { u } ( 1 - \rho - \epsilon ) ^ { u } ( \rho + \epsilon ) ^ { - u } . } \end{array}$
+
+We present two lemma that analyze the two alternative steps of our algorithm. The following lemma gives a bound on the excess risk of function $\widehat { f }$ learnt by minimizing the modified loss function $\ell _ { \widehat { \pi } }$ .
+
+Lemma A.1. Under the assumptions of Theorem 4.1, the excess risk of function $\hat { f }$ in Equation (6), computed with posterior distribution $\mathbb { P } _ { \widehat { \pi } }$ (5) using $n$ training examples is bounded by
+
+$$
+R _ { \ell , D } ( \widehat f ) - \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f ) \ \leq \ \frac { C } { 1 - 2 \beta _ { \widehat \pi } } \left( \sqrt { \frac { V } { n } } + \sqrt { \frac { \log ( 1 / \delta _ { 1 } ) } { n } } \right) ,
+$$
+
+with probability at least $1 - \delta _ { 1 }$ , where $C$ is a universal constant. When $\mathbb { P } _ { \widehat { \pi } }$ is computed using majority vote, while initializing the iterative Algorithm $^ { l }$ b, the above bound holds with $\beta _ { \widehat { \pi } }$ replaced by $\alpha$ .
+
+The following lemma gives an $\ell _ { \infty }$ norm bound on confusion matrices $\widehat { \pi }$ estimated using model prediction ${ \widehat { f } } ( X )$ as the ground truth labels. In the analysis, we assume fresh samples are used for estimating confusion matrices in step 3, Algorithm 1. Therefore the function $\widehat { f }$ is independent of the samples $X _ { i }$ ’s on which $\widehat { \pi }$ is estimated. Let $K = | { \cal K } |$ .
+
+Lemma A.2. Under the assumptions of Theorem 4.1, $\ell _ { \infty }$ error in estimated confusion matrices $\widehat { \pi }$ as computed in Equation (7), using $n$ samples and a predictor function $\widehat { f }$ with risk $R _ { \ell , \mathcal { D } } \ \leq \ \delta$ , is bounded by
+
+$$
+\left. \widehat { \pi } _ { k s } ^ { ( a ) } - \pi _ { k s } ^ { ( a ) } \right. ~ \leq ~ \frac { 2 \delta + 1 6 \sqrt { m \log ( 4 m K ^ { 2 } \delta _ { 1 } ) / ( n r ) } } { 1 / K - \delta - 8 \sqrt { m \log ( 4 m K ^ { 2 } / \delta _ { 1 } ) / ( n r ) } } , \qquad \forall a \in [ m ] , \ k , s \in { \mathcal K } ,
+$$
+
+with probability at least $1 - \delta _ { 1 }$ .
+
+First we apply Lemma A.1 with $\mathbb { P } _ { \widehat { \pi } }$ computed using majority vote. We get a bound on the risk of function $\widehat { f }$ computed in the first round. With this $\widehat { f }$ , we apply Lemma A.2. When $n$ is sufficiently large such that Equation (8) holds, the denominator in Equation (18), $1 / K - \delta -$ $8 \sqrt { m \log ( 4 m K ^ { 2 } / \delta _ { 1 } ) / ( n r ) } \ge 1 / 8$ . Therefore, in the first round, the error in confusion matrix estimation is bounded by $\epsilon$ , which is defined in the Theorem.
+
+For the second round: we apply Lemma A.1 with $\mathbb { P } _ { \widehat { \pi } }$ computed as the posterior distribution (5). Where $\ell _ { \infty }$ error in $\widehat { \pi }$ is bounded by $\epsilon$ b. This gives the desired bound in (9). With this $\hat { f }$ , we apply bLemma A.2 and obtain $\ell _ { \infty }$ error in $\widehat { \pi }$ bounded by $\epsilon _ { 1 }$ , which is defined in the Theorem.
+
+For the given probability of error $\delta$ in the Theorem, we chose $\delta _ { 1 }$ in both the lemma to be $\delta / 4$ such that with union bound we get the desired probability of $\delta$ .
+
+# A.1 PROOF OF LEMMA A.1
+
+Let $\begin{array} { r } { f ^ { * } : = \arg \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \ell , \mathcal { D } } ( f ) } \end{array}$ . Let’s denote the distribution of $( \boldsymbol { X } , \boldsymbol { Z } ^ { ( r ) } , \boldsymbol { w } ^ { ( r ) } )$ by $\mathcal { D } _ { W , \pi , r }$ . For ease of notation, we denote $\mathcal { D } _ { W , \pi , r }$ by $\mathcal { D } _ { \pi }$ . Similar to $R _ { \ell , \mathcal { D } }$ , risk of decision function $f$ with respect to the modified loss function $\ell _ { \widehat { \pi } }$ is characterized by the following quantities:
+
+1. $\ell _ { \widehat { \pi } }$ -risk unde $\mathcal D _ { \pi } \colon R _ { \ell _ { \widehat { \pi } } , \mathcal D _ { \pi } } ( f ) : = \mathbb E _ { ( X , Z ^ { ( r ) } , w ^ { ( r ) } ) \sim \mathcal D _ { \pi } } \left[ \ell _ { \widehat { \pi } } \big ( f ( X ) , Z ^ { ( r ) } , w ^ { ( r ) } \big ) \right] .$   
+2. Empirical $\ell _ { \widehat { \pi } }$ -risk on samples: $\begin{array} { r } { \widehat { R } _ { \ell _ { \widehat { \pi } } , \mathcal { D } _ { \pi } } ( f ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell _ { \widehat { \pi } } ( f ( X _ { i } ) , Z _ { i } ^ { ( r ) } , w _ { i } ^ { ( r ) } ) . } \end{array}$
+
+With the above definitions, we have the following,
+
+$$
+\begin{array} { r l r } & { \ } & { R _ { \ell , D } ( \widehat f ) - R _ { \ell , D } ( f ^ { * } ) } \\ & { = } & { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) + ( R _ { \ell , D } ( \widehat f ) - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) ) - ( R _ { \ell , D } ( f ^ { * } ) - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) ) } \\ & { = } & { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) + 2 \beta _ { \overline { { \pi } } } ( R _ { \ell , D } ( \widehat f ) - R _ { \ell , D } ( f ^ { * } ) ) } \\ & { \leq } & { \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) } - \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } } ( f ^ { * } ) + 2 \beta _ { \overline { { \pi } } } ( R _ { \ell , D } ( \widehat f ) - R _ { \ell , D } ( f ^ { * } ) ) } \\ & { = } & { \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) } - \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) } + ( R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) - \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( \widehat f ) } ) + ( \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) } - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ^ { * } ) ) } \\ & { } & { + 2 \beta _ { \overline { { \pi } } } ( R _ { \ell , D } ( \widehat f ) - R _ { \ell , D } ( f ^ { * } ) ) } \\ & { \leq } &  2 \operatorname* { m a x } | \widehat { R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } } ( f ) - R _ { \ell , \widehat \sigma , \mathcal { P } _ { \ell } } ( f ) \ \end{array}
+$$
+
+where (19) follows from Equation (24). (20) follows from the fact that $\widehat { f }$ is the minimizer of $\widehat { R } _ { \ell _ { \widehat { \pi } } , { \mathcal { D } } _ { \pi } }$ as computed in (6). (21) follows from the basic excess-risk bound. $V$ bis the VC dimension of hypothesis class $\mathcal { F }$ , and $C$ is a universal constant.
+
+Following shows the inequality used in Equation (19). For binary classification, we denote the two classes by $Y , - Y$ .
+
+$$
+\begin{array} { r l r } { = } & { { \cal R } _ { \ell , \mathcal { D } } ( \widehat { f } ) - { \cal R } _ { \ell _ { \widehat { \pi } } , \mathcal { D } _ { \pi } } ( \widehat { f } ) - ( { \cal R } _ { \ell , \mathcal { D } } ( f ^ { * } ) - { \cal R } _ { \ell _ { \widehat { \pi } } , \mathcal { D } _ { \pi } } ( f ^ { * } ) ) } \\ { = } & { \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } [ \beta _ { \widehat { \pi } } ( Y ) ( ( \ell ( \widehat { f } ( X ) , Y ) - \ell ( f ^ { * } ( X ) , Y ) ) - ( \ell ( \widehat { f } ( X ) , - Y ) - \ell ( f ^ { * } ( X ) , - Y ) ) ) ] ) } \\ { = } & { 2 \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } [ \beta _ { \widehat { \pi } } ( Y ) ( \ell ( \widehat { f } ( X ) , Y ) - \ell ( f ^ { * } ( X ) , Y ) ) ] } & { ( 2 3 ) } \\ { \leq } & { 2 \beta _ { \widehat { \pi } } ( { \cal R } _ { \ell , \mathcal { D } } ( \widehat { f } ) - { \cal R } _ { \ell , \mathcal { D } } ( f ^ { * } ) ) , } & { ( 2 4 ) } \end{array}
+$$
+
+where (22) follows from Equation (26). (23) follows from the fact that for 0-1 loss function $\ell ( f ( X ) , Y ) + \ell ( f ( X ) , - Y ) = 1$ . (24) follows from the definition of $\beta _ { \widehat { \pi } }$ defined in Equation (12). When $\ell _ { \widehat { \pi } }$ bis computed using weighted majority vote of the workers then (24) holds with $\beta _ { \widehat { \pi } }$ replaced by $\alpha , \alpha$ bis defined in (14).
+
+Following shows the equality used in Equation (22). Using the notations $\rho _ { \widehat { \pi } }$ and $\tau _ { \pi }$ , in the following, for any function $f \in { \mathcal { F } }$ b, we compute the excess risk due to the unbiasedness of the modified loss function $\ell _ { \widehat { \pi } }$ .
+
+$$
+\begin{array} { r l } & { \displaystyle R _ { \xi , \mathcal { D } } ( f ) - R _ { \xi _ { \Psi } , \mathcal { D } _ { \pi } } ( f ) } \\ { = } & { \displaystyle \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } \left[ \ell ( f ( X ) , Y ) \right] - \mathbb { E } _ { ( X , Z ^ { ( r ) } , w ^ { ( r ) } ) \sim \mathcal { D } _ { \mathbb { P } } } [ \ell _ { \widetilde { \pi } } ( f ( X ) , Z ^ { ( r ) } , w ^ { ( r ) } ) ] } \\ { = } & { \displaystyle \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } \left[ \ell ( f ( X ) , Y ) \right] } \\ & { \displaystyle - \mathbb { E } _ { ( X , Y , w ^ { ( r ) } ) \sim \mathcal { D } _ { \pi } } \Bigg [ \sum _ { Z ^ { ( r ) } \in \{ \pm 1 \} ^ { r } } \Big ( ( 1 - \rho _ { \widetilde { \pi } } ( - Y , Z ^ { ( r ) } , w ^ { ( r ) } ) ) \ell ( f ( X ) , Y ) } \\ & { \displaystyle + \rho _ { \widetilde { \pi } } ( - Y , Z ^ { ( r ) } , w ^ { ( r ) } ) \ell ( f ( X ) , - Y ) \Big ) \tau _ { \pi } ( Y , Z ^ { ( r ) } , w ^ { ( r ) } ) \Bigg ] } \\ { = } & { \displaystyle \mathbb { E } _ { ( X , Y ) \sim \mathcal { D } } \left[ \beta _ { \widetilde { \pi } } ( Y ) \left( \ell ( f ( X ) , Y ) - \ell ( f ( X ) , - Y ) \right) \right] , } \end{array}
+$$
+
+where $\beta _ { \widehat { \pi } } ( Y )$ is defined in (11). Where (25) follows from the definition of $\ell _ { \widehat { \pi } }$ given in Equation (4). bObserve that when $\ell _ { \widehat { \pi } }$ bis computed using weighted majority vote of the workers then Equation (26) holds with $\beta _ { \widehat { \pi } } ( Y )$ b replaced by $\alpha ( y ) . \alpha ( \bar { y } )$ is defined in (13).
+
+# A.2 PROOF OF LEMMA A.2
+
+Recall that we have
+
+$$
+\begin{array} { r l r } { \widehat { \pi } _ { k s } ^ { ( a ) } } & { = } & { \frac { \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { r } \mathbb { I } [ w _ { i j } = a ] \mathbb { I } [ t _ { i } = k ] \mathbb { I } [ Z _ { i j } = s ] } { \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { r } \mathbb { I } [ w _ { i j } = a ] \mathbb { I } [ t _ { i } = k ] } } \end{array}
+$$
+
+Let $t _ { i }$ denote ${ \widehat { f } } ( X _ { i } )$ . By the definition of risk, for any $k \in \mathcal { K }$ , we have
+
+$$
+\mathbb { P } \Big [ \big | \mathbb { I } [ Y _ { i } = k ] - \mathbb { I } [ t _ { i } = k ] \big | = 1 \Big ] = \delta .
+$$
+
+Let $| { \cal K } | = { \cal K }$ . Define, for fixed $a \in [ m ]$ , and $k , s \in \mathcal { K }$ ,
+
+$$
+\begin{array} { r l } { A } & { \displaystyle : = \begin{array} { l } { \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { n } [ [ w _ { i j } = a ] [ [ t _ { i } = k ] ] [ Z _ { i j } = s ] , } \end{array} \quad \bar { A } = \frac { n r \pi _ { k i } } { m K } } \\ { B } &  \displaystyle : = \begin{array} { l } { \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { n } [ [ w _ { i j } = a ] [ [ t _ { i } = k ] ] , \quad \quad \bar { B } : = \frac { n r } { m K } } \\ { \displaystyle C } \end{array} \end{array}
+$$
+
+Note that $A , B , C , D , E$ depend upon $a \in [ m ] , k , s \in \mathcal { K }$ . However, for ease of notations, we have not included the subscripts. We have,
+
+$$
+\begin{array} { r l r } { \left| \widehat { \pi } _ { k s } ^ { ( a ) } - \pi _ { k s } ^ { ( a ) } \right| = \frac { A - B \pi _ { k s } } { B } } & { = } & { \frac { \left| \left( A - \bar { A } \right) - \left( B - \bar { B } \right) \pi _ { k s } \right| } { \left| \bar { B } + \left( B - \bar { B } \right) \right| } } \\ & { \leq } & { \frac { \left| A - \bar { A } \right| + \left| \left( B - \bar { B } \right) \right| \pi _ { k s } } { \left| \bar { B } \right| - \left| B - \bar { B } \right| } } \end{array}
+$$
+
+Now, we have,
+
+$$
+\begin{array} { l r c l } { { | A - \bar { A } | } } & { { \le } } & { { | A - D | + | D - \bar { A } | } } \\ { { } } & { { \le } } & { { C + | D - \bar { A } | . } } \end{array}
+$$
+
+We have,
+
+$$
+\begin{array} { l l l } { | B - \bar { B } | } & { \leq } & { | B - E | \ + \ | E - \bar { B } | } \\ & { \leq } & { C + \ | E - \bar { B } | } \end{array}
+$$
+
+Observe that $C$ is a sum of $n r$ i.i.d. Bernoulli random variables with mean $\delta / m$ . Using Chernoff bound we get that
+
+$$
+C ~ \leq ~ \frac { n r \delta } { m } + \sqrt { \frac { 3 n r \delta \log ( 2 m K / \delta _ { 1 } ) } { m } } ,
+$$
+
+for all $a \in [ m ]$ , and $k \in \mathcal { K }$ with probability at least $1 - \delta _ { 1 }$ . Similarly, $D$ is a sum of $n r$ i.i.d. Bernoulli random variables with mean $\pi _ { k s } / ( m k )$ . Again, using Chernoff bound we get that
+
+$$
+\begin{array} { r l r } { \left| D - \bar { A } \right| } & { \leq } & { \sqrt { \frac { 3 n r \pi _ { k s } \log ( 2 m K ^ { 2 } / \delta _ { 1 } ) } { m K } } , } \end{array}
+$$
+
+for all $a \in [ m ] , k , s \in \mathcal { K }$ with probability at least $1 - \delta _ { 1 }$ . From the bound on $| D - { \bar { A } } |$ , it follows that
+
+$$
+| E - \bar { B } | \leq \sqrt { \frac { 3 n r \log ( 2 m K ^ { 2 } / \delta _ { 1 } ) } { m } }
+$$
+
+Collecting Equations (33)-(38), we have for all $a \in [ m ] , k , s \in \mathcal { K }$
+
+$$
+\left| { \widehat \pi } _ { k s } ^ { ( a ) } - \pi _ { k s } ^ { ( a ) } \right| ~ \leq ~ { \frac { 2 \delta + 1 6 \sqrt { m \log ( 2 m K ^ { 2 } \delta _ { 1 } / ( n r ) } } { 1 / K - \delta - 8 \sqrt { m \log ( 2 m K ^ { 2 } / \delta _ { 1 } ) / ( n r ) } } } ,
+$$
+
+with probability at least $1 - 2 \delta _ { 1 }$ .

parse/train/Hkc-TeZ0W/Hkc-TeZ0W.md

@@ -0,0 +1,251 @@
+# A HIERARCHICAL MODEL FOR DEVICE PLACEMENT
+
+Azalia Mirhoseini\*, Anna Goldie∗, Hieu Pham, Benoit Steiner, Quoc V. Le and Jeff Dean {azalia,agoldie,hyhieu,bsteiner,qvl,jeff}@google.com
+
+# ABSTRACT
+
+We introduce a hierarchical model for efficient placement of computational graphs onto hardware devices, especially in heterogeneous environments with a mixture of CPUs, GPUs, and other computational devices. Our method learns to assign graph operations to groups and to allocate those groups to available devices. The grouping and device allocations are learned jointly. The proposed method is trained with policy gradient and requires no human intervention. Experiments with widely-used computer vision and natural language models show that our algorithm can find optimized, non-trivial placements for TensorFlow computational graphs with over 80,000 operations. In addition, our approach outperforms placements by human experts as well as a previous state-of-the-art placement method based on deep reinforcement learning. Our method achieves runtime reductions of up to $6 0 . 6 \%$ per training step when applied to models such as Neural Machine Translation.
+
+# 1 INTRODUCTION & RELATED WORK
+
+Deep neural networks have been successfully applied to many practical problems, such as image classification (LeCun et al., 1998; Krizhevsky et al., 2012; Taigman et al., 2014; Szegedy et al., 2015), speech recognition (Hinton et al., 2012; Hannun et al., 2014), and machine translation (Sutskever et al., 2014; Bahdanau et al., 2015; Wu et al., 2016b). These successes have lead to a surge in demand for the computational resources needed to train and infer with neural networks. A common approach to addressing this demand is to use a distributed environment with a combination of CPUs and GPUs. In this environment, it is typical for a machine learning practitioner to explicitly place the operations of their neural network onto particular computing devices for model parallelism and data parallelism. For example, one might distribute the computation of the first layer in a translation network onto the first GPU and the computation of the second layer onto the second GPU (Sutskever et al., 2014; Wu et al., 2016b). Although these decisions can be made by a human practitioner, such an approach does not scale well or produce optimal results, especially in the case of more complicated networks (Szegedy et al., 2016b;a). Given the growing diversity of hardware devices (e.g., Google TPUs, Intel Nervana, etc.) and recent trends toward automated neural architecture search (Zoph & Le, 2017; Real et al., 2017; Baker et al., 2016), where new models are generated, trained and evaluated in an entirely end-to-end fashion, it seems natural to move toward more automated solutions for efficiently distributing computation.
+
+Device placement can be framed as the problem of learning to partition a graph across available devices. Given that graph partitioning is a well-studied subject in computer science (Fiduccia & Mattheyses, 1988; Karypis & Kumar, 1995b; Pellegrini, 2009b), traditional graph partitioning methods represent a natural baseline for automated device placement. We ran experiments using Scotch (Pellegrini, 2009b), a well-established open source library for graph partitioning, which includes optimizations such as k-way Fiduccia-Mattheyses (Fiduccia & Mattheyses, 1988), Multilevel methods (Barnard & Simon, 1994; Hendrickson & Leland, 1993; Karypis & Kumar, 1995a), the Band Method (Chevalier & Pellegrini, 2006), the Diffusion Method (Pellegrini, 2007), and Dual Recursive Bipartitioning Mapping (Pellegrini & Roman, 1996). The objective was to balance the computational load across a set of connected processing nodes, while colocating neighboring nodes to minimize communication cost. Despite its promise, this approach yielded disappointing results, likely due to the non-stationarity of the cost function. We target a distributed environment where we use a shared cluster of CPUs and GPUs, and our CPUs may also serve other jobs at the same time. Thus, while cost-based models such as (Matthias Boehm & Tian, 2014) provide a strong baseline for memory optimizations, since memory usage is deterministic, they cannot be directly applied to environments with dynamic costs.
+
+Using deep networks and reinforcement learning for combinatorial optimization has already been proposed (Vinyals et al., 2015; Bello et al., 2016; Mirhoseini et al., 2017). Recent work (Mirhoseini et al., 2017) uses a recurrent neural network (RNN) policy network to predict the placement of operations in a computational graph, optimizing for speed of computation using policy gradient methods. While this approach outperforms traditional graph partitioning heuristics and human expert placements, it is prohibitively expensive for the RNN policy to learn when the number of operations is large. This method is therefore limited to small graphs (with fewer than 1000 nodes) and requires human experts to manually partition the graph into collocation groups as a pre-processing step in order to scale to larger graphs. We refer to the method in (Mirhoseini et al., 2017) as ColocRL.
+
+In this paper, we propose a more flexible approach which learns to optimize device placement for training neural networks that have tens of thousands of operations with no need for manual grouping. Our method consists of a two-level hierarchical network, in which the first model groups the operations of the graph (the Grouper) and the second model places those groups onto devices (the Placer). The Grouper is a feed forward network which reads in information about each operation and its context within the graph, in order to predict the group to which that operation should be assigned. The Placer is a sequence-to-sequence model (Sutskever et al., 2014) that reads in the embedding of the group and predicts the device placement for that group. The entire two-level network is trained jointly using reinforcement learning to optimize for speed of computation and for feasibility (e.g., having sufficient memory available on each device for the computation assigned). Unlike the previous work, our method is end-to-end and does not require human experts to manually group operations as a pre-processing step, making it a fully automated solution to optimizing device placement.
+
+Our main result is that our model effectively handles very large graphs and finds non-trivial placements on multiple devices for models such as Inception-V3 (Szegedy et al., 2016b), ResNet (He et al., 2016), Language Modeling (Jozefowicz et al., 2016), and Neural Machine Translation (Wu et al., 2016b). The placements found by our model outperform TensorFlow’s default placements (Abadi et al., 2016), the Scotch algorithm’s placements, and human expert placements, as well as those of ColocRL (Mirhoseini et al., 2017). Our results demonstrate that the proposed approach learns the properties of the environment, including the complex tradeoff between computation and communication in hardware. For example, on a Neural Machine Translation model, our method achieves a $6 0 . 6 \%$ reduction in training time per iteration.
+
+# 2 METHOD
+
+An overview of our hierarchical model for device placement is shown in Figure 1. Our model consists of two sub-networks: A Grouper that assigns operations to groups and a Placer that assigns groups to target devices. The two models are trained jointly.
+
+The objective of the proposed approach, which we refer to as the Hierarchical Planner, is to predict a placement that speeds up the training of neural network graphs. The runtime we are optimizing for is the time taken to conduct one forward pass, one back-propagation pass, and one parameter update on the target neural network. To measure the runtime, the predicted placement is run on actual hardware. Since the reward (runtime) in this problem is non-differentiable, we use policy gradients to train the Hierarchical Planner. Moreover, the policy gradients flow through to train both the feed forward Grouper and the recurrent Placer.
+
+The Grouper assigns each operation to a group. Once all the operations are grouped, we use information about each member operation to generate an embedding for that group. We then pass these embeddings as input to the Placer, which computes device placements for each group. The Placer assigns zero or more groups to each available device. The final placement is determined by placing each operation on the device that its group was assigned to.
+
+In our implementation, the Grouper is a feed forward model followed by a softmax layer with an output size equal to the number of groups. The Placer is a sequence-to-sequence model (Sutskever et al., 2014) with Long Short-Term Memory (Hochreiter & Schmidhuber, 1997) and a content-based attention mechanism (Bahdanau et al., 2015) to predict the placements.
+
+![](images/c56fddd4bd4622ffa38b903a4d0ab98f0a71a48323668391d2d32def81c9d41c.jpg)  
+Figure 1: Hierarchical model for device placement (see text for more details).
+
+We first generate operation embeddings to pass as input to the Grouper. Each operation embedding consists of 3 vectors: 1) A vector that embeds operation type information (e.g., MatMul, Conv2d, Sum, etc.). We treat this as a language modeling task and learn an operation type embedding of size 20 with a vocabulary of the 200 most commonly used TF operations. 2) A vector that contains output sizes and number of outputs for each operation. We limit the number of output edges to 6 and the size of each of these outputs to 4 elements. We populate this vector by reading the outputs of an operation one by one and inserting the output operations shapes. We pad the vector with -1 if we have fewer outgoing edges or smaller sizes. 3) A vector that contains adjacency information for that operation. We index the graph by traversing it in a BFS manner and set the maximum number of incoming and outgoing edges to 12 (6 for each direction). We then fill the vector with the index of the incoming and outgoing operations. We pad the vector with -1, in cases where the number of incoming or outgoing edges is less than 6.
+
+To generate input for the Placer, we take each group and create its group embedding by concatenating 3 vectors: 1) A vector containing the count of each operation type in the group. 2) A vector that counts the total number of output shapes of all the operations in that group. This vector is created by concatenating all the operation output shape embeddings described above (not including the -1) and is of size 16. 3) A vector that contains group adjacency information. The size of this vector is the number of groups (256 in our experiments), and its i-th value is 1 if the group has edges to the i-th group and 0 otherwise.
+
+The Placer’s RNN encoder reads the group embeddings one at a time and produces $M$ hidden states. We treat $M$ , which is equal to the number of groups, as a hyperparameter. The Placer’s decoder RNN predicts one device per time step. The devices are returned in the same order as the input group embeddings, i.e., the operations in the first group will be placed on the device returned by the first decoder step, and so on. Each device has its own trainable embedding, which is then fed as input to the next decoder time step.
+
+At each step $t$ (where $1 \leq t \leq M )$ ), the decoder uses an attention mechanism to attend over the encoder states. We use the attention mechanism from Vinyals et al. (2015). At training time, the decoder samples one device $d _ { t }$ per step from the Placer’s softmax. To make the activations $l _ { t }$ less steep and to allow the model to explore, we follow Bello et al. (2016) and use a temperature $T$ and apply a tanh constant $C$ to $l _ { t }$ . Thus, we sample $d _ { t }$ as follows:
+
+$$
+d _ { t } \sim \operatorname { s o f t m a x } ( C \operatorname { t a n h } \left( l _ { t } / T \right) )
+$$
+
+The placement decisions are then used to place the model. In the following section, we describe a policy gradient method to train the Hierarchical Planner, such that it improves its decisions over time.
+
+Training with REINFORCE: The planner optimizes the training time for a target model (e.g., a TensorFlow graph) given the decisions made by the Grouper and the Placer. Let $r _ { d }$ be the runtime
+
+per training step for a predicted device placement $d$ . We define the reward for placement $d$ as $R _ { d } = - s q r t ( r )$ . The planner should try to maximize the expectation of $R _ { d }$ given its decisions. As such, the cost function we are optimizing for is:
+
+$$
+J ( \theta _ { g } , \theta _ { d } ) = \mathbf { E _ { P ( d ; \theta _ { \mathbf { g } } , \theta _ { \mathbf { d } } ) } } [ R _ { d } ] = \sum _ { g \sim \pi _ { g } } \sum _ { d \sim \pi _ { d } } p ( g ; \theta _ { g } ) p ( d | g ; \theta _ { d } ) R _ { d }
+$$
+
+Let $\theta _ { g }$ and $\theta _ { d }$ be parameters of the Grouper and Placer, respectively. Here, $p ( g ; \theta _ { g } )$ is the probability of a sample group assignment $g$ drawn from the Grouper softmax distribution $\sim \pi _ { g }$ and $p ( d ; \theta _ { d } )$ is the probability of a sample device placement $d$ drawn from the Placer softmax distribution $\sim \pi d$ . We can write the derivative of the cost function defined in Eq. 2 w.r.t. $\theta _ { g }$ and $\theta _ { d }$ as follows:
+
+$$
+\begin{array} { l } { { \displaystyle \nabla _ { \theta g } J ( \theta _ { g } , \theta _ { d } ) = \sum _ { g \sim \pi _ { g } } \nabla _ { \theta g } p ( g ; \theta _ { g } ) \sum _ { d \sim \pi _ { d } } p ( d | g ; \theta _ { d } ) R _ { d } } } \\ { { \displaystyle \approx \frac { 1 } { m } \sum _ { g _ { i } \sim \pi _ { g } } ^ { 1 \le i \le m } \nabla _ { \theta g } \log p ( g _ { i } ; \theta _ { g } ) . \frac { 1 } { k } ( \sum _ { d _ { j } \sim \pi _ { d } } ^ { 1 \le j \le k } R _ { d _ { j } } ) } } \end{array}
+$$
+
+$$
+\begin{array} { l l } { { \displaystyle \nabla _ { \theta d } J ( \theta _ { g } , \theta _ { d } ) = \sum _ { d \sim \pi _ { d } } \sum _ { g \sim \pi _ { g } } p ( g ; \theta _ { g } ) \nabla _ { \theta d } p ( d | g ; \theta _ { d } ) R _ { d } } } \\ { { \displaystyle \approx \frac { 1 } { k } \sum _ { d _ { j } \sim \pi _ { d } } ^ { 1 \le j \le k } \frac { 1 } { m } ( \sum _ { g _ { i } \sim \pi _ { g } } ^ { 1 \le i \le m } \nabla _ { \theta d } \log p ( d _ { j } | g _ { i } ; \theta _ { d } ) R _ { d _ { j } } ) } } \end{array}
+$$
+
+Deriving Eqs. 3 and 5 from the cost function is straightforward. We use the REINFORCE rule (Williams, 1992) and approximate expectation values with samples $g _ { i }$ and $d _ { j }$ drawn from the Grouper and Placer to arrive at Eqs. 4 and 6.
+
+In our implementation, the Grouper makes independent predictions when assigning operations to groups. The Placer, however, conditions the placement of groups on those that have already been placed. To reduce the variance, we also subtract a baseline $B$ from $R$ . In our experiments, we found that the exponential moving average of the reward was an effective baseline.
+
+Distributed Training: Our policy is trained in a distributed manner. Our framework has a parameter server that is shared among several controllers. All controllers use the same set of parameters and update the policy asynchronously. Each controller communicates with $k$ worker nodes, where $k$ is as shown in Eqs. 4 and 6. Each worker interacts with only one controller.
+
+Each worker executes the placement given by its controller and reports the runtime. In our experiments, we use 4 controllers and 16 workers (4 per controller). For example, if we are optimizing a placement on 2 GPUs, each worker needs 2 GPUs to measure runtime. Each controller is hosted on a single GPU. Therefore, we use a total of 36 GPUs for this example. The workers run the placements in parallel. Once all workers have finished running the placements, the controller computes the gradients using the measured runtimes. To reduce the variance of runtime measurements across workers (different machines), each controller maintains a separate baseline. While we get our best results by using many workers, we show in Section 3 that it is possible to train the policy and achieve comparable results with far fewer resources.
+
+# 3 EXPERIMENTS
+
+In this section, we apply Hierarchical Planner to widely used machine learning models in computer vision and natural language processing. We compare our results to heuristic and RL-based graph optimization baselines and demonstrate our approach’s ability to find performant placements. We also compare our method with two simpler alternatives: 1). no grouping (a feed forward model that directly places each operation) and 2). random grouping (a random Grouper feeding into a learned Placer), to demonstrate that our hierarchical architecture allows us to learn better placements.
+
+Models: We evaluate our approach on four widely used deep neural networks:
+
+• Inception-V3 (Szegedy et al., 2016b) is a model used for a variety of computer vision tasks, including classification, recognition, or generation (Khetan & Oh, 2016; Esteva et al., 2016). The network consists of multiple blocks, each of which is made up of several convolutional and pooling layers. Within a block, the layers can be executed in parallel. However, since the outputs of each block are concatenated together to form the input to the next block, the blocks must be executed sequentially. We use a batch size of 1. The TensorFlow graph encoding this model contains 24,713 operations.
+
+• ResNet (He et al., 2016) is a popular model for image classification. It is a deep convolutional network that uses residual connections to avoid the vanishing gradient problem. We use batch size 128. The TensorFlow implementation of this model has 20,586 operations.
+
+• RNNLM (Zaremba et al., 2014; Jozefowicz et al., 2016) Recurrent Neural Network Language Model is made of many LSTM cells organized in a grid structure. The processing of each LSTM cell only depends on the results of 2 other cells, which make the concurrent execution of many LSTM cells possible given enough hardware resources. We use batch size 64. The corresponding TensorFlow graph contains 9,021 operations.
+
+• NMT (Bahdanau et al., 2015; Wu et al., 2016a) Neural Machine Translation with attention mechanism has an architecture similar to that of RNNLM, but its many hidden states make it far more computationally expensive. To decrease the training time, both Sutskever et al. (2014) and Wu et al. (2016a) propose placing each LSTM layer, as well as the attention and the softmax layer, on a separate device. While this strategy results in meaningful speed improvements, we show that our Hierarchical Planner can find significantly better placements. We use batch size 64. We evaluated 3 versions of the NMT model:
+
+– The original 2-layer encoder-decoder consisting of 28,044 operations.   
+– An extended 4-layer version consisting of 46,600 operations.   
+– An even larger 8-layer version consisting of 83,712 operations.
+
+For a fair comparison to previous state-of-the-art deep RL methods (Mirhoseini et al., 2017), we use the same model architectures (Inception-V3, RNNLM, and 2-Layer NMT models), hyperparameters and input data. In addition, we evaluate our model on a 152-layer ResNet (He et al., 2016) with ImageNet data (Deng et al., 2009), as well as more complex NMT models with 4 and 8 layers.
+
+Baselines: We compare the placements found by our approach to the following baselines:
+
+• CPU Only. Here, we execute the model on a single CPU. While this is generally slow, we find that some large models are very hard to fit on GPUs, given their memory limitations, leaving a CPU-only placement as the only naive option.   
+• GPU Only. In cases where it is possible to fit the entire model on a single GPU, this is a strong baseline as most graph operations run fastest on GPU and this placement incurs no cross-device communication cost. For operations that are not implemented on GPU, TensorFlow automatically places them on CPU.   
+Scotch. We use the Scotch static mapper (Pellegrini, 2009a) which takes as input the graph, the computational cost of each operation, the volume of data flowing through each edge, and the compute and communication capacities of the pertinent devices.   
+MinCut. We use the Scotch optimizer, but we only consider GPUs as our devices. The objective is to balance computation across all the available devices while minimizing interdevice communication.   
+Human Expert. We use hand-crafted placements from previous publications. For InceptionV3 (Szegedy et al., 2016b) and Resnet (He et al., 2016), where it is difficult to exploit model parallelism, human experts place the graph on a single GPU. For RNNLM and NMT, existing work (Sutskever et al., 2014; Wu et al., 2016a) places each LSTM layer on a separate GPU. For NMT, the attention and softmax layers are placed on the same device as the final LSTM layer, while the embedding layer is colocated with the first LSTM layer.   
+ColocRL. This method (Mirhoseini et al., 2017) uses policy gradient to train a recurrent neural network that reads in hand-crafted colocation groups and then places each group on a device.
+
+Measuring Reward: Our reward is the negative square root of the runtime for one training step of the target TensorFlow model (lower runtimes are better). We assign a large negative reward of -10 to invalid placements (e.g., due to memory limitations). We define runtime as the time in seconds required to complete one training step of the target model (i.e. one forward pass, one back-propagation pass, and one parameter update). To reduce measurement variance, we run each predicted placement of the model for 10 steps. We discard the first 5 steps (to avoid warmup variations) and use the median value of the next 5 steps to calculate the reward. We found empirically that calculating the reward with square root yielded better placements than identity or logarithm. Note that by altering the reward, we can use our proposed method for optimizing other metrics, such as inference speed, throughput, and network congestion.
+
+Devices and Software: Our experiments are run on machines with 1 Intel Haswell 2300 CPU and up to 8 Nvidia Tesla K40 GPUs. We use TensorFlow r1.3 to run our experiments.
+
+Architecture of the Policy Network: In the Hierarchical Planner, the Grouper is a feed forward network with a hidden size of 64 and the Placer is a sequence-to-sequence (Sutskever et al., 2014) model with an LSTM hidden size of 256. For the encoder of the sequence-to-sequence model, we used two layers of LSTM to form a bi-LSTM similar to (Wu et al., 2016b). We used a uni-directional LSTM for the decoder. The Grouper’s softmax output size is equal to the number of groups, which we set to 256 in our experiments. We also experimented with a range of group sizes (64 to 1024), but got the best results with group size 256. The number of unrolled steps in the Placer is equal to the number of groups. The Placer’s softmax output size in both models is equal to the number of available hardware devices.
+
+Training Details: We train both policies using Adam (Kingma & Ba, 2015) optimizer with a fixed learning rate of 0.1, gradient clipping of norm 1.0, tanh constant $C = 5 . 0$ , and temperature $T = 1 0 . 0$ The number of Grouper and Placer samples in Eqs. 4 and 6 are $m = 1$ and $k = 4$ , respectively.
+
+To encourage more exploration, we added noise to the logits of both the Grouper and the Placer networks for the first 500 policy training steps. The noise was sampled from the normal distribution and modulated to have a max amplitude of 0.1.
+
+Given that the vast majority of placements are invalid, especially for more complex models such as NMT, we update the policy only for valid placements after the first 500 steps. By updating the baseline and the policy only for samples that give valid placements, we prevent the policy from converging to the reward associated with invalid placements.
+
+<table><tr><td>Tasks</td><td>CPU Only</td><td>GPU Only</td><td>#GPUs</td><td>Human Expert</td><td>Scotch</td><td>MinCut</td><td>Hierarchical Planner</td><td>Runtime Reduction</td></tr><tr><td>Inception-V3</td><td>0.61</td><td>0.15</td><td>2</td><td>0.15</td><td>0.93</td><td>0.82</td><td>0.13</td><td>16.3%</td></tr><tr><td>ResNet</td><td>-</td><td>1.18</td><td>2</td><td>1.18</td><td>6.27</td><td>2.92</td><td>1.18</td><td>0%</td></tr><tr><td>RNNLM</td><td>6.89</td><td>1.57</td><td>2</td><td>1.57</td><td>5.62</td><td>5.21</td><td>1.57</td><td>0%</td></tr><tr><td>NMT (2-layer)</td><td>6.46</td><td>0OM</td><td>2</td><td>2.13</td><td>3.21</td><td>5.34</td><td>0.84</td><td>60.6%</td></tr><tr><td>NMT (4-layer)</td><td>10.68</td><td>0OM</td><td>4</td><td>3.64</td><td>11.18</td><td>11.63</td><td>1.69</td><td>53.7%</td></tr><tr><td>NMT (8-layer)</td><td>11.52</td><td>OOM</td><td>8</td><td>3.88</td><td>17.85</td><td>19.01</td><td>4.07</td><td>-4.9%</td></tr></table>
+
+Table 1: Model Runtimes (seconds) for different placements (lower is better). OOM: Out Of Memory.
+
+Results Compared with Graph Partitioning Heuristics: In Table 1, we report the performance of the Hierarchical Planner and compare it to the aforementioned baselines. The only information available to our models is the TensorFlow graph and a list of devices. The reduction percentages are computed by taking the difference between the runtime achieved by the Hierarchical Planner and that of the best prior placement, and then dividing it by that best prior runtime.
+
+For ResNet and RNNLM, our model learns that it is more efficient to use a single GPU, as this minimizes communication cost. For Inception-V3, the Hierarchical Planner learns to distribute the model across 2 GPUs, achieving a $1 6 . 3 \%$ reduction in runtime over placing the model on a single GPU.
+
+![](images/05c5e6e0ca836f2165ff98aa796f2ac741410c430f98974eb75d5f834387925f.jpg)  
+Figure 2: The Hierarchical Planner’s placement of a NMT (4-layer) model. White denotes CPU and the four colors each represent one of the GPUs. Note that every step of every layer is allocated across multiple GPUs. This placement is $5 3 . 7 \%$ faster than that generated by a human expert.
+
+For NMT with 2, 4, and 8 layers, we ran experiments with 2, 4, and 8 GPUs, respectively. We outperform the best prior results by $6 0 . 6 \%$ for NMT (2-layer) and $5 3 . 7 \%$ for NMT (4-layer). For further insight into the model’s behavior, we visualize its placement for NMT (4-layer) in Figure 2.
+
+For NMT (8-layer), the Hierarchical Planner finds a placement that is $4 . 9 \%$ slower than that of human experts. Even in this one case where the method slightly underperforms, it is still useful to have an automated method of finding placements that are comparable to those of human experts.
+
+Results associated with both Scotch and MinCut were significantly worse than human expert baselines, which is consistent with results reported in (Mirhoseini et al., 2017).
+
+Results Compared with ColocRL: A fair comparison with ColocRL would require us to run the models using exactly the same software (TensorFlow version) and hardware (CPU and GPU types). Although our runtimes are considerably faster, they are not directly comparable to those reported in (Mirhoseini et al., 2017), because we ran with different GPU types (the slower $\mathbf { k } 4 0$ for us vs. their $\mathbf { k } 8 0$ ) and TensorFlow versions (our r1.3 vs. their unspecified but presumably earlier version). We will discuss the relative improvements achieved by our approach.
+
+For NMT (2-layer), our improvement over best heuristics is $6 0 . 6 \%$ , compared to $1 9 . 0 \%$ for ColocRL. For NMT (4-layer) and NMT (8-layer), no results were reported for ColocRL, which we suspect is due to the model being unable to handle the large number of operations in these graphs.
+
+Unlike our method, ColocRL makes the strong assumption that certain operations must be colocated. Figure 2 shows the high granularity of the Hierarchical Planner’s placements, a degree of parallelism that would be infeasible for prior methods. For example, the Hierarchical Planner places each step of an unrolled LSTM layer across multiple GPUs, whereas ColocRL colocates all operations in a step.
+
+Analysis: Here, we want to understand and analyze the placements generated by the RL model. In Figure 2, we show a portion of the placement found by the Hierarchical Planner for NMT (4-layer). With this placement, the runtime per training iteration is $5 3 . 7 \%$ faster than that of a hand-crafted placement. As shown by the figure, the generated placement is non-trivial and highly parallelized. In particular, all of the unrolled steps of the LSTM, attention, and softmax layers are distributed across multiple GPUs. Note that it is impossible for an approach like ColocRL (Mirhoseini et al., 2017) to find such a placement, as this method forces all operations within an unrolled LSTM step to be placed on the same device. Our method also learns to place the sparse embedding lookup operations on CPU. In this placement, the policy search space is incredibly large, i.e., $5 ^ { 4 6 , 6 0 \breve { 0 } }$ (5 devices and 46,600 operations). The automated placement enabled by jointly learned grouping not only outperforms previous methods, but unlike ColocRL, it is deployable with no human effort (e.g. manual grouping).
+
+In our experiments, we set the number of groups to 256. While training the policy, we observed that initially the operations are assigned nearly uniformly across all the 256 groups, but that the Grouper ultimately converges to using only a small subset of groups across all models. This suggests that the feed forward Grouper has learned to partition the computational graph such that operations that should be placed on the same device are grouped together.
+
+We cast the device placement problem as a sequential decision-making task. Since there is no canonical order for a TensorFlow graph, we randomized the order of the groups that we fed into the the Placer, with the Placer’s bi-LSTM architecture enabling it to look at a graph more holistically. We ran 10 experiments on our NMT (4-layer) baseline, shuffling the order of group embeddings passed to the Placer. The difference between the fastest and slowest placements was less than $7 \%$ .
+
+Overhead of Training Hierarchical Planner: For each of the models, we train a new policy which learns to optimize placements for that particular model. All results for the Hierarchical Planner are after 1000 iterations of updating the policy. In practice, this takes at most three hours for our largest benchmark. The runtime per policy update is dominated by the time it takes to measure the reward for a sampled placement. To calculate the reward, we run the target model according to the predicted placement for 5 training steps and use the median runtime. The policy itself is a lightweight network that is trained on a single GPU. For measuring the reward, however, we use actual runtime measurements for the given placements. Measuring runtime is done by worker nodes. For example, if we are optimizing a model placement on 5 devices (1 CPU and 4 GPUs), we need at least one worker with that many devices that can run the input model for the predicted placements and report the runtime.
+
+As described in Section 2, we used 16 workers to train our policy. However, in this section, we will show that we can generate good results even in limited hardware settings. We consider training the policy to optimize placement of our 4-layer NMT benchmark model on 5 devices (1 Intel Haswell 2300 and 4 Nvidia Tesla K40s). The goal is to show that the policy can be trained efficiently even when we have only one worker. Figure 3 demonstrates the policy loss reduction as a function of policy training time, in 2 different scenarios where we have access to 1 and 4 workers. The policy is hosted on a single K40 GPU which sends 1 placement at a time to the worker(s) and applies a gradient step for each reward collected from worker(s). While, more workers can reduce the policy training time, as seen in Figure 3, we still get reasonable training times with only one worker. In this case, it takes less than 2.5 hours for the policy to achieve a placement with training step time of 1.94 seconds.
+
+Given that we train NMT models for hundreds of thousands of steps, the overhead of policy optimization is more than justified. For example, to train WMT’14 En->Fr dataset which has more than 36 million examples for one epoch (with batch-size ${ \boldsymbol { \mathsf { \Omega } } } { = } 6 4$ ), we need to run the NMT model for approximately 562500 steps. Since we reduce the runtime per step by approximately $4 6 . 7 \%$ (from 3.64 to 1.94 seconds), this saves us 265 GPU-hours, which is a significant savings even if we consider the 12.5 GPU-hours we spent on training the policy on a single worker.
+
+![](images/9c472e8d083086362413a9ab80ff21df602bae192aaf8cc1f5ac6235b098704f.jpg)  
+Figure 3: Training the policy with 1 and 4 workers to measure the reward. Each worker is a platform with 1 Intel Haswell 2300 and 4 Nvidia Tesla K40s.
+
+For simplicity, we first train the Hierarchical Planner and then use its best placement to train the target model. For greater efficiency, however, we could interleave the training of the Hierarchical Planner with the training of the target model, using the runtime of actual training steps as our reward signal.
+
+Alternative Policy Architectures: We compared the Hierarchical Planner against two alternative policy architectures described below.
+
+The first alternative we considered was a simpler model consisting of a single feed forward network. This model, which we refer to as the Simple Planner, independently predicts the placement for each operation in the input model, given information about that operation and its connectivity to others. This is equivalent to having only a feed forward Grouper which predicts placements rather than groups (i.e. the number of groups is equal to number of available devices).
+
+The Simple Planner finds placements for Inception-V3 that are within $20 \%$ of the Hierarchical Planner, and it successfully learns to place RNNLM and ResNet benchmarks on a single GPU. Its learned placement for NMT (2-layer), however, is more than twice as slow as that of the Hierarchical Planner. The Simple Planner also fails to find any valid placements for larger benchmarks, such as NMT with 4 or 8 layers. The Hierarchical Planner which breaks the problem into grouping and placing sub-tasks is able to scale to much larger models. The Hierarchical Planner’s sequence-to-sequence model also enables conditioning placement of an operation on those previously placed.
+
+Another architecture we considered was a Hierarchical Planner with randomized grouping. To verify that the Grouper was contributing meaningfully to the performance of the Hierarchical Planner, we compared its performance to a baseline where we fed randomized groupings into the Placer. We ran this experiment with 10 different randomized group assignments for 1000 iterations. As can be seen in Table 2, there is significant variance across different trials and the best result is worse than that of the Hierarchical Planner. This suggests that end-to-end learning of grouping operations and placing groups does indeed improve the performance.
+
+<table><tr><td>Benchmark</td><td>Best</td><td>Median</td><td>Worst</td><td>Improvement with Hierarchical Planner</td></tr><tr><td>Inception-V3</td><td>0.22</td><td>0.51</td><td>0.65</td><td>40.9%</td></tr><tr><td>ResNet</td><td>1.18</td><td>1.18</td><td>1.18</td><td>0%</td></tr><tr><td>RNNLM</td><td>1.57</td><td>1.57</td><td>1.57</td><td>0%</td></tr><tr><td>NMT (2-layer)</td><td>2.25</td><td>3.72</td><td>4.45</td><td>62.7%</td></tr><tr><td>NMT (4-layer)</td><td>3.20</td><td>3.42</td><td>6.91</td><td>47.2%</td></tr><tr><td>NMT (8-layer)</td><td>6.35</td><td>6.86</td><td>7.23</td><td>35.9%</td></tr></table>
+
+Table 2: Best, median and worst runtimes for 10 trials each with a different randomized grouping of operations. These results demonstrate that the Hierarchical Planner, which uses learned groupings rather than random ones, is able to significantly reduce runtime.
+
+# 4 CONCLUSION
+
+In this paper, we present a hierarchical method for efficiently placing the operations of a computational graph onto devices. Our approach consists of a hierarchical model that first assigns the operations to groups and then places those groups onto devices. We use a policy gradient method to optimize the parameters of the planner. The proposed method enables us to scale to computational graphs containing over 80,000 operations. Unlike previous work, our method is end-to-end and requires no manual effort. On a range of tasks including image classification, language modeling, and machine translation, our method surpasses placements designed by human experts as well as those of previous state-of-the-art deep RL methods. Our approach finds highly granular parallelism within the graph, enabling us to outperform prior methods by up to $6 0 . 6 \%$ .
+
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+        "text": "We introduce a hierarchical model for efficient placement of computational graphs onto hardware devices, especially in heterogeneous environments with a mixture of CPUs, GPUs, and other computational devices. Our method learns to assign graph operations to groups and to allocate those groups to available devices. The grouping and device allocations are learned jointly. The proposed method is trained with policy gradient and requires no human intervention. Experiments with widely-used computer vision and natural language models show that our algorithm can find optimized, non-trivial placements for TensorFlow computational graphs with over 80,000 operations. In addition, our approach outperforms placements by human experts as well as a previous state-of-the-art placement method based on deep reinforcement learning. Our method achieves runtime reductions of up to $6 0 . 6 \\%$ per training step when applied to models such as Neural Machine Translation. ",
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+        "text": "1 INTRODUCTION & RELATED WORK ",
+        "text_level": 1,
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+        "text": "Deep neural networks have been successfully applied to many practical problems, such as image classification (LeCun et al., 1998; Krizhevsky et al., 2012; Taigman et al., 2014; Szegedy et al., 2015), speech recognition (Hinton et al., 2012; Hannun et al., 2014), and machine translation (Sutskever et al., 2014; Bahdanau et al., 2015; Wu et al., 2016b). These successes have lead to a surge in demand for the computational resources needed to train and infer with neural networks. A common approach to addressing this demand is to use a distributed environment with a combination of CPUs and GPUs. In this environment, it is typical for a machine learning practitioner to explicitly place the operations of their neural network onto particular computing devices for model parallelism and data parallelism. For example, one might distribute the computation of the first layer in a translation network onto the first GPU and the computation of the second layer onto the second GPU (Sutskever et al., 2014; Wu et al., 2016b). Although these decisions can be made by a human practitioner, such an approach does not scale well or produce optimal results, especially in the case of more complicated networks (Szegedy et al., 2016b;a). Given the growing diversity of hardware devices (e.g., Google TPUs, Intel Nervana, etc.) and recent trends toward automated neural architecture search (Zoph & Le, 2017; Real et al., 2017; Baker et al., 2016), where new models are generated, trained and evaluated in an entirely end-to-end fashion, it seems natural to move toward more automated solutions for efficiently distributing computation. ",
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+        "text": "Device placement can be framed as the problem of learning to partition a graph across available devices. Given that graph partitioning is a well-studied subject in computer science (Fiduccia & Mattheyses, 1988; Karypis & Kumar, 1995b; Pellegrini, 2009b), traditional graph partitioning methods represent a natural baseline for automated device placement. We ran experiments using Scotch (Pellegrini, 2009b), a well-established open source library for graph partitioning, which includes optimizations such as k-way Fiduccia-Mattheyses (Fiduccia & Mattheyses, 1988), Multilevel methods (Barnard & Simon, 1994; Hendrickson & Leland, 1993; Karypis & Kumar, 1995a), the Band Method (Chevalier & Pellegrini, 2006), the Diffusion Method (Pellegrini, 2007), and Dual Recursive Bipartitioning Mapping (Pellegrini & Roman, 1996). The objective was to balance the computational load across a set of connected processing nodes, while colocating neighboring nodes to minimize communication cost. Despite its promise, this approach yielded disappointing results, likely due to the non-stationarity of the cost function. We target a distributed environment where we use a shared cluster of CPUs and GPUs, and our CPUs may also serve other jobs at the same time. Thus, while cost-based models such as (Matthias Boehm & Tian, 2014) provide a strong baseline for memory optimizations, since memory usage is deterministic, they cannot be directly applied to environments with dynamic costs. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Using deep networks and reinforcement learning for combinatorial optimization has already been proposed (Vinyals et al., 2015; Bello et al., 2016; Mirhoseini et al., 2017). Recent work (Mirhoseini et al., 2017) uses a recurrent neural network (RNN) policy network to predict the placement of operations in a computational graph, optimizing for speed of computation using policy gradient methods. While this approach outperforms traditional graph partitioning heuristics and human expert placements, it is prohibitively expensive for the RNN policy to learn when the number of operations is large. This method is therefore limited to small graphs (with fewer than 1000 nodes) and requires human experts to manually partition the graph into collocation groups as a pre-processing step in order to scale to larger graphs. We refer to the method in (Mirhoseini et al., 2017) as ColocRL. ",
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+        "text": "In this paper, we propose a more flexible approach which learns to optimize device placement for training neural networks that have tens of thousands of operations with no need for manual grouping. Our method consists of a two-level hierarchical network, in which the first model groups the operations of the graph (the Grouper) and the second model places those groups onto devices (the Placer). The Grouper is a feed forward network which reads in information about each operation and its context within the graph, in order to predict the group to which that operation should be assigned. The Placer is a sequence-to-sequence model (Sutskever et al., 2014) that reads in the embedding of the group and predicts the device placement for that group. The entire two-level network is trained jointly using reinforcement learning to optimize for speed of computation and for feasibility (e.g., having sufficient memory available on each device for the computation assigned). Unlike the previous work, our method is end-to-end and does not require human experts to manually group operations as a pre-processing step, making it a fully automated solution to optimizing device placement. ",
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+        "text": "Our main result is that our model effectively handles very large graphs and finds non-trivial placements on multiple devices for models such as Inception-V3 (Szegedy et al., 2016b), ResNet (He et al., 2016), Language Modeling (Jozefowicz et al., 2016), and Neural Machine Translation (Wu et al., 2016b). The placements found by our model outperform TensorFlow’s default placements (Abadi et al., 2016), the Scotch algorithm’s placements, and human expert placements, as well as those of ColocRL (Mirhoseini et al., 2017). Our results demonstrate that the proposed approach learns the properties of the environment, including the complex tradeoff between computation and communication in hardware. For example, on a Neural Machine Translation model, our method achieves a $6 0 . 6 \\%$ reduction in training time per iteration. ",
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+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "An overview of our hierarchical model for device placement is shown in Figure 1. Our model consists of two sub-networks: A Grouper that assigns operations to groups and a Placer that assigns groups to target devices. The two models are trained jointly. ",
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+        "text": "The objective of the proposed approach, which we refer to as the Hierarchical Planner, is to predict a placement that speeds up the training of neural network graphs. The runtime we are optimizing for is the time taken to conduct one forward pass, one back-propagation pass, and one parameter update on the target neural network. To measure the runtime, the predicted placement is run on actual hardware. Since the reward (runtime) in this problem is non-differentiable, we use policy gradients to train the Hierarchical Planner. Moreover, the policy gradients flow through to train both the feed forward Grouper and the recurrent Placer. ",
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+        "text": "The Grouper assigns each operation to a group. Once all the operations are grouped, we use information about each member operation to generate an embedding for that group. We then pass these embeddings as input to the Placer, which computes device placements for each group. The Placer assigns zero or more groups to each available device. The final placement is determined by placing each operation on the device that its group was assigned to. ",
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+        "text": "In our implementation, the Grouper is a feed forward model followed by a softmax layer with an output size equal to the number of groups. The Placer is a sequence-to-sequence model (Sutskever et al., 2014) with Long Short-Term Memory (Hochreiter & Schmidhuber, 1997) and a content-based attention mechanism (Bahdanau et al., 2015) to predict the placements. ",
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+    {
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+        "img_path": "images/c56fddd4bd4622ffa38b903a4d0ab98f0a71a48323668391d2d32def81c9d41c.jpg",
+        "image_caption": [
+            "Figure 1: Hierarchical model for device placement (see text for more details). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "We first generate operation embeddings to pass as input to the Grouper. Each operation embedding consists of 3 vectors: 1) A vector that embeds operation type information (e.g., MatMul, Conv2d, Sum, etc.). We treat this as a language modeling task and learn an operation type embedding of size 20 with a vocabulary of the 200 most commonly used TF operations. 2) A vector that contains output sizes and number of outputs for each operation. We limit the number of output edges to 6 and the size of each of these outputs to 4 elements. We populate this vector by reading the outputs of an operation one by one and inserting the output operations shapes. We pad the vector with -1 if we have fewer outgoing edges or smaller sizes. 3) A vector that contains adjacency information for that operation. We index the graph by traversing it in a BFS manner and set the maximum number of incoming and outgoing edges to 12 (6 for each direction). We then fill the vector with the index of the incoming and outgoing operations. We pad the vector with -1, in cases where the number of incoming or outgoing edges is less than 6. ",
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+        "type": "text",
+        "text": "To generate input for the Placer, we take each group and create its group embedding by concatenating 3 vectors: 1) A vector containing the count of each operation type in the group. 2) A vector that counts the total number of output shapes of all the operations in that group. This vector is created by concatenating all the operation output shape embeddings described above (not including the -1) and is of size 16. 3) A vector that contains group adjacency information. The size of this vector is the number of groups (256 in our experiments), and its i-th value is 1 if the group has edges to the i-th group and 0 otherwise. ",
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+        "text": "The Placer’s RNN encoder reads the group embeddings one at a time and produces $M$ hidden states. We treat $M$ , which is equal to the number of groups, as a hyperparameter. The Placer’s decoder RNN predicts one device per time step. The devices are returned in the same order as the input group embeddings, i.e., the operations in the first group will be placed on the device returned by the first decoder step, and so on. Each device has its own trainable embedding, which is then fed as input to the next decoder time step. ",
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+        "type": "text",
+        "text": "At each step $t$ (where $1 \\leq t \\leq M )$ ), the decoder uses an attention mechanism to attend over the encoder states. We use the attention mechanism from Vinyals et al. (2015). At training time, the decoder samples one device $d _ { t }$ per step from the Placer’s softmax. To make the activations $l _ { t }$ less steep and to allow the model to explore, we follow Bello et al. (2016) and use a temperature $T$ and apply a tanh constant $C$ to $l _ { t }$ . Thus, we sample $d _ { t }$ as follows: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/f7415bb4e73a530bdec96d04a55078f6fbcd4f0fff896e5002790b9d105816ea.jpg",
+        "text": "$$\nd _ { t } \\sim \\operatorname { s o f t m a x } ( C \\operatorname { t a n h } \\left( l _ { t } / T \\right) )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The placement decisions are then used to place the model. In the following section, we describe a policy gradient method to train the Hierarchical Planner, such that it improves its decisions over time. ",
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+    {
+        "type": "text",
+        "text": "Training with REINFORCE: The planner optimizes the training time for a target model (e.g., a TensorFlow graph) given the decisions made by the Grouper and the Placer. Let $r _ { d }$ be the runtime ",
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+        "type": "text",
+        "text": "per training step for a predicted device placement $d$ . We define the reward for placement $d$ as $R _ { d } = - s q r t ( r )$ . The planner should try to maximize the expectation of $R _ { d }$ given its decisions. As such, the cost function we are optimizing for is: ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/cd42f458a2168c3ed1acdd4a44ac865216db917cc10b23cab3f86283680f0daf.jpg",
+        "text": "$$\nJ ( \\theta _ { g } , \\theta _ { d } ) = \\mathbf { E _ { P ( d ; \\theta _ { \\mathbf { g } } , \\theta _ { \\mathbf { d } } ) } } [ R _ { d } ] = \\sum _ { g \\sim \\pi _ { g } } \\sum _ { d \\sim \\pi _ { d } } p ( g ; \\theta _ { g } ) p ( d | g ; \\theta _ { d } ) R _ { d }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Let $\\theta _ { g }$ and $\\theta _ { d }$ be parameters of the Grouper and Placer, respectively. Here, $p ( g ; \\theta _ { g } )$ is the probability of a sample group assignment $g$ drawn from the Grouper softmax distribution $\\sim \\pi _ { g }$ and $p ( d ; \\theta _ { d } )$ is the probability of a sample device placement $d$ drawn from the Placer softmax distribution $\\sim \\pi d$ . We can write the derivative of the cost function defined in Eq. 2 w.r.t. $\\theta _ { g }$ and $\\theta _ { d }$ as follows: ",
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+        "type": "equation",
+        "img_path": "images/43bd891bbb836352b3c46b1dae4a609b1a9b33b65c5666c66d4c6b9e8ef53f42.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\nabla _ { \\theta g } J ( \\theta _ { g } , \\theta _ { d } ) = \\sum _ { g \\sim \\pi _ { g } } \\nabla _ { \\theta g } p ( g ; \\theta _ { g } ) \\sum _ { d \\sim \\pi _ { d } } p ( d | g ; \\theta _ { d } ) R _ { d } } } \\\\ { { \\displaystyle \\approx \\frac { 1 } { m } \\sum _ { g _ { i } \\sim \\pi _ { g } } ^ { 1 \\le i \\le m } \\nabla _ { \\theta g } \\log p ( g _ { i } ; \\theta _ { g } ) . \\frac { 1 } { k } ( \\sum _ { d _ { j } \\sim \\pi _ { d } } ^ { 1 \\le j \\le k } R _ { d _ { j } } ) } } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b2a9142fe1a771f0dfb2c047d5a743165e5ed2ab999d4e7c8e685466de2b4b1c.jpg",
+        "text": "$$\n\\begin{array} { l l } { { \\displaystyle \\nabla _ { \\theta d } J ( \\theta _ { g } , \\theta _ { d } ) = \\sum _ { d \\sim \\pi _ { d } } \\sum _ { g \\sim \\pi _ { g } } p ( g ; \\theta _ { g } ) \\nabla _ { \\theta d } p ( d | g ; \\theta _ { d } ) R _ { d } } } \\\\ { { \\displaystyle \\approx \\frac { 1 } { k } \\sum _ { d _ { j } \\sim \\pi _ { d } } ^ { 1 \\le j \\le k } \\frac { 1 } { m } ( \\sum _ { g _ { i } \\sim \\pi _ { g } } ^ { 1 \\le i \\le m } \\nabla _ { \\theta d } \\log p ( d _ { j } | g _ { i } ; \\theta _ { d } ) R _ { d _ { j } } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+        "type": "text",
+        "text": "Deriving Eqs. 3 and 5 from the cost function is straightforward. We use the REINFORCE rule (Williams, 1992) and approximate expectation values with samples $g _ { i }$ and $d _ { j }$ drawn from the Grouper and Placer to arrive at Eqs. 4 and 6. ",
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+    {
+        "type": "text",
+        "text": "In our implementation, the Grouper makes independent predictions when assigning operations to groups. The Placer, however, conditions the placement of groups on those that have already been placed. To reduce the variance, we also subtract a baseline $B$ from $R$ . In our experiments, we found that the exponential moving average of the reward was an effective baseline. ",
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+    {
+        "type": "text",
+        "text": "Distributed Training: Our policy is trained in a distributed manner. Our framework has a parameter server that is shared among several controllers. All controllers use the same set of parameters and update the policy asynchronously. Each controller communicates with $k$ worker nodes, where $k$ is as shown in Eqs. 4 and 6. Each worker interacts with only one controller. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Each worker executes the placement given by its controller and reports the runtime. In our experiments, we use 4 controllers and 16 workers (4 per controller). For example, if we are optimizing a placement on 2 GPUs, each worker needs 2 GPUs to measure runtime. Each controller is hosted on a single GPU. Therefore, we use a total of 36 GPUs for this example. The workers run the placements in parallel. Once all workers have finished running the placements, the controller computes the gradients using the measured runtimes. To reduce the variance of runtime measurements across workers (different machines), each controller maintains a separate baseline. While we get our best results by using many workers, we show in Section 3 that it is possible to train the policy and achieve comparable results with far fewer resources. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we apply Hierarchical Planner to widely used machine learning models in computer vision and natural language processing. We compare our results to heuristic and RL-based graph optimization baselines and demonstrate our approach’s ability to find performant placements. We also compare our method with two simpler alternatives: 1). no grouping (a feed forward model that directly places each operation) and 2). random grouping (a random Grouper feeding into a learned Placer), to demonstrate that our hierarchical architecture allows us to learn better placements. ",
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+    {
+        "type": "text",
+        "text": "Models: We evaluate our approach on four widely used deep neural networks: ",
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+    {
+        "type": "text",
+        "text": "• Inception-V3 (Szegedy et al., 2016b) is a model used for a variety of computer vision tasks, including classification, recognition, or generation (Khetan & Oh, 2016; Esteva et al., 2016). The network consists of multiple blocks, each of which is made up of several convolutional and pooling layers. Within a block, the layers can be executed in parallel. However, since the outputs of each block are concatenated together to form the input to the next block, the blocks must be executed sequentially. We use a batch size of 1. The TensorFlow graph encoding this model contains 24,713 operations. ",
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+    {
+        "type": "text",
+        "text": "• ResNet (He et al., 2016) is a popular model for image classification. It is a deep convolutional network that uses residual connections to avoid the vanishing gradient problem. We use batch size 128. The TensorFlow implementation of this model has 20,586 operations. ",
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+    {
+        "type": "text",
+        "text": "• RNNLM (Zaremba et al., 2014; Jozefowicz et al., 2016) Recurrent Neural Network Language Model is made of many LSTM cells organized in a grid structure. The processing of each LSTM cell only depends on the results of 2 other cells, which make the concurrent execution of many LSTM cells possible given enough hardware resources. We use batch size 64. The corresponding TensorFlow graph contains 9,021 operations. ",
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+    {
+        "type": "text",
+        "text": "• NMT (Bahdanau et al., 2015; Wu et al., 2016a) Neural Machine Translation with attention mechanism has an architecture similar to that of RNNLM, but its many hidden states make it far more computationally expensive. To decrease the training time, both Sutskever et al. (2014) and Wu et al. (2016a) propose placing each LSTM layer, as well as the attention and the softmax layer, on a separate device. While this strategy results in meaningful speed improvements, we show that our Hierarchical Planner can find significantly better placements. We use batch size 64. We evaluated 3 versions of the NMT model: ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "– The original 2-layer encoder-decoder consisting of 28,044 operations.   \n– An extended 4-layer version consisting of 46,600 operations.   \n– An even larger 8-layer version consisting of 83,712 operations. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "For a fair comparison to previous state-of-the-art deep RL methods (Mirhoseini et al., 2017), we use the same model architectures (Inception-V3, RNNLM, and 2-Layer NMT models), hyperparameters and input data. In addition, we evaluate our model on a 152-layer ResNet (He et al., 2016) with ImageNet data (Deng et al., 2009), as well as more complex NMT models with 4 and 8 layers. ",
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+    {
+        "type": "text",
+        "text": "Baselines: We compare the placements found by our approach to the following baselines: ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "• CPU Only. Here, we execute the model on a single CPU. While this is generally slow, we find that some large models are very hard to fit on GPUs, given their memory limitations, leaving a CPU-only placement as the only naive option.   \n• GPU Only. In cases where it is possible to fit the entire model on a single GPU, this is a strong baseline as most graph operations run fastest on GPU and this placement incurs no cross-device communication cost. For operations that are not implemented on GPU, TensorFlow automatically places them on CPU.   \nScotch. We use the Scotch static mapper (Pellegrini, 2009a) which takes as input the graph, the computational cost of each operation, the volume of data flowing through each edge, and the compute and communication capacities of the pertinent devices.   \nMinCut. We use the Scotch optimizer, but we only consider GPUs as our devices. The objective is to balance computation across all the available devices while minimizing interdevice communication.   \nHuman Expert. We use hand-crafted placements from previous publications. For InceptionV3 (Szegedy et al., 2016b) and Resnet (He et al., 2016), where it is difficult to exploit model parallelism, human experts place the graph on a single GPU. For RNNLM and NMT, existing work (Sutskever et al., 2014; Wu et al., 2016a) places each LSTM layer on a separate GPU. For NMT, the attention and softmax layers are placed on the same device as the final LSTM layer, while the embedding layer is colocated with the first LSTM layer.   \nColocRL. This method (Mirhoseini et al., 2017) uses policy gradient to train a recurrent neural network that reads in hand-crafted colocation groups and then places each group on a device. ",
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+    {
+        "type": "text",
+        "text": "Measuring Reward: Our reward is the negative square root of the runtime for one training step of the target TensorFlow model (lower runtimes are better). We assign a large negative reward of -10 to invalid placements (e.g., due to memory limitations). We define runtime as the time in seconds required to complete one training step of the target model (i.e. one forward pass, one back-propagation pass, and one parameter update). To reduce measurement variance, we run each predicted placement of the model for 10 steps. We discard the first 5 steps (to avoid warmup variations) and use the median value of the next 5 steps to calculate the reward. We found empirically that calculating the reward with square root yielded better placements than identity or logarithm. Note that by altering the reward, we can use our proposed method for optimizing other metrics, such as inference speed, throughput, and network congestion. ",
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+        "type": "text",
+        "text": "Devices and Software: Our experiments are run on machines with 1 Intel Haswell 2300 CPU and up to 8 Nvidia Tesla K40 GPUs. We use TensorFlow r1.3 to run our experiments. ",
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+        "text": "Architecture of the Policy Network: In the Hierarchical Planner, the Grouper is a feed forward network with a hidden size of 64 and the Placer is a sequence-to-sequence (Sutskever et al., 2014) model with an LSTM hidden size of 256. For the encoder of the sequence-to-sequence model, we used two layers of LSTM to form a bi-LSTM similar to (Wu et al., 2016b). We used a uni-directional LSTM for the decoder. The Grouper’s softmax output size is equal to the number of groups, which we set to 256 in our experiments. We also experimented with a range of group sizes (64 to 1024), but got the best results with group size 256. The number of unrolled steps in the Placer is equal to the number of groups. The Placer’s softmax output size in both models is equal to the number of available hardware devices. ",
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+    {
+        "type": "text",
+        "text": "Training Details: We train both policies using Adam (Kingma & Ba, 2015) optimizer with a fixed learning rate of 0.1, gradient clipping of norm 1.0, tanh constant $C = 5 . 0$ , and temperature $T = 1 0 . 0$ The number of Grouper and Placer samples in Eqs. 4 and 6 are $m = 1$ and $k = 4$ , respectively. ",
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+    },
+    {
+        "type": "text",
+        "text": "To encourage more exploration, we added noise to the logits of both the Grouper and the Placer networks for the first 500 policy training steps. The noise was sampled from the normal distribution and modulated to have a max amplitude of 0.1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given that the vast majority of placements are invalid, especially for more complex models such as NMT, we update the policy only for valid placements after the first 500 steps. By updating the baseline and the policy only for samples that give valid placements, we prevent the policy from converging to the reward associated with invalid placements. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/18622e6ffe8b936f0f07e1994b524bd28f99a5904a08e4326eb7d9d2098314df.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Tasks</td><td>CPU Only</td><td>GPU Only</td><td>#GPUs</td><td>Human Expert</td><td>Scotch</td><td>MinCut</td><td>Hierarchical Planner</td><td>Runtime Reduction</td></tr><tr><td>Inception-V3</td><td>0.61</td><td>0.15</td><td>2</td><td>0.15</td><td>0.93</td><td>0.82</td><td>0.13</td><td>16.3%</td></tr><tr><td>ResNet</td><td>-</td><td>1.18</td><td>2</td><td>1.18</td><td>6.27</td><td>2.92</td><td>1.18</td><td>0%</td></tr><tr><td>RNNLM</td><td>6.89</td><td>1.57</td><td>2</td><td>1.57</td><td>5.62</td><td>5.21</td><td>1.57</td><td>0%</td></tr><tr><td>NMT (2-layer)</td><td>6.46</td><td>0OM</td><td>2</td><td>2.13</td><td>3.21</td><td>5.34</td><td>0.84</td><td>60.6%</td></tr><tr><td>NMT (4-layer)</td><td>10.68</td><td>0OM</td><td>4</td><td>3.64</td><td>11.18</td><td>11.63</td><td>1.69</td><td>53.7%</td></tr><tr><td>NMT (8-layer)</td><td>11.52</td><td>OOM</td><td>8</td><td>3.88</td><td>17.85</td><td>19.01</td><td>4.07</td><td>-4.9%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Model Runtimes (seconds) for different placements (lower is better). OOM: Out Of Memory. ",
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+        "type": "text",
+        "text": "Results Compared with Graph Partitioning Heuristics: In Table 1, we report the performance of the Hierarchical Planner and compare it to the aforementioned baselines. The only information available to our models is the TensorFlow graph and a list of devices. The reduction percentages are computed by taking the difference between the runtime achieved by the Hierarchical Planner and that of the best prior placement, and then dividing it by that best prior runtime. ",
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+    {
+        "type": "text",
+        "text": "For ResNet and RNNLM, our model learns that it is more efficient to use a single GPU, as this minimizes communication cost. For Inception-V3, the Hierarchical Planner learns to distribute the model across 2 GPUs, achieving a $1 6 . 3 \\%$ reduction in runtime over placing the model on a single GPU. ",
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+    {
+        "type": "image",
+        "img_path": "images/05c5e6e0ca836f2165ff98aa796f2ac741410c430f98974eb75d5f834387925f.jpg",
+        "image_caption": [
+            "Figure 2: The Hierarchical Planner’s placement of a NMT (4-layer) model. White denotes CPU and the four colors each represent one of the GPUs. Note that every step of every layer is allocated across multiple GPUs. This placement is $5 3 . 7 \\%$ faster than that generated by a human expert. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "For NMT with 2, 4, and 8 layers, we ran experiments with 2, 4, and 8 GPUs, respectively. We outperform the best prior results by $6 0 . 6 \\%$ for NMT (2-layer) and $5 3 . 7 \\%$ for NMT (4-layer). For further insight into the model’s behavior, we visualize its placement for NMT (4-layer) in Figure 2. ",
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+    {
+        "type": "text",
+        "text": "For NMT (8-layer), the Hierarchical Planner finds a placement that is $4 . 9 \\%$ slower than that of human experts. Even in this one case where the method slightly underperforms, it is still useful to have an automated method of finding placements that are comparable to those of human experts. ",
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+    {
+        "type": "text",
+        "text": "Results associated with both Scotch and MinCut were significantly worse than human expert baselines, which is consistent with results reported in (Mirhoseini et al., 2017). ",
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+        "text": "Results Compared with ColocRL: A fair comparison with ColocRL would require us to run the models using exactly the same software (TensorFlow version) and hardware (CPU and GPU types). Although our runtimes are considerably faster, they are not directly comparable to those reported in (Mirhoseini et al., 2017), because we ran with different GPU types (the slower $\\mathbf { k } 4 0$ for us vs. their $\\mathbf { k } 8 0$ ) and TensorFlow versions (our r1.3 vs. their unspecified but presumably earlier version). We will discuss the relative improvements achieved by our approach. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For NMT (2-layer), our improvement over best heuristics is $6 0 . 6 \\%$ , compared to $1 9 . 0 \\%$ for ColocRL. For NMT (4-layer) and NMT (8-layer), no results were reported for ColocRL, which we suspect is due to the model being unable to handle the large number of operations in these graphs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unlike our method, ColocRL makes the strong assumption that certain operations must be colocated. Figure 2 shows the high granularity of the Hierarchical Planner’s placements, a degree of parallelism that would be infeasible for prior methods. For example, the Hierarchical Planner places each step of an unrolled LSTM layer across multiple GPUs, whereas ColocRL colocates all operations in a step. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis: Here, we want to understand and analyze the placements generated by the RL model. In Figure 2, we show a portion of the placement found by the Hierarchical Planner for NMT (4-layer). With this placement, the runtime per training iteration is $5 3 . 7 \\%$ faster than that of a hand-crafted placement. As shown by the figure, the generated placement is non-trivial and highly parallelized. In particular, all of the unrolled steps of the LSTM, attention, and softmax layers are distributed across multiple GPUs. Note that it is impossible for an approach like ColocRL (Mirhoseini et al., 2017) to find such a placement, as this method forces all operations within an unrolled LSTM step to be placed on the same device. Our method also learns to place the sparse embedding lookup operations on CPU. In this placement, the policy search space is incredibly large, i.e., $5 ^ { 4 6 , 6 0 \\breve { 0 } }$ (5 devices and 46,600 operations). The automated placement enabled by jointly learned grouping not only outperforms previous methods, but unlike ColocRL, it is deployable with no human effort (e.g. manual grouping). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our experiments, we set the number of groups to 256. While training the policy, we observed that initially the operations are assigned nearly uniformly across all the 256 groups, but that the Grouper ultimately converges to using only a small subset of groups across all models. This suggests that the feed forward Grouper has learned to partition the computational graph such that operations that should be placed on the same device are grouped together. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We cast the device placement problem as a sequential decision-making task. Since there is no canonical order for a TensorFlow graph, we randomized the order of the groups that we fed into the the Placer, with the Placer’s bi-LSTM architecture enabling it to look at a graph more holistically. We ran 10 experiments on our NMT (4-layer) baseline, shuffling the order of group embeddings passed to the Placer. The difference between the fastest and slowest placements was less than $7 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Overhead of Training Hierarchical Planner: For each of the models, we train a new policy which learns to optimize placements for that particular model. All results for the Hierarchical Planner are after 1000 iterations of updating the policy. In practice, this takes at most three hours for our largest benchmark. The runtime per policy update is dominated by the time it takes to measure the reward for a sampled placement. To calculate the reward, we run the target model according to the predicted placement for 5 training steps and use the median runtime. The policy itself is a lightweight network that is trained on a single GPU. For measuring the reward, however, we use actual runtime measurements for the given placements. Measuring runtime is done by worker nodes. For example, if we are optimizing a model placement on 5 devices (1 CPU and 4 GPUs), we need at least one worker with that many devices that can run the input model for the predicted placements and report the runtime. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As described in Section 2, we used 16 workers to train our policy. However, in this section, we will show that we can generate good results even in limited hardware settings. We consider training the policy to optimize placement of our 4-layer NMT benchmark model on 5 devices (1 Intel Haswell 2300 and 4 Nvidia Tesla K40s). The goal is to show that the policy can be trained efficiently even when we have only one worker. Figure 3 demonstrates the policy loss reduction as a function of policy training time, in 2 different scenarios where we have access to 1 and 4 workers. The policy is hosted on a single K40 GPU which sends 1 placement at a time to the worker(s) and applies a gradient step for each reward collected from worker(s). While, more workers can reduce the policy training time, as seen in Figure 3, we still get reasonable training times with only one worker. In this case, it takes less than 2.5 hours for the policy to achieve a placement with training step time of 1.94 seconds. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given that we train NMT models for hundreds of thousands of steps, the overhead of policy optimization is more than justified. For example, to train WMT’14 En->Fr dataset which has more than 36 million examples for one epoch (with batch-size ${ \\boldsymbol { \\mathsf { \\Omega } } } { = } 6 4$ ), we need to run the NMT model for approximately 562500 steps. Since we reduce the runtime per step by approximately $4 6 . 7 \\%$ (from 3.64 to 1.94 seconds), this saves us 265 GPU-hours, which is a significant savings even if we consider the 12.5 GPU-hours we spent on training the policy on a single worker. ",
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+    {
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+        "img_path": "images/9c472e8d083086362413a9ab80ff21df602bae192aaf8cc1f5ac6235b098704f.jpg",
+        "image_caption": [
+            "Figure 3: Training the policy with 1 and 4 workers to measure the reward. Each worker is a platform with 1 Intel Haswell 2300 and 4 Nvidia Tesla K40s. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For simplicity, we first train the Hierarchical Planner and then use its best placement to train the target model. For greater efficiency, however, we could interleave the training of the Hierarchical Planner with the training of the target model, using the runtime of actual training steps as our reward signal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Alternative Policy Architectures: We compared the Hierarchical Planner against two alternative policy architectures described below. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The first alternative we considered was a simpler model consisting of a single feed forward network. This model, which we refer to as the Simple Planner, independently predicts the placement for each operation in the input model, given information about that operation and its connectivity to others. This is equivalent to having only a feed forward Grouper which predicts placements rather than groups (i.e. the number of groups is equal to number of available devices). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Simple Planner finds placements for Inception-V3 that are within $20 \\%$ of the Hierarchical Planner, and it successfully learns to place RNNLM and ResNet benchmarks on a single GPU. Its learned placement for NMT (2-layer), however, is more than twice as slow as that of the Hierarchical Planner. The Simple Planner also fails to find any valid placements for larger benchmarks, such as NMT with 4 or 8 layers. The Hierarchical Planner which breaks the problem into grouping and placing sub-tasks is able to scale to much larger models. The Hierarchical Planner’s sequence-to-sequence model also enables conditioning placement of an operation on those previously placed. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Another architecture we considered was a Hierarchical Planner with randomized grouping. To verify that the Grouper was contributing meaningfully to the performance of the Hierarchical Planner, we compared its performance to a baseline where we fed randomized groupings into the Placer. We ran this experiment with 10 different randomized group assignments for 1000 iterations. As can be seen in Table 2, there is significant variance across different trials and the best result is worse than that of the Hierarchical Planner. This suggests that end-to-end learning of grouping operations and placing groups does indeed improve the performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f0f6f8d9cf62d6161c32aea194454690ca8afe636435c3763cc166fcd6a054bd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Benchmark</td><td>Best</td><td>Median</td><td>Worst</td><td>Improvement with Hierarchical Planner</td></tr><tr><td>Inception-V3</td><td>0.22</td><td>0.51</td><td>0.65</td><td>40.9%</td></tr><tr><td>ResNet</td><td>1.18</td><td>1.18</td><td>1.18</td><td>0%</td></tr><tr><td>RNNLM</td><td>1.57</td><td>1.57</td><td>1.57</td><td>0%</td></tr><tr><td>NMT (2-layer)</td><td>2.25</td><td>3.72</td><td>4.45</td><td>62.7%</td></tr><tr><td>NMT (4-layer)</td><td>3.20</td><td>3.42</td><td>6.91</td><td>47.2%</td></tr><tr><td>NMT (8-layer)</td><td>6.35</td><td>6.86</td><td>7.23</td><td>35.9%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Best, median and worst runtimes for 10 trials each with a different randomized grouping of operations. These results demonstrate that the Hierarchical Planner, which uses learned groupings rather than random ones, is able to significantly reduce runtime. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a hierarchical method for efficiently placing the operations of a computational graph onto devices. Our approach consists of a hierarchical model that first assigns the operations to groups and then places those groups onto devices. We use a policy gradient method to optimize the parameters of the planner. The proposed method enables us to scale to computational graphs containing over 80,000 operations. Unlike previous work, our method is end-to-end and requires no manual effort. On a range of tasks including image classification, language modeling, and machine translation, our method surpasses placements designed by human experts as well as those of previous state-of-the-art deep RL methods. Our approach finds highly granular parallelism within the graph, enabling us to outperform prior methods by up to $6 0 . 6 \\%$ . ",
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+    {
+        "type": "text",
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parse/train/Hke3gyHYwH/Hke3gyHYwH.md

@@ -0,0 +1,607 @@
+# SIMPLE AND EFFECTIVE REGULARIZATION METHODS FOR TRAINING ON NOISILY LABELED DATA WITH GENERALIZATION GUARANTEE
+
+Wei Hu Zhiyuan Li Dingli Yu   
+Princeton University   
+{huwei,zhiyuanli,dingliy}@cs.princeton.edu
+
+# ABSTRACT
+
+Over-parameterized deep neural networks trained by simple first-order methods are known to be able to fit any labeling of data. Such over-fitting ability hinders generalization when mislabeled training examples are present. On the other hand, simple regularization methods like early-stopping can often achieve highly nontrivial performance on clean test data in these scenarios, a phenomenon not theoretically understood. This paper proposes and analyzes two simple and intuitive regularization methods: (i) regularization by the distance between the network parameters to initialization, and (ii) adding a trainable auxiliary variable to the network output for each training example. Theoretically, we prove that gradient descent training with either of these two methods leads to a generalization guarantee on the clean data distribution despite being trained using noisy labels. Our generalization analysis relies on the connection between wide neural network and neural tangent kernel (NTK). The generalization bound is independent of the network size, and is comparable to the bound one can get when there is no label noise. Experimental results verify the effectiveness of these methods on noisily labeled datasets.
+
+# 1 INTRODUCTION
+
+Modern deep neural networks are trained in a highly over-parameterized regime, with many more trainable parameters than training examples. It is well-known that these networks trained with simple first-order methods can fit any labels, even completely random ones (Zhang et al., 2017). Although training on properly labeled data usually leads to good generalization performance, the ability to over-fit the entire training dataset is undesirable for generalization when noisy labels are present. Therefore preventing over-fitting is crucial for robust performance since mislabeled data are ubiquitous in very large datasets (Krishna et al., 2016).
+
+In order to prevent over-fitting to mislabeled data, some form of regularization is necessary. A simple such example is early stopping, which has been observed to be surprisingly effective for this purpose (Rolnick et al., 2017; Guan et al., 2018; Li et al., 2019). For instance, training ResNet-34 with early stopping can achieve $8 4 \%$ test accuracy on CIFAR-10 even when $6 0 \%$ of the training labels are corrupted (Table 1). This is nontrivial since the test error is much smaller than the error rate in training data. How to explain such generalization phenomenon is an intriguing theoretical question.
+
+As a step towards a theoretical understanding of the generalization phenomenon for overparameterized neural networks when noisy labels are present, this paper proposes and analyzes two simple regularization methods as alternatives of early stopping:
+
+1. Regularization by distance to initialization. Denote by $\pmb \theta$ the network parameters and by $\pmb \theta ( 0 )$ its random initialization. This method adds a regularizer $\lambda \| \pmb \theta - \pmb \theta ( 0 ) \| ^ { 2 }$ to the training objective.   
+2. Adding an auxiliary variable for each training example. Let $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ be the $i$ -th training example and $f ( \pmb \theta , \cdot )$ represent the neural net. This method adds a trainable variable $b _ { i }$ and tries to fit the $i$ -th label using $f ( \pmb { \theta } , \pmb { x } _ { i } ) + \lambda \pmb { b } _ { i }$ . At test time, only the neural net $f ( \theta , \cdot )$ is used and the auxiliary variables are discarded.
+
+These two choices of regularization are well motivated with clear intuitions. First, distance to initialization has been observed to be very related to generalization in deep learning (Neyshabur et al., 2019; Nagarajan and Kolter, 2019), so regularizing by distance to initialization can potentially help generalization. Second, the effectiveness of early stopping indicates that clean labels are somewhat easier to fit than wrong labels; therefore, adding an auxiliary variable could help “absorb” the noise in the labels, thus making the neural net itself not over-fitting.
+
+We provide theoretical analysis of the above two regularization methods for a class of sufficiently wide neural networks by proving a generalization bound for the trained network on clean data distribution when the training dataset contains noisy labels. Our generalization bound depends on the (unobserved) clean labels, and is comparable to the bound one can get when there is no label noise, therefore indicating that the proposed regularization methods are robust to noisy labels.
+
+Our theoretical analysis is based on the recently established connection between wide neural net and neural tangent kernel (Jacot et al., 2018; Lee et al., 2019; Arora et al., 2019a). In this line of work, parameters in a wide neural net are shown to stay close to their initialization during gradient descent training, and as a consequence, the neural net can be effectively approximated by its first-order Taylor expansion with respect to its parameters at initialization. This leads to tractable linear dynamics under $\ell _ { 2 }$ loss, and the final solution can be characterized by kernel regression using a particular kernel named neural tangent kernel (NTK). In fact, we show that for wide neural nets, both of our regularization methods, when trained with gradient descent to convergence, correspond to kernel ridge regression using the NTK, which is often regarded as an alternative to early stopping in kernel literature. This viewpoint makes explicit the connection between our methods and early stopping.
+
+The effectiveness of these two regularization methods is verified empirically – on MNIST and CIFAR-10, they are able to achieve highly nontrivial test accuracy, on a par with or even better than early stopping. Furthermore, with our regularization, the validation accuracy is almost monotone increasing throughout the entire training process, indicating their resistance to over-fitting.
+
+# 2 RELATED WORK
+
+Neural tangent kernel was first explicitly studied and named by Jacot et al. (2018), with several further refinements and extensions by Lee et al. (2019); Yang (2019); Arora et al. (2019a). Using the similar idea that weights stay close to initialization and that the neural network is approximated by a linear model, a series of theoretical papers studied the optimization and generalization issues of very wide deep neural nets trained by (stochastic) gradient descent (Du et al., 2019b; 2018b; Li and Liang, 2018; Allen-Zhu et al., 2018a;b; Zou et al., 2018; Arora et al., 2019b; Cao and Gu, 2019). Empirically, variants of NTK on convolutional neural nets and graph neural nets exhibit strong practical performance (Arora et al., 2019a; Du et al., 2019a), thus suggesting that ultra-wide (or infinitely wide) neural nets are at least not irrelevant.
+
+Our methods are closely related to kernel ridge regression, which is one of the most common kernel methods and has been widely studied. It was shown to perform comparably to early-stopped gradient descent (Bauer et al., 2007; Gerfo et al., 2008; Raskutti et al., 2014; Wei et al., 2017). Accordingly, we indeed observe in our experiments that our regularization methods perform similarly to gradient descent with early stopping in neural net training.
+
+In another theoretical study relevant to ours, Li et al. (2019) proved that gradient descent with early stopping is robust to label noise for an over-parameterized two-layer neural net. Under a clustering assumption on data, they showed that gradient descent fits the correct labels before starting to over-fit wrong labels. Their result is different from ours from several aspects: they only considered two-layer nets while we allow arbitrarily deep nets; they required a clustering assumption on data while our generalization bound is general and data-dependent; furthermore, they did not address the question of generalization, but only provided guarantees on the training data.
+
+A large body of work proposed various methods for training with mislabeled examples, such as estimating noise distribution (Liu and Tao, 2015) or confusion matrix (Sukhbaatar et al., 2014), using surrogate loss functions (Ghosh et al., 2017; Zhang and Sabuncu, 2018), meta-learning (Ren et al., 2018), using a pre-trained network (Jiang et al., 2017), and training two networks simultaneously (Malach and Shalev-Shwartz, 2017; Han et al., 2018; Yu et al., 2019). While our methods are not necessarily superior to these methods in terms of performance, our methods are arguably simpler (with minimal change to normal training procedure) and come with formal generalization guarantee.
+
+# 3 PRELIMINARIES
+
+Notation. We use bold-faced letters for vectors and matrices. We use $\left. \cdot \right.$ to denote the Euclidean norm of a vector or the spectral norm of a matrix, and $\left\| \cdot \right\| _ { F }$ to denote the Frobenius norm of a matrix. $\langle \cdot , \cdot \rangle$ represents the standard inner product. Let $\pmb { I }$ be the identity matrix of appropriate dimension. Let $[ n ] ^ { - } = \{ 1 , 2 , \dots , n \}$ . Let $\mathbb { I } [ A ]$ be the indicator of event $A$ .
+
+# 3.1 SETTING: LEARNING FROM NOISILY LABELED DATA
+
+Now we formally describe the setting considered in this paper. We first describe the binary classification setting as a warm-up, and then describe the more general setting of multi-class classification.
+
+Binary classification. Suppose that there is an underlying data distribution $\mathcal { D }$ over $\mathbb { R } ^ { d } \times \{ \pm 1 \}$ , where 1 and $- 1$ are labels corresponding to two classes. However, we only have access to samples from a noisily labeled version of $\mathcal { D }$ . Formally, the data generation process is: draw $( \boldsymbol { x } , \boldsymbol { y } ) \sim \mathcal { D }$ , and flip the sign of label $y$ with probability $p$ $( 0 \leq p < \frac { 1 } { 2 } )$ ; let $\tilde { y } \in \{ \pm 1 \bar { \} }$ be the resulting noisy label.
+
+Let $\{ ( \pmb { x } _ { i } , \tilde { y } _ { i } ) \} _ { i = 1 } ^ { n }$ be i.i.d. samples generated from the above process. Although we only have access to these noisily labeled data, the goal is still to learn a function (in the form of a neural net) that can predict the true label well on the clean distribution $\mathcal { D }$ . For binary classification, it suffices to learn a single-output function $f :  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ whose sign is used to predict the class, and thus the classification error of $f$ on $\mathcal { D }$ is defined as $\mathrm { P r } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } } [ \mathrm { s g n } \bar { ( } f ( \pmb { x } ) ) \pmb { \operatorname { \mu } } _ { { { \bf \mu } } } y ]$ .
+
+Multi-class classification. When there are $K$ classes $K > 2$ ), let the underlying data distribution $\mathcal { D }$ be over $\mathbb { R } ^ { d } \times [ K ]$ . We describe the noise generation process as a matrix $\overset { \cdot } { P } \in \mathbb { R } ^ { K \times K }$ , whose entry $p _ { c ^ { \prime } , c }$ is the probability that the label $c$ is transformed into $\mathrm { { \sf c } } ^ { \prime } \left( \forall \mathrm { { \boldsymbol { c } } } , \boldsymbol { c } ^ { \prime } \in \left[ K \right] \right)$ . Therefore the data generation process is: draw $( \pmb { x } , c ) \sim \mathcal { D }$ , and replace the label $c$ with $\tilde { c }$ from the distribution $\mathrm { P r } [ \tilde { c } = c ^ { \prime } | c ] = p _ { c ^ { \prime } , c }$ $\forall c ^ { \prime } \in [ K ] )$ .
+
+Let $\left\{ \left( \pmb { x } _ { i } , \tilde { c } _ { i } \right) \right\} _ { i = 1 } ^ { n }$ be i.i.d. samples from the above process. Again we would like to learn a neural net with low classification error on the clean distribution $\mathcal { D }$ . For $K$ -way classification, it is common to use a neural net with $K$ outputs, and the index of the maximum output is used to predict the class. Thus for $f : \mathbb { R } ^ { d }  \mathbb { R } ^ { K }$ , its (top-1) classification error on $\mathcal { D }$ is $\mathrm { P r } _ { ( \pmb { x } , c ) \sim \mathcal { D } } [ c \not \in \mathrm { a r g m a x } _ { h \in [ K ] } f ^ { ( h ) } ( \pmb { x } ) ]$ , where $f ^ { ( h ) } : \mathbb { R } ^ { d }  \mathbb { R }$ is the function computed by the $h$ -th output of $f$ .
+
+As standard practice, a class label $c ~ \in ~ [ K ]$ is also treated as its one-hot encoding $\begin{array} { r l } { e ^ { ( c ) } } & { { } = } \end{array}$ $( 0 , 0 , \cdot \cdot \cdot , 0 , \bar { 1 } , 0 , \cdot \cdot \cdot , 0 ) \in \mathbb { R } ^ { K }$ (the $c$ -th coordinate being 1), which can be paired with the $K$ outputs of the network and fed into a loss function during training.
+
+Note that it is necessary to assume $p _ { c , c } > p _ { c ^ { \prime } , c }$ for all $c \neq c ^ { \prime }$ , i.e., the probability that a class label $c$ is transformed into another particular label must be smaller than the label $c$ being correct – otherwise it is impossible to identify class $c$ correctly from noisily labeled data.
+
+# 3.2 RECAP OF NEURAL TANGENT KERNEL
+
+Now we briefly and informally recap the theory of neural tangent kernel (NTK) (Jacot et al., 2018; Lee et al., 2019; Arora et al., 2019a), which establishes the equivalence between training a wide neural net and a kernel method.
+
+We first consider a neural net with a scalar output, defined as $f ( { \pmb \theta } , { \pmb x } ) \in \mathbb { R }$ , where $\pmb \theta \in \mathbb { R } ^ { N }$ is all the parameters in the net and $\pmb { x } \in \mathbb { R } ^ { d }$ is the input. Suppose that the net is trained by minimizing theř $\ell _ { 2 }$ loss over a training dataset $\begin{array} { r } { \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times \mathbb { R } \colon L ( \pmb \theta ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \left( f ( \pmb \theta , x _ { i } ) - y _ { i } \right) ^ { 2 } } \end{array}$ . Let the random initial parameters be $\pmb \theta ( 0 )$ , and the parameters be updated according to gradient descent on $L ( \theta )$ . It is shown that if the network is sufficiently wide1, the parameters $\pmb { \theta }$ will stay close to the initialization $\pmb \theta ( 0 )$ during training so that the following first-order approximation is accurate:
+
+$$
+f ( \pmb \theta , \pmb x ) \approx f ( \pmb \theta ( 0 ) , \pmb x ) + \langle \nabla _ { \pmb \theta } f ( \pmb \theta ( 0 ) , \pmb x ) , \pmb \theta - \pmb \theta ( 0 ) \rangle .
+$$
+
+This approximation is exact in the infinite width limit, but can also be shown when the width is finite but sufficiently large. When approximation (1) holds, we say that we are in the NTK regime.
+
+Define ${ \boldsymbol { \phi } } ( { \boldsymbol { \mathbf { \mathit { x } } } } ) ~ = ~ \nabla _ { \boldsymbol { \theta } } f ( \boldsymbol { \theta } ( 0 ) , { \boldsymbol { \mathbf { \mathit { x } } } } )$ for any $\pmb { x } \in \mathbb { R } ^ { d }$ . The right hand side in (1) is linear in $\pmb { \theta }$ . As a consequence, training on the $\ell _ { 2 }$ loss with gradient descent leads to the kernel regression solution with respect to the kernel induced by (random) features $\phi ( { \pmb x } )$ , which is defined as $k ( { \pmb x } , { \pmb x } ^ { \prime } ) =$ $\langle \phi ( { \pmb x } ) , \bar { \phi } ( { \pmb x } ^ { \prime } ) \rangle$ for $\pmb { x } , \pmb { x } ^ { \prime } \in \mathbb { R } ^ { d }$ . This kernel was named the neural tangent kernel (NTK) by Jacot et al. (2018). Although this kernel is random, it is shown that when the network is sufficiently wide, this random kernel converges to a deterministic limit in probability (Arora et al., 2019a). If we additionally let the neural net and its initialization be defined so that the initial output is small, i.e., $f ( \pmb \theta ( 0 ) , \pmb x ) \overset { \cdot } { \approx } \ 0$ ,2 then the network at the end of training approximately computes the following function:
+
+$$
+{ \pmb x } \mapsto k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) \right) ^ { - 1 } { \pmb y } ,
+$$
+
+where $\pmb { X } = \left( \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { n } \right)$ is the training inputs, $\pmb { y } = ( y _ { 1 } , \ldots , y _ { n } ) ^ { \top }$ is the training targets, $k ( { \pmb x } , { \pmb X } ) =$ ${ \big ( } k ( \mathbf { { \pmb x } } , \mathbf { { \pmb x } } _ { 1 } ) , \dots , k ( \mathbf { { \pmb x } } , \mathbf { { \pmb x } } _ { n } ) { \big ) } ^ { \top } \in \mathbb { R } ^ { n }$ , and $k ( X , X ) \in \mathbb { R } ^ { n \times n }$ with $( i , j )$ -th entry being $k ( \pmb { x } _ { i } , \pmb { x } _ { j } )$ .
+
+Multiple outputs. The NTK theory above can also be generalized straightforwardly to the case of multiple outputs (Jacot et al., 2018; Lee et al., 2019). Suppose we train a neural net with $K$ outputs, $\bar { f } ( \pmb \theta , \bar { \pmb x ) }$ , by minimizing the $\ell _ { 2 }$ loss over a training dataset $\{ ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times \mathbb { R } ^ { K }$ $\begin{array} { r } { L ( \pmb { \theta } ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \left\| \pmb { f } ( \pmb { \theta } , \pmb { x } _ { i } ) - \pmb { y } _ { i } \right\| ^ { 2 } } \end{array}$ . When the hidden layers are sufficiently wide such that we are in the NTK regime, at the end of gradient descent, each output of $f$ also attains the kernel regression solution with respect to the same NTK as before, using the corresponding dimension in the training targets $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \mathbf { \psi } _ { 2 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 }$ . Namely, the $h$ -th output of the network computes the function
+
+$$
+f ^ { ( h ) } ( { \pmb x } ) = k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) \right) ^ { - 1 } { \pmb y } ^ { ( h ) } ,
+$$
+
+where $\pmb { y } ^ { ( h ) } \in \mathbb { R } ^ { n }$ whose $i$ -th coordinate is the $h$ -th coordinate of $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \psi _ { i }$
+
+# 4 REGULARIZATION METHODS
+
+In this section we describe two simple regularization methods for training with noisy labels, and show that if the network is sufficiently wide, both methods lead to kernel ridge regression using the NTK.3
+
+We first consider the case of scalar target and single-output network. The generalization to multiple outputs is straightforward and is treated at the end of this section. Given a noisily labeled training dataset $\{ ( \pmb { x } _ { i } , \tilde { y } _ { i } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times \mathbb { R }$ , let $f ( \pmb \theta , \cdot )$ be a neural net to be trained. A direct, unregularized train-ř ing method would involve minimizing an objective function like $\begin{array} { r } { L ( \pmb { \theta } ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \left( f ( \pmb { \theta } , \pmb { x } _ { i } ) - \tilde { y } _ { i } \right) ^ { 2 } } \end{array}$ To prevent over-fitting, we suggest the following simple regularization methods that slightly modify this objective:
+
+• Method 1: Regularization using Distance to Initialization (RDI). We let the initial parameters $\pmb \theta ( 0 )$ be randomly generated, and minimize the following regularized objective:
+
+$$
+L _ { \lambda } ^ { \mathsf { R D I } } ( \pmb \theta ) = \frac 1 2 \sum _ { i = 1 } ^ { n } \left( f ( \pmb \theta , \pmb x _ { i } ) - \tilde { y } _ { i } \right) ^ { 2 } + \frac { \lambda ^ { 2 } } { 2 } \left\| \pmb \theta - \pmb \theta ( 0 ) \right\| ^ { 2 } .
+$$
+
+• Method 2: adding an AUXiliary variable for each training example (AUX). We add an auxiliary trainable parameter $b _ { i } \in \mathbb { R }$ for each $i \in [ n ]$ , and minimize the following objective:
+
+$$
+L _ { \lambda } ^ { \mathsf { A U X } } ( \pmb \theta , \pmb b ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \left( f ( \pmb \theta , \pmb x _ { i } ) + \lambda b _ { i } - \tilde { y } _ { i } \right) ^ { 2 } ,
+$$
+
+where $\pmb { b } = ( b _ { 1 } , \dots , b _ { n } ) ^ { \top } \in \mathbb { R } ^ { n }$ is initialized to be 0.
+
+Equivalence to kernel ridge regression in wide neural nets. Now we assume that we are in the NTK regime described in Section 3.2, where the neural net architecture is sufficiently wide so that the first-order approximation (1) is accurate during gradient descent: $f ( \pmb \theta , \pmb x ) \approx f ( \pmb \theta ( 0 ) , \pmb x ) +$ $\phi ( { \pmb x } ) ^ { \top } ( { \pmb \theta } - { \pmb \theta } ( 0 ) )$ . Recall that we have ${ \boldsymbol { \phi } } ( { \boldsymbol { \mathbf { \mathit { x } } } } ) = \nabla _ { \boldsymbol { \theta } } f ( \boldsymbol { \theta } ( 0 ) , { \boldsymbol { \mathbf { \mathit { x } } } } )$ which induces the NTK $k ( \pmb { x } , \pmb { x } ^ { \prime } ) =$ $\langle \phi ( { \pmb x } ) , \phi ( { \pmb x } ^ { \prime } ) \rangle$ . Also recall that we can assume near-zero initial output: $f ( { \pmb \theta } ( 0 ) , { \pmb x } ) ~ \approx ~ 0$ (see Footnote 2). Therefore we have the approximation:
+
+$$
+f ( \pmb \theta , \pmb x ) \approx \phi ( \pmb x ) ^ { \top } ( \pmb \theta - \pmb \theta ( 0 ) ) .
+$$
+
+Under the approximation (5), it suffices to consider gradient descent on the objectives (3) and (4) using the linearized model instead:
+
+$$
+\begin{array} { r l } & { \tilde { L } _ { \lambda } ^ { \mathrm { R D I } } ( \pmb \theta ) = \displaystyle \frac 1 2 \sum _ { i = 1 } ^ { n } \Big ( \phi ( \pmb x _ { i } ) ^ { \top } ( \pmb \theta - \pmb \theta ( 0 ) ) - \tilde { y } _ { i } \Big ) ^ { 2 } + \frac { \lambda ^ { 2 } } { 2 } \left. \pmb \theta - \pmb \theta ( 0 ) \right. ^ { 2 } , } \\ & { \tilde { L } _ { \lambda } ^ { \mathrm { A U X } } ( \pmb \theta , \pmb b ) = \displaystyle \frac 1 2 \sum _ { i = 1 } ^ { n } \Big ( \phi ( \pmb x _ { i } ) ^ { \top } ( \pmb \theta - \pmb \theta ( 0 ) ) + \lambda b _ { i } - \tilde { y } _ { i } \Big ) ^ { 2 } . } \end{array}
+$$
+
+The following theorem shows that in either case, gradient descent leads to the same dynamics and converges to the kernel ridge regression solution using the NTK.
+
+Theorem 4.1. Fix a learning rate $\eta > 0$ . Consider gradient descent on $\tilde { L } _ { \lambda } ^ { \mathsf { R D I } }$ with initialization $\pmb \theta ( 0 )$
+
+$$
+\pmb { \theta } ( t + 1 ) = \pmb { \theta } ( t ) - \eta \nabla _ { \pmb { \theta } } \tilde { L } _ { \lambda } ^ { \mathrm { R D 1 } } ( \pmb { \theta } ( t ) ) , \quad t = 0 , 1 , 2 , . . .
+$$
+
+and gradient descent on $\tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \pmb \theta , \pmb b )$ with initialization $\pmb \theta ( 0 )$ and $\mathbf { \delta } \mathbf { b } ( 0 ) = \mathbf { 0 }$ :
+
+$$
+\begin{array} { r l } { \bar { \pmb \theta } ( 0 ) = \pmb \theta ( 0 ) , \quad \bar { \pmb \theta } ( t + 1 ) = \bar { \pmb \theta } ( t ) - \eta \nabla _ { \pmb \theta } \tilde { L } _ { \lambda } ^ { \mathrm { A U X } } ( \bar { \pmb \theta } ( t ) , { \pmb b } ( t ) ) , \quad t = 0 , 1 , 2 , . . . } \\ { { \pmb b } ( 0 ) = { \bf 0 } , \quad { \pmb b } ( t + 1 ) = { \pmb b } ( t ) - \eta \nabla _ { b } \tilde { L } _ { \lambda } ^ { \mathrm { A U X } } ( \bar { \pmb \theta } ( t ) , { \pmb b } ( t ) ) , \quad t = 0 , 1 , 2 , . . . } \end{array}
+$$
+
+Then we must have θptq “ θ¯ptq for all t. Furthermore, if the learning rate satisfies η ď 1}kpX,Xq}\`λ2 , then $\{ \pmb \theta ( t ) \}$ converges linearly to a limit solution $\pmb { \theta } ^ { * }$ such that:
+
+$$
+\phi ( { \pmb x } ) ^ { \top } ( { \pmb \theta } ^ { * } - { \pmb \theta } ( 0 ) ) = k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } { \pmb \tilde { y } } , \quad \forall { \pmb x } ,
+$$
+
+where $\tilde { \pmb { y } } = ( \tilde { y } _ { 1 } , \dots , \tilde { y } _ { n } ) ^ { \top } \in \mathbb { R } ^ { n }$
+
+Proof sketch. The proof is given in Appendix B. A key step is to observe $\begin{array} { r } { \bar { \pmb \theta } ( t ) = \pmb \theta ( 0 ) + \sum _ { i = 1 } ^ { n } \frac { 1 } { \lambda } b _ { i } ( t ) } \end{array}$ $\phi ( { \pmb x } _ { i } )$ , from which we can show that $\{ \pmb \theta ( t ) \}$ and $\{ \bar { \pmb { \theta } } ( t ) \}$ follow the same update rule.
+
+Theorem 4.1 indicates that gradient descent on the regularized objectives (3) and (4) both learn approximately the following function at the end of training when the neural net is sufficiently wide:\` ˘
+
+If no regularization were used, the labels $\tilde { \pmb { y } }$ would be fitted perfectly and the learned function would be $k ( { \pmb x } , { \bf { \bar { X } } } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) \right) ^ { - 1 } { \tilde { \pmb y } }$ (c.f. (2)). Therefore the effect of regularization is to add $\lambda ^ { 2 } I$ to the kernel matrix, and (8) is known as the solution to kernel ridge regression in kernel literature. In Section 5, we give a generalization bound of this solution on the clean data distribution, which is comparable to the bound one can obtain even when clean labels are used in training.
+
+Extension to multiple outputs. Suppose that the training dataset is $\{ ( \pmb { x } _ { i } , \tilde { \pmb { y } } _ { i } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times \mathbb { R } ^ { K }$ , an the neural net ${ f } ( \pmb \theta , \pmb x )$ has $K$ outputs. On top of the vanilla loss $\begin{array} { r } { L ( \pmb { \theta } ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \| \pmb { f } ( \pmb { \theta } , \pmb { x } _ { i } ) - \tilde { \pmb { y } } _ { i } \| ^ { 2 } } \end{array}$ the two regularization methods RDI and AUX give the following objectives similar to (3) and (4):
+
+$$
+L _ { \lambda } ^ { \mathsf { R D I } } ( \pmb \theta ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \| \pmb f ( \pmb \theta , \pmb x _ { i } ) - \tilde { \pmb y } _ { i } \| ^ { 2 } + \frac { \lambda ^ { 2 } } { 2 } \left\| \pmb \theta - \pmb \theta ( 0 ) \right\| ^ { 2 } ,
+$$
+
+$$
+L _ { \lambda } ^ { \mathrm { A u x } } ( \pmb \theta , \pmb B ) = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { n } \| \pmb f ( \pmb \theta , \pmb x _ { i } ) + \lambda \pmb b _ { i } - \tilde { \pmb y } _ { i } \| ^ { 2 } , \quad \pmb B = \left( \pmb b _ { 1 } , \dots , \pmb b _ { n } \right) \in \mathbb { R } ^ { K \times n } .
+$$
+
+In the NTK regime, both methods lead to the kernel ridge regression solution at each output. Namely, letting $\tilde { \pmb { Y } } = \left( \tilde { \pmb { y } } _ { 1 } , \dots , \tilde { \pmb { y } } _ { n } \right) \in \mathbb { R } ^ { K \times n }$ be the training target matrix and $\tilde { \pmb y } ^ { ( h ) } \in \mathbb { R } ^ { n }$ be the $h$ -th row of $\tilde { Y }$ at the end of training the $h$ -th output of the network learns the following function:\` ˘
+
+$$
+\overset { \circ } { f } ^ { ( h ) } ( \pmb { x } ) = k ( \overset { \cdot } { \pmb { x } } , \pmb { X } ) ^ { \top } \left( k ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } \right) ^ { - 1 } \tilde { \pmb { y } } ^ { ( h ) } , \quad \overset { \circ } { h } \in [ K ] .
+$$
+
+# 5 GENERALIZATION GUARANTEE ON CLEAN DATA DISTRIBUTION
+
+We show that gradient descent training on noisily labeled data with our regularization methods RDI or AUX leads to a generalization guarantee on the clean data distribution. As in Section 4, we consider the NTK regime and let $k ( \cdot , \cdot )$ be the NTK corresponding to the neural net. It suffices to analyze the kernel ridge regression predictor, i.e., (8) for single output and (9) for multiple outputs.
+
+We start with a regression setting where labels are real numbers and the noisy label is the true label plus an additive noise (Theorem 5.1). Built on this result, we then provide generalization bounds for the classification settings described in Section 3.1. Omitted proofs in this section are in Appendix C.
+
+Theorem 5.1 (Additive label noise). Let $\mathcal { D }$ be a distribution over $\mathbb { R } ^ { d } \times [ - 1 , 1 ]$ . Consider the following data generation process: $( i )$ draw $( \boldsymbol { x } , \boldsymbol { y } ) \sim \mathcal { D }$ , (ii) conditioned on $( { \pmb x } , y )$ , let $\varepsilon$ be drawn from a noise distribution $\mathcal { E } _ { x , y }$ over $\mathbb { R }$ that may depend on $_ { \textbf { \em x } }$ and $y$ , and (iii) let $\tilde { y } = y + \varepsilon$ . Suppose that $\mathcal { E } _ { x , y }$ has mean 0 and is subgaussian with parameter $\sigma > 0$ , for any $( { \pmb x } , y )$ .
+
+Let $\{ ( \pmb { x } _ { i } , y _ { i } , \tilde { y } _ { i } ) \} _ { i = 1 } ^ { n }$ be i.i.d. samples from the above process. Denote $\pmb { X } = \left( \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { n } \right)$ , ${ \textbf { \em y } } =$ $( y _ { 1 } , \ldots , y _ { n } ) ^ { \top }$ and $\tilde { \pmb { y } } = ( \tilde { y } _ { 1 } , \dots , \tilde { y } _ { n } ) ^ { \top }$ . Consider the kernel ridge regression solution in (8): $f ^ { * } ( x ) =$ $k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } { \tilde { \pmb y } } .$ . Suppose that the kernel matrix satisfies $\operatorname { t r } [ k ( X , X ) ] = O ( n )$ . Then for any loss function $\ell : \mathbb { R } \times \mathbb { R } \to [ 0 , 1 ]$ that is 1-Lipschitz in the first argument such that $\ell ( y , \dot { y } ) = 0$ , with probability at least $1 - \delta$ we havec
+
+$$
+\mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } } \left[ \ell ( \pmb { f } ^ { * } ( \pmb { x } ) , \pmb { y } ) \right] \leqslant \frac { \lambda + O ( 1 ) } { 2 } \sqrt { \frac { \pmb { y } ^ { \top } ( k ( \pmb { X } , \pmb { X } ) ) ^ { - 1 } \pmb { y } } { n } } + O \left( \frac { \sigma } { \lambda } \right) + \Delta ,
+$$
+
+$\begin{array} { r } { \Delta = O \left( \sigma \sqrt { \frac { \log \left( 1 / \delta \right) } { n } } + \frac { \sigma } { \lambda } \sqrt { \frac { \log \left( 1 / \delta \right) } { n } } + \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) . } \end{array}$
+
+Remark 5.1. As the number of samples $n \to \infty$ , we have $\Delta  0$ . In order for the second term $O ( { \frac { \sigma } { \lambda } } )$ in (10) to go to $0$ , we need to choose $\lambda$ to grow with $n$ , e.g., $\lambda = n ^ { c }$ for some small constant $c > 0$ . Then, the only remaining term in (10) to worry about is ${ \frac { \lambda } { 2 } } { \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } }$ . Notice that it depends on the (unobserved) clean labels $\textbf {  { y } }$ , instead of the noisy labels $\tilde { y }$ . By a very similar proof, one can show that training on the clean labels ´b ¯ $\textbf {  { y } }$ (without regularization) leads to a population loss bound $O { \Big ( } { \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } } { \Big ) }$ . In comparison, we can see that even when there is label noise, we only lose a factor of $O ( \lambda )$ in the population loss on the clean distribution, which can be chosen as any slow-growing function of $n$ . If $\pmb { y } ^ { \top } ( k ( \pmb { X } , \pmb { X } ) ) ^ { - 1 } \pmb { y }$ grows much slower than $n$ , by choosing an appropriate $\lambda$ , our result indicates that the underlying distribution is learnable in presence of additive label noise. See Remark 5.2 for an example.
+
+Remark 5.2. Arora et al. (2019b) proved that two-layer ReLU neural nets trained with gradient descent can learn a class of smooth functions on the unit sphere. Their proof is by showing $\pmb { y } ^ { \top } ( k ( \pmb { X } , \pmb { X } ) ) ^ { - 1 } \pmb { y } = O ( 1 )$ if $y _ { i } = g ( \pmb { x } _ { i } )$ ${ \check { \forall } } i \in [ n ] )$ for certain function $g$ , where $k ( \cdot , \cdot )$ is the NTK corresponding to two-layer ReLU nets. Combined with their result, Theorem 5.1 implies that the same class of functions can be learned by the same network even if the labels are noisy.
+
+Next we use Theorem 5.1 to provide generalization bounds for the classification settings described in Section 3.1. For binary classification, we treat the labels as $\pm 1$ and consider a single-output neural net; for $K$ -class classification, we treat the labels as their one-hot encodings (which are $K$ standard unit vectors in $\mathbb { R } ^ { K }$ ) and consider a $K$ -output neural net. Again, we use $\ell _ { 2 }$ loss and wide neural nets so that it suffices to consider the kernel ridge regression solution ((8) or (9)).
+
+Theorem 5.2 (Binary classification). Consider the binary classification setting stated in Section 3.1. Let $\{ ( \pmb { x } _ { i } , y _ { i } , \tilde { y } _ { i } ) \} _ { i = 1 } ^ { n } \overset { \cdot } { \subset } \mathbb { R } ^ { d } \times \{ \pm 1 \} \times \{ \pm 1 \}$ be i.i.d. samples from that process. Recall that $\mathrm { P r } [ \tilde { y } _ { i } \neq$ $y _ { i } | y _ { i } ] = p$ $\begin{array} { r } { 0 \leqslant p < \frac { 1 } { 2 } , } \end{array}$ ). Denote $\pmb { X } = \left( \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { n } \right)$ , $\pmb { y } = ( y _ { 1 } , \dots , y _ { n } ) ^ { \top }$ and \` $\tilde { \pmb { y } } = ( \tilde { y } _ { 1 } , \dots , \tilde { y } _ { n } ) ^ { \top }$ . Consider the kernel ridge regression solution in (8): $f ^ { * } ( { \pmb x } ) = k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } \tilde { \pmb y }$ . Suppose that the kernel matrix satisfies $\operatorname { t r } [ k ( X , X ) ] = O ( n )$ . Then with probability at least $1 - \delta$ , the classification error of $f ^ { * }$ on the clean distribution c $\mathcal { D }$ satisfies
+
+$$
+\operatorname* { P r } _ { \substack { v , y \sim \widetilde { D } } } [ \mathrm { s g n } ( f ^ { * } ( x ) ) \neq y ] \leqslant \frac { \lambda + O ( 1 ) } { 2 } \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } + \frac { 1 } { 1 - 2 p } O \left( \frac { \sqrt { p } } { \lambda } + \sqrt { \frac { p \log \frac { 1 } { \delta } } { n } } + \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) .
+$$
+
+Theorem 5.3 (Multi-class classification). Consider the $K$ -class classification setting stated in Section 3.1. Let $\{ ( \pmb { x } _ { i } , \pmb { c } _ { i } , \tilde { \pmb { c } _ { i } } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times [ K ] \times [ K ]$ be i.i.d. samples from that process. Recall that $\mathrm { P r } [ \tilde { c } _ { i } = c ^ { \prime } | c _ { i } = c ] = p _ { c ^ { \prime } , c } ( \forall c , c ^ { \prime } \in [ K ] ) ,$ , where the transition probabilities form a matrix $\pmb { P } \in \mathbb { R } ^ { K \times K }$ . $\begin{array} { r } { L e t \mathsf { g a p } = \operatorname* { m i n } _ { c , c ^ { \prime } \in [ K ] , c \neq c ^ { \prime } } ( p _ { c , c } - p _ { c ^ { \prime } , c } ) . } \end{array}$ .
+
+Let $\pmb { X } = ( \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { n } )$ , and let $\pmb { y } _ { i } = \pmb { e } ^ { ( c _ { i } ) } \in \mathbb { R } ^ { K }$ , $\tilde { \pmb { y } } _ { i } = \pmb { e } ^ { ( \tilde { c } _ { i } ) } \in \mathbb { R } ^ { K }$ be one-hot label encodings. Denote $\pmb { Y } = ( \pmb { y } _ { 1 } , \dots , \pmb { y } _ { n } ) \in \mathbb { R } ^ { K \times n }$ , $\tilde { \pmb { Y } } = ( \tilde { \pmb { y } } _ { 1 } , \dots , \tilde { \pmb { y } } _ { n } ) \in \mathbb { R } ^ { K \times n }$ , and let $\tilde { \pmb y } ^ { ( h ) } \in \mathbb { R } ^ { n }$ be the $h$ -th row of $\tilde { \mathbf { Y } }$ . Define a matrix $Q = P \cdot Y \in \mathbb { R } ^ { K \times n }$ , and let $\pmb q ^ { ( h ) } \in \mathbb { R } ^ { n }$ be the $h$ -th row of $Q$ .
+
+Consider the kernel ridge regression solution in (9): $f ^ { ( h ) } ( { \pmb x } ) = k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } \tilde { \pmb y } ^ { ( h ) }$ . Suppose that the kernel matrix satisfies $\operatorname { t r } [ k ( X , X ) ] = O ( n )$ . Then with probability at least $1 - \delta$ , the classification error of $\pmb { f } = ( f ^ { ( h ) } ) _ { k = 1 } ^ { K }$ on the clean data distribution $\mathcal { D }$ is bounded as
+
+![](images/7537c94aaa9bd3f1bfd70c5a6a1e5facc5cf753d7b19bfbf79046d57acdf19ed.jpg)
+
+(a) Test error vs. noise level for Setting 1. For each noise level, we do a grid search for $\lambda$ and report the best accuracy.
+
+![](images/3b1ecffa19f60959b0e172d180726aceb4d7bc33072f485ffe56f456abf01a10.jpg)  
+(b) Training (dashed) & test (solid) errors vs. epoch for Setting 2. Noise rate $= 2 0 \%$ , $\lambda = 4$ . Training error of AUX is measured with auxiliary variables.
+
+Figure 1: Performance on binary classification using $\ell _ { 2 }$ loss. Setting 1: MNIST; Setting 2: CIFAR.
+
+$$
+\begin{array} { c } { { \displaystyle \operatorname* { P r } _ { ( \alpha , c ) \sim \mathcal { D } } \left[ c \not \in \mathrm { a r g m a x } _ { h \in [ K ] } f ^ { ( h ) } ( x ) \right] } } \\ { { \displaystyle \leqslant \frac { 1 } { \mathsf { g a p } } \left( \frac { \lambda + O ( 1 ) } { 2 } \sum _ { h = 1 } ^ { K } \sqrt { \frac { ( q ^ { ( h ) } ) ^ { \top } ( k ( X , X ) ) ^ { - 1 } q ^ { ( h ) } } { n } } + K \cdot O \left( \frac { 1 } { \lambda } + \sqrt { \frac { \log \frac { 1 } { \delta ^ { \prime } } } { n } } + \sqrt { \frac { \log \frac { n } { \delta ^ { \prime } \lambda } } { n } } \right) \right) . } } \end{array}
+$$
+
+Note that the bounds in Theorems 5.2 and 5.3 only depend on the clean labels instead of the noisy labels, similar to Theorem 5.1.
+
+# 6 EXPERIMENTS
+
+In this section, we empirically verify the effectiveness of our regularization methods RDI and AUX, and compare them against gradient descent or stochastic gradient descent (GD/SGD) with or without early stopping. We experiment with three settings of increasing complexities:
+
+Setting 1: Binary classification on MNIST (“5” vs. “8”) using a two-layer wide fully-connected net.
+
+Setting 2: Binary classification on CIFAR (“airplanes” vs. “automobiles”) using a 11-layer convolutional neural net (CNN).
+
+Setting 3: CIFAR-10 classification (10 classes) using standard ResNet-34.
+
+For detailed description see Appendix D. We obtain noisy labels by randomly corrupting correct labels, where noise rate/level is the fraction of corrupted labels (for CIFAR-10, a corrupted label is chosen uniformly from the other 9 classes.)
+
+# 6.1 PERFORMANCE OF REGULARIZATION METHODS
+
+For Setting 1 (binary MNIST), we plot the test errors of different methods under different noise rates in Figure 1a. We observe that both methods $\mathrm { G D + A U X }$ and $\mathrm { G D + R D I }$ consistently achieve much lower test error than vanilla GD which over-fits the noisy dataset, and they achieve similar test error to GD with early stopping. We see that $\mathtt { G D + A U X }$ and $\mathrm { G D + R D I }$ have essentially the same performance, which verifies our theory of their equivalence in wide networks (Theorem 4.1).
+
+For Setting 2 (binary CIFAR), Figure 1b shows the learning curves (training and test errors) of SGD, $\mathrm { S G D + A U X }$ and $\mathrm { S G D + R D I }$ for noise rate $2 0 \%$ and $\lambda = 4$ . Additional figures for other choices of $\lambda$ are in Figure 5. We again observe that both $\mathrm { S G D + A U X }$ and $\mathrm { S G D + R D I }$ outperform vanilla SGD and are comparable to SGD with early stopping. We also observe a discrepancy between $\mathrm { S G D + A U X }$ and $\mathrm { S G D + R D I }$ , possibly due to the noise in SGD or the finite width.
+
+Finally, for Setting 3 (CIFAR-10), Table 1 shows the test accuracies of training with and without AUX. We train with both mean square error $\mathrm { ( M S E / \ell _ { 2 } }$ loss) and categorical cross entropy (CCE) loss. For normal training without AUX, we report the test accuracy at the epoch where validation accuracy is maximum (early stopping). For training with AUX, we report the test accuracy at the last epoch as well as the best epoch. Figure 2 shows the training curves for noise rate 0.4. We observe that training with AUX achieves very good test accuracy – even better than the best accuracy of normal training with early stopping, and better than the recent method of Zhang and Sabuncu (2018) using the same architecture (ResNet-34). Furthermore, AUX does not over-fit (the last epoch performs similarly to the best epoch). In addition, we find that in this setting classification performance is insensitive of whether MSE or CCE is used as the loss function.
+
+Table 1: CIFAR-10 test accuracies of different methods under different noise rates.   
+
+<table><tr><td rowspan=1 colspan=1>Noise rate</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>0.6</td></tr><tr><td rowspan=1 colspan=1>Normal CCE (earlystop)</td><td rowspan=1 colspan=1>94.05±0.07</td><td rowspan=1 colspan=1>89.73±0.43</td><td rowspan=1 colspan=1>86.35±0.47</td><td rowspan=1 colspan=1>79.13±0.41</td></tr><tr><td rowspan=1 colspan=1>Normal MSE (early stop)</td><td rowspan=1 colspan=1>93.88±0.37</td><td rowspan=1 colspan=1>89.96±0.13</td><td rowspan=1 colspan=1>85.92±0.32</td><td rowspan=1 colspan=1>78.68±0.56</td></tr><tr><td rowspan=1 colspan=1>CCE+AUX (last)</td><td rowspan=1 colspan=1>94.22±0.10</td><td rowspan=1 colspan=1>92.07±0.10</td><td rowspan=1 colspan=1>87.81±0.37</td><td rowspan=1 colspan=1>82.60±0.29</td></tr><tr><td rowspan=1 colspan=1>CCE+AUX (best)</td><td rowspan=1 colspan=1>94.30±0.09</td><td rowspan=1 colspan=1>92.16±0.08</td><td rowspan=1 colspan=1>88.61±0.14</td><td rowspan=1 colspan=1>82.91±0.22</td></tr><tr><td rowspan=1 colspan=1>MSE+AUX (last)</td><td rowspan=1 colspan=1>94.25±0.10</td><td rowspan=1 colspan=1>92.31±0.18</td><td rowspan=1 colspan=1>88.92±0.30</td><td rowspan=1 colspan=1>83.90±0.30</td></tr><tr><td rowspan=1 colspan=1>MSE+AUX(best)</td><td rowspan=1 colspan=1>94.32±0.06</td><td rowspan=1 colspan=1>92.40±0.18</td><td rowspan=1 colspan=1>88.95±0.31</td><td rowspan=1 colspan=1>83.95±0.30</td></tr><tr><td rowspan=1 colspan=1>(Zhang and Sabuncu, 2018)</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>89.83±0.20</td><td rowspan=1 colspan=1>87.62±0.26</td><td rowspan=1 colspan=1>82.70±0.23</td></tr></table>
+
+![](images/dafd1e57d46a3bf8c0ca3876b747c6b30bc67ce38ed01a130159a468c82a6938.jpg)  
+Figure 2: Test accuracy during CIFAR-10 training (noise 0.4).
+
+![](images/222b9d81796a0a530a4307273a0af4ab819b89c86dad6ad4a37c0bd98634fa9f.jpg)  
+Figure 3: Setting 2, $\| \mathbf { W } ^ { ( 4 ) } \| _ { F }$ and $\lVert W ^ { ( 4 ) } - W ^ { ( 4 ) } ( 0 ) \rVert _ { F }$ during training. Noise $= 2 0 \%$ , $\lambda = 4$
+
+# 6.2 DISTANCE OF WEIGHTS TO INITIALIZATION, VERIFICATION OF THE NTK REGIME
+
+We also track how much the weights move during training as a way to see whether the neural net is in the NTK regime. For Settings 1 and 2, we find that the neural nets are likely in or close to the NTK regime because the weight movements are small during training. Figure 3 shows in Setting 2 how much the 4-th layer weights move during training. Additional figures are provided as Figures 6 to 8.
+
+Table 2 summarizes the relationship between the distance to initialization and other hyper-parameters that we observe from various experiments. Note that the weights tend to move more with larger noise level, and AUX and RDI can reduce the moving distance as expected (as shown in Figure 3).
+
+The ResNet-34 in Setting 3 is likely not operating in the NTK regime, so its effectiveness cannot yet be explained by our theory. This is an intriguing direction of future work.
+
+Table 2: Relationship between distance to initialization at convergence and other hyper-parameters. “Õ”: positive correlation; $\tilde { \mathbf { \Gamma } } ^ { 6 6 } \dot { \mathbf { \Gamma } } \searrow \ '$ : negative correlation; ‘—’: no correlation as long as width is sufficiently large and learning rate is sufficiently small.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1># samples</td><td rowspan=1 colspan=1>noise level</td><td rowspan=1 colspan=1>width</td><td rowspan=1 colspan=1>regularizationstrength 入</td><td rowspan=1 colspan=1>learning rate</td></tr><tr><td rowspan=1 colspan=1>Distance</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>/</td><td rowspan=1 colspan=1></td></tr></table>
+
+# 7 CONCLUSION
+
+Towards understanding generalization of deep neural networks in presence of noisy labels, this paper presents two simple regularization methods and shows that they are theoretically and empirically effective. The theoretical analysis relies on the correspondence between neural networks and NTKs. We believe that a better understanding of such correspondence could help the design of other principled methods in practice. We also observe that our methods can be effective outside the NTK regime. Explaining this theoretically is left for future work.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Mozilla Research, Amazon Research, DARPA and SRC. The authors thank Sanjeev Arora for helpful discussions and suggestions. The authors thank Amazon Web Services for cloud computing time.
+
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+
+# A DIFFERENCE TRICK
+
+We give a quick analysis of the “difference trick” described in Footnote 2, i.e., let $f ( { \pmb \theta } , { \pmb x } ) ~ =$ $\begin{array} { r l r } {  { \frac { \sqrt { 2 } } { 2 } g ( \pmb { \theta } _ { 1 } , \pmb { x } ) - \frac { \sqrt { 2 } } { 2 } g ( \pmb { \theta } _ { 2 } , \pmb { x } ) } } \end{array}$ where mma im $\pmb { \theta } = ( \pmb { \theta } _ { 1 } , \pmb { \theta } _ { 2 } )$ , and initi NTKs for ize an $\pmb { \theta } _ { 1 }$ and are t $\pmb { \theta } _ { 2 }$ to be the same (and still same. $f$ $g$
+
+Lemma A.1. If $\pmb { \theta } _ { 1 } ( 0 ) = \pmb { \theta } _ { 2 } ( 0 )$ , then
+
+(i) $\begin{array} { r } { f ( \pmb { \theta } ( 0 ) , \pmb { x } ) = 0 , \forall \pmb { x } , } \end{array}$
+
+$$
+\begin{array} { r } { ( i i ) \ \langle \nabla _ { \theta } f ( \theta ( 0 ) , x ) , \nabla _ { \theta } f ( \theta ( 0 ) , x ^ { \prime } ) \rangle = \langle \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ) , \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ^ { \prime } ) \rangle , \forall x , x ^ { \prime } . } \end{array}
+$$
+
+Proof. (i) holds by definition. For (ii), we can calculate that
+
+$$
+\begin{array} { r l } & { \quad \left. \nabla _ { \theta } f ( \theta ( 0 ) , x ) , \nabla _ { \theta } f ( \theta ( 0 ) , x ^ { \prime } ) \right. } \\ & { = \left. \nabla _ { \theta _ { 1 } } f ( \theta ( 0 ) , x ) , \nabla _ { \theta _ { 1 } } f ( \theta ( 0 ) , x ^ { \prime } ) \right. + \left. \nabla _ { \theta _ { 2 } } f ( \theta ( 0 ) , x ) , \nabla _ { \theta _ { 2 } } f ( \theta ( 0 ) , x ^ { \prime } ) \right. } \\ & { = \displaystyle \frac { 1 } { 2 } \left. \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ) , \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ) \right. + \displaystyle \frac { 1 } { 2 } \left. \nabla _ { \theta _ { 2 } } g ( \theta _ { 2 } ( 0 ) , x ) , \nabla _ { \theta _ { 2 } } g ( \theta _ { 2 } ( 0 ) , x ^ { \prime } ) \right. } \\ & { = \left. \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ) , \nabla _ { \theta _ { 1 } } g ( \theta _ { 1 } ( 0 ) , x ^ { \prime } ) \right. . } \end{array}
+$$
+
+The above lemma allows us to ensure zero output at initialization while preserving NTK. As a comparison, Chizat and Bach (2018) proposed the following "doubling trick": neurons in the last layer are duplicated, with the new neurons having the same input weights and opposite output weights. This satisfies zero output at initialization, but destroys the NTK. To see why, note that with the “doubling trick", the network will output 0 at initialization no matter what the input to its second to last layer is. Thus the gradients with respect to all parameters that are not in the last two layers are 0.
+
+In our experiments, we observe that the performance of the neural net improves with the “difference trick.” See Figure 4. This intuitively makes sense, since the initial network output is independent of the label (only depends on the input) and thus can be viewed as noise. When the width of the neural net is infinity, the initial network output is actually a zero-mean Gaussian process, whose covariance matrix is equal to the NTK contributed by the gradients of parameters in its last layer. Therefore, learning an infinitely wide neural network with nonzero initial output is equivalent to doing kernel regression with an additive correlated Gaussian noise on training and testing labels.
+
+![](images/582a5be651ca698a49cd2364c39a2a899c51c35bca97ccc115d50f6c39ed7b14.jpg)
+
+Figure 4: Plot of test error for fully connected two-layer network on MNIST (binary classification between $\ddot { } 5 \ '$ and “8”) with difference trick and different mixing coefficients b $\alpha$ , where ${ \bf \dot { \boldsymbol { f } } } ( \pmb { \theta } _ { 1 } , \pmb { \theta } _ { 2 } , \pmb { x } ) =$ $\sqrt { \textstyle \frac { \alpha } { 2 } } g ( \pmb { \theta } _ { 1 } , \pmb { x } ) - \sqrt { \textstyle \frac { 1 - \alpha } { 2 } } g ( \pmb { \theta } _ { 1 } , \pmb { x } )$ . Note that this parametrization preserves the NTK. The network has 10,000 hidden neurons and we train both layers with gradient descent with fixed learning rate for 5,000 steps. The training loss is less than 0.0001 at the time of stopping. We observe that when $\alpha$ increases, the test error drops because the scale of the initial output of the network goes down.
+
+# B MISSING PROOF IN SECTION 4
+
+Proof of Theorem 4.1. The gradient of $\tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \pmb \theta , \pmb b )$ can be written as
+
+$$
+\begin{array} { r } { \nabla _ { \pmb { \theta } } \tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \pmb { \theta } , \pmb { b } ) = \sum _ { i = 1 } ^ { n } a _ { i } \phi ( \pmb { x } _ { i } ) , \quad \nabla _ { b _ { i } } \tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \pmb { \theta } , \pmb { b } ) = \lambda a _ { i } , \quad i = 1 , \ldots , n , } \end{array}
+$$
+
+where ř $a _ { i } \ = \ \phi ( { \pmb x } _ { i } ) ^ { \top } ( { \pmb \theta } \ - \ { \pmb \theta } ( 0 ) ) + \lambda b _ { i } - \tilde { y } _ { i } ( i \ \in \ [ n ] )$ . Therefore we have $\nabla _ { \pmb { \theta } } \tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \pmb { \theta } , \pmb { b } ) \ =$ $\begin{array} { r } { \sum _ { i = 1 } ^ { n } \frac { 1 } { \lambda } \nabla _ { b _ { i } } \tilde { L } _ { \lambda } ^ { \mathsf { A U X } } ( \theta , b ) \cdot \phi ( { \boldsymbol x } ) } \end{array}$ . Then, according to the gradient descent update rule (7), we know thatřn $ { \bar { \theta } } ( t )$ and $\mathbf { } _ { \mathbf { } } \mathbf { } _ { \mathbf { } } \mathbf { } _ { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \Psi \Psi \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf $ can always be related by $\begin{array} { r } { \bar { \pmb \theta } ( t ) = \pmb \theta ( 0 ) + \sum _ { i = 1 } ^ { n } \frac { 1 } { \lambda } b _ { i } ( t ) \cdot \pmb \phi ( \pmb x _ { i } ) } \end{array}$ . It follows that
+
+$$
+\begin{array} { r l } & { \bar { \theta } ( t + 1 ) = \bar { \theta } ( t ) - \eta \sum _ { i = 1 } ^ { n } \left( \phi ( { \pmb x } _ { i } ) ^ { \top } ( \bar { \theta } ( t ) - \theta ( 0 ) ) + \lambda b _ { i } ( t ) - \tilde { y } _ { i } \right) \phi ( { \pmb x } _ { i } ) } \\ & { \qquad = \bar { \theta } ( t ) - \eta \sum _ { i = 1 } ^ { n } \left( \phi ( { \pmb x } _ { i } ) ^ { \top } ( \bar { \theta } ( t ) - \theta ( 0 ) ) - \tilde { y } _ { i } \right) \phi ( { \pmb x } _ { i } ) - \eta \lambda \sum _ { i = 1 } ^ { n } b _ { i } ( t ) \phi ( { \pmb x } _ { i } ) } \\ & { \qquad = \bar { \theta } ( t ) - \eta \sum _ { i = 1 } ^ { n } \left( \phi ( { \pmb x } _ { i } ) ^ { \top } ( \bar { \theta } ( t ) - \theta ( 0 ) ) - \tilde { y } _ { i } \right) \phi ( { \pmb x } _ { i } ) - \eta \lambda ^ { 2 } ( \bar { \theta } ( t ) - \theta ( 0 ) ) . } \end{array}
+$$
+
+On the other hand, from (6) we have
+
+$$
+\theta ( t + 1 ) = \theta ( t ) - \eta \sum _ { i = 1 } ^ { n } \left( \phi ( x _ { i } ) ^ { \top } ( \theta ( t ) - \theta ( 0 ) ) - \tilde { y } _ { i } \right) \phi ( x _ { i } ) - \eta \lambda ^ { 2 } ( \theta ( t ) - \theta ( 0 ) ) .
+$$
+
+Comparing the above two equations, we find that $\{ \pmb \theta ( t ) \}$ and $\{ \bar { \pmb { \theta } } ( t ) \}$ have the same update rule. Since $\pmb { \theta } ( 0 ) \overset { \cdot } { = } \bar { \pmb { \theta } } ( 0 )$ , this proves $\pmb { \theta } ( \mathbf { \hat { \eta } } _ { t } ) = \bar { \pmb { \theta } } ( t )$ for all $t$ .
+
+Now we prove the second part of the theorem. Notice that ř $\tilde { L } _ { \lambda } ^ { \mathsf { R D I } } ( \pmb { \theta } )$ is a strongly convex quadratic function with Hessian $\begin{array} { r } { \nabla _ { \pmb { \theta } } ^ { 2 } \tilde { L } _ { \lambda } ^ { \mathrm { R D I } } ( \pmb { \theta } ) \ = \ \sum _ { i = 1 } ^ { n } \phi ( { \pmb x } _ { i } ) \phi ( { \pmb x } _ { i } ) ^ { \top } \ + \ \lambda ^ { 2 } \pmb { I } \ = \ Z Z ^ { \top } + \ \lambda ^ { 2 } \pmb { I } , } \end{array}$ , where $z =$ $( \phi ( \pmb { x } _ { 1 } ) , \dots , \phi ( \pmb { x } _ { n } ) )$ . From the classical convex optimization theory, as long as $\begin{array} { r } { \eta \leqslant \frac { 1 } { \| Z Z ^ { \top } + \lambda ^ { 2 } I \| } = } \end{array}$ $\frac { 1 } { \| Z ^ { \top } Z + \lambda ^ { 2 } I \| } ~ = ~ \frac { 1 } { \| k ( X , X ) \| + \lambda ^ { 2 } } ~$ “ 1}kpX,Xq}\`λ2 , gradient descent converges linearly to the unique optimum θ˚ of $\tilde { L } _ { \lambda } ^ { \mathsf { R D I } } ( \pmb { \theta } )$ , which can be easily obtained:
+
+$$
+\pmb { \theta } ^ { * } = \pmb { \theta } ( 0 ) + \pmb { Z } ( k ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } ) ^ { - 1 } \tilde { \pmb { y } } .
+$$
+
+Then we have
+
+$$
+\phi ( { \pmb x } ) ^ { \top } ( { \pmb \theta } ^ { * } - { \pmb \theta } ( { \bf 0 } ) ) = \phi ( { \pmb x } ) ^ { \top } { \pmb Z } ( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } ) ^ { - 1 } { \tilde { \pmb y } } = k ( { \pmb x } , { \pmb X } ) ^ { \top } \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } { \tilde { \pmb y } } ,
+$$
+
+finishing the proof.
+
+# C MISSING PROOFS IN SECTION 5
+
+# C.1 PROOF OF THEOREM 5.1
+
+Define $\varepsilon _ { i } = \tilde { y } _ { i } - y _ { i }$ $( i \in [ n ] )$ , and $\pmb { \varepsilon } = ( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } ) ^ { \top } = \tilde { \pmb { y } } - \pmb { y }$
+
+We first prove two lemmas.
+
+Lemma C.1. With probability at least $1 - \delta$ , we have
+
+$$
+\sqrt { \sum _ { i = 1 } ^ { n } \left( f ^ { * } ( { \pmb x } _ { i } ) - y _ { i } \right) ^ { 2 } \leqslant \frac { \lambda } { 2 } \sqrt { { \pmb y } ^ { \top } ( k ( { \pmb X } , { \pmb X } ) ) ^ { - 1 } { \pmb y } } + \frac { \sigma } { 2 \lambda } \sqrt { \mathrm { t r } [ k ( { \pmb X } , { \pmb X } ) ] } + \sigma \sqrt { 2 \log ( 1 / \delta ) } } .
+$$
+
+Proof. In this proof we are conditioned on $\boldsymbol { X }$ and $\textbf {  { y } }$ , and only consider the randomness in $\tilde { \pmb { y } }$ given $\boldsymbol { X }$ and $\textbf {  { y } }$ .
+
+First of all, we can write
+
+$$
+\left( f ^ { * } ( \pmb { x } _ { 1 } ) , \ldots , f ^ { * } ( \pmb { x } _ { n } ) \right) ^ { \top } = k ( \pmb { X } , \pmb { X } ) \left( k ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } \right) ^ { - 1 } \pmb { \tilde { y } } ,
+$$
+
+so we can write the training loss on true labels $\textbf {  { y } }$ as
+
+$$
+\begin{array} { r l } {  { \sqrt { \sum _ { i = 1 } ^ { n } ( f ^ { * } ( \boldsymbol { x } _ { i } ) - y _ { i } ) ^ { 2 } } = \| k ( \boldsymbol { X } , \boldsymbol { X } ) ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } \tilde { \boldsymbol { y } } - \boldsymbol { y } \| } } \\ & { = \| k ( \boldsymbol { X } , \boldsymbol { X } ) ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } ( \boldsymbol { y } + \varepsilon ) - \boldsymbol { y } \| } \\ & { = \| k ( \boldsymbol { X } , \boldsymbol { X } ) ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } \varepsilon - \lambda ^ { 2 } ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } \boldsymbol { y } \| } \\ & { \leqslant \| k ( \boldsymbol { X } , \boldsymbol { X } ) ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } \varepsilon \| + \lambda ^ { 2 } \| ( k ( \boldsymbol { X } , \boldsymbol { X } ) + \lambda ^ { 2 } \boldsymbol { I } ) ^ { - 1 } \boldsymbol { y } \| . } \end{array}
+$$
+
+Next, since $\varepsilon$ , conditioned on $\boldsymbol { X }$ and $\textbf {  { y } }$ , has independent and subgaussian entries (with parameter $\sigma$ ), by (Hsu et al., 2012), for any symmetric matrix $\pmb { A }$ , with probability at least $1 - \delta$ ,
+
+$$
+\| A \pmb { \varepsilon } \| \leqslant \sigma \sqrt { \mathrm { t r } [ \pmb { A } ^ { 2 } ] + 2 \sqrt { \mathrm { t r } [ \pmb { A } ^ { 4 } ] \log ( 1 / \delta ) } + 2 \| \pmb { A } ^ { 2 } \| \log ( 1 / \delta ) } .
+$$
+
+Let $\pmb { A } = k ( \pmb { X } , \pmb { X } ) \left( k ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } \right) ^ { - 1 }$ and let $\lambda _ { 1 } , \ldots , \lambda _ { n } > 0$ be the eigenvalues of $k ( X , X )$
+
+$$
+\begin{array} { r l } & { \mathrm { t r } [ A ^ { 2 } ] = \displaystyle \sum _ { i = 1 } ^ { n } \frac { \lambda _ { i } ^ { 2 } } { ( \lambda _ { i } + \lambda ^ { 2 } ) ^ { 2 } } \leqslant \displaystyle \sum _ { i = 1 } ^ { n } \frac { \lambda _ { i } ^ { 2 } } { 4 \lambda _ { i } \cdot \lambda ^ { 2 } } = \frac { \mathrm { t r } [ k ( X , X ) ] } { 4 \lambda ^ { 2 } } , } \\ & { \mathrm { t r } [ A ^ { 4 } ] = \displaystyle \sum _ { i = 1 } ^ { n } \frac { \lambda _ { i } ^ { 4 } } { ( \lambda _ { i } + \lambda ^ { 2 } ) ^ { 4 } } \leqslant \displaystyle \sum _ { i = 1 } ^ { n } \frac { \lambda _ { i } ^ { 4 } } { 4 ^ { 4 } \lambda ^ { 2 } \left( \frac { \lambda _ { i } } { 3 } \right) ^ { 3 } } \leqslant \frac { \mathrm { t r } [ k ( X , X ) ] } { 9 \lambda ^ { 2 } } , } \\ & { \| A ^ { 2 } \| \leqslant 1 . } \end{array}
+$$
+
+Therefore,
+
+$$
+\begin{array} { r l } & { k \big ( { \pmb X } , { \pmb X } \big ) \left( k ( { \pmb X } , { \pmb X } ) + \lambda ^ { 2 } { \pmb I } \right) ^ { - 1 } \varepsilon \Big \| \leqslant \sigma \sqrt { \frac { \mathrm { t r } \big [ k ( { \pmb X } , { \pmb X } ) \big ] } { 4 \lambda ^ { 2 } } + 2 \sqrt { \frac { \mathrm { t r } \big [ k ( { \pmb X } , { \pmb X } ) \big ] \log ( 1 / \delta ) } { 9 \lambda ^ { 2 } } } + 2 \log ( 1 / \delta ) } } \\ & { \leqslant \sigma \left( \sqrt { \frac { \mathrm { t r } \big [ k ( { \pmb X } , { \pmb X } ) \big ] } { 4 \lambda ^ { 2 } } } + \sqrt { 2 \log ( 1 / \delta ) } \right) . } \end{array}
+$$
+
+Finally, since $\big ( k ( X , X ) + \lambda ^ { 2 } I \big ) ^ { - 2 } \preceq \frac { 1 } { 4 \lambda ^ { 2 } } \left( k ( X , X ) \right) ^ { - 1 }$ (note $( \lambda _ { i } + \lambda ^ { 2 } ) ^ { 2 } \geqslant 4 \lambda _ { i } \cdot \lambda ^ { 2 } )$ , we have
+
+$$
+\lambda ^ { 2 } \left\| \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } y \right\| = \lambda ^ { 2 } { \sqrt { y ^ { \top } \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 2 } y } } \leqslant { \frac { \lambda } { 2 } } { \sqrt { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } } .
+$$
+
+The proof is finished by combining (11), (13) and (14).
+
+Let $\mathcal { H }$ be the reproducing kernel Hilbert space (RKHS) corresponding to the kernel $k ( \cdot , \cdot )$ . Recall that the RKHS norm of a function $f ( { \pmb x } ) = \mathbf { \dot { \alpha } } { } ^ { \top } k ( { \pmb x } , { \pmb X } )$ is
+
+$$
+\| f \| _ { \mathcal { H } } = \sqrt { \pmb { \alpha } ^ { \top } k ( \pmb { X } , \pmb { X } ) \pmb { \alpha } } .
+$$
+
+Lemma C.2. With probability at least $1 - \delta$ , we have
+
+$$
+\| f ^ { * } \| _ { \mathcal { H } } \leqslant \sqrt { y ^ { \top } \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } y } + \frac { \sigma } { \lambda } \left( \sqrt { n } + \sqrt { 2 \log ( 1 / \delta ) } \right)
+$$
+
+Proof. In this proof we are still conditioned on $\boldsymbol { X }$ and $\textbf {  { y } }$ , and only consider the randomness in\` ˘ $\tilde { \pmb { y } }$ given $\boldsymbol { X }$ and $\textbf {  { y } }$ . Note that $f ^ { * } ( { \pmb x } ) \ = \ { \pmb \alpha } ^ { \top } k ( { \pmb x } , { \pmb X } )$ with $\pmb { \alpha } = \left( \bar { k } ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } \right) ^ { - 1 } \tilde { \pmb { y } }$ . Since $\left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } \preceq \left( k ( X , X ) \right) ^ { - 1 }$ and $\begin{array} { r } { \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } \le \frac { 1 } { \lambda ^ { 2 } } I , } \end{array}$ , we can bound
+
+$$
+\begin{array} { r l } & { \| f ^ { * } \| _ { \mathcal H } = \sqrt { \tilde { y } ^ { \top } \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } k ( X , X ) \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } \tilde { y } } } \\ & { \qquad \leqslant \sqrt { ( y + \varepsilon ) ^ { \top } \left( k ( X , X ) + \lambda ^ { 2 } I \right) ^ { - 1 } ( y + \varepsilon ) } } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { \leqslant \sqrt { { { y } ^ { \top } } \left( k ( { \boldsymbol { X } } , { \boldsymbol { X } } ) + { { \lambda } ^ { 2 } } { { I } } \right) ^ { - 1 } { { y } } } + \sqrt { { { \varepsilon } ^ { \top } } \left( k ( { \boldsymbol { X } } , { \boldsymbol { X } } ) + { { \lambda } ^ { 2 } } { { I } } \right) ^ { - 1 } { { \varepsilon } } } } \\ & { \leqslant \sqrt { { { y } ^ { \top } } \left( k ( { \boldsymbol { X } } , { \boldsymbol { X } } ) + { { \lambda } ^ { 2 } } { { I } } \right) ^ { - 1 } { { y } } } + \frac { \sqrt { { { \varepsilon } ^ { \top } } { { \varepsilon } } } } { \lambda } . } \end{array}
+$$
+
+Since $\varepsilon$ has independent and subgaussian (with parameter $\sigma$ ) coordinates, using (12) with $A = I$ with probability at least $1 - \delta$ we have
+
+$$
+\sqrt { \pmb { \varepsilon } ^ { \top } \pmb { \varepsilon } } \leqslant \sigma \left( \sqrt { n } + \sqrt { 2 \log ( 1 / \delta ) } \right) .
+$$
+
+Now we prove Theorem 5.1.
+
+Proof of Theorem 5.1. First, by Lemma C.1, with probability $1 - \delta / 3$ ,
+
+$$
+\sqrt { \sum _ { i = 1 } ^ { n } \left( f ^ { * } ( { \pmb x } _ { i } ) - y _ { i } \right) ^ { 2 } } \leqslant \frac { \lambda } { 2 } \sqrt { { \pmb y } ^ { \top } ( k ( { \pmb X } , { \pmb X } ) ) ^ { - 1 } { \pmb y } } + \frac { \sigma } { 2 \lambda } \sqrt { \mathrm { t r } [ k ( { \pmb X } , { \pmb X } ) ] } + \sigma \sqrt { 2 \log ( 3 / \delta ) } ,
+$$
+
+which implies that the training error on the true labels under loss function $\ell$ is bounded as
+
+$$
+\begin{array} { l } { \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ^ { * } ( x _ { i } ) , y _ { i } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \ell ( f ^ { * } ( x _ { i } ) , y _ { i } ) - \ell ( y _ { i } , y _ { i } ) ) } \\ { \displaystyle \leqslant \frac { 1 } { n } \sum _ { i = 1 } ^ { n } | f ^ { * } ( x _ { i } ) - y _ { i } | } \\ { \displaystyle \leqslant \frac { 1 } { \sqrt { n } } \sqrt { \sum _ { i = 1 } ^ { n } | f ^ { * } ( x _ { i } ) - y _ { i } | ^ { 2 } } } \\ { \displaystyle \leqslant \frac { \lambda } { 2 } \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } + \frac { \sigma } { 2 \lambda } \sqrt { \frac { \mathrm { t r } [ k ( X , X ) ] } { n } } + \sigma \sqrt { \frac { 2 \log ( 3 / \delta ) } { n } } . } \end{array}
+$$
+
+Next, for function class ${ \mathcal { F } } _ { B } = \{ f ( { \pmb x } ) = \pmb { \alpha } ^ { \top } k ( { \pmb x } , { \pmb X } ) : \| f \| _ { \mathcal { H } } \leqslant B \}$ , Bartlett and Mendelson (2002) showed that its empirical Rademacher complexity can be bounded as
+
+$$
+{ \hat { \mathcal { R } } } _ { S } ( { \mathcal { F } } _ { B } ) \triangleq { \frac { 1 } { n } } \mathop { \mathbb { E } } _ { \gamma \sim \{ \pm 1 \} ^ { n } } \left[ \operatorname* { s u p } _ { f \in { \mathcal { F } } _ { B } } \sum _ { i = 1 } ^ { n } f ( { \pmb x } _ { i } ) \gamma _ { i } \right] \leqslant { \frac { B { \sqrt { \operatorname { t r } [ k ( { \pmb X } , { \pmb X } ) ] } } } { n } } .
+$$
+
+By Lemma C.2, with probability at least $1 - \delta / 3$ we have
+
+$$
+\| f ^ { * } \| _ { \mathcal { H } } \leqslant \sqrt { \pmb { y } ^ { \top } \left( k ( \pmb { X } , \pmb { X } ) + \lambda ^ { 2 } \pmb { I } \right) ^ { - 1 } \pmb { y } } + \frac { \sigma } { \lambda } \left( \sqrt { n } + \sqrt { 2 \log ( 3 / \delta ) } \right) \triangleq B ^ { \prime } .
+$$
+
+We also recall the standard generalization bound from Rademacher complexity (see e.g. (Mohri et al., 2012)): with probability at least $1 - \delta$ , we have
+
+$$
+\operatorname* { s u p } _ { f \in \mathcal { F } } \left\{ \mathbb { E } _ { ( \alpha , y ) \sim \mathcal { D } } [ \ell ( f ( \boldsymbol { x } ) , y ) ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( \boldsymbol { x } _ { i } ) , y _ { i } ) \right\} \leqslant 2 \hat { \mathcal { R } } _ { S } ( \mathcal { F } ) + 3 \sqrt { \frac { \log ( 2 / \delta ) } { 2 n } } .
+$$
+
+Then it is tempting to apply the above bound on the function class $\mathcal { F } _ { B ^ { \prime } }$ which contains $f ^ { * }$ . However, we are not yet able to do so, because $B ^ { \prime }$ depends on the data $\boldsymbol { X }$ and $\textbf {  { y } }$ . To deal with this, we use a standard $\epsilon$ -net argument on the interval that $B ^ { \prime }$ must lie in: (note that $\| { \pmb y } \| = O ( { \sqrt { n } } ) ,$ )
+
+$$
+B ^ { \prime } \in \left[ \frac { \sigma } { \lambda } \left( \sqrt { n } + \sqrt { 2 \log ( 3 / \delta ) } \right) , \frac { \sigma } { \lambda } \left( \sqrt { n } + \sqrt { 2 \log ( 3 / \delta ) } \right) + O \left( \frac { n } { \lambda } \right) \right] .
+$$
+
+The above interval has length ${ \it O } \left( \frac { n } { \lambda } \right)$ , so its $\epsilon$ -net $\mathcal { N }$ has size $\begin{array} { r } { O \left( \frac { n } { \epsilon \lambda } \right) } \end{array}$ . Using a union bound, we apply the generalization bound (15) simultaneously on $\mathcal { F } _ { B }$ for all $B \in { \mathcal { N } }$ : with probability at least $1 - \delta / 3$ we have
+
+$$
+\operatorname* { s u p } _ { f _ { B } \in \mathcal { F } } \left\{ \mathbb { E } _ { ( \alpha , y ) \sim \mathcal { D } } [ \ell ( f ( \alpha ) , y ) ] - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( f ( x _ { i } ) , y _ { i } ) \right\} \leqslant 2 \hat { \mathcal { R } } _ { S } ( \mathcal { F } _ { B } ) + O \left( \sqrt { \frac { \log \frac { n } { \epsilon \delta \lambda } } { n } } \right) , \quad \forall B \in \mathcal { N }
+$$
+
+By definition there exists $B \in \mathcal N$ such that $B ^ { \prime } \leqslant B \leqslant B ^ { \prime } + \epsilon .$ . Then we also have $f ^ { * } \in \mathcal { F } _ { B }$ . Using the above bound on this particular $B$ , and putting all parts together, we know that with probability at least $1 - \delta$ ,
+
+$$
+\begin{array} { r l } & { \quad \forall _ { \{ u , v \} = 1 , \ldots } [ f _ { v } ^ { ( 1 ) } ( \overline { { \rho } } ( \overline { { \rho } } ( x , \overline { { x } } ) ) ^ { - 1 } ) ] } \\ & { \quad \in \mathbb { I } _ { \frac { 1 } { 2 } \leq \frac { 1 } { \pi } } \frac { \sum _ { j } ^ { \prime } \Gamma \left( f _ { v } ^ { ( 1 ) } ( x , \overline { { x } } ) \right) + \mathcal { B } _ { c } \zeta _ { j } [ P _ { v } ] + \mathcal { O } \left( \sqrt { \frac { \| \overline { { \theta } } ( \overline { { x } } , \overline { { x } } )  } { \pi } } } { \pi\right) } } \\ & { \quad \in \frac { \mathcal { B } _ { c } } { \pi } \frac { 1 } { \pi } \frac { \sum _ { j } ^ { \prime } \Gamma \left( j _ { v } ^ { ( 1 ) } ( \overline { { \rho } } ( x , \overline { { x } } ) ) \right) - \mathcal { B } _ { c } } { \pi } + \frac { \alpha } { 2 \sqrt { N } } \sqrt { \frac { \| \mathrm { d } ( \overline { { \nu } } ( x , \overline { { x } } ) ) \| } { \pi } } + \sigma \sqrt { \frac { \| \mathrm { d } ( \overline { { \nu } } ( \overline { { \rho } } ( \overline { { \rho } } ( \overline { { \rho } } ( \overline { { \rho } } ) ) ) ) } { \pi } } } \\ & { \quad \quad + \frac { 2 ( l ^ { ( 1 ) } + \mathcal { C } ) \sqrt { \pi } \Gamma \left( \overline { { \mathrm { R } } } ( ( \overline { { X } } , \overline { { x } } ) ) \right) } { \pi } + \sigma \left( \sqrt { \frac { \| \mathrm { d } ( \overline { { \nu } } ( \overline { { \rho } } , \overline { { x } } ) ) \| } { \pi } } \right) } \\ &  \quad \in \frac { \mathcal { B } _ { c } } { \pi } \frac { 1 } { \sqrt { \pi } ^ { \prime } ( N ( \overline { { \rho } } ( \overline { { \rho } } ( \overline { { X } } , \overline { { X } } ) ) ) - \frac { \rho } { 2 } } + \frac  \mathcal { O } \left( \sqrt { \frac { \| \mathrm { d } ( \overline { { X } } , \overline { { X } } ) \| } { \pi } } { \pi } \right) + \sigma \sqrt  \frac { \| \mathrm { d } ( \overline { { X } } ) \| }   \end{array}
+$$
+
+Then, using $\operatorname { t r } [ k ( X , X ) ] = O ( n )$ and choosing $\epsilon = 1$ , we obtain
+
+$$
+\begin{array} { r l } & { \quad \mathbb { E } _ { ( \alpha , y ) \sim \mathcal { D } } [ \ell ( f ^ { * } ( x ) , y ) ] } \\ & { \leqslant \displaystyle \frac { \lambda } { 2 } \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } + O \left( \frac { \sigma } { \lambda } \right) + \sigma \sqrt { \frac { 2 \log ( 3 / \delta ) } { n } } } \\ & { \quad \quad + O \left( \sqrt { \frac { y ^ { \top } \left( k ( X , X ) \right) ^ { - 1 } y } { n } } \right) + O \left( \frac { \sigma } { \lambda \sqrt { n } } \sqrt { 2 \log ( 3 / \delta ) } \right) + O \left( \frac { 1 } { \sqrt { n } } \right) + O \left( \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) } \\ & { \leqslant \displaystyle \frac { \lambda + O ( 1 ) } { 2 } \sqrt { \frac { y ^ { \top } \left( k ( X , X ) \right) ^ { - 1 } y } { n } } + O \left( \frac { \sigma } { \lambda } + \sigma \sqrt { \frac { \log ( 1 / \delta ) } { n } } + \frac { \sigma } { \lambda } \sqrt { \frac { \log ( 1 / \delta ) } { n } } + \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) . } \end{array}
+$$
+
+# C.2 PROOF OF THEOREM 5.2
+
+Proof of Theorem 5.2. We will apply Theorem 5.1 in this proof. Note that we cannot directly apply it on $\{ ( \pmb { x } _ { i } , y _ { i } , \tilde { y } _ { i } ) \}$ because the mean of $\tilde { y } _ { i } - y _ { i }$ is non-zero (conditioned on $( x _ { i } , y _ { i } ) )$ . Nevertheless, this issue can be resolved by considering $\{ ( { \pmb x } _ { i } , ( 1 - 2 p ) y _ { i } , \tilde { y } _ { i } ) \}$ instead.4 Then we can easily check that conditioned on $y _ { i } , \tilde { y } _ { i } - ( 1 - 2 p ) y _ { i }$ has mean 0 and is subgaussian with parameter $\sigma = O ( { \sqrt { p } } )$ . With this change, we are ready to apply Theorem 5.1.
+
+Define the following ramp loss for $u \in \mathbb { R } , \bar { y } \in \{ \pm ( 1 - 2 p ) \}$ :
+
+$$
+\ell ^ { \mathrm { r a m p } } ( u , \bar { y } ) = \left\{ \begin{array} { l l } { ( 1 - 2 p ) , } & { u \bar { y } \leqslant 0 , } \\ { ( 1 - 2 p ) - \frac { 1 } { 1 - 2 p } u \bar { y } , } & { 0 < u \bar { y } < ( 1 - 2 p ) ^ { 2 } , } \\ { 0 , } & { u \bar { y } \geqslant ( 1 - 2 p ) ^ { 2 } . } \end{array} \right.
+$$
+
+It is easy to see that $\ell ^ { \mathrm { r a m p } } ( u , \bar { y } )$ is 1-Lipschitz in $u$ for $\bar { y } \in \{ \pm ( 1 - 2 p ) \}$ , and satisfies $\ell ^ { \mathrm { r a m p } } ( \bar { y } , \bar { y } ) = 0$ for $\bar { y } \in \{ \pm ( 1 - 2 p ) \}$ . Then by Theorem 5.1, with probability at least $1 - \delta$ ,
+
+$$
+\mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } } \left[ \ell ^ { \mathrm { r a m p } } ( f ^ { * } ( \pmb { x } ) , ( 1 - 2 p ) \pmb { y } ) \right]
+$$
+
+$$
+\begin{array} { r l } & { \leqslant \frac { \lambda + O ( 1 ) } { 2 } \sqrt { \frac { ( 1 - 2 p ) y ^ { \top } ( k ( X , X ) ) ^ { - 1 } \cdot ( 1 - 2 p ) y } { n } } + O \left( \frac { \sqrt { p } } { \lambda } + \sqrt { \frac { p \log ( 1 / \delta ) } { n } } + \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) } \\ & { = \frac { ( 1 - 2 p ) ( \lambda + O ( 1 ) ) } { 2 } \sqrt { \frac { y ^ { \top } ( k ( X , X ) ) ^ { - 1 } y } { n } } + O \left( \frac { \sqrt { p } } { \lambda } + \sqrt { \frac { p \log ( 1 / \delta ) } { n } } + \sqrt { \frac { \log \frac { n } { \delta \lambda } } { n } } \right) . } \end{array}
+$$
+
+Note that $\ell ^ { \mathrm { { r a m p } } }$ also bounds the 0-1 loss (classification error) as
+
+$$
+\begin{array} { r } { \mathrm { r a m p } ( u , ( 1 - 2 p ) y ) \geqslant ( 1 - 2 p ) \mathbb { I } [ \mathrm { s g n } ( u ) \neq \mathrm { s g n } ( y ) ] = ( 1 - 2 p ) \mathbb { I } [ \mathrm { s g n } ( u ) \neq y ] , \quad \forall u \in \mathbb { R } , y \in \{ \pm 1 \} . } \end{array}
+$$
+
+Then the conclusion follows.
+
+# C.3 PROOF OF THEOREM 5.3
+
+Proof of Theorem 5.3. By definition, we have $\mathbb { E } \big [ \tilde { \pmb { y } } _ { i } \big | c _ { i } \big ] = \mathbb { E } \big [ { e } ^ { ( \tilde { c } _ { i } ) } \big | c _ { i } \big ] = { \pmb { p } } _ { c _ { i } }$ (the $c _ { i }$ -th column of $P _ { \mathrm { ~ , ~ } }$ ). This means $\mathbb { E } [ \tilde { Y } | Q ] = Q$ . Therefore we can view $Q$ as an encoding of clean labels, and then the observed noisy labels $\tilde { Y }$ is $Q$ plus a zero-mean noise. (The noise is always bounded by 1 so is subgaussian with parameter 1.) This enables us to apply Theorem 5.1 to each $f ^ { ( h ) }$ , which says that with probability at least $1 - \delta ^ { \prime }$ ,
+
+$$
+\mathfrak { L } _ { ( \mathbf { x } , c ) \sim \mathcal { D } } \left| f ^ { ( h ) } ( \mathbf { x } ) - p _ { h , c } \right| \leqslant \frac { \lambda + O ( 1 ) } { 2 } \sqrt { \frac { ( g ^ { ( h ) } ) ^ { \top } ( k ( \mathbf { X } , \mathbf { X } ) ) ^ { - 1 } q ^ { ( h ) } } { n } } + O \left( \frac { 1 } { \lambda } + \sqrt { \frac { \log \frac { 1 } { \delta ^ { \prime } } } { n } } + \sqrt { \frac { \log \frac { 1 } { \delta ^ { \prime } } } { n } } \right)
+$$
+
+Lettinevery $\begin{array} { r } { \delta ^ { \prime } = \frac { \delta } { K } } \end{array}$ and taking a union bound over neously with probability at least $h \in [ K ]$ , we know that the above bound holds for $h$ $1 - \delta$
+
+Now we proceed to bound the classification error. Note that ˇ ˇ $c \not \in \mathrm { a r g m a x } _ { h \in [ K ] } f ^ { ( h ) } ( { \pmb x } )$ implies $\begin{array} { r } { \sum _ { h = 1 } ^ { K } \left| f ^ { ( h ) } ( { \pmb x } ) - p _ { h , c } \right| \geqslant \mathtt { g a p } } \end{array}$ . Therefore the classification error can be bounded as
+
+$$
+\begin{array} { r l } & { \qquad \underset { ( \alpha , c ) \sim \mathcal { D } } { \operatorname* { P r } } \left[ c \neq \mathrm { a r g m a x } _ { h \in [ K ] } f ^ { ( h ) } ( \boldsymbol { x } ) \right] \leqslant \underset { ( \alpha , c ) \sim \mathcal { D } } { \operatorname* { P r } } \left[ \underset { h = 1 } { \overset { K } { \sum } } \left| f ^ { ( h ) } ( \boldsymbol { x } ) - p _ { h , c } \right| \geqslant \mathrm { g a p } \right] } \\ & { \leqslant \frac { 1 } { \mathrm { g a p } } \mathbb { E } _ { ( \boldsymbol { x } , c ) \sim \mathcal { D } } \left[ \underset { h = 1 } { \overset { K } { \sum } } \left| f ^ { ( h ) } ( \boldsymbol { x } ) - p _ { h , c } \right| \right] = \frac { 1 } { \mathrm { g a p } } \underset { h = 1 } { \overset { K } { \sum } } \mathbb { E } _ { ( \boldsymbol { x } , c ) \sim \mathcal { D } } \left[ \left| f ^ { ( h ) } ( \boldsymbol { x } ) - p _ { h , c } \right| \right] } \\ & { \leqslant \frac { 1 } { \mathrm { g a p } } \left( \frac { \lambda + O ( 1 ) } { 2 } \underset { h = 1 } { \overset { K } { \sum } } \sqrt { \frac { ( \boldsymbol { q } ^ { ( h ) } ) ^ { \top } ( k ( \boldsymbol { X } , \boldsymbol { X } ) ) ^ { - 1 } \boldsymbol { q } ^ { ( h ) } } { n } } + K \cdot O \left( \frac { 1 } { \lambda } + \sqrt { \frac { \log \frac { 1 } { \delta ^ { \prime } } } { n } } + \sqrt { \frac { \log \frac { n } { \delta ^ { \prime } \lambda } } { n } } \right) \right) , } \end{array}
+$$
+
+completing the proof.
+
+# D EXPERIMENT DETAILS AND ADDITIONAL FIGURES
+
+In Setting 1, we train a two-layer neural network with 10,000 hidden neurons on MNIST (“5” vs “8”). In Setting 2, we train a CNN, which has 192 channels for each of its 11 layers, on CIFAR (“airplanes” vs “automobiles”). We do not have biases in these two networks. In Setting 3, we use the standard ResNet-34.
+
+In Settings 1 and 2, we use a fixed learning rate for GD or SGD, and we do not use tricks like batch normalization, data augmentation, dropout, etc., except the difference trick in Appendix A. We also freeze the first and the last layer of the CNN and the second layer of the fully-connected net.5
+
+In Setting 3, we use SGD with 0.9 momentum, weight decay of $5 \times 1 0 ^ { - 4 }$ , and batch size 128. The learning rate is 0.1 initially and is divided by 10 after 82 and 123 epochs (164 in total). Since we observe little over-fitting to noise in the first stage of training (before learning rate decay), we restrict the regularization power of AUX by applying weight decay on auxiliary variables, and dividing their weight decay factor by 10 after each learning rate decay.
+
+See Figures 5 to 8 for additional results for Setting 2 with different $\lambda$ ’s.
+
+![](images/baa534318edc0fa0e4d388e697b70cd2933f9f6b35ff2e223f6e1819eea3ef55.jpg)  
+Figure 5: Training (dashed) $\&$ test (solid) errors vs. epoch for Setting 2. Noise rate $= ~ 2 0 \%$ , $\lambda \in \{ 0 . 2 5 , 0 . 5 , 1 , 2 , 4 , 8 \}$ . Training error of AUX is measured with auxiliary variables.
+
+![](images/d1a90e8d9dab198702b5eafc86cadf3829ef9f98763f359bc26e4e01e98f5690.jpg)  
+Figure 6: Setting 2, $\| \mathbf { W } ^ { ( 7 ) } \| _ { F }$ and $\lVert W ^ { ( 7 ) } - W ^ { ( 7 ) } ( 0 ) \rVert _ { F }$ during training. Noise rate $= 2 0 \%$ , $\lambda = 4$
+
+![](images/ab524ac44e1c735189a643d68f5519f6dc2f25e3e64c90a5295f91faa7fb961c.jpg)  
+Figure 7: Setting 2, $\| \pmb { W } ^ { ( 4 ) } \| _ { F }$ and $\lVert W ^ { ( 4 ) } - W ^ { ( 4 ) } ( 0 ) \rVert _ { F }$ during training. Noise rate $= 0$ , $\lambda = 2$
+
+![](images/24f1aa5f167e4a079e4a6a51d1ea9b5442a0d212a3f6f81e798683fbf3f19a4e.jpg)  
+Figure 8: Setting 2, $\| \mathbf { W } ^ { ( 7 ) } \| _ { F }$ and $\lVert W ^ { ( 7 ) } - W ^ { ( 7 ) } ( 0 ) \rVert _ { F }$ during training. Noise rate $= 0$ , $\lambda = 2$

parse/train/HyeVtoRqtQ/HyeVtoRqtQ.md

@@ -0,0 +1,443 @@
+# TRELLIS NETWORKS FOR SEQUENCE MODELING
+
+Shaojie Bai Carnegie Mellon University
+
+J. Zico Kolter Carnegie Mellon University and Bosch Center for AI
+
+Vladlen Koltun Intel Labs
+
+# ABSTRACT
+
+We present trellis networks, a new architecture for sequence modeling. On the one hand, a trellis network is a temporal convolutional network with special structure, characterized by weight tying across depth and direct injection of the input into deep layers. On the other hand, we show that truncated recurrent networks are equivalent to trellis networks with special sparsity structure in their weight matrices. Thus trellis networks with general weight matrices generalize truncated recurrent networks. We leverage these connections to design high-performing trellis networks that absorb structural and algorithmic elements from both recurrent and convolutional models. Experiments demonstrate that trellis networks outperform the current state of the art methods on a variety of challenging benchmarks, including word-level language modeling and character-level language modeling tasks, and stress tests designed to evaluate long-term memory retention. The code is available here1.
+
+# 1 INTRODUCTION
+
+What is the best architecture for sequence modeling? Recent research has produced significant progress on multiple fronts. Recurrent networks, such as LSTMs, continue to be optimized and extended (Merity et al., 2018b; Melis et al., 2018; Yang et al., 2018; Trinh et al., 2018). Temporal convolutional networks have demonstrated impressive performance, particularly in modeling longrange context (van den Oord et al., 2016; Dauphin et al., 2017; Bai et al., 2018). And architectures based on self-attention are gaining ground (Vaswani et al., 2017; Santoro et al., 2018).
+
+In this paper, we introduce a new architecture for sequence modeling, the Trellis Network. We aim to both improve empirical performance on sequence modeling benchmarks and shed light on the relationship between two existing model families: recurrent and convolutional networks.
+
+On the one hand, a trellis network is a special temporal convolutional network, distinguished by two unusual characteristics. First, the weights are tied across layers. That is, weights are shared not only by all time steps but also by all network layers, tying them into a regular trellis pattern. Second, the input is injected into all network layers. That is, the input at a given time-step is provided not only to the first layer, but directly to all layers in the network. So far, this may seem merely as a peculiar convolutional network for processing sequences, and not one that would be expected to perform particularly well.
+
+Yet on the other hand, we show that trellis networks generalize truncated recurrent networks (recurrent networks with bounded memory horizon). The precise derivation of this connection is one of the key contributions of our work. It allows trellis networks to serve as bridge between recurrent and convolutional architectures, benefitting from algorithmic and architectural techniques developed in either context. We leverage these relationships to design high-performing trellis networks that absorb ideas from both architectural families. Beyond immediate empirical gains, these connections may serve as a step towards unification in sequence modeling.
+
+We evaluate trellis networks on challenging benchmarks, including word-level language modeling on the standard Penn Treebank (PTB) and the much larger WikiText-103 (WT103) datasets; character-level language modeling on Penn Treebank; and standard stress tests (e.g. sequential MNIST, permuted MNIST, etc.) designed to evaluate long-term memory retention. On word-level
+
+Penn Treebank, a trellis network outperforms by more than a unit of perplexity the recent architecture search work of Pham et al. (2018), as well as the recent results of Melis et al. (2018), which leveraged the Google Vizier service for exhaustive hyperparameter search. On character-level Penn Treebank, a trellis network outperforms the thorough optimization work of Merity et al. (2018a). On word-level WikiText-103, a trellis network outperforms by $7 . 6 \%$ in perplexity the contemporaneous self-attention-based Relational Memory Core (Santoro et al., 2018), and by $1 1 . 5 \%$ the work of Merity et al. (2018a). (Concurrently with our work, Dai et al. (2019) employ a transformer and achieve even better results on WikiText-103.) On stress tests, trellis networks outperform recent results achieved by recurrent networks and self-attention (Trinh et al., 2018). It is notable that the prior state of the art across these benchmarks was held by models with sometimes dramatic mutual differences.
+
+# 2 BACKGROUND
+
+Recurrent networks (Elman, 1990; Werbos, 1990; Graves, 2012), particularly with gated cells such as LSTMs (Hochreiter & Schmidhuber, 1997) and GRUs (Cho et al., 2014), are perhaps the most popular architecture for modeling temporal sequences. Recurrent architectures have been used to achieve breakthrough results in natural language processing and other domains (Sutskever et al., 2011; Graves, 2013; Sutskever et al., 2014; Bahdanau et al., 2015; Vinyals et al., 2015; Karpathy & Li, 2015). Convolutional networks have also been widely used for sequence processing (Waibel et al., 1989; Collobert et al., 2011). Recent work indicates that convolutional networks are effective on a variety of sequence modeling tasks, particularly ones that demand long-range information propagation (van den Oord et al., 2016; Kalchbrenner et al., 2016; Dauphin et al., 2017; Gehring et al., 2017; Bai et al., 2018). A third notable approach to sequence processing that has recently gained ground is based on self-attention (Vaswani et al., 2017; Santoro et al., 2018; Chen et al., 2018). Our work is most closely related to the first two approaches. In particular, we establish a strong connection between recurrent and convolutional networks and introduce a model that serves as a bridge between the two. A related recent theoretical investigation showed that under a certain stability condition, recurrent networks can be well-approximated by feed-forward models (Miller & Hardt, 2018).
+
+There have been many combinations of convolutional and recurrent networks (Sainath et al., 2015). For example, convolutional LSTMs combine convolutional and recurrent units (Donahue et al., 2015; Venugopalan et al., 2015; Shi et al., 2015). Quasi-recurrent neural networks interleave convolutional and recurrent layers (Bradbury et al., 2017). Techniques introduced for convolutional networks, such as dilation, have been applied to RNNs (Chang et al., 2017). Our work establishes a deeper connection, deriving a direct mapping across the two architectural families and providing a structural bridge that can incorporate techniques from both sides.
+
+# 3 SEQUENCE MODELING AND TRELLIS NETWORKS
+
+Sequence modeling. Given an input $x _ { 1 : T } = x _ { 1 } , . . . , x _ { T }$ with sequence length $T$ , a sequence model is any function $G : { \overset { \smile } { \chi } } ^ { T } \to { \mathcal { V } } ^ { T }$ such that
+
+$$
+y _ { 1 : T } = y _ { 1 } , \ldots , y _ { T } = G ( x _ { 1 } , \ldots , x _ { T } ) ,
+$$
+
+where $y _ { t }$ should only depend on $x _ { 1 : t }$ and not on $x _ { t + 1 : T }$ (i.e. no leakage of information from the future). This causality constraint is essential for autoregressive modeling.
+
+In this section, we describe a new architecture for sequence modeling, referred to as a trellis network or TrellisNet. In particular, we provide an atomic view of TrellisNet, present its fundamental features, and highlight the relationship to convolutional networks. Section 4 will then elaborate on the relationship of trellis networks to convolutional and recurrent models.
+
+Notation. We use $\boldsymbol { x } _ { 1 : T } = ( x _ { 1 } , \dots , x _ { T } )$ to denote a length- $\mathcal { T }$ input sequence, where vector $\boldsymbol { x } _ { t } \in \mathbb { R } ^ { p }$ is the input at time step $t$ . Thus $\boldsymbol { x } _ { 1 : T } \in \mathbb { R } ^ { T \times p }$ . We use $\boldsymbol { z } _ { t } ^ { ( i ) } \in \mathbb { R } ^ { q }$ to represent the hidden unit at time $t$ in layer $i$ of the network. We use $\mathrm { C o n v 1 D } ( x ; W )$ to denote a 1D convolution with a kernel $W$ applied to input $x = x _ { 1 : T }$ .
+
+A basic trellis network. At the most basic level, a feature vector z(i+1)t+1 at time step $t + 1$ and level $i + 1$ of TrellisNet is computed via three steps, illustrated in Figure 1a:
+
+![](images/3d673cc8be5729c42eb3369d3ed5014f3cdb07ac0651d4b7ff926242e384655a.jpg)  
+Figure 1: The interlayer transformation of TrellisNet, at an atomic level (time steps $t$ and $t + 1$ layers $i$ and $i + 1$ ) and on a longer sequence (time steps 1 to 8, layers $i$ and $i + 1$ ).
+
+1. The hidden input comprises the hidden outputs z(i)t , z(i)t+1 $z _ { t + 1 } ^ { ( i ) } \in \mathbb { R } ^ { q }$ from the previous layer $i$ , as well as an injection of the input vectors $x _ { t } , x _ { t + 1 }$ . At level 0, we initialize to $z _ { t } ^ { ( 0 ) } = \mathbf { 0 }$ .
+
+2. A pre-activation output $\hat { z } _ { t + 1 } ^ { ( i + 1 ) } \in \mathbb { R } ^ { r }$ is produced by a feed-forward linear transformation:
+
+$$
+\hat { z } _ { t + 1 } ^ { ( i + 1 ) } = W _ { 1 } \left[ { x _ { t } } ^ { } \atop { z _ { t } ^ { ( i ) } } \right] + W _ { 2 } \left[ { x _ { t + 1 } } ^ { } \right] ,
+$$
+
+where W1, W2 ∈ Rr×(p+q) are weights, and r is the size of the pre-activation output zˆ(i+1)t+1 ， (Here and throughout the paper, all linear transformations can include additive biases. We omit these for clarity.)
+
+3. The output z(i+1)t+1 is produced by a nonlinear activation function $f : \mathbb { R } ^ { r } \times \mathbb { R } ^ { q } \to \mathbb { R } ^ { q }$ applied to the pre-activation output $\hat { z } _ { t + 1 } ^ { ( i + 1 ) }$ and the output $ { \boldsymbol { z } } _ { t } ^ { ( i ) }$ × →from the previous layer. More formally, $z _ { t + 1 } ^ { ( i + 1 ) } = f \left( \hat { z } _ { t + 1 } ^ { ( i + 1 ) } , z _ { t } ^ { ( i ) } \right)$ .
+
+A full trellis network can be built by tiling this elementary procedure across time and depth. Given   
+an input sequence $x _ { 1 : T }$ , we apply the same production procedure across all time steps and all layers,   
+using the same weights. The transformation is the same for all elements in the temporal dimen  
+sion and in the depth dimension. This is illustrated in Figure 1b. Note that since we inject the   
+same input sequence at every layer of the TrellisNet, we can precompute the linear transformation   
+$\tilde { x } _ { t + 1 } = W _ { 1 } ^ { x } x _ { t } + W _ { 2 } ^ { x } x _ { t + 1 }$ l layers propria $i$ . This identical linear combination of th linear combination of the hidden units, $i$ $\bar { W _ { 1 } ^ { z } z _ { t } ^ { ( i ) } + W _ { 2 } ^ { z } z _ { t + 1 } ^ { ( i ) } }$ $W _ { j } ^ { x } \in \mathbb { R } ^ { r \times p }$ $W _ { j } ^ { z } \in \mathbb { R } ^ { r \times q }$
+
+Now observe that in each level of the network, we are in effect performing a 1D convolution over the hidden units $z _ { 1 : T } ^ { ( i ) }$ . The output of this convolution is then passed through the activation function $f$ . Formally, with $\ddot { W } \in \mathbb { R } ^ { r \times q }$ as the kernel weight matrix, the computation in layer $i$ can be summarized as follows (Figure 1b):
+
+$$
+\hat { z } _ { 1 : T } ^ { ( i + 1 ) } = \mathrm { C o n v } \ln \left( z _ { 1 : T } ^ { ( i ) } ; W \right) + \tilde { x } _ { 1 : T } , \qquad z _ { 1 : T } ^ { ( i + 1 ) } = f \left( \hat { z } _ { 1 : T } ^ { ( i + 1 ) } , z _ { 1 : T - 1 } ^ { ( i ) } \right) .
+$$
+
+The resulting network operates in feed-forward fashion, with deeper elements having progressively larger receptive fields. There are, however, important differences from typical (temporal) convolutional networks. Notably, the filter matrix is shared across all layers. That is, the weights are tied not only across time but also across depth. (Vogel & Pock (2017) have previously tied weights across depth in image processing.) Another difference is that the transformed input sequence $\tilde { x } _ { 1 : T }$ is directly injected into each hidden layer. These differences and their importance will be analyzed further in Section 4.
+
+The activation function $f$ in Equation (3) can be any nonlinearity that processes the pre-activation output $\hat { z } _ { 1 : T } ^ { ( i + 1 ) }$ and the output from the previous layer $z _ { 1 : T - 1 } ^ { ( i ) }$ . We will later describe an activation function based on the LSTM cell. The rationale for its use will become clearer in light of the analysis presented in the next section.
+
+# 4 TRELLISNET, TCN, AND RNN
+
+In this section, we analyze the relationships between trellis networks, convolutional networks, and recurrent networks. In particular, we show that trellis networks can serve as a bridge between convolutional and recurrent networks. On the one hand, TrellisNet is a special form of temporal convolutional networks (TCN); this has already been clear in Section 3 and will be discussed further in Section 4.1. On the other hand, any truncated RNN can be represented as a TrellisNet with special structure in the interlayer transformations; this will be the subject of Section 4.2. These connections allow TrellisNet to harness architectural elements and regularization techniques from both TCNs and RNNs; this will be summarized in Section 4.3.
+
+# 4.1 TRELLISNET AND TCN
+
+We briefly introduce TCNs here, and refer the readers to Bai et al. (2018) for a more thorough discussion. Briefly, a temporal convolutional network (TCN) is a ConvNet that uses one-dimensional convolutions over the sequence. The convolutions are causal, meaning that, at each layer, the transformation at time $t$ can only depend on previous layer units at times $t$ or earlier, not from later points in time. Such approaches were used going back to the late 1980s, under the name of “time-delay neural networks” (Waibel et al., 1989), and have received significant interest in recent years due to their application in architectures such as WaveNet (van den Oord et al., 2016).
+
+In essence, TrellisNet is a special kind of temporal convolutional network. TCNs have two distinctive characteristics: 1) causal convolution in each layer to satisfy the causality constraint and 2) deep stacking of layers to increase the effective history length (i.e. receptive field). Trellis networks have both of these characteristics. The basic model presented in Section 3 can easily be elaborated with larger kernel sizes, dilated convolutions, and other architectural elements used in TCNs; some of these are reviewed further in Section 4.3.
+
+However, TrellisNet is not a general TCN. As mentioned in Section 3, two important differences are: 1) the weights are tied across layers and 2) the linearly transformed input $\tilde { x } _ { 1 : T }$ is injected into each layer. Weight tying can be viewed as a form of regularization that can stabilize training, support generalization, and significantly reduce the size of the model. Input injection mixes deep features with the original sequence. These structural characteristics will be further illuminated by the connection between trellis networks and recurrent networks, presented next.
+
+# 4.2 TRELLISNET AND RNN
+
+Recurrent networks appear fundamentally different from convolutional networks. Instead of operating on all elements of a sequence in parallel in each layer, an RNN processes one input element at a time and unrolls in the time dimension. Given a non-linearity $g$ (which could be a sigmoid or a more elaborate cell), we can summarize the transformations in an $L$ -layer RNN at time-step $t$ as follows:
+
+$$
+h _ { t } ^ { ( i ) } = g \left( W _ { h x } ^ { ( i ) } h _ { t } ^ { ( i - 1 ) } + W _ { h h } ^ { ( i ) } h _ { t - 1 } ^ { ( i ) } \right) \quad \mathrm { f o r } 1 \leq i \leq L , \qquad h _ { t } ^ { ( 0 ) } = x _ { t } .
+$$
+
+Despite the apparent differences, we will now show that any RNN unrolled to a finite length is equivalent to a TrellisNet with special sparsity structure in the kernel matrix $W$ . We begin by formally defining the notion of a truncated (i.e. finite-horizon) RNN.
+
+Definition 1. Given an RNN $\rho ,$ , a corresponding M-truncated RNN $\rho ^ { M }$ , applied to the sequence $x _ { 1 : T }$ , produces at time step $t$ the output $y _ { t }$ by applying $\rho$ to the sequence $x _ { t - M + 1 : t }$ (here $x _ { < 0 } = 0$ ).
+
+Theorem 1. Let $\rho ^ { M }$ be an $M$ -truncated RNN with $L$ layers and hidden unit dimensionality $d .$ . Then there exists an equivalent TrellisNet $\tau$ with depth $( M + L - 1 )$ and layer width (i.e. number of channels in each hidden layer) Ld. Specifically, for any $x _ { 1 : T }$ , $\rho ^ { M } ( x _ { 1 : T } ) = \tau _ { L ( d - 1 ) + 1 : L d } ( x _ { 1 : T } )$ (i.e. the TrellisNet outputs contain the RNN outputs).
+
+Theorem 1 states that any $M$ -truncated RNN can be represented as a TrellisNet. How severe of a restriction is $M$ -truncation? Note that $M$ -truncation is intimately related to truncated backpropagation-through-time (BPTT), used pervasively in training recurrent networks on long sequences. While RNNs can in principle retain unlimited history, there is both empirical and theoretical evidence that the memory horizon of RNNs is bounded (Bai et al., 2018; Khandelwal et al., 2018;
+
+Miller & Hardt, 2018). Furthermore, if desired, TrellisNets can recover exactly a common method of applying RNNs to long sequences – hidden state repackaging, i.e. copying the hidden state across subsequences. This is accomplished using an analogous form of hidden state repackaging, detailed in Appendix B.
+
+Proof of Theorem 1. Let $h _ { t , t ^ { \prime } } ^ { ( i ) } \in \mathbb { R } ^ { d }$ be the hidden state at time $t$ and layer $i$ of the truncated RNN $\rho ^ { t - t ^ { \prime } + 1 }$ (i.e., the RNN begun at time $t ^ { \prime }$ and run until time ). Note that without truncation, history starts at tiwe define $t ^ { \prime } = 1$ , so the hidden state(i.e. no history info $h _ { t } ^ { ( i ) }$ ofion $\rho$ can be equivalently expressed as f the clock starts in the future). $h _ { t , 1 } ^ { ( i ) }$ . When $t ^ { \prime } > t$ $h _ { t , t ^ { \prime } } = 0$
+
+By assumption, $\rho ^ { M }$ is an RNN defined by the following parameters: $\{ W _ { h x } ^ { ( i ) } , W _ { h h } ^ { ( i ) } , g , M \}$ , where $W _ { h h } ^ { ( i ) } \in \mathbb { R } ^ { w \times d }$ for all $i$ , $W _ { h x } ^ { ( 1 ) } \in \mathbb { R } ^ { w \times p }$ , and $W _ { h x } ^ { ( i ) } \in \mathbb { R } ^ { w \times d }$ for all $i = 2 , \ldots , L$ are the weight matrices at each layer ( $w$ is the dimension of pre-activation output). We now construct a TrellisNet $\tau$ according to the exact definition in Section 3, with parameters $\{ W _ { 1 } , W _ { 2 } , f \}$ , where
+
+$$
+\begin{array} { r } { W _ { 1 } = \left[ \begin{array} { c c c c c } { 0 } & { W _ { h h } ^ { ( 1 ) } } & { 0 } & { \cdots } & { 0 } \\ { 0 } & { 0 } & { W _ { h h } ^ { ( 2 ) } } & { \cdots } & { 0 } \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { 0 } & { 0 } & { \cdots } & { W _ { h h } ^ { ( L ) } } \end{array} \right] , W _ { 2 } = \left[ \begin{array} { c c c c c } { W _ { h x } ^ { ( 1 ) } } & { 0 } & { \cdots } & { 0 } & { 0 } \\ { 0 } & { W _ { h x } ^ { ( 2 ) } } & { \cdots } & { 0 } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } & { \vdots } \\ { 0 } & { 0 } & { \cdots } & { W _ { h x } ^ { ( L ) } } & { 0 } \end{array} \right] , } \end{array}
+$$
+
+such that $W _ { 1 } , W _ { 2 } \in \mathbb { R } ^ { L w \times ( p + L d ) }$ . We define a nonlinearity $f$ by $f ( \alpha , \beta ) = g ( \alpha )$ (i.e. applying $g$ only on the first entry).
+
+Let $t \in [ T ] , j \geq 0$ be arbitrary and fixed. We now claim that the hidden unit at time $t$ and layer $j$ of TrellisNet $\tau$ can be expressed in terms of hidden units at time $t$ in truncated forms of $\rho$ :
+
+$$
+z _ { t } ^ { ( j ) } = \left[ h _ { t , t - j + 1 } ^ { ( 1 ) } \quad h _ { t , t - j + 2 } ^ { ( 2 ) } \quad \ldots \quad h _ { t , t - j + L } ^ { ( L ) } \right] ^ { \top } \in \mathbb { R } ^ { L d } ,
+$$
+
+where z(jt is the time- $\cdot t$ hidden state at layer $j$ of $\tau$ and $h _ { t , t ^ { \prime } } ^ { ( i ) }$ is the time- $t$ hidden state at layer $i$ of $\rho ^ { t - t ^ { \prime } + 1 }$ .
+
+We prove Eq. (6) by induction on $j$ . As a base case, consider $j = 0$ ; i.e. the input layer of $\tau$ . Since $h _ { t , t ^ { \prime } } = 0$ when $t ^ { \prime } > t$ , we have that $z _ { j } ^ { ( 0 ) } = [ 0 \ : \ : 0 \ : \ : \ : . . \ : \ : \ : 0 ] ^ { \bar { \top } }$ . (Recall that in the input layer of TrellisNet we initialize $z _ { t } ^ { ( 0 ) } = \mathbf { 0 } .$ .) For the inductive step, suppose Eq. (6) holds for layer $j$ , and consider layer $j + 1$ . By the feed-forward transformation of TrellisNet defined in Eq. (2) and the nonlinearity $f$ we defined above, we have:
+
+$$
+\begin{array} { r l } & { \hat { z } _ { t } ^ { ( j + 1 ) } = W _ { 1 } \left[ \begin{array} { c } { x _ { t - 1 } } \\ { z _ { t - 1 } ^ { ( j ) } } \end{array} \right] + W _ { 2 } \left[ \begin{array} { c } { x _ { t } } \\ { z _ { t } ^ { ( j ) } } \end{array} \right] } \\ & { \quad = \left[ \begin{array} { c c c c c } { 0 } & { W _ { h h } ^ { ( 1 ) } } & { 0 } & { \cdots } & { 0 } \\ { 0 } & { W _ { h h } ^ { ( 2 ) } } & { \cdots } & { \vdots } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { 0 } & { \cdots } & { W _ { h h } ^ { ( L ) } } \end{array} \right] \left[ \begin{array} { c } { x _ { t - 1 } } \\ { h _ { t - 1 , t - j } ^ { ( 1 ) } } \\ { \vdots } \\ { h _ { t - 1 , t - j + L - 1 } ^ { ( L ) } } \end{array} \right] + \left[ \begin{array} { c c c c c } { W _ { h x } ^ { ( 1 ) } } & { 0 } & { \cdots } & { 0 } & { 0 } \\ { 0 } & { W _ { h x } ^ { ( 2 ) } } & { \cdots } & { 0 } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { 0 } & { \cdots } & { W _ { h x } ^ { ( L ) } } & { 0 } \end{array} \right] \left[ \begin{array} { c } { x _ { t } } \\ { h _ { t , t - j + 1 } ^ { ( 1 ) } } \\ { \vdots } \\ { h _ { t , t - j + L - 1 } ^ { ( L ) } } \end{array} \right] } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { \quad = \left[ \begin{array} { c c c c } { W _ { h h } ^ { ( 1 ) } h _ { t - 1 , t - j } ^ { ( 1 ) } + W _ { h x } ^ { ( 1 ) } x _ { t } } & & \\ & { \vdots } & \\ { W _ { h h } ^ { ( L ) } h _ { t - 1 , t - j + L - 1 } ^ { ( L ) } + W _ { h x } ^ { ( L ) } h _ { t , t - j + L - 1 } ^ { ( L - 1 ) } } \end{array} \right] } \\ & { z _ { t } ^ { ( j + 1 ) } = f ( \hat { z } _ { t } ^ { ( j + 1 ) } , z _ { t - 1 } ^ { ( j ) } ) = g ( \hat { z } _ { t } ^ { ( j + 1 ) } ) = \left[ h _ { t , t - j } ^ { ( 1 ) } \quad h _ { t , t - j + 1 } ^ { ( 2 ) } \quad . . . \quad h _ { t , t - j + L - 1 } ^ { ( L ) } \right] ^ { \top } } \end{array}
+$$
+
+where in Eq. (10) we apply the RNN non-linearity $g$ following Eq. (4). Therefore, by induction, we have shown that Eq. (6) holds for all $j \geq 0$ .
+
+If TrellisNet $\tau$ has $M + L - 1$ layers, then at the final layer we have $\boldsymbol { z } _ { t } ^ { ( M + L - 1 ) } = \left[ \cdot \cdot \cdot \ \cdot \ \cdot \ h _ { t , t + 1 - M } ^ { ( L ) } \right] ^ { \top }$ Since $\rho ^ { M }$ is an $L$ -layer $M$ -truncated RNN, this (taking the last $d$ channels of z(M+L−1)) is exactly the output of $\rho ^ { M }$ at time $t$ .
+
+![](images/423da487a84995f6d8a91f2310ae894f78327132c9bac036c255e90994733ad1.jpg)  
+Figure 2: Representing a truncated 2-layer RNN $\rho ^ { M }$ as a trellis network $\tau$ . (a) Each unit of $\tau$ has three groups, which house the input, first-layer hidden vector, and second-layer hidden vector of $\rho ^ { M }$ , respectively. (b) Each group in the hidden unit of $\tau$ in level $i + 1$ at time step $t + 1$ is computed by a linear combination of appropriate groups of hidden units in level $i$ at time steps $t$ and $t + 1$ . The linear transformations form a mixed group convolution that reproduces computation in $\rho ^ { M }$ . (Nonlinearities not shown for clarity.)
+
+In other words, we have shown that $\rho ^ { M }$ is equivalent to a TrellisNet with sparse kernel matrices $W _ { 1 } , W _ { 2 }$ . This completes the proof.
+
+Note that the convolutions in the TrellisNet $\tau$ constructed in Theorem 1 are sparse, as shown in Eq. (5). They are related to group convolutions (Krizhevsky et al., 2012), but have an unusual form because group $k$ at time $t$ is convolved with group $k - 1$ at time $t + 1$ . We refer to these as mixed group convolutions. Moreover, while Theorem 1 assumes that all layers of $\rho ^ { M }$ have the same dimensionality $d$ for clarity, the proof easily generalizes to cases where each layer has different widths.
+
+For didactic purposes, we recap and illustrate the construction in the case of a 2-layer RNN. The key challenge is that a na¨ıve unrolling of the RNN into a feed-forward network does not produce a convolutional network, since the linear transformation weights are not constant across a layer. The solution, illustrated in Figure 2a, is to organize each hidden unit into groups of channels, such that each TrellisNet unit represents 3 RNN units simultaneously (for $x _ { t } , h _ { t } ^ { ( 1 ) } , h _ { t } ^ { ( 2 ) } )$ . Each TrellisNet unit thus has $( p + 2 d )$ channels. The interlayer transformation can then be expressed as a mixed group convolution, illustrated in Figure 2b. This can be represented as a sparse convolution with the structure given in Eq. (5) (with $L = 2$ ). Applying the nonlinearity $g$ on the pre-activation output, this exactly reproduces the transformations in the original 2-layer RNN.
+
+The TrellisNet that emerges from this construction has special sparsity structure in the weight matrix. It stands to reason that a general TrellisNet with an unconstrained (dense) weight matrix $W$ may have greater expressive power: it can model a broader class of transformations than the original RNN $\mathsf { \bar { \rho } } ^ { M }$ . Note that while the hidden channels of the TrellisNet $\tau$ constructed in the proof of Theorem 1 are naturally arranged into groups that represent different layers of the RNN $\bar { \rho } ^ { M }$ (Eq. (6)), an unconstrained dense weight matrix $W$ no longer admits such an interpretation. A model defined by a dense weight matrix is fundamentally distinct from the RNN $\rho ^ { M }$ that served as our point of departure. We take advantage of this expressivity and use general weight matrices $W$ , as presented in Section 3, in our experiments. Our ablation analysis will show that such generalized dense transformations are beneficial, even when model capacity is controlled for.
+
+The proof of Theorem 1 did not delve into the inner structure of the nonlinear transformation $g$ in RNN (or $f$ in the constructed TrellisNet). For a vanilla RNN, for instance, $f$ is usually an elementwise sigmoid or tanh function. But the construction in Theorem 1 applies just as well to RNNs with structured cells, such as LSTMs and GRUs. We adopt LSTM cells for the TrellisNets in our experiments and provide a detailed treatment of this nonlinearity in Section 5.1 and Appendix A.
+
+# 4.3 TRELLISNET AS A BRIDGE BETWEEN RECURRENT AND CONVOLUTIONAL MODELS
+
+In Section 4.1 we concluded that TrellisNet is a special kind of TCN, characterized by weight tying and input injection. In Section 4.2 we established that TrellisNet is a generalization of truncated
+
+RNNs. These connections along with the construction in our proof of Theorem 1 allow TrellisNets to benefit significantly from techniques developed originally for RNNs, while also incorporating architectural and algorithmic motifs developed for convolutional networks. We summarize a number of techniques here. From recurrent networks, we can integrate 1) structured nonlinear activations (e.g. LSTM and GRU gates); 2) variational RNN dropout (Gal & Ghahramani, 2016); 3) recurrent DropConnect (Merity et al., 2018b); and 4) history compression and repackaging. From convolutional networks, we can adapt 1) larger kernels and dilated convolutions (Yu & Koltun, 2016); 2) auxiliary losses at intermediate layers (Lee et al., 2015; Xie & Tu, 2015); 3) weight normalization (Salimans & Kingma, 2016); and 4) parallel convolutional processing. Being able to directly incorporate techniques from both streams of research is one of the benefits of trellis networks. We leverage this in our experiments and provide a more comprehensive treatment of these adaptations in Appendix B.
+
+# 5 EXPERIMENTS
+
+# 5.1 A TRELLISNET WITH GATED ACTIVATION
+
+In our description of generic trellis networks in Section 3, the activation function $f$ can be any nonlinearity that computes $z _ { 1 : T } ^ { ( i + 1 ) }$ based on $\hat { z } _ { 1 : T } ^ { ( i + 1 ) }$ and $z _ { 1 : T - 1 } ^ { ( i ) }$ . In experiments, we use a gated activation based on the LSTM cell. Gated activations have been used before in convolutional networks for sequence modeling (van den Oord et al., 2016; Dauphin et al., 2017). Our choice is inspired directly by Theorem 1, which suggests incorporating an existing RNN cell into TrellisNet. We use the LSTM cell due to its effectiveness in recurrent networks (Jozefowicz et al., 2015; Greff et al., 2017; Melis et al., 2018). We summarize the construction here; a more detailed treatment can be found in Appendix A.
+
+![](images/e408b37a58a2bfe0accb91f0e2b58a49cfdcf7bb3081fb3c4589ee86334ac414.jpg)  
+Figure 3: A gated activation based on the LSTM cell.
+
+In an LSTM cell, three information-controlling gates are computed at time $t$ . Moreover, there is a cell state that does not participate in the hidden-to-hidden transformations but is updated in every step using the result from the gated activations. We integrate the LSTM cell into the TrellisNet as follows (Figure 3):
+
+$$
+\begin{array} { r l r } & { \hat { z } _ { t + 1 } ^ { ( i + 1 ) } = W _ { 1 } \left[ z _ { t , 2 } ^ { ( i ) } \right] + W _ { 2 } \left[ z _ { t + 1 , 2 } ^ { ( i + 1 } \right] = \left[ \hat { z } _ { t + 1 , 1 } \quad \hat { z } _ { t + 1 , 2 } \quad \hat { z } _ { t + 1 , 3 } \quad \hat { z } _ { t + 1 , 4 } \right] ^ { \top } } & \\ & { z _ { t + 1 , 1 } ^ { ( i + 1 ) } = \sigma ( \hat { z } _ { t + 1 , 1 } ) \circ z _ { t , 1 } ^ { ( i ) } + \sigma ( \hat { z } _ { t + 1 , 2 } ) \circ \operatorname { t a n h } ( \hat { z } _ { t + 1 , 3 } ) } & \\ & { z _ { t + 1 , 2 } ^ { ( i + 1 ) } = \sigma ( \hat { z } _ { t + 1 , 4 } ) \circ \operatorname { t a n h } ( z _ { t + 1 , 1 } ^ { ( i + 1 ) } ) } & { ( 1 2 ; \operatorname { G a t e c } } \end{array}
+$$
+
+d activation $f$ )
+
+Thus the linear transformation in each layer of the TrellisNet produces a pre-activation feature $\hat { z } _ { t + 1 }$ with $r = 4 q$ feature channels, which are then processed by elementwise transformations and Hadamard products to yield the final output z(i+1)t+1 $\boldsymbol { z } _ { t + 1 } ^ { ( i + 1 ) } = \left( \boldsymbol { z } _ { t + 1 , 1 } ^ { ( i + 1 ) } , \boldsymbol { z } _ { t + 1 , 2 } ^ { ( i + 1 ) } \right)$ of the layer.
+
+# 5.2 RESULTS
+
+We evaluate trellis networks on word-level and character-level language modeling on the standard Penn Treebank (PTB) dataset (Marcus et al., 1993; Mikolov et al., 2010), large-scale word-level modeling on WikiText-103 (WT103) (Merity et al., 2017), and standard stress tests used to study long-range information propagation in sequence models: sequential MNIST, permuted MNIST (PMNIST), and sequential CIFAR-10 (Chang et al., 2017; Bai et al., 2018; Trinh et al., 2018). Note that these tasks are on very different scales, with unique properties that challenge sequence models in different ways. For example, word-level PTB is a small dataset that a typical model easily overfits, so judicious regularization is essential. WT103 is a hundred times larger, with less danger of overfitting, but with a vocabulary size of 268K that makes training more challenging (and precludes the application of techniques such as mixture of softmaxes (Yang et al., 2018)). A more complete description of these tasks and their characteristics can be found in Appendix C.
+
+Table 1: Test perplexities (ppl) on word-level language modeling with the PTB corpus. \` means lower is better.   
+
+<table><tr><td colspan="3">Word-level Penn Treebank (PTB)</td></tr><tr><td>Model</td><td>Size</td><td>Test perplexitye</td></tr><tr><td>Generic TCN (Bai et al., 2018)</td><td>13M</td><td>88.68</td></tr><tr><td>Variational LSTM (Gal &amp; Ghahramani,2016)</td><td>66M</td><td>73.4</td></tr><tr><td>NAS Cell (Zoph &amp; Le,2017)</td><td>54M</td><td>62.4</td></tr><tr><td>AWD-LSTM (Merity et al.,2018b)</td><td>24M</td><td>58.8</td></tr><tr><td>(Black-box tuned) NAS (Melis et al.,2018)</td><td>24M</td><td>59.7</td></tr><tr><td>(Black-box tuned) LSTM+ skip conn. (Melis et al.,2018)</td><td>24M</td><td>58.3</td></tr><tr><td>AWD-LSTM-MoC (Yang et al.,2018)</td><td>22M</td><td>57.55</td></tr><tr><td>DARTS (Liu et al., 2018)</td><td>23M</td><td>56.10</td></tr><tr><td>AWD-LSTM-MoS (Yang et al., 2018)</td><td>24M</td><td>55.97</td></tr><tr><td>ENAS (Pham et al., 2018)</td><td>24M</td><td>55.80</td></tr><tr><td>Ours - TrellisNet</td><td>24M</td><td>56.97</td></tr><tr><td>Ours - TrellisNet (1.4x larger)</td><td>33M</td><td>56.80</td></tr><tr><td>Ours-TrellisNet-MoS</td><td>25M</td><td>54.67</td></tr><tr><td>Ours - TrellisNet-MoS (1.4x larger)</td><td>34M</td><td>54.19</td></tr></table>
+
+Table 2: Test perplexities (ppl) on word-level language modeling with the WT103 corpus.   
+
+<table><tr><td colspan="3">Word-level WikiText-103(WT103)</td></tr><tr><td>Model</td><td>Size</td><td>Test perplexityl</td></tr><tr><td>LSTM (Grave et al., 2017b)</td><td>-</td><td>48.7</td></tr><tr><td>LSTM+continuous cache (Grave et al., 2017b)</td><td>=</td><td>40.8</td></tr><tr><td>Generic TCN(Bai et al., 2018)</td><td>150M</td><td>45.2</td></tr><tr><td>Gated Linear ConvNet (Dauphin et al., 2017)</td><td>230M</td><td>37.2</td></tr><tr><td>AWD-QRNN (Merity et al., 2018a)</td><td>159M</td><td>33.0</td></tr><tr><td>Relational Memory Core (Santoro et al., 2018)</td><td>195M</td><td>31.6</td></tr><tr><td>Ours - TrellisNet</td><td>180M</td><td>29.19</td></tr></table>
+
+The prior state of the art on these tasks was set by completely different models, such as AWD-LSTM on character-level PTB (Merity et al., 2018a), neural architecture search on word-level PTB (Pham et al., 2018), and the self-attention-based Relational Memory Core on WikiText-103 (Santoro et al., 2018). We use trellis networks on all tasks and outperform the respective state-of-the-art models on each. For example, on word-level Penn Treebank, TrellisNet outperforms by a good margin the recent results of Melis et al. (2018), which used the Google Vizier service for exhaustive hyperparameter tuning, as well as the recent neural architecture search work of Pham et al. (2018). On WikiText-103, a trellis network outperforms by $7 . 6 \%$ the Relational Memory Core (Santoro et al., 2018) and by $1 1 . 5 \%$ the thorough optimization work of Merity et al. (2018a).
+
+Many hyperparameters we use are adapted directly from prior work on recurrent networks. (As highlighted in Section 4.3, many techniques can be carried over directly from RNNs.) For others, we perform a basic grid search. We decay the learning rate by a fixed factor once validation error plateaus. All hyperparameters are reported in Appendix D, along with an ablation study.
+
+Word-level language modeling. For word-level language modeling, we use PTB and WT103. The results on PTB are listed in Table 1. TrellisNet sets a new state of the art on PTB, both with and without mixture of softmaxes (Yang et al., 2018), outperforming all previously published results by more than one unit of perplexity.
+
+WT103 is 110 times larger than PTB, with vocabulary size 268K. We follow prior work and use the adaptive softmax (Grave et al., 2017a), which improves memory efficiency by assigning higher capacity to more frequent words. The results are listed in Table 2. TrellisNet sets a new state of the art on this dataset as well, with perplexity 29.19: about $7 . 6 \%$ better than the contemporaneous self-attention-based Relational Memory Core (RMC) (Santoro et al., 2018). TrellisNet achieves this better accuracy with much faster convergence: 25 epochs, versus 90 for RMC.
+
+Table 3: Test bits-per-character (bpc) on character-level language modeling with the PTB corpus.   
+
+<table><tr><td colspan="3">Char-levelPTB</td></tr><tr><td>Model</td><td>Size</td><td>Test bpcl</td></tr><tr><td>Generic TCN (Bai et al.,2018)</td><td>3.0M</td><td>1.31</td></tr><tr><td>Independently RNN (Li et al., 2018)</td><td>12.0M</td><td>1.23</td></tr><tr><td>Hyper LSTM (Ha et al., 2017)</td><td>14.4M</td><td>1.219</td></tr><tr><td>NAS Cell (Zoph &amp; Le,2017)</td><td>16.3M</td><td>1.214</td></tr><tr><td>Fast-Slow-LSTM-2 (Mujika et al., 2017)</td><td>7.2M</td><td>1.19</td></tr><tr><td>Quasi-RNN (Merity et al.,2018a)</td><td>13.8M</td><td>1.187</td></tr><tr><td>AWD-LSTM (Merity et al.,2018a)</td><td>13.8M</td><td>1.175</td></tr><tr><td>Ours- TrellisNet</td><td>13.4M</td><td>1.158</td></tr></table>
+
+Table 4: Test accuracies on long-range modeling benchmarks. h means higher is better.   
+
+<table><tr><td>Model</td><td>Seq. MNIST Test acc.h</td><td>Permuted MNIST Test acc.h</td><td>Seq. CIFAR-10 Test acc.h</td></tr><tr><td>Dilated GRU (Chang et al., 2017)</td><td>99.0</td><td>94.6</td><td></td></tr><tr><td>IndRNN (Li et al., 2018)</td><td>99.0</td><td>96.0</td><td>=</td></tr><tr><td>Generic TCN (Bai et al., 2018)</td><td>99.0</td><td>97.2</td><td>-</td></tr><tr><td>r-LSTM w/ Aux.Loss (Trinh et al.,2018)</td><td>98.4</td><td>95.2</td><td>72.2</td></tr><tr><td>Transformer (self-attention) (Trinh et al.,2018)</td><td>98.9</td><td>97.9</td><td>62.2</td></tr><tr><td>Ours- TrellisNet</td><td>99.20</td><td>98.13</td><td>73.42</td></tr></table>
+
+Character-level language modeling. When used for character-level modeling, PTB is a mediumscale dataset with stronger long-term dependencies between characters. We thus use a deeper network as well as techniques such as weight normalization (Salimans & Kingma, 2016) and deep supervision (Lee et al., 2015; Xie & Tu, 2015). The results are listed in Table 3. TrellisNet sets a new state of the art with 1.158 bpc, outperforming the recent results of Merity et al. (2018a) by a comfortable margin.
+
+Long-range modeling with Sequential MNIST, PMNIST, and CIFAR-10. We also evaluate the TrellisNet for ability to model long-term dependencies. In the Sequential MNIST, PMNIST, and CIFAR-10 tasks, images are processed as long sequences, one pixel at a time (Chang et al., 2017; Bai et al., 2018; Trinh et al., 2018). Our model has 8M parameters, in alignment with prior work. To cover the larger context, we use dilated convolutions in intermediate layers, adopting a common architectural element from TCNs (Yu & Koltun, 2016; van den Oord et al., 2016; Bai et al., 2018). The results are listed in Table 4. Note that the performance of prior models is inconsistent. The Transformer works well on MNIST but fairs poorly on CIFAR-10, while $r$ -LSTM with unsupervised auxiliary losses achieves good results on CIFAR-10 but underperforms on Permuted MNIST. TrellisNet outperforms all these models on all three tasks.
+
+# 6 DISCUSSION
+
+We presented trellis networks, a new architecture for sequence modeling. Trellis networks form a structural bridge between convolutional and recurrent models. This enables direct assimilation of many techniques designed for either of these two architectural families. We leverage these connections to train high-performing trellis networks that set a new state of the art on highly competitive language modeling benchmarks. Beyond the empirical gains, we hope that trellis networks will serve as a step towards deeper and more unified understanding of sequence modeling.
+
+There are many exciting opportunities for future work. First, we have not conducted thorough performance optimizations on trellis networks. For example, architecture search on the structure of the gated activation $f$ may yield a higher-performing activation function than the classic LSTM cell we used (Zoph & Le, 2017; Pham et al., 2018). Likewise, principled hyperparameter tuning will likely improve modeling accuracy beyond the levels we have observed (Melis et al., 2018). Future work can also explore acceleration schemes that speed up training and inference.
+
+Another significant opportunity is to establish connections between trellis networks and selfattention-based architectures (Transformers) (Vaswani et al., 2017; Santoro et al., 2018; Chen et al., 2018), thus unifying all three major contemporary approaches to sequence modeling. Finally, we look forward to seeing applications of trellis networks to industrial-scale challenges such as machine translation.
+
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+# A EXPRESSING AN LSTM AS A TRELLISNET
+
+![](images/7ece8640e8a4d3bf96a598d1ac638717113d4e343e5a1b715fd063ecd12e8905.jpg)  
+Figure 4: A TrellisNet with an LSTM nonlinearity, at an atomic level and on a longer sequence.
+
+Here we trace in more detail the transformation of an LSTM into a TrellisNet. This is an application of Theorem 1. The nonlinear activation has been examined in Section 5.1. We will walk through the construction again here.
+
+In each time step, an LSTM cell computes the following:
+
+$$
+\begin{array} { r l } & { f _ { t } ^ { ( \ell ) } = \sigma ( W _ { f } h _ { t } ^ { ( \ell - 1 ) } + U _ { f } h _ { t - 1 } ^ { ( \ell ) } ) \quad i _ { t } ^ { ( \ell ) } = \sigma ( W _ { i } h _ { t } ^ { ( \ell - 1 ) } + U _ { i } h _ { t - 1 } ^ { ( \ell ) } ) \quad g _ { t } ^ { ( \ell ) } = \operatorname { t a n h } ( W _ { g } h _ { t } ^ { ( \ell - 1 ) } + U _ { g } h _ { t - 1 } ^ { ( \ell ) } ) } \\ & { o _ { t } ^ { ( \ell ) } = \sigma ( W _ { o } h _ { t } ^ { ( \ell - 1 ) } + U _ { o } h _ { t - 1 } ^ { ( \ell ) } ) \quad c _ { t } ^ { ( \ell ) } = f _ { t } ^ { ( \ell ) } \circ c _ { t - 1 } ^ { ( \ell ) } + i _ { t } ^ { ( \ell ) } \circ g _ { t } ^ { ( \ell ) } \quad h _ { t } ^ { ( \ell ) } = o _ { t } ^ { ( \ell ) } \circ \operatorname { t a n h } ( c _ { t } ^ { ( \ell ) } ) } \end{array}
+$$
+
+where $h _ { t } ^ { ( 0 ) } = x _ { t }$ , and $f _ { t } , i _ { t } , o _ { t }$ are typically called the forget, input, and output gates. By a similar construction to how we defined $\tau$ in Theorem 1, to recover an LSTM the mixed group convolution needs to produce $3 q$ more channels for these gated outputs, which have the form $f _ { t , t ^ { \prime } } , i _ { t , t ^ { \prime } }$ and $g _ { t , t ^ { \prime } }$ (see Figure 5 for an example). In addition, at each layer of the mixed group convolution, the network also needs to maintain a group of channels for cell states $c _ { t , t ^ { \prime } }$ . Note that in an LSTM network, $c _ { t }$ is updated “synchronously” with $h _ { t }$ , so we can similarly write
+
+$$
+c _ { t , t ^ { \prime } } ^ { ( 1 ) } = f _ { t , t ^ { \prime } } ^ { ( 1 ) } \circ c _ { t - 1 , t ^ { \prime } } ^ { ( 1 ) } + i _ { t , t ^ { \prime } } ^ { ( 1 ) } \circ g _ { t , t ^ { \prime } } ^ { ( 1 ) } \qquad h _ { t , t ^ { \prime } } ^ { ( 1 ) } = o _ { t , t ^ { \prime } } ^ { ( 1 ) } \circ \operatorname { t a n h } ( c _ { t , t ^ { \prime } } ^ { ( 1 ) } )
+$$
+
+Based on these changes, we show in Figure 4 an atomic and a sequence view of TrellisNet with the LSTM activation. The hidden units $z _ { 1 : T }$ consist of two parts: $z _ { 1 : T , 1 }$ , which gets updated directly via the gated activations (akin to LSTM cell states), and $z _ { 1 : T , 2 }$ , which is processed by parameterized convolutions (akin to LSTM hidden states). Formally, in layer $i$ :
+
+$$
+\begin{array} { r l } & { \hat { z } _ { 1 : T } ^ { ( i + 1 ) } =  { \operatorname { C o n v } }  { \operatorname { l D } } ( z _ { 1 : T , 2 } ^ { ( i ) } ; W ) + \tilde { x } _ { 1 : T } = [ \hat { z } _ { 1 : T , 1 } \quad \hat { z } _ { 1 : T , 2 } \quad \hat { z } _ { 1 : T , 3 } \quad \hat { z } _ { 1 : T , 4 } ] ^ { \top } } \\ & { z _ { 1 : T , 1 } ^ { ( i + 1 ) } = \sigma ( \hat { z } _ { 1 : T , 1 } ) \circ z _ { 0 : T - 1 , 1 } ^ { ( i ) } + \sigma ( \hat { z } _ { 1 : T , 2 } ) \circ  { \operatorname { t a n h } } ( \hat { z } _ { 1 : T , 3 } ) } \\ & { z _ { 1 : T , 2 } ^ { ( i + 1 ) } = \sigma ( \hat { z } _ { 1 : T , 4 } ) \circ  { \operatorname { t a n h } } ( z _ { 1 : T , 1 } ^ { ( i + 1 ) } ) } \end{array}
+$$
+
+![](images/e46cbe818ada0818d04fd45cf9d757cb8dd00bf6b857a3d0b387c29cfac98024.jpg)  
+Figure 5: A 2-layer LSTM is expressed as a trellis network with mixed group convolutions on four groups of feature channels. (Partial view.)
+
+# B OPTIMIZING AND REGULARIZING TRELLISNET WITH RNN AND TCN METHODS
+
+(a) History repackaging between truncated sequences in recurrent networks.
+
+![](images/c5287e8e00351701cefdeed9dfdbc38cca2d7fb23f9ebf8e4c0b2b8686084246.jpg)  
+(b) History repackaging in mixed group convolutions, where we write out $z _ { t }$ explicitly by Eq. (6).
+
+![](images/20fcbbe53f278f1fc12c9c5866ff02383bad4d6c664f6a6bdc448d892cfca736.jpg)
+
+Figure 6: Using the equivalence established by Theorem 1, we can transfer the notion of history repackaging in recurrent networks to trellis networks.
+
+In Section 4, we formally described the relationship between TrellisNets, RNNs, and temporal convolutional networks (TCN). On the one hand, TrellisNet is a special TCN (with weight-tying and input injection), while on the other hand it can also express any structured RNN via a sparse convolutional kernel. These relationships open clear paths for applying techniques developed for either recurrent or convolutional networks. We summarize below some of the techniques that can be applied in this way to TrellisNet, categorizing them as either inspired by RNNs or TCNs.
+
+# B.1 FROM RECURRENT NETWORKS
+
+History repackaging. One theoretical advantage of RNNs is their ability to represent a history of infinite length. However, in many applications, sequence lengths are too long for infinite backpropagation during training. A typical solution is to partition the sequence into smaller subsequences and perform truncated backpropagation through time (BPTT) on each. At sequence boundaries, the hidden state $h _ { t }$ is “repackaged” and passed onto the next RNN sequence. Thus gradient flow stops at sequence boundaries (see Figure 6a). Such repackaging is also sometimes used at test time.
+
+We can now map this repackaging procedure to trellis networks. As shown in Figure 6, the notion of passing the compressed history vector $h _ { t }$ in an RNN corresponds to specific non-zero padding in the mixed group convolution of the corresponding TrellisNet. The padding is simply the channels from the last step of the final layer applied on the previous sequence (see Figure 6b, where without the repackaging padding, at layer 2 we will have $h _ { T + 1 , T + 1 } ^ { ( 1 ) }$ instead of $h _ { T + 1 , 1 } ^ { ( 1 ) } ,$ ). We illustrate this in Figure 6b, where we have written out $\boldsymbol { z } _ { t } ^ { ( i ) }$ in TrellisNet explicitly in the form of $h _ { t , t ^ { \prime } }$ according to Eq. (6). This suggests that instead of storing all effective history in memory, we can compress history in a feed-forward network to extend its history as well. For a general TrellisNet that employs a dense kernel, similarly, we can pass the hidden channels of the last step of the final layer in the previous sequence as the “history” padding for the next TrellisNet sequence (this works in both training and testing).
+
+Gated activations. In general, the structured gates in RNN cells can be translated to gated activations in temporal convolutions, as we did in Appendix A in the case of an LSTM. While in the experiments we adopted the LSTM gating, other activations (e.g. GRUs (Cho et al., 2014) or activations found via architecture search (Zoph & Le, 2017)) can also be applied in trellis networks via the equivalence established in Theorem 1.
+
+RNN variational dropout. Variational dropout (VD) for RNNs (Gal & Ghahramani, 2016) is a useful regularization scheme that applies the same mask at every time step within a layer (see Figure 7a). A direct translation of this technique from RNN to the group temporal convolution implies that we need to create a different mask for each diagonal of the network (i.e. each history starting point), as well as for each group of the mixed group convolution. We propose an alternative (and extremely simple) dropout scheme for TrellisNet, which is inspired by VD in RNNs as well as Theo(a) Left: variational dropout (VD) in an RNN. Right: VD in a (b) Auxiliary loss on intermediate layers TrellisNet. Each color indicates a different dropout mask. in a TrellisNet.
+
+![](images/4f3f5e3ccce36f93ab23fcfeda845345ce8a0ad4429f97601f8ed55d7792a48c.jpg)  
+Figure 7: (a) RNN-inspired variational dropout. (b) ConvNet-inspired auxiliary losses.
+
+rem 1. In each iteration, we apply the same mask on the post-activation outputs, at every time step in both the temporal dimension and depth dimension. That is, based on Eq. (6) in Theorem 1, we adapt VD to the TrellisNet setting by assuming $h _ { t , t ^ { \prime } \pm \delta } \approx h _ { t , t ^ { \prime } }$ ; see Figure 7a. Empirically, we found this dropout to work significantly better than other dropout schemes (e.g. drop certain channels entirely).
+
+Recurrent weight dropout/DropConnect. We apply DropConnect on the TrellisNet kernel. Merity et al. (2018b) showed that regularizing hidden-to-hidden weights $W _ { h h }$ can be useful in optimizing LSTM language models, and we carry this scheme over to trellis networks.
+
+# B.2 FROM CONVOLUTIONAL NETWORKS
+
+Dense convolutional kernel. Generalizing the convolution from a mixed group (sparse) convolution to a general (dense) one means the connections are no longer recurrent and we are computing directly on the hidden units with a large kernel, just like any temporal ConvNet.
+
+Deep supervision. Recall that for sparse TrellisNet to recover truncated RNN, at each level the hidden units are of the form $h _ { t , t ^ { \prime } }$ , representing the state at time $t$ if we assume that history started at time $t ^ { \prime }$ (Eq. (6)). We propose to inject the loss function at intermediate layers of the convolutional network (e.g. after every $\ell$ layers of transformations, where we call $\ell$ the auxiliary loss frequency). For example, during training, to predict an output at time $t$ with a $L$ -layer TrellisNet, besides $ { \boldsymbol { z } } _ { t } ^ { ( L ) }$ in the last layer, we can also apply the loss function on $\boldsymbol { z } _ { t } ^ { ( L - \ell ) }$ , $z _ { t } ^ { ( L - 2 \bar { \ell } ) }$ , etc. – where hidden units will predict with a shorter history because they are at lower levels of the network. This had been introduced for convolutional models in computer vision (Lee et al., 2015; Xie & Tu, 2015). The eventual loss of the network will be
+
+$$
+\mathcal { L } _ { \mathrm { t o t a l } } = \mathcal { L } _ { \mathrm { o r i g } } + \lambda \cdot \mathcal { L } _ { \mathrm { a u x } } ,
+$$
+
+where $\lambda$ is a fixed scaling factor that controls the weight of the auxiliary loss.
+
+Note that this technique is not directly transferable (or applicable) to RNNs.
+
+Larger kernel and dilations (Yu & Koltun, 2016). These techniques have been used in convolutional networks to more quickly increase the receptive field. They can be immediately applied to trellis networks. Note that the activation function $f$ of TrellisNet may need to change if we change the kernel size or dilation settings (e.g. with dilation $d$ and kernel size 2, the activation will be $f ( \hat { z } _ { 1 : T } ^ { ( i ) } , z _ { 1 : T - d } ^ { ( i ) } ) )$ .
+
+Weight normalization (Salimans & Kingma, 2016). Weight normalization (WN) is a technique that learns the direction and the magnitude of the weight matrix independently. Applying WN on the convolutional kernel was used in some prior works on temporal convolutional architectures (Dauphin et al., 2017; Bai et al., 2018), and have been found useful in regularizing the convolutional filters and boosting convergence.
+
+Parallelism. Because TrellisNet is convolutional in nature, it can easily leverage the parallel processing in the convolution operation (which slides the kernel across the input features). We note that when the input sequence is relatively long, the predictions of the first few time steps will have insufficient history context compared to the predictions later in the sequence. This can be addressed by either history padding (mentioned in Appendix B.1) or chopping off the loss incurred by the first few time steps.
+
+# C BENCHMARK TASKS
+
+Word-level language modeling on Penn Treebank (PTB). The original Penn Treebank (PTB) dataset selected 2,499 stories from a collection of almost 100K stories published in Wall Street Journal (WSJ) (Marcus et al., 1993). After Mikolov et al. (2010) processed the corpus, the PTB dataset contains 888K words for training, 70K for validation and 79K for testing, where each sentence is marked with an $< \ominus \hphantom { . 0 0 0 }$ tag at its end. All of the numbers (e.g. in financial news) were replaced with a ? symbol with many punctuations removed. Though small, PTB has been a highly studied dataset in the domain of language modeling (Miyamoto & Cho, 2016; Zilly et al., 2017; Merity et al., 2018b; Melis et al., 2018; Yang et al., 2018). Due to its relatively small size, many computational models can easily overfit on word-level PTB. Therefore, good regularization methods and optimization techniques designed for sequence models are especially important on this benchmark task (Merity et al., 2018b).
+
+Word-level language modeling on WikiText-103. WikiText-103 (WT103) is 110 times larger than PTB, containing a training corpus from 28K lightly processed Wikipedia articles (Merity et al., 2017). In total, WT103 features a vocabulary size of about $2 6 8 \mathrm { K } ^ { 2 }$ , with 103M words for training, 218K words for validation, and 246K words for testing/evaluation. The WT103 corpus also retains the original case, punctuation and numbers in the raw data, all of which were removed from the PTB corpus. Moreover, since WT103 is composed of full articles (whereas PTB is sentence-based), it is better suited for testing long-term context retention. For these reasons, WT103 is typically considered much more representative and realistic than PTB (Merity et al., 2018a).
+
+Character-level language modeling on Penn Treebank (PTB). When used for character-level language modeling, PTB is a medium size dataset that contains 5M chracters for training, 396K for validation, and 446K for testing, with an alphabet size of 50 (note: the $< \ominus \hphantom { . 0 0 0 }$ tag that marks the end of a sentence in word-level tasks is now considered one character). While the alphabet size of char-level PTB is much smaller compared to the word-level vocabulary size (10K), there is much longer sequential token dependency because a sentence contains many more characters than words.
+
+Sequential and permuted MNIST classification. The MNIST handwritten digits dataset (LeCun et al., 1989) contains 60K normalized training images and 10K testing images, all of size $2 8 \times 2 8$ . In the sequential MNIST task, MNIST images are presented to the sequence model as a flattened $7 8 4 \times 1$ sequence for digit classification. Accurate predictions therefore require good long-term memory of the flattened pixels – longer than in most language modeling tasks. In the setting of permuted MNIST (PMNIST), the order of the sequence is permuted at random, so the network can no longer rely on local pixel features for classification.
+
+Sequential CIFAR-10 classification. The CIFAR-10 dataset (Krizhevsky & Hinton, 2009) contains 50K images for training and 10K for testing, all of size $3 2 \times 3 2$ . In the sequential CIFAR-10 task, these images are passed into the model one at each time step, flattended as in the MNIST tasks. Compared to sequential MNIST, this task is more challenging. For instance, CIFAR-10 contains more complex image structures and intra-class variations, and there are 3 channels to the input. Moreover, as the images are larger, a sequence model needs to have even longer memory than in sequential MNIST or PMNIST (Trinh et al., 2018).
+
+# D HYPERPARAMETERS AND ABLATION STUDY
+
+Table 5 specifies the trellis networks used for the various tasks. There are a few things to note while reading the table. First, in training, we decay the learning rate once the validation error plateaus for a while (or according to some fixed schedule, such as after 100 epochs). Second, for auxiliary loss (see Appendix B for more details), we insert the loss function after every fixed number of layers in the network. This “frequency” is included below under the “Auxiliary Frequency” entry. Finally, the hidden dropout in the Table refers to the variational dropout we translated from RNNs (see Appendix B), which is applied at all hidden layers of the TrellisNet. Due to the insight from Theorem 1, many techniques in TrellisNet were translated directly from RNNs or TCNs. Thus, most of the hyperparameters were based on the numbers reported in prior works (e.g. embedding size, embedding dropout, hidden dropout, output dropout, optimizer, weight-decay, etc.) with minor adjustments (Merity et al., 2018b; Yang et al., 2018; Bradbury et al., 2017; Merity et al., 2018a; Trinh et al., 2018; Bai et al., 2018; Santoro et al., 2018). For factors such as auxiliary loss weight and frequency, we perform a basic grid search.
+
+Table 5: Models and hyperparameters used in experiments. “–” means not applicable/used.   
+
+<table><tr><td></td><td>Word-PTB (w/o MoS)</td><td>Word-PTB(w/MoS)</td><td>Word-WT103</td><td>Char-PTB</td><td>(P)MNIST/CIFAR-10</td></tr><tr><td>Optimizer</td><td>SGD</td><td>SGD</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Initial Learning Rate</td><td>20</td><td>20</td><td>1e-3</td><td>2e-3</td><td>2e-3</td></tr><tr><td>Hidden Size (i.e. ht)</td><td>1000</td><td>1000</td><td>2000</td><td>1000</td><td>100</td></tr><tr><td>Output Size (only for MoS)</td><td>1</td><td>480</td><td>1</td><td>1</td><td>1</td></tr><tr><td># of Experts (only for MoS)</td><td>1</td><td>15</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Embedding Size</td><td>400</td><td>280</td><td>512</td><td>200</td><td>1</td></tr><tr><td>Embedding Dropout</td><td>0.1</td><td>0.05</td><td>0.0</td><td>0.0</td><td>1</td></tr><tr><td>Hidden (VD-based) Dropout</td><td>0.28</td><td>0.28</td><td>0.1</td><td>0.3</td><td>0.2</td></tr><tr><td>Output Dropout</td><td>0.45</td><td>0.4</td><td>0.1</td><td>0.1</td><td>0.2</td></tr><tr><td>Weight Dropout</td><td>0.5</td><td>0.45</td><td>0.1</td><td>0.25</td><td>0.1</td></tr><tr><td># of Layers</td><td>55</td><td>55</td><td>70</td><td>125</td><td>16</td></tr><tr><td>Auxiliary Loss 入</td><td>0.05</td><td>0.05</td><td>0.08</td><td>0.3</td><td>1</td></tr><tr><td>Auxiliary Frequency</td><td>16</td><td>16</td><td>25</td><td>70</td><td></td></tr><tr><td>Weight Normalization</td><td>1</td><td>1</td><td>√</td><td>√</td><td>√</td></tr><tr><td>Gradient Clip</td><td>0.225</td><td>0.2</td><td>0.1</td><td>0.2</td><td>0.5</td></tr><tr><td>Weight Decay</td><td>1e-6</td><td>1e-6</td><td>0.0</td><td>1e-6</td><td>1e-6</td></tr><tr><td>Model Size</td><td>24M</td><td>25M</td><td>180M</td><td>13.4M</td><td>8M</td></tr></table>
+
+We have also performed an ablation study on TrellisNet to study the influence of various ingredients and techniques on performance. The results are reported in Table 6. We conduct the study on wordlevel PTB using a TrellisNet with 24M parameters. When we study one factor (e.g. removing hidden dropout), all hyperparameters and settings remain the same as in column 1 of Table 5 (except for “Dense Kernel”, where we adjust the number of hidden units so that the model size remains the same).
+
+Table 6: Ablation study on word-level PTB (w/o MoS)   
+
+<table><tr><td></td><td>Model Size</td><td>Test ppl</td><td>△SOTA</td></tr><tr><td>TrellisNet</td><td>24.1M</td><td>56.97</td><td>1 ↓7.72</td></tr><tr><td>-Hidden(VD-based) Dropout</td><td>24.1M</td><td>64.69</td><td>↓6.85</td></tr><tr><td>- Weight Dropout - Auxiliary Losses</td><td>24.1M 24.1M</td><td>63.82 57.99</td><td>↓1.02</td></tr><tr><td>-Long Seq.Parallelism</td><td>24.1M</td><td>57.35</td><td>↓0.38</td></tr><tr><td>-Dense Kernel (i.e.mixed group conv)</td><td>24.1M</td><td>59.18</td><td>↓2.21</td></tr><tr><td>-Injected Input (every 2 layers instead)</td><td>24.1M</td><td>57.44</td><td>↓0.47</td></tr><tr><td>Injected Input (every 5 layers instead)</td><td>24.1M</td><td>59.75</td><td>↓2.78</td></tr><tr><td>-Injected Input (every1O layers instead)</td><td>24.1M</td><td></td><td></td></tr><tr><td>-Injected Input (every 2O layers instead)</td><td>24.1M</td><td>60.70 74.91</td><td>↓3.73 ↓17.94</td></tr></table>

parse/train/HyrCWeWCb/HyrCWeWCb.md

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+# TRUST-PCL: AN OFF-POLICY TRUST REGION METHOD FOR CONTINUOUS CONTROL
+
+Ofir Nachum, Mohammad Norouzi, Kelvin Xu, & Dale Schuurmans∗ {ofirnachum,mnorouzi,kelvinxx,schuurmans}@google.com Google Brain
+
+# ABSTRACT
+
+Trust region methods, such as TRPO, are often used to stabilize policy optimization algorithms in reinforcement learning (RL). While current trust region strategies are effective for continuous control, they typically require a large amount of on-policy interaction with the environment. To address this problem, we propose an off-policy trust region method, Trust-PCL, which exploits an observation that the optimal policy and state values of a maximum reward objective with a relative-entropy regularizer satisfy a set of multi-step pathwise consistencies along any path. The introduction of relative entropy regularization allows Trust-PCL to maintain optimization stability while exploiting off-policy data to improve sample efficiency. When evaluated on a number of continuous control tasks, Trust-PCL significantly improves the solution quality and sample efficiency of TRPO.1
+
+# 1 INTRODUCTION
+
+The goal of model-free reinforcement learning (RL) is to optimize an agent’s behavior policy through trial and error interaction with a black box environment. Value-based RL algorithms such as Q-learning (Watkins, 1989) and policy-based algorithms such as actor-critic (Konda & Tsitsiklis, 2000) have achieved well-known successes in environments with enumerable action spaces and predictable but possibly complex dynamics, e.g., as in Atari games (Mnih et al., 2013; Van Hasselt et al., 2016; Mnih et al., 2016). However, when applied to environments with more sophisticated action spaces and dynamics (e.g., continuous control and robotics), success has been far more limited.
+
+In an attempt to improve the applicability of Q-learning to continuous control, Silver et al. (2014) and Lillicrap et al. (2015) developed an off-policy algorithm DDPG, leading to promising results on continuous control environments. That said, current off-policy methods including DDPG often improve data efficiency at the cost of optimization stability. The behaviour of DDPG is known to be highly dependent on hyperparameter selection and initialization (Metz et al., 2017); even when using optimal hyperparameters, individual training runs can display highly varying outcomes.
+
+On the other hand, in an attempt to improve the stability and convergence speed of policy-based RL methods, Kakade (2002) developed a natural policy gradient algorithm based on Amari (1998), which subsequently led to the development of trust region policy optimization (TRPO) (Schulman et al., 2015). TRPO has shown strong empirical performance on difficult continuous control tasks often outperforming value-based methods like DDPG. However, a major drawback is that such methods are not able to exploit off-policy data and thus require a large amount of on-policy interaction with the environment, making them impractical for solving challenging real-world problems.
+
+Efforts at combining the stability of trust region policy-based methods with the sample efficiency of value-based methods have focused on using off-policy data to better train a value estimate, which can be used as a control variate for variance reduction (Gu et al., 2017a;b).
+
+In this paper, we investigate an alternative approach to improving the sample efficiency of trust region policy-based RL methods. We exploit the key fact that, under entropy regularization, the optimal policy and value function satisfy a set of pathwise consistency properties along any sampled path (Nachum et al., 2017), which allows both on and off-policy data to be incorporated in an actor-critic algorithm, PCL. The original PCL algorithm optimized an entropy regularized maximum reward objective and was evaluated on relatively simple tasks. Here we extend the ideas of PCL to achieve strong results on standard, challenging continuous control benchmarks. The main observation is that by alternatively augmenting the maximum reward objective with a relative entropy regularizer, the optimal policy and values still satisfy a certain set of pathwise consistencies along any sampled trajectory. The resulting objective is equivalent to maximizing expected reward subject to a penalty-based constraint on divergence from a reference (i.e., previous) policy.
+
+We exploit this observation to propose a new off-policy trust region algorithm, Trust-PCL, that is able to exploit off-policy data to train policy and value estimates. Moreover, we present a simple method for determining the coefficient on the relative entropy regularizer to remain agnostic to reward scale, hence ameliorating the task of hyperparameter tuning. We find that the incorporation of a relative entropy regularizer is crucial for good and stable performance. We evaluate TrustPCL against TRPO, and observe that Trust-PCL is able to solve difficult continuous control tasks, while improving the performance of TRPO both in terms of the final reward achieved as well as sample-efficiency.
+
+# 2 RELATED WORK
+
+Trust Region Methods. Gradient descent is the predominant optimization method for neural networks. A gradient descent step is equivalent to solving a trust region constrained optimization,
+
+$$
+\mathrm { m i n i m i z e } ~ { \boldsymbol { \ell } } ( { \boldsymbol { \theta } } + \mathrm { d } { \boldsymbol { \theta } } ) \approx { \boldsymbol { \ell } } ( { \boldsymbol { \theta } } ) + { \boldsymbol { \nabla } } { \boldsymbol { \ell } } ( { \boldsymbol { \theta } } ) ^ { \top } \mathrm { d } { \boldsymbol { \theta } } \qquad { \mathrm { s . t . } } \quad \mathrm { d } { \boldsymbol { \theta } } ^ { \top } \mathrm { d } { \boldsymbol { \theta } } \leq \epsilon ,
+$$
+
+which yields the locally optimal update $\mathrm { d } \theta = - \eta \nabla \ell ( \theta )$ such that $\eta ~ = ~ { \sqrt { \epsilon } } / \| \nabla \ell ( \theta ) \|$ ; hence by considering a Euclidean ball, gradient descent assumes the parameters lie in a Euclidean space.
+
+However, in machine learning, particularly in the context of multi-layer neural network training, Euclidean geometry is not necessarily the best way to characterize proximity in parameter space. It is often more effective to define an appropriate Riemannian metric that respects the loss surface (Amari, 2012), which allows much steeper descent directions to be identified within a local neighborhood (e.g., Amari (1998); Martens & Grosse (2015)). Whenever the loss is defined in terms of a Bregman divergence between an (unknown) optimal parameter $\theta ^ { * }$ and model parameter $\theta$ , i.e., $\ell ( \theta ) \equiv D _ { \mathrm { { F } } } ( \theta ^ { * } , \theta )$ , it is natural to use the same divergence to form the trust region:
+
+$$
+\mathrm { m i n i m i z e } D _ { \mathrm { F } } ( \theta ^ { * } , \theta + \mathrm { d } \theta ) \mathrm { s . t . } D _ { \mathrm { F } } ( \theta , \theta + \mathrm { d } \theta ) \leq \epsilon .
+$$
+
+The natural gradient (Amari, 1998) is a generalization of gradient descent where the Fisher information matrix $F ( \theta )$ is used to define the local geometry of the parameter space around $\theta$ . If a parameter update is constrained by $\mathrm { d } \theta ^ { \mathsf { T } } F ( \theta ) \mathrm { d } \theta \leq \epsilon$ , a descent direction of $\mathrm { d } \theta \equiv - \eta F ( \theta ) ^ { - 1 } \nabla \ell ( \theta )$ is obtained. This geometry is especially effective for optimizing the log-likelihood of a conditional probabilistic model, where the objective is in fact the KL divergence $D _ { \mathrm { K L } } ( \theta ^ { * } , \theta )$ . The local optimization is,
+
+$$
+\mathrm { m i n i m i z e } D _ { \mathrm { K L } } ( \theta ^ { * } , \theta + \mathrm { d } \theta ) \quad \mathrm { ~ s . t . ~ } \quad D _ { \mathrm { K L } } ( \theta , \theta + \mathrm { d } \theta ) \approx \mathrm { d } \theta ^ { \mathsf { T } } F ( \theta ) \mathrm { d } \theta \leq \epsilon .
+$$
+
+Thus, natural gradient approximates the trust region by $D _ { \mathrm { { K L } } } ( a , b ) \approx ( a - b ) ^ { \mathsf { T } } F ( a ) ( a - b )$ , which is accurate up to a second order Taylor approximation. Previous work (Kakade, 2002; Bagnell $\&$ Schneider, 2003; Peters & Schaal, 2008; Schulman et al., 2015) has applied natural gradient to policy optimization, locally improving expected reward subject to variants of $\mathrm { d } \theta ^ { \mathsf { T } } F ( \theta ) \mathrm { d } \theta \leq \epsilon$ . Recently, TRPO (Schulman et al., 2015; 2016) has achieved state-of-the-art results in continuous control by adding several approximations to the natural gradient to make nonlinear policy optimization feasible.
+
+Another approach to trust region optimization is given by proximal gradient methods (Parikh et al., 2014). The class of proximal gradient methods most similar to our work are those that replace the hard constraint in (2) with a penalty added to the objective. These techniques have recently become popular in RL (Wang et al., 2016; Heess et al., 2017; Schulman et al., 2017b), although in terms of final reward performance on continuous control benchmarks, TRPO is still considered to be the state-of-the-art.
+
+Norouzi et al. (2016) make the observation that entropy regularized expected reward may be expressed as a reversed KL divergence $D _ { \mathrm { K L } } ( \theta , \theta ^ { * } )$ , which suggests that an alternative to the constraint in (3) should be used when such regularization is present:
+
+$$
+D _ { \mathrm { K L } } ( \boldsymbol \theta + \mathrm { d } \boldsymbol \theta , \boldsymbol \theta ^ { * } ) \qquad \mathrm { s . t . } \qquad D _ { \mathrm { K L } } ( \boldsymbol \theta + \mathrm { d } \boldsymbol \theta , \boldsymbol \theta ) \approx \mathrm { d } \boldsymbol \theta ^ { \mathsf { T } } \boldsymbol F ( \boldsymbol \theta + \mathrm { d } \boldsymbol \theta ) \mathrm { d } \boldsymbol \theta \leq \epsilon .
+$$
+
+Unfortunately, this update requires computing the Fisher matrix at the endpoint of the update. The use of $F ( \theta )$ in previous work can be considered to be an approximation when entropy regularization is present, but it is not ideal, particularly if $\mathrm { d } \theta$ is large. In this paper, by contrast, we demonstrate that the optimal $\mathrm { d } \theta$ under the reverse KL constraint $\bar { D _ { \mathrm { K L } } } ( \theta + \mathrm { d } \bar { \theta } , \bar { \theta } ) \leq \dot { \epsilon }$ can indeed be characterized. Defining the constraint in this way appears to be more natural and effective than that of TRPO.
+
+Softmax Consistency. To comply with the information geometry over policy parameters, previous work has used the relative entropy (i.e., KL divergence) to regularize policy optimization; resulting in a softmax relationship between the optimal policy and state values (Peters et al., 2010; Azar et al., 2012; 2011; Fox et al., 2016; Rawlik et al., 2013) under single-step rollouts. Our work is unique in that we leverage consistencies over multi-step rollouts.
+
+The existence of multi-step softmax consistencies has been noted by prior work—first by Nachum et al. (2017) in the presence of entropy regularization. The existence of the same consistencies with relative entropy has been noted by Schulman et al. (2017a). Our work presents multi-step consistency relations for a hybrid relative entropy plus entropy regularized expected reward objective, interpreting relative entropy regularization as a trust region constraint. This work is also distinct from prior work in that the coefficient of relative entropy can be automatically determined, which we have found to be especially crucial in cases where the reward distribution changes dramatically during training.
+
+Most previous work on softmax consistency (e.g., Fox et al. (2016); Azar et al. (2012); Nachum et al. (2017)) have only been evaluated on relatively simple tasks, including grid-world and discrete algorithmic environments. Rawlik et al. (2013) conducted evaluations on simple variants of the CartPole and Pendulum continuous control tasks. More recently, Haarnoja et al. (2017) showed that soft Qlearning (a single-step special case of PCL) can succeed on more challenging environments, such as a variant of the Swimmer task we consider below. By contrast, this paper presents a successful application of the softmax consistency concept to difficult and standard continuous-control benchmarks, resulting in performance that is competitive with and in some cases beats the state-of-the-art.
+
+# 3 NOTATION & BACKGROUND
+
+We model an agent’s behavior by a policy distribution $\pi ( a | s )$ over a set of actions (possibly discrete or continuous). At iteration $t$ , the agent encounters a state $s _ { t }$ and performs an action $a _ { t }$ sampled from $\pi ( \boldsymbol { a } \mid \boldsymbol { s } _ { t } )$ . The environment then returns a scalar reward $\displaystyle r _ { t } \sim r ( s _ { t } , a _ { t } )$ and transitions to the next state $s _ { t + 1 } \sim \rho ( s _ { t } , a _ { t } )$ . When formulating expectations over actions, rewards, and state transitions we will often omit the sampling distributions, $\pi , r$ , and $\rho$ , respectively.
+
+Maximizing Expected Reward. The standard objective in RL is to maximize expected future discounted reward. We formulate this objective on a per-state basis recursively as
+
+$$
+O _ { \mathrm { E R } } ( s , \pi ) = \mathbb { E } _ { a , r , s ^ { \prime } } \left[ r + \gamma O _ { \mathrm { E R } } ( s ^ { \prime } , \pi ) \right] .
+$$
+
+The overall, state-agnostic objective is the expected per-state objective when states are sampled from interactions with the environment:
+
+$$
+O _ { \mathrm { E R } } ( \pi ) = \mathbb { E } _ { s } [ O _ { \mathrm { E R } } ( s , \pi ) ] .
+$$
+
+Most policy-based algorithms, including REINFORCE (Williams & Peng, 1991) and actorcritic (Konda & Tsitsiklis, 2000), aim to optimize $O _ { \mathrm { E R } }$ given a parameterized policy.
+
+Path Consistency Learning (PCL). Inspired by Williams & Peng (1991), Nachum et al. (2017) augment the objective $O _ { \mathrm { E R } }$ in (5) with a discounted entropy regularizer to derive an objective,
+
+$$
+O _ { \mathrm { E N T } } ( s , \pi ) = O _ { \mathrm { E R } } ( s , \pi ) + \tau \mathbb { H } ( s , \pi ) ,
+$$
+
+where $\tau \geq 0$ is a user-specified temperature parameter that controls the degree of entropy regularization, and the discounted entropy $\mathbb { H } ( s , \pi )$ is recursively defined as
+
+$$
+\begin{array} { r } { \mathbb { H } ( s , \pi ) = \mathbb { E } _ { a , s ^ { \prime } } [ - \log \pi ( a \mid s ) + \gamma \mathbb { H } ( s ^ { \prime } , \pi ) ] . } \end{array}
+$$
+
+Note that the objective $O _ { \mathrm { E N T } } ( s , \pi )$ can then be re-expressed recursively as,
+
+$$
+O _ { \mathrm { E N T } } ( s , \pi ) = \mathbb { E } _ { a , r , s ^ { \prime } } [ r - \tau \log \pi ( a \mid s ) + \gamma O _ { \mathrm { E N T } } ( s ^ { \prime } , \pi ) ] .
+$$
+
+Nachum et al. (2017) show that the optimal policy $\pi ^ { * }$ for $O _ { \mathrm { E N T } }$ and $V ^ { * } ( s ) = O _ { \mathrm { E N T } } ( s , \pi ^ { * } )$ mutually satisfy a softmax temporal consistency constraint along any sequence of states $s _ { 0 } , \ldots , s _ { d }$ starting at $s _ { 0 }$ and a corresponding sequence of actions $a _ { 0 } , \ldots , a _ { d - 1 }$ :
+
+$$
+V ^ { * } ( s _ { 0 } ) = \underset { r _ { i } , s _ { i } } { \mathbb { E } } \left[ \gamma ^ { d } V ^ { * } ( s _ { d } ) + \sum _ { i = 0 } ^ { d - 1 } \gamma ^ { i } ( r _ { i } - \tau \log \pi ^ { * } ( a _ { i } | s _ { i } ) ) \right] .
+$$
+
+This observation led to the development of the PCL algorithm, which attempts to minimize squared error between the LHS and RHS of (10) to simultaneously optimize parameterized $\pi _ { \theta }$ and $V _ { \phi }$ . Importantly, PCL is applicable to both on-policy and off-policy trajectories.
+
+Trust Region Policy Optimization (TRPO). As noted, standard policy-based algorithms for maximizing $O _ { \mathrm { E R } }$ can be unstable and require small learning rates for training. To alleviate this issue, Schulman et al. (2015) proposed to perform an iterative trust region optimization to maximize $O _ { \mathrm { E R } }$ . At each step, a prior policy $\tilde { \pi }$ is used to sample a large batch of trajectories, then $\pi$ is subsequently optimized to maximize $O _ { \mathrm { E R } }$ while remaining within a constraint defined by the average per-state KL-divergence with $\tilde { \pi }$ . That is, at each iteration TRPO solves the constrained optimization problem,
+
+$$
+\begin{array} { r } { \operatornamewithlimits { m a x i m i z e } O _ { \mathrm { E R } } ( \pi ) \mathrm { s . t . } \mathbb { E } _ { s \sim \pi , \rho } [ \mathrm { K L } \left( \pi ( - | s ) \| \pi ( - | s ) \right) ] \leq \epsilon . } \end{array}
+$$
+
+The prior policy is then replaced with the new policy $\pi$ , and the process is repeated.
+
+# 4 METHOD
+
+To enable more stable training and better exploit the natural information geometry of the parameter space, we propose to augment the entropy regularized expected reward objective $O _ { \mathrm { E N T } }$ in (7) with a discounted relative entropy trust region around a prior policy $\tilde { \pi }$ ,
+
+$$
+\operatorname* { m a x i m i z e } _ { \pi } \mathbb { E } _ { s } [ O _ { \mathrm { E N T } } ( \pi ) ] \mathrm { ~ s . t . ~ } \mathbb { E } _ { s } [ \mathbb { G } ( s , \pi , { \tilde { \pi } } ) ] \leq \epsilon ,
+$$
+
+where the discounted relative entropy is recursively defined as
+
+$$
+\mathbb { G } ( s , \pi , \tilde { \pi } ) = \mathbb { E } _ { a , s ^ { \prime } } \left[ \log \pi ( a | s ) - \log \tilde { \pi } ( a | s ) + \gamma \mathbb { G } ( s ^ { \prime } , \pi , \tilde { \pi } ) \right] .
+$$
+
+This objective attempts to maximize entropy regularized expected reward while maintaining natural proximity to the previous policy. Although previous work has separately proposed to use relative entropy and entropy regularization, we find that the two components serve different purposes, each of which is beneficial: entropy regularization helps improve exploration, while the relative entropy improves stability and allows for a faster learning rate. This combination is a key novelty.
+
+Using the method of Lagrange multipliers, we cast the constrained optimization problem in (13) into maximization of the following objective,
+
+$$
+{ \cal O } _ { \mathrm { R E L E N T } } ( s , \pi ) = { \cal O } _ { \mathrm { E N T } } ( s , \pi ) - \lambda \mathbb { G } ( s , \pi , { \tilde { \pi } } ) .
+$$
+
+Again, the environment-wide objective is the expected per-state objective when states are sampled from interactions with the environment,
+
+$$
+O _ { \mathrm { R E L E N T } } ( \pi ) = \mathbb { E } _ { s } [ O _ { \mathrm { R E L E N T } } ( s , \pi ) ] .
+$$
+
+# 4.1 PATH CONSISTENCY WITH RELATIVE ENTROPY
+
+A key technical observation is that the $O _ { \mathrm { R E L E N T } }$ objective has a similar decomposition structure to $O _ { \mathrm { E N T } }$ , and one can cast $O$ RELENT as an entropy regularized expected reward objective with a set of transformed rewards, i.e.,
+
+$$
+O _ { \mathrm { R E L E N T } } ( s , \pi ) = \widetilde { O } _ { \mathtt { E R } } ( s , \pi ) + ( \tau + \lambda ) \mathbb { H } ( s , \pi ) ,
+$$
+
+where $\tilde { O } _ { \mathrm { E R } } ( s , \pi )$ is an expected reward objective on a transformed reward distribution function $\tilde { r } ( s , a ) = r ( s , a ) + \lambda \log \tilde { \pi } ( a | s )$ . Thus, in what follows, we derive a corresponding form of the multi-step path consistency in (10).
+
+Let $\pi ^ { * }$ denote the optimal policy, defined as $\pi ^ { * } = \operatorname { a r g m a x } _ { \pi } O _ { \mathrm { R E L E N T } } ( \pi )$ . As in PCL (Nachum et al., 2017), this optimal policy may be expressed as
+
+$$
+\pi ^ { * } ( a _ { t } | s _ { t } ) = \exp \left\{ \frac { \mathbb { E } _ { \tilde { r } _ { t } \sim \tilde { r } ( s _ { t } , a _ { t } ) , s _ { t + 1 } } [ \tilde { r } _ { t } + \gamma V ^ { * } ( s _ { t + 1 } ) ] - V ^ { * } ( s _ { t } ) } { \tau + \lambda } \right\} ,
+$$
+
+where $V ^ { * }$ are the softmax state values defined recursively as
+
+$$
+V ^ { * } ( s _ { t } ) = ( \tau + \lambda ) \log \int _ { A } \exp \left. \frac { \mathbb { E } _ { \tilde { r } _ { t } \sim \tilde { r } ( s _ { t } , a ) , s _ { t + 1 } } [ \tilde { r } _ { t } + \gamma V ^ { * } ( s _ { t + 1 } ) ] } { \tau + \lambda } \right. \mathrm { d } a .
+$$
+
+We may re-arrange (17) to yield
+
+$$
+\begin{array} { r c l } { V ^ { * } ( s _ { t } ) } & { = } & { \mathbb { E } _ { \tilde { r } _ { t } \sim \tilde { r } ( s _ { t } , a _ { t } ) , s _ { t + 1 } } [ \tilde { r } _ { t } - ( \tau + \lambda ) \log \pi ^ { * } ( a _ { t } | s _ { t } ) + \gamma V ^ { * } ( s _ { t + 1 } ) ] } \\ & { = } & { \mathbb { E } _ { r _ { t } , s _ { t + 1 } } [ r _ { t } - ( \tau + \lambda ) \log \pi ^ { * } ( a _ { t } | s _ { t } ) + \lambda \log \tilde { \pi } ( a _ { t + i } | s _ { t + i } ) + \gamma V ^ { * } ( s _ { t + 1 } ) ] . } \end{array}
+$$
+
+This is a single-step temporal consistency which may be extended to multiple steps by further expanding $V ^ { \ast } ( s _ { t + 1 } )$ on the RHS using the same identity. Thus, in general we have the following softmax temporal consistency constraint along any sequence of states defined by a starting state $s _ { t }$ and a sequence of actions $a _ { t } , \ldots , a _ { t + d - 1 }$ :
+
+$$
+r ^ { * } ( s _ { t } ) = \underset { r _ { t + i } , s _ { t + i } } { \mathbb { E } } \left[ \gamma ^ { d } V ^ { * } ( s _ { t + d } ) + \sum _ { i = 0 } ^ { d - 1 } \gamma ^ { i } \left( r _ { t + i } - ( \tau + \lambda ) \log \pi ^ { * } ( a _ { t + i } | s _ { t + i } ) + \lambda \log \tilde { \pi } ( a _ { t + i } | s _ { t + i } ) \right) \right] .
+$$
+
+# 4.2 TRUST-PCL
+
+We propose to train a parameterized policy $\pi _ { \theta }$ and value estimate $V _ { \phi }$ to satisfy the multi-step consistencies in (21). Thus, we define a consistency error for a sequence of states, actions, and rewards st:t+d ≡ (st, at, rt, . . . , st+d−1, at+d−1, rt+d−1, st+d) sampled from the environment as
+
+$$
+\begin{array} { r c l } { { C ( s _ { t : t + d } , \theta , \phi ) ~ { = } ~ - { V _ { \phi } ( s _ { t } ) + \gamma ^ { d } V _ { \phi } ( s _ { t + d } ) ~ + } } } & { { } } & { { } } \\ { { } } & { { } } & { { { \displaystyle \sum _ { i = 0 } ^ { d - 1 } \gamma ^ { i } \left( r _ { t + i } - ( { \tau + } \lambda ) \log { \pi _ { \theta } ( a _ { t + i } | s _ { t + i } ) } + \lambda \log { \pi _ { \bar { \theta } } ( a _ { t + i } | s _ { t + i } ) } \right) ~ . } } } \end{array}
+$$
+
+loss for a given batch of episodes (or sub-episodes) We aim to minimize the squared consistency error on every sub-trajectory of length $S = \{ s _ { 0 : T _ { k } } ^ { ( k ) } \} _ { k = 1 } ^ { B }$ is $d$ . That is, the
+
+$$
+\mathcal { L } ( S , \theta , \phi ) = \sum _ { k = 1 } ^ { B } \sum _ { t = 0 } ^ { T _ { k } - 1 } C ( s _ { t : t + d } ^ { ( k ) } , \theta , \phi ) ^ { 2 } .
+$$
+
+We perform gradient descent on $\theta$ and $\phi$ to minimize this loss. In practice, we have found that it is beneficial to learn the parameter $\phi$ at least as fast as $\theta$ , and accordingly, given a mini-batch of episodes we perform a single gradient update on $\theta$ and possibly multiple gradient updates on $\phi$ (see Appendix for details).
+
+In principle, the mini-batch $S$ may be taken from either on-policy or off-policy trajectories. In our implementation, we utilized a replay buffer prioritized by recency. As episodes (or sub-episodes) are sampled from the environment they are placed in a replay buffer and a priority $p { \big ( } s _ { 0 : T } )$ is given to a trajectory $s _ { 0 : T }$ equivalent to the current training step. Then, to sample a batch for training, $B$ episodes are sampled from the replay buffer proportional to exponentiated priority $\exp \{ \beta p ( s _ { 0 : T } ) \}$ for some hyperparameter $\beta \geq 0$ .
+
+For the prior policy $\pi _ { \tilde { \theta } }$ , we use a lagged geometric mean of the parameters. At each training step, we update $\tilde { \theta }  \alpha \tilde { \theta } + ( 1 - \alpha ) \theta$ . Thus on average our training scheme attempts to maximize entropy regularized expected reward while penalizing divergence from a policy roughly $1 / ( 1 - \alpha )$ training steps in the past.
+
+# 4.3 AUTOMATIC TUNING OF THE LAGRANGE MULTIPLIER $\lambda$
+
+The use of a relative entropy regularizer as a penalty rather than a constraint introduces several difficulties. The hyperparameter $\lambda$ must necessarily adapt to the distribution of rewards. Thus, $\lambda$ must be tuned not only to each environment but also during training on a single environment, since the observed reward distribution changes as the agent’s behavior policy improves. Using a constraint form of the regularizer is more desirable, and others have advocated its use in practice (Schulman et al., 2015) specifically to robustly allow larger updates during training.
+
+To this end, we propose to redirect the hyperparameter tuning from $\lambda$ to $\epsilon$ . Specifically, we present a method which, given a desired hard constraint on the relative entropy defined by $\epsilon$ , approximates the equivalent penalty coefficient $\lambda ( \epsilon )$ . This is a key novelty of our work and is distinct from previous attempts at automatically tuning a regularizing coefficient, which iteratively increase and decrease the coefficient based on observed training behavior (Schulman et al., 2017b; Heess et al., 2017).
+
+We restrict our analysis to the undiscounted setting $\gamma = 1$ with entropy regularizer $\tau = 0$ . Additionally, we assume deterministic, finite-horizon environment dynamics. An additional assumption we make is that the expected KL-divergence over states is well-approximated by the KL-divergence starting from the unique initial state $s _ { 0 }$ . Although in our experiments these restrictive assumptions are not met, we still found our method to perform well for adapting $\lambda$ during training.
+
+In this setting the optimal policy of (14) is proportional to exponentiated scaled reward. Specifically, for a full episode $s _ { 0 : T } = ( s _ { 0 } , a _ { 0 } , r _ { 0 } , \ldots , s _ { T - 1 } , a _ { T - 1 } , r _ { T - 1 } , s _ { T } )$ , we have
+
+$$
+\pi ^ { * } ( s _ { 0 : T } ) \propto \tilde { \pi } ( s _ { 0 : T } ) \exp \left\{ \frac { R ( s _ { 0 : T } ) } { \lambda } \right\} ,
+$$
+
+where $\begin{array} { r } { \pi ( s _ { 0 : T } ) = \prod _ { i = 0 } ^ { T - 1 } \pi ( a _ { i } | s _ { i } ) } \end{array}$ and $\begin{array} { r } { R ( s _ { 0 : T } ) = \sum _ { i = 0 } ^ { T - 1 } r _ { i } } \end{array}$ . The normalization factor of $\pi ^ { * }$
+
+$$
+Z = \mathbb { E } _ { s _ { 0 : T } \sim \tilde { \pi } } \left[ \exp \left\{ \frac { R ( s _ { 0 : T } ) } { \lambda } \right\} \right] .
+$$
+
+We would like to approximate the trajectory-wide KL-divergence between $\pi ^ { * }$ and $\tilde { \pi }$ . We may express the KL-divergence analytically:
+
+$$
+\begin{array} { r l } & { K L ( \pi ^ { * } | | \tilde { \pi } ) = \mathbb { E } _ { s _ { 0 : T } \sim \pi ^ { * } } \left[ \log \left( \frac { \pi ^ { * } \left( s _ { 0 : T } \right) } { \tilde { \pi } \left( s _ { 0 : T } \right) } \right) \right] } \\ & { \qquad = \mathbb { E } _ { s _ { 0 : T } \sim \pi ^ { * } } \left[ \frac { R \left( s _ { 0 : T } \right) } { \lambda } - \log Z \right] } \\ & { \qquad = - \log Z + \mathbb { E } _ { s _ { 0 : T } \sim \tilde { \pi } } \left[ \frac { R \left( s _ { 0 : T } \right) } { \lambda } \cdot \frac { \pi ^ { * } \left( s _ { 0 : T } \right) } { \tilde { \pi } \left( s _ { 0 : T } \right) } \right] } \\ & { \qquad = - \log Z + \mathbb { E } _ { s _ { 0 : T } \sim \tilde { \pi } } \left[ \frac { R \left( s _ { 0 : T } \right) } { \lambda } \exp \{ R ( s _ { 0 : T } ) / \lambda - \log Z \} \right] . } \end{array}
+$$
+
+Since all expectations are with respect to $\tilde { \pi }$ , this quantity is tractable to approximate given episodes sampled from $\tilde { \pi }$
+
+Therefore, in Trust-PCL, given a set of episodes sampled from the prior policy $\pi _ { \tilde { \theta } }$ and a desired maximum divergence $\epsilon$ , we can perform a simple line search to find a suitable $\lambda ( \epsilon )$ which yields $K L ( \pi ^ { * } | | \pi _ { \tilde { \theta } } )$ as close as possible to $\epsilon$ .
+
+The preceding analysis provided a method to determine $\lambda ( \epsilon )$ given a desired maximum divergence $\epsilon$ . However, there is still a question of whether $\epsilon$ should change during training. Indeed, as episodes may possibly increase in length, $K L ( \pi ^ { * } | | \tilde { \pi } )$ naturally increases when compared to the average perstate $K L ( \pi ^ { * } ( - | s ) | | \tilde { \pi } ( - | s ) )$ , and vice versa for decreasing length. Thus, in practice, given an $\epsilon$ and a set of sampled episodes $\dot { S } = \{ s _ { 0 : T _ { k } } ^ { ( k ) } \} _ { k = 1 } ^ { N }$ , we approximate the best $\lambda$ which yields a maximum divergence of $\begin{array} { r } { \frac { \epsilon } { N } \sum _ { k = 1 } ^ { N } T _ { k } } \end{array}$ . This makes it so that $\epsilon$ corresponds more to a constraint on the lengthaveraged KL-divergence.
+
+To avoid incurring a prohibitively large number of interactions with the environment for each parameter update, in practice we use the last 100 episodes as the set of sampled episodes $S$ . While this is not exactly the same as sampling episodes from $\pi _ { \tilde { \theta } }$ , it is not too far off since $\pi _ { \tilde { \theta } }$ is a lagged version of the online policy $\pi _ { \theta }$ . Moreover, we observed this protocol to work well in practice. A more sophisticated and accurate protocol may be derived by weighting the episodes according to the importance weights corresponding to their true sampling distribution.
+
+# 5 EXPERIMENTS
+
+We evaluate Trust-PCL against TRPO on a number of benchmark tasks. We choose TRPO as a baseline since it is a standard algorithm known to achieve state-of-the-art performance on the continuous control tasks we consider (see e.g., leaderboard results on the OpenAI Gym website (Brockman et al., 2016)). We find that Trust-PCL can match or improve upon TRPO’s performance in terms of both average reward and sample efficiency.
+
+# 5.1 SETUP
+
+We chose a number of control tasks available from OpenAI Gym (Brockman et al., 2016). The first task, Acrobot, is a discrete-control task, while the remaining tasks (HalfCheetah, Swimmer, Hopper, Walker2d, and Ant) are well-known continuous-control tasks utilizing the MuJoCo environment (Todorov et al., 2012).
+
+For TRPO we trained using batches of $Q \ = \ 2 5 , 0 0 0$ steps (12, 500 for Acrobot), which is the approximate batch size used by other implementations (Duan et al., 2016; Schulman, 2017). Thus, at each training iteration, TRPO samples 25, 000 steps using the policy $\pi _ { \tilde { \theta } }$ and then takes a single step within a KL-ball to yield a new $\pi _ { \theta }$ .
+
+Trust-PCL is off-policy, so to evaluate its performance we alternate between collecting experience and training on batches of experience sampled from the replay buffer. Specifically, we alternate between collecting $P = 1 0$ steps from the environment and performing a single gradient step based on a batch of size $Q = 6 4$ sub-episodes of length $P$ from the replay buffer, with a recency weight of $\beta = 0 . 0 0 1$ on the sampling distribution of the replay buffer. To maintain stability we use $\alpha = 0 . 9 9$ and we modified the loss from squared loss to Huber loss on the consistency error. Since our policy is parameterized by a unimodal Gaussian, it is impossible for it to satisfy all path consistencies, and so we found this crucial for stability.
+
+For each of the variants and for each environment, we performed a hyperparameter search to find the best hyperparameters. The plots presented here show the reward achieved during training on the best hyperparameters averaged over the best 4 seeds of 5 randomly seeded training runs. Note that this reward is based on greedy actions (rather than random sampling).
+
+Experiments were performed using Tensorflow (Abadi et al., 2016). Although each training step of Trust-PCL (a simple gradient step) is considerably faster than TRPO, we found that this does not have an overall effect on the run time of our implementation, due to a combination of the fact that each environment step is used in multiple training steps of Trust-PCL and that a majority of the run time is spent interacting with the environment. A detailed description of our implementation and hyperparameter search is available in the Appendix.
+
+# 5.2 RESULTS
+
+We present the reward over training of Trust-PCL and TRPO in Figure 1. We find that Trust-PCL can match or beat the performance of TRPO across all environments in terms of both final reward and sample efficiency. These results are especially significant on the harder tasks (Walker2d and Ant). We additionally present our results compared to other published results in Table 1. We find that even when comparing across different implementations, Trust-PCL can match or beat the state-of-the-art.
+
+# 5.2.1 HYPERPARAMETER ANALYSIS
+
+The most important hyperparameter in our method is $\epsilon$ , which determines the size of the trust region and thus has a critical role in the stability of the algorithm. To showcase this effect, we present the reward during training for several different values of $\epsilon$ in Figure 2. As $\epsilon$ increases, instability increases as well, eventually having an adverse effect on the agent’s ability to achieve optimal reward.
+
+![](images/2819973078574959af663e0a5bd2f625284a6841ecbb7974799c6d7e0e859717.jpg)  
+Figure 1: The results of Trust-PCL against a TRPO baseline. Each plot shows average greedy reward with single standard deviation error intervals capped at the min and max across 4 best of 5 randomly seeded training runs after choosing best hyperparameters. The $\mathbf { X }$ -axis shows millions of environment steps. We observe that Trust-PCL is consistently able to match and, in many cases, beat TRPO’s performance both in terms of reward and sample efficiency.
+
+![](images/34d09bb3531e483b9be242882f2de4c101ffb45016e355dc9b7df44bf18184cb.jpg)  
+Figure 2: The results of Trust-PCL across several values of $\epsilon$ , defining the size of the trust region. Each plot shows average greedy reward across 4 best of 5 randomly seeded training runs after choosing best hyperparameters. The $\mathbf { X }$ -axis shows millions of environment steps. We observe that instability increases with $\epsilon$ , thus concluding that the use of trust region is crucial.
+
+Note that standard PCL (Nachum et al., 2017) corresponds to $\epsilon  \infty$ (that is, $\lambda = 0$ ). Therefore, standard PCL would fail in these environments, and the use of trust region is crucial.
+
+The main advantage of Trust-PCL over existing trust region methods for continuous control is its ability to learn in an off-policy manner. The degree to which Trust-PCL is off-policy is determined by a combination of the hyparparameters $\alpha , \beta$ , and $P$ . To evaluate the importance of training off-policy, we evaluate Trust-PCL with a hyperparameter setting that is more on-policy. We set $\alpha = 0 . 9 5$ , $\beta = 0 . 1$ , and $P = 1 , 0 0 0$ . In this setting, we also use large batches of $Q = 2 5$ episodes of length $P$ (a total of 25, 000 environment steps per batch). Figure 3 shows the results of Trust-PCL with our original parameters and this new setting. We note a dramatic advantage in sample efficiency when using off-policy training. Although Trust-PCL (on-policy) can achieve state-of-the-art reward performance, it requires an exorbitant amount of experience. On the other hand, Trust-PCL (off
+
+![](images/b0d97e378156ad5edbbab6dfd5281f228dce1b8dcd57a11bb07f63cb417ae41b.jpg)  
+Figure 3: The results of Trust-PCL varying the degree of on/off-policy. We see that Trust-PCL (on-policy) has a behavior similar to TRPO, achieving good final reward but requiring an exorbitant number of experience collection. When collecting less experience per training step in Trust-PCL (off-policy), we are able to improve sample efficiency while still achieving a competitive final reward.
+
+<table><tr><td>Domain</td><td>TRPO-GAE</td><td>TRPO (rllab)</td><td>TRPO (ours)</td><td>Trust-PCL</td><td>IPG</td></tr><tr><td>HalfCheetah</td><td>4871.36</td><td>2889</td><td>4343.6</td><td>7057.1</td><td>4767</td></tr><tr><td>Swimmer</td><td>137.25</td><td>1</td><td>288.1</td><td>297.0</td><td></td></tr><tr><td>Hopper</td><td>3765.78</td><td>1</td><td>3516.7</td><td>3804.9</td><td>一</td></tr><tr><td>Walker2d</td><td>6028.73</td><td>1487</td><td>2838.4</td><td>5027.2</td><td>3047</td></tr><tr><td>Ant</td><td>2918.25</td><td>1520</td><td>4347.5</td><td>6104.2</td><td>4415</td></tr></table>
+
+Table 1: Results for best average reward in the first 10M steps of training for our implementations (TRPO (ours) and Trust-PCL) and external implementations. TRPO-GAE are results of Schulman (2017) available on the OpenAI Gym website. TRPO (rllab) and IPG are taken from Gu et al. (2017b). These results are each on different setups with different hyperparameter searches and in some cases different evaluation protocols (e.g.,TRPO (rllab) and IPG were run with a simple linear value network instead of the two-hidden layer network we use). Thus, it is not possible to make any definitive claims based on this data. However, we do conclude that our results are overall competitive with state-of-the-art external implementations.
+
+policy) can be competitive in terms of reward while providing a significant improvement in sample efficiency.
+
+One last hyperparameter is $\tau$ , determining the degree of exploration. Anecdotally, we found $\tau$ to not be of high importance for the tasks we evaluated. Indeed many of our best results use $\tau =$ 0. Including $\tau > 0$ had a marginal effect, at best. The reason for this is likely due to the tasks themselves. Indeed, other works which focus on exploration in continuous control have found the need to propose exploration-advanageous variants of these standard benchmarks (Haarnoja et al., 2017; Houthooft et al., 2016).
+
+# 6 CONCLUSION
+
+We have presented Trust-PCL, an off-policy algorithm employing a relative-entropy penalty to impose a trust region on a maximum reward objective. We found that Trust-PCL can perform well on a set of standard control tasks, improving upon TRPO both in terms of average reward and sample efficiency. Our best results on Trust-PCL are able to maintain the stability and solution quality of TRPO while approaching the sample-efficiency of value-based methods (see e.g., Metz et al. (2017)). This gives hope that the goal of achieving both stability and sample-efficiency without trading-off one for the other is attainable in a single unifying RL algorithm.
+
+# 7 ACKNOWLEDGMENT
+
+We thank Matthew Johnson, Luke Metz, Shane Gu, and the Google Brain team for insightful comments and discussions.
+
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+# A IMPLEMENTATION BENEFITS OF TRUST-PCL
+
+We have already highlighted the ability of Trust-PCL to use off-policy data to stably train both a parameterized policy and value estimate, which sets it apart from previous methods. We have also noted the ease with which exploration can be incorporated through the entropy regularizer. We elaborate on several additional benefits of Trust-PCL.
+
+Compared to TRPO, Trust-PCL is much easier to implement. Standard TRPO implementations perform second-order gradient calculations on the KL-divergence to construct a Fisher information matrix (more specifically a vector product with the inverse Fisher information matrix). This yields a vector direction for which a line search is subsequently employed to find the optimal step. Compare this to Trust-PCL which employs simple gradient descent. This makes implementation much more straightforward and easily realizable within standard deep learning frameworks.
+
+Even if one replaces the constraint on the average KL-divergence of TRPO with a simple regularization penalty (as in proximal policy gradient methods (Schulman et al., 2017b; Wang et al., 2016)), optimizing the resulting objective requires computing the gradient of the KL-divergence. In Trust-PCL, there is no such necessity. The per-state KL-divergence need not have an analytically computable gradient. In fact, the KL-divergence need not have a closed form at all. The only requirement of Trust-PCL is that the log-density be analytically computable. This opens up the possible policy parameterizations to a much wider class of functions. While continuous control has traditionally used policies parameterized by unimodal Gaussians, with Trust-PCL the policy can be replaced with something much more expressive—for example, mixtures of Gaussians or autoregressive policies as in Metz et al. (2017).
+
+We have yet to fully explore these additional benefits in this work, but we hope that future investigations can exploit the flexibility and ease of implementation of Trust-PCL to further the progress of RL in continuous control environments.
+
+# B EXPERIMENTAL SETUP
+
+We describe in detail the experimental setup regarding implementation and hyperparameter search.
+
+# B.1 ENVIRONMENTS
+
+In Acrobot, episodes were cut-off at step 500. For the remaining environments, episodes were cutoff at step $1 , 0 0 0$ .
+
+Acrobot, HalfCheetah, and Swimmer are all non-terminating environments. Thus, for these environments, each episode had equal length and each batch contained the same number of episodes. Hopper, Walker2d, and Ant are environments that can terminate the agent. Thus, for these environments, the batch size throughout training remained constant in terms of steps but not in terms of episodes.
+
+There exists an additional common MuJoCo task called Humanoid. We found that neither our implementation of TRPO nor Trust-PCL could make more than negligible headway on this task, and so omit it from the results. We are aware that TRPO with the addition of GAE and enough finetuning can be made to achieve good results on Humanoid (Schulman et al., 2016). We decided to not pursue a GAE implementation to keep a fair comparison between variants. Trust-PCL can also be made to incorporate an analogue to GAE (by maintaining consistencies at varying time scales), but we leave this to future work.
+
+# B.2 IMPLEMENTATION DETAILS
+
+We use fully-connected feed-forward neural networks to represent both policy and value.
+
+The policy $\pi _ { \theta }$ is represented by a neural network with two hidden layers of dimension 64 with tanh activations. At time step $t$ , the network is given the observation $s _ { t }$ . It produces a vector $\mu _ { t }$ , which is combined with a learnable (but $t$ -agnostic) parameter $\xi$ to parametrize a unimodal Gaussian with mean $\mu _ { t }$ and standard deviation $\exp ( \xi )$ . The next action $a _ { t }$ is sampled randomly from this Gaussian.
+
+The value network $V _ { \phi }$ is represented by a neural network with two hidden layers of dimension 64 with tanh activations. At time step $t$ the network is given the observation $s _ { t }$ and the component-wise squared observation $s _ { t } \odot s _ { t }$ . It produces a single scalar value.
+
+# B.2.1 TRPO LEARNING
+
+At each training iteration, both the policy and value parameters are updated. The policy is trained by performing a trust region step according to the procedure described in Schulman et al. (2015).
+
+The value parameters at each step are solved using an LBFGS optimizer. To avoid instability, the value parameters are solved to fit a mixture of the empirical values and the expected values. That is, we determine $\phi$ to minimize $\begin{array} { r } { \sum _ { s \in \mathrm { b a t c h } } ( V _ { \phi } ( s ) - \kappa V _ { \tilde { \phi } } ( s ) - ( 1 - \kappa ) \hat { V } _ { \tilde { \phi } } ( s ) ) ^ { 2 } } \end{array}$ , where again $\tilde { \phi }$ is the previous value parameterization. We use $\kappa = 0 . 9$ . This method for training $\phi$ is according to that used in Schulman (2017).
+
+# B.2.2 TRUST-PCL LEARNING
+
+At each training iteration, both the policy and value parameters are updated. The specific updates are slightly different between Trust-PCL (on-policy) and Trust-PCL (off-policy).
+
+For Trust-PCL (on-policy), the policy is trained by taking a single gradient step using the Adam optimizer (Kingma & Ba, 2015) with learning rate 0.001. The value network update is inspired by that used in TRPO we perform 5 gradients steps with learning rate 0.001, calculated with regards to a mix between the empirical values and the expected values according to the previous $\tilde { \phi }$ . We use $\kappa = 0 . 9 5$ .
+
+For Trust-PCL (off-policy), both the policy and value parameters are updated in a single step using the Adam optimizer with learning rate 0.0001. For this variant, we also utilize a target value network (lagged at the same rate as the target policy network) to replace the value estimate at the final state for each path. We do not mix between empirical and expected values.
+
+# B.3 HYPERPARAMETER SEARCH
+
+We found the most crucial hyperparameters for effective learning in both TRPO and TrustPCL to be $\epsilon$ (the constraint defining the size of the trust region) and $d$ (the rollout determining how to evaluate the empirical value of a state). For TRPO we performed a grid search over $\epsilon \in \{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 \} , d \in \{ 1 0 , 5 0 \}$ . For Trust-PCL we performed a grid search over $\epsilon \in \{ 0 . 0 0 1 , 0 . 0 0 2 , 0 . 0 0 5 , 0 . 0 1 \} , d \in \{ 1 0$ $d \in \{ 1 0 , 5 0 \}$ . For Trust-PCL we also experimented with the value of $\tau$ , either keeping it at a constant 0 (thus, no exploration) or decaying it from 0.1 to 0.0 by a smoothed exponential rate of 0.1 every 2,500 training iterations.
+
+We fix the discount to $\gamma = 0 . 9 9 5$ for all environments.
+
+# C PSEUDOCODE
+
+A simplified pseudocode for Trust-PCL is presented in Algorithm 1.
+
+# Algorithm 1 Trust-PCL
+
+Input: Environment $E N V$ , trust region constraint $\epsilon$ , learning rates $\eta _ { \pi } , \eta _ { v }$ , discount factor $\gamma$ , rollout $d$ , batch size $Q$ , collect steps per train step $P$ , number of training steps $N$ , replay buffer $R B$ with exponential lag $\beta$ , lag on prior policy $\alpha$ .
+
+function Gradients $( \{ s _ { t : t + P } ^ { ( k ) } \} _ { k = 1 } ^ { B } )$ $/ / C$ is thpute $\begin{array} { r } { \Delta \theta = \sum _ { k = 1 } ^ { B } \sum _ { p = 0 } ^ { P - 1 } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \theta , \phi ) \nabla _ { \theta } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \theta , \phi ) . } \end{array}$ $\begin{array} { r } { \Delta \phi = \sum _ { k = 1 } ^ { B } \sum _ { p = 0 } ^ { P - 1 } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \theta , \phi ) \nabla _ { \phi } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \theta , \phi ) . } \end{array}$ Return $\Delta \theta , \Delta \phi$   
+end function   
+Initialize $\theta , \phi , \lambda$ , set $\tilde { \theta } = \theta$ .   
+Initialize empty replay buffer $R B ( \beta )$ .   
+for $i = 0$ to $N - 1$ do // Collect Sample $P$ steps $s _ { t : t + P } \sim \pi _ { \theta }$ on $E N V$ . Insert $s _ { t : t + P }$ to $R B$ . // Train   
+Sample batch $\{ s _ { t : t + P } ^ { ( k ) } \} _ { k = 1 } ^ { B }$ from $R B$ to contain a total of $Q$ transitions $( B \approx Q / P )$ . $\Delta \theta , \Delta \phi = \mathrm { G r a d i e n t s } \big ( \{ s _ { t : t + P } ^ { ( k ) } \} _ { k = 1 } ^ { B } \big )$ . Update $\theta  \theta - \eta _ { \pi } \Delta \theta$ . Update $\phi  \phi - \eta _ { v } \Delta \phi$ . // Update auxiliary variables Update $\tilde { \theta } = \alpha \tilde { \theta } + \mathrm { \Gamma } ( 1 - \alpha ) \theta$ . Update $\lambda$ in terms of $\epsilon$ according to Section 4.3.   
+end for

parse/train/HyrCWeWCb/HyrCWeWCb_content_list.json

@@ -0,0 +1,2149 @@
+[
+    {
+        "type": "text",
+        "text": "TRUST-PCL: AN OFF-POLICY TRUST REGION METHOD FOR CONTINUOUS CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ofir Nachum, Mohammad Norouzi, Kelvin Xu, & Dale Schuurmans∗ {ofirnachum,mnorouzi,kelvinxx,schuurmans}@google.com Google Brain ",
+        "bbox": [
+            184,
+            170,
+            687,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Trust region methods, such as TRPO, are often used to stabilize policy optimization algorithms in reinforcement learning (RL). While current trust region strategies are effective for continuous control, they typically require a large amount of on-policy interaction with the environment. To address this problem, we propose an off-policy trust region method, Trust-PCL, which exploits an observation that the optimal policy and state values of a maximum reward objective with a relative-entropy regularizer satisfy a set of multi-step pathwise consistencies along any path. The introduction of relative entropy regularization allows Trust-PCL to maintain optimization stability while exploiting off-policy data to improve sample efficiency. When evaluated on a number of continuous control tasks, Trust-PCL significantly improves the solution quality and sample efficiency of TRPO.1 ",
+        "bbox": [
+            233,
+            281,
+            764,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            463,
+            336,
+            479
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The goal of model-free reinforcement learning (RL) is to optimize an agent’s behavior policy through trial and error interaction with a black box environment. Value-based RL algorithms such as Q-learning (Watkins, 1989) and policy-based algorithms such as actor-critic (Konda & Tsitsiklis, 2000) have achieved well-known successes in environments with enumerable action spaces and predictable but possibly complex dynamics, e.g., as in Atari games (Mnih et al., 2013; Van Hasselt et al., 2016; Mnih et al., 2016). However, when applied to environments with more sophisticated action spaces and dynamics (e.g., continuous control and robotics), success has been far more limited. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            593
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In an attempt to improve the applicability of Q-learning to continuous control, Silver et al. (2014) and Lillicrap et al. (2015) developed an off-policy algorithm DDPG, leading to promising results on continuous control environments. That said, current off-policy methods including DDPG often improve data efficiency at the cost of optimization stability. The behaviour of DDPG is known to be highly dependent on hyperparameter selection and initialization (Metz et al., 2017); even when using optimal hyperparameters, individual training runs can display highly varying outcomes. ",
+        "bbox": [
+            174,
+            599,
+            823,
+            684
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, in an attempt to improve the stability and convergence speed of policy-based RL methods, Kakade (2002) developed a natural policy gradient algorithm based on Amari (1998), which subsequently led to the development of trust region policy optimization (TRPO) (Schulman et al., 2015). TRPO has shown strong empirical performance on difficult continuous control tasks often outperforming value-based methods like DDPG. However, a major drawback is that such methods are not able to exploit off-policy data and thus require a large amount of on-policy interaction with the environment, making them impractical for solving challenging real-world problems. ",
+        "bbox": [
+            174,
+            690,
+            825,
+            789
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Efforts at combining the stability of trust region policy-based methods with the sample efficiency of value-based methods have focused on using off-policy data to better train a value estimate, which can be used as a control variate for variance reduction (Gu et al., 2017a;b). ",
+        "bbox": [
+            174,
+            795,
+            825,
+            837
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we investigate an alternative approach to improving the sample efficiency of trust region policy-based RL methods. We exploit the key fact that, under entropy regularization, the optimal policy and value function satisfy a set of pathwise consistency properties along any sampled path (Nachum et al., 2017), which allows both on and off-policy data to be incorporated in an actor-critic algorithm, PCL. The original PCL algorithm optimized an entropy regularized maximum reward objective and was evaluated on relatively simple tasks. Here we extend the ideas of PCL to achieve strong results on standard, challenging continuous control benchmarks. The main observation is that by alternatively augmenting the maximum reward objective with a relative entropy regularizer, the optimal policy and values still satisfy a certain set of pathwise consistencies along any sampled trajectory. The resulting objective is equivalent to maximizing expected reward subject to a penalty-based constraint on divergence from a reference (i.e., previous) policy. ",
+        "bbox": [
+            176,
+            844,
+            821,
+            872
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We exploit this observation to propose a new off-policy trust region algorithm, Trust-PCL, that is able to exploit off-policy data to train policy and value estimates. Moreover, we present a simple method for determining the coefficient on the relative entropy regularizer to remain agnostic to reward scale, hence ameliorating the task of hyperparameter tuning. We find that the incorporation of a relative entropy regularizer is crucial for good and stable performance. We evaluate TrustPCL against TRPO, and observe that Trust-PCL is able to solve difficult continuous control tasks, while improving the performance of TRPO both in terms of the final reward achieved as well as sample-efficiency. ",
+        "bbox": [
+            174,
+            236,
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+            348
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            367,
+            344,
+            383
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Trust Region Methods. Gradient descent is the predominant optimization method for neural networks. A gradient descent step is equivalent to solving a trust region constrained optimization, ",
+        "bbox": [
+            173,
+            400,
+            825,
+            429
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7299773d66c3516e87bf42904a77ee55d6965d9fa9a633be53acddc1b4c596d2.jpg",
+        "text": "$$\n\\mathrm { m i n i m i z e } ~ { \\boldsymbol { \\ell } } ( { \\boldsymbol { \\theta } } + \\mathrm { d } { \\boldsymbol { \\theta } } ) \\approx { \\boldsymbol { \\ell } } ( { \\boldsymbol { \\theta } } ) + { \\boldsymbol { \\nabla } } { \\boldsymbol { \\ell } } ( { \\boldsymbol { \\theta } } ) ^ { \\top } \\mathrm { d } { \\boldsymbol { \\theta } } \\qquad { \\mathrm { s . t . } } \\quad \\mathrm { d } { \\boldsymbol { \\theta } } ^ { \\top } \\mathrm { d } { \\boldsymbol { \\theta } } \\leq \\epsilon ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            436,
+            710,
+            457
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "which yields the locally optimal update $\\mathrm { d } \\theta = - \\eta \\nabla \\ell ( \\theta )$ such that $\\eta ~ = ~ { \\sqrt { \\epsilon } } / \\| \\nabla \\ell ( \\theta ) \\|$ ; hence by considering a Euclidean ball, gradient descent assumes the parameters lie in a Euclidean space. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, in machine learning, particularly in the context of multi-layer neural network training, Euclidean geometry is not necessarily the best way to characterize proximity in parameter space. It is often more effective to define an appropriate Riemannian metric that respects the loss surface (Amari, 2012), which allows much steeper descent directions to be identified within a local neighborhood (e.g., Amari (1998); Martens & Grosse (2015)). Whenever the loss is defined in terms of a Bregman divergence between an (unknown) optimal parameter $\\theta ^ { * }$ and model parameter $\\theta$ , i.e., $\\ell ( \\theta ) \\equiv D _ { \\mathrm { { F } } } ( \\theta ^ { * } , \\theta )$ , it is natural to use the same divergence to form the trust region: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bedb492bf85445eecd0787b4ef439976acf11b4736b8e33e1aff7b23597e8111.jpg",
+        "text": "$$\n\\mathrm { m i n i m i z e } D _ { \\mathrm { F } } ( \\theta ^ { * } , \\theta + \\mathrm { d } \\theta ) \\mathrm { s . t . } D _ { \\mathrm { F } } ( \\theta , \\theta + \\mathrm { d } \\theta ) \\leq \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            604,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The natural gradient (Amari, 1998) is a generalization of gradient descent where the Fisher information matrix $F ( \\theta )$ is used to define the local geometry of the parameter space around $\\theta$ . If a parameter update is constrained by $\\mathrm { d } \\theta ^ { \\mathsf { T } } F ( \\theta ) \\mathrm { d } \\theta \\leq \\epsilon$ , a descent direction of $\\mathrm { d } \\theta \\equiv - \\eta F ( \\theta ) ^ { - 1 } \\nabla \\ell ( \\theta )$ is obtained. This geometry is especially effective for optimizing the log-likelihood of a conditional probabilistic model, where the objective is in fact the KL divergence $D _ { \\mathrm { K L } } ( \\theta ^ { * } , \\theta )$ . The local optimization is, ",
+        "bbox": [
+            173,
+            636,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fff2dcea30febf56cfce9a6bacd3c55322886614c47674fb50c65304e46990e.jpg",
+        "text": "$$\n\\mathrm { m i n i m i z e } D _ { \\mathrm { K L } } ( \\theta ^ { * } , \\theta + \\mathrm { d } \\theta ) \\quad \\mathrm { ~ s . t . ~ } \\quad D _ { \\mathrm { K L } } ( \\theta , \\theta + \\mathrm { d } \\theta ) \\approx \\mathrm { d } \\theta ^ { \\mathsf { T } } F ( \\theta ) \\mathrm { d } \\theta \\leq \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            718,
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+            737
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Thus, natural gradient approximates the trust region by $D _ { \\mathrm { { K L } } } ( a , b ) \\approx ( a - b ) ^ { \\mathsf { T } } F ( a ) ( a - b )$ , which is accurate up to a second order Taylor approximation. Previous work (Kakade, 2002; Bagnell $\\&$ Schneider, 2003; Peters & Schaal, 2008; Schulman et al., 2015) has applied natural gradient to policy optimization, locally improving expected reward subject to variants of $\\mathrm { d } \\theta ^ { \\mathsf { T } } F ( \\theta ) \\mathrm { d } \\theta \\leq \\epsilon$ . Recently, TRPO (Schulman et al., 2015; 2016) has achieved state-of-the-art results in continuous control by adding several approximations to the natural gradient to make nonlinear policy optimization feasible. ",
+        "bbox": [
+            173,
+            746,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another approach to trust region optimization is given by proximal gradient methods (Parikh et al., 2014). The class of proximal gradient methods most similar to our work are those that replace the hard constraint in (2) with a penalty added to the objective. These techniques have recently become popular in RL (Wang et al., 2016; Heess et al., 2017; Schulman et al., 2017b), although in terms of final reward performance on continuous control benchmarks, TRPO is still considered to be the state-of-the-art. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Norouzi et al. (2016) make the observation that entropy regularized expected reward may be expressed as a reversed KL divergence $D _ { \\mathrm { K L } } ( \\theta , \\theta ^ { * } )$ , which suggests that an alternative to the constraint in (3) should be used when such regularization is present: ",
+        "bbox": [
+            173,
+            103,
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+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57aba5a060ffe5c586141be01042086645da4d8884fa37241cef19f01ef1d16f.jpg",
+        "text": "$$\nD _ { \\mathrm { K L } } ( \\boldsymbol \\theta + \\mathrm { d } \\boldsymbol \\theta , \\boldsymbol \\theta ^ { * } ) \\qquad \\mathrm { s . t . } \\qquad D _ { \\mathrm { K L } } ( \\boldsymbol \\theta + \\mathrm { d } \\boldsymbol \\theta , \\boldsymbol \\theta ) \\approx \\mathrm { d } \\boldsymbol \\theta ^ { \\mathsf { T } } \\boldsymbol F ( \\boldsymbol \\theta + \\mathrm { d } \\boldsymbol \\theta ) \\mathrm { d } \\boldsymbol \\theta \\leq \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            150,
+            769,
+            169
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, this update requires computing the Fisher matrix at the endpoint of the update. The use of $F ( \\theta )$ in previous work can be considered to be an approximation when entropy regularization is present, but it is not ideal, particularly if $\\mathrm { d } \\theta$ is large. In this paper, by contrast, we demonstrate that the optimal $\\mathrm { d } \\theta$ under the reverse KL constraint $\\bar { D _ { \\mathrm { K L } } } ( \\theta + \\mathrm { d } \\bar { \\theta } , \\bar { \\theta } ) \\leq \\dot { \\epsilon }$ can indeed be characterized. Defining the constraint in this way appears to be more natural and effective than that of TRPO. ",
+        "bbox": [
+            173,
+            171,
+            825,
+            242
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Softmax Consistency. To comply with the information geometry over policy parameters, previous work has used the relative entropy (i.e., KL divergence) to regularize policy optimization; resulting in a softmax relationship between the optimal policy and state values (Peters et al., 2010; Azar et al., 2012; 2011; Fox et al., 2016; Rawlik et al., 2013) under single-step rollouts. Our work is unique in that we leverage consistencies over multi-step rollouts. ",
+        "bbox": [
+            173,
+            248,
+            825,
+            319
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The existence of multi-step softmax consistencies has been noted by prior work—first by Nachum et al. (2017) in the presence of entropy regularization. The existence of the same consistencies with relative entropy has been noted by Schulman et al. (2017a). Our work presents multi-step consistency relations for a hybrid relative entropy plus entropy regularized expected reward objective, interpreting relative entropy regularization as a trust region constraint. This work is also distinct from prior work in that the coefficient of relative entropy can be automatically determined, which we have found to be especially crucial in cases where the reward distribution changes dramatically during training. ",
+        "bbox": [
+            173,
+            325,
+            825,
+            438
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Most previous work on softmax consistency (e.g., Fox et al. (2016); Azar et al. (2012); Nachum et al. (2017)) have only been evaluated on relatively simple tasks, including grid-world and discrete algorithmic environments. Rawlik et al. (2013) conducted evaluations on simple variants of the CartPole and Pendulum continuous control tasks. More recently, Haarnoja et al. (2017) showed that soft Qlearning (a single-step special case of PCL) can succeed on more challenging environments, such as a variant of the Swimmer task we consider below. By contrast, this paper presents a successful application of the softmax consistency concept to difficult and standard continuous-control benchmarks, resulting in performance that is competitive with and in some cases beats the state-of-the-art. ",
+        "bbox": [
+            173,
+            443,
+            825,
+            555
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 NOTATION & BACKGROUND",
+        "text_level": 1,
+        "bbox": [
+            176,
+            574,
+            439,
+            590
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We model an agent’s behavior by a policy distribution $\\pi ( a | s )$ over a set of actions (possibly discrete or continuous). At iteration $t$ , the agent encounters a state $s _ { t }$ and performs an action $a _ { t }$ sampled from $\\pi ( \\boldsymbol { a } \\mid \\boldsymbol { s } _ { t } )$ . The environment then returns a scalar reward $\\displaystyle r _ { t } \\sim r ( s _ { t } , a _ { t } )$ and transitions to the next state $s _ { t + 1 } \\sim \\rho ( s _ { t } , a _ { t } )$ . When formulating expectations over actions, rewards, and state transitions we will often omit the sampling distributions, $\\pi , r$ , and $\\rho$ , respectively. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Maximizing Expected Reward. The standard objective in RL is to maximize expected future discounted reward. We formulate this objective on a per-state basis recursively as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e92d3528327f5c6feb27f80d5de42be710a854f8349136b6871b145c20d71a6.jpg",
+        "text": "$$\nO _ { \\mathrm { E R } } ( s , \\pi ) = \\mathbb { E } _ { a , r , s ^ { \\prime } } \\left[ r + \\gamma O _ { \\mathrm { E R } } ( s ^ { \\prime } , \\pi ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The overall, state-agnostic objective is the expected per-state objective when states are sampled from interactions with the environment: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30d347d2fef35e1ac423b56070e6549b4c59b95c0fbda7742e55574a49f71bc3.jpg",
+        "text": "$$\nO _ { \\mathrm { E R } } ( \\pi ) = \\mathbb { E } _ { s } [ O _ { \\mathrm { E R } } ( s , \\pi ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Most policy-based algorithms, including REINFORCE (Williams & Peng, 1991) and actorcritic (Konda & Tsitsiklis, 2000), aim to optimize $O _ { \\mathrm { E R } }$ given a parameterized policy. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Path Consistency Learning (PCL). Inspired by Williams & Peng (1991), Nachum et al. (2017) augment the objective $O _ { \\mathrm { E R } }$ in (5) with a discounted entropy regularizer to derive an objective, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f318ae0770265f10a6ce2028b2c0c6cb781e1d0b44c240f11260b5b15f1f3674.jpg",
+        "text": "$$\nO _ { \\mathrm { E N T } } ( s , \\pi ) = O _ { \\mathrm { E R } } ( s , \\pi ) + \\tau \\mathbb { H } ( s , \\pi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau \\geq 0$ is a user-specified temperature parameter that controls the degree of entropy regularization, and the discounted entropy $\\mathbb { H } ( s , \\pi )$ is recursively defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d89d8a964f30e72a66ed6e03b4d51b5f804f9a099d84501300f3ee384b2938ad.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { H } ( s , \\pi ) = \\mathbb { E } _ { a , s ^ { \\prime } } [ - \\log \\pi ( a \\mid s ) + \\gamma \\mathbb { H } ( s ^ { \\prime } , \\pi ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the objective $O _ { \\mathrm { E N T } } ( s , \\pi )$ can then be re-expressed recursively as, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad6fd278e46d9326230444d93bae18e8213b2bc8b02cd6f5ee6b5e08ffcf2fcd.jpg",
+        "text": "$$\nO _ { \\mathrm { E N T } } ( s , \\pi ) = \\mathbb { E } _ { a , r , s ^ { \\prime } } [ r - \\tau \\log \\pi ( a \\mid s ) + \\gamma O _ { \\mathrm { E N T } } ( s ^ { \\prime } , \\pi ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Nachum et al. (2017) show that the optimal policy $\\pi ^ { * }$ for $O _ { \\mathrm { E N T } }$ and $V ^ { * } ( s ) = O _ { \\mathrm { E N T } } ( s , \\pi ^ { * } )$ mutually satisfy a softmax temporal consistency constraint along any sequence of states $s _ { 0 } , \\ldots , s _ { d }$ starting at $s _ { 0 }$ and a corresponding sequence of actions $a _ { 0 } , \\ldots , a _ { d - 1 }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a34440ce67fdd4ff211efe5e959dbdb724f460d7d19891149b1cc907a65992e.jpg",
+        "text": "$$\nV ^ { * } ( s _ { 0 } ) = \\underset { r _ { i } , s _ { i } } { \\mathbb { E } } \\left[ \\gamma ^ { d } V ^ { * } ( s _ { d } ) + \\sum _ { i = 0 } ^ { d - 1 } \\gamma ^ { i } ( r _ { i } - \\tau \\log \\pi ^ { * } ( a _ { i } | s _ { i } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This observation led to the development of the PCL algorithm, which attempts to minimize squared error between the LHS and RHS of (10) to simultaneously optimize parameterized $\\pi _ { \\theta }$ and $V _ { \\phi }$ . Importantly, PCL is applicable to both on-policy and off-policy trajectories. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Trust Region Policy Optimization (TRPO). As noted, standard policy-based algorithms for maximizing $O _ { \\mathrm { E R } }$ can be unstable and require small learning rates for training. To alleviate this issue, Schulman et al. (2015) proposed to perform an iterative trust region optimization to maximize $O _ { \\mathrm { E R } }$ . At each step, a prior policy $\\tilde { \\pi }$ is used to sample a large batch of trajectories, then $\\pi$ is subsequently optimized to maximize $O _ { \\mathrm { E R } }$ while remaining within a constraint defined by the average per-state KL-divergence with $\\tilde { \\pi }$ . That is, at each iteration TRPO solves the constrained optimization problem, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ec4d359eb657b71cb019d0e5c026f0d7345d61724d4c0f299d9151c15fd9e85.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatornamewithlimits { m a x i m i z e } O _ { \\mathrm { E R } } ( \\pi ) \\mathrm { s . t . } \\mathbb { E } _ { s \\sim \\pi , \\rho } [ \\mathrm { K L } \\left( \\pi ( - | s ) \\| \\pi ( - | s ) \\right) ] \\leq \\epsilon . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The prior policy is then replaced with the new policy $\\pi$ , and the process is repeated. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            282,
+            479
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To enable more stable training and better exploit the natural information geometry of the parameter space, we propose to augment the entropy regularized expected reward objective $O _ { \\mathrm { E N T } }$ in (7) with a discounted relative entropy trust region around a prior policy $\\tilde { \\pi }$ , ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe8df21a4d205d6235194deeed2c880b81fc8998d020d48d8732667e01ddaa92.jpg",
+        "text": "$$\n\\operatorname* { m a x i m i z e } _ { \\pi } \\mathbb { E } _ { s } [ O _ { \\mathrm { E N T } } ( \\pi ) ] \\mathrm { ~ s . t . ~ } \\mathbb { E } _ { s } [ \\mathbb { G } ( s , \\pi , { \\tilde { \\pi } } ) ] \\leq \\epsilon ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            545,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the discounted relative entropy is recursively defined as ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7d7c510878a19012a03194105cd802dd2ee9aed8a60c2d9267246b6379bd1a3.jpg",
+        "text": "$$\n\\mathbb { G } ( s , \\pi , \\tilde { \\pi } ) = \\mathbb { E } _ { a , s ^ { \\prime } } \\left[ \\log \\pi ( a | s ) - \\log \\tilde { \\pi } ( a | s ) + \\gamma \\mathbb { G } ( s ^ { \\prime } , \\pi , \\tilde { \\pi } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This objective attempts to maximize entropy regularized expected reward while maintaining natural proximity to the previous policy. Although previous work has separately proposed to use relative entropy and entropy regularization, we find that the two components serve different purposes, each of which is beneficial: entropy regularization helps improve exploration, while the relative entropy improves stability and allows for a faster learning rate. This combination is a key novelty. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using the method of Lagrange multipliers, we cast the constrained optimization problem in (13) into maximization of the following objective, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9e8d74d96603ad40723f98aa42c83762d1ce045f6ebff16dfd697d0164818c5d.jpg",
+        "text": "$$\n{ \\cal O } _ { \\mathrm { R E L E N T } } ( s , \\pi ) = { \\cal O } _ { \\mathrm { E N T } } ( s , \\pi ) - \\lambda \\mathbb { G } ( s , \\pi , { \\tilde { \\pi } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Again, the environment-wide objective is the expected per-state objective when states are sampled from interactions with the environment, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82340d13708fe541c61d0e7ac47e13d0ce6beecb492886d0a15779d0711278e5.jpg",
+        "text": "$$\nO _ { \\mathrm { R E L E N T } } ( \\pi ) = \\mathbb { E } _ { s } [ O _ { \\mathrm { R E L E N T } } ( s , \\pi ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 PATH CONSISTENCY WITH RELATIVE ENTROPY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A key technical observation is that the $O _ { \\mathrm { R E L E N T } }$ objective has a similar decomposition structure to $O _ { \\mathrm { E N T } }$ , and one can cast $O$ RELENT as an entropy regularized expected reward objective with a set of transformed rewards, i.e., ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b80f5158c2cb54762d820136461113e3a80c70aedc60c0484aeaebc67f23619.jpg",
+        "text": "$$\nO _ { \\mathrm { R E L E N T } } ( s , \\pi ) = \\widetilde { O } _ { \\mathtt { E R } } ( s , \\pi ) + ( \\tau + \\lambda ) \\mathbb { H } ( s , \\pi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { O } _ { \\mathrm { E R } } ( s , \\pi )$ is an expected reward objective on a transformed reward distribution function $\\tilde { r } ( s , a ) = r ( s , a ) + \\lambda \\log \\tilde { \\pi } ( a | s )$ . Thus, in what follows, we derive a corresponding form of the multi-step path consistency in (10). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $\\pi ^ { * }$ denote the optimal policy, defined as $\\pi ^ { * } = \\operatorname { a r g m a x } _ { \\pi } O _ { \\mathrm { R E L E N T } } ( \\pi )$ . As in PCL (Nachum et al., 2017), this optimal policy may be expressed as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/979ffb451eead80e8fc748a3f09b0134b020abc49384a5830499c872fedc2342.jpg",
+        "text": "$$\n\\pi ^ { * } ( a _ { t } | s _ { t } ) = \\exp \\left\\{ \\frac { \\mathbb { E } _ { \\tilde { r } _ { t } \\sim \\tilde { r } ( s _ { t } , a _ { t } ) , s _ { t + 1 } } [ \\tilde { r } _ { t } + \\gamma V ^ { * } ( s _ { t + 1 } ) ] - V ^ { * } ( s _ { t } ) } { \\tau + \\lambda } \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $V ^ { * }$ are the softmax state values defined recursively as ",
+        "bbox": [
+            176,
+            233,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d8495a1c78f24ec10af1adcac211e1d258f6dccd3d7d7000875688e8b2ba05ee.jpg",
+        "text": "$$\nV ^ { * } ( s _ { t } ) = ( \\tau + \\lambda ) \\log \\int _ { A } \\exp \\left. \\frac { \\mathbb { E } _ { \\tilde { r } _ { t } \\sim \\tilde { r } ( s _ { t } , a ) , s _ { t + 1 } } [ \\tilde { r } _ { t } + \\gamma V ^ { * } ( s _ { t + 1 } ) ] } { \\tau + \\lambda } \\right. \\mathrm { d } a .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We may re-arrange (17) to yield ",
+        "bbox": [
+            174,
+            289,
+            387,
+            304
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/439adf9eff512b687cdf20e744a793bf8f82d8d7ce0fd0b06ce58bd40ab56b2b.jpg",
+        "text": "$$\n\\begin{array} { r c l } { V ^ { * } ( s _ { t } ) } & { = } & { \\mathbb { E } _ { \\tilde { r } _ { t } \\sim \\tilde { r } ( s _ { t } , a _ { t } ) , s _ { t + 1 } } [ \\tilde { r } _ { t } - ( \\tau + \\lambda ) \\log \\pi ^ { * } ( a _ { t } | s _ { t } ) + \\gamma V ^ { * } ( s _ { t + 1 } ) ] } \\\\ & { = } & { \\mathbb { E } _ { r _ { t } , s _ { t + 1 } } [ r _ { t } - ( \\tau + \\lambda ) \\log \\pi ^ { * } ( a _ { t } | s _ { t } ) + \\lambda \\log \\tilde { \\pi } ( a _ { t + i } | s _ { t + i } ) + \\gamma V ^ { * } ( s _ { t + 1 } ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This is a single-step temporal consistency which may be extended to multiple steps by further expanding $V ^ { \\ast } ( s _ { t + 1 } )$ on the RHS using the same identity. Thus, in general we have the following softmax temporal consistency constraint along any sequence of states defined by a starting state $s _ { t }$ and a sequence of actions $a _ { t } , \\ldots , a _ { t + d - 1 }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3606945042fe5a19a07a336bb6ea79de35757c671371fa7b46a27905c2d8720.jpg",
+        "text": "$$\nr ^ { * } ( s _ { t } ) = \\underset { r _ { t + i } , s _ { t + i } } { \\mathbb { E } } \\left[ \\gamma ^ { d } V ^ { * } ( s _ { t + d } ) + \\sum _ { i = 0 } ^ { d - 1 } \\gamma ^ { i } \\left( r _ { t + i } - ( \\tau + \\lambda ) \\log \\pi ^ { * } ( a _ { t + i } | s _ { t + i } ) + \\lambda \\log \\tilde { \\pi } ( a _ { t + i } | s _ { t + i } ) \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 TRUST-PCL ",
+        "text_level": 1,
+        "bbox": [
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+            478,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose to train a parameterized policy $\\pi _ { \\theta }$ and value estimate $V _ { \\phi }$ to satisfy the multi-step consistencies in (21). Thus, we define a consistency error for a sequence of states, actions, and rewards st:t+d ≡ (st, at, rt, . . . , st+d−1, at+d−1, rt+d−1, st+d) sampled from the environment as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f092576ac4f465cf906b13fdf21a507f87ff4ac8ea6dad47a68a795d061e7f04.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { C ( s _ { t : t + d } , \\theta , \\phi ) ~ { = } ~ - { V _ { \\phi } ( s _ { t } ) + \\gamma ^ { d } V _ { \\phi } ( s _ { t + d } ) ~ + } } } & { { } } & { { } } \\\\ { { } } & { { } } & { { { \\displaystyle \\sum _ { i = 0 } ^ { d - 1 } \\gamma ^ { i } \\left( r _ { t + i } - ( { \\tau + } \\lambda ) \\log { \\pi _ { \\theta } ( a _ { t + i } | s _ { t + i } ) } + \\lambda \\log { \\pi _ { \\bar { \\theta } } ( a _ { t + i } | s _ { t + i } ) } \\right) ~ . } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "loss for a given batch of episodes (or sub-episodes) We aim to minimize the squared consistency error on every sub-trajectory of length $S = \\{ s _ { 0 : T _ { k } } ^ { ( k ) } \\} _ { k = 1 } ^ { B }$ is $d$ . That is, the ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c982c50804668652f721e1593b886fd05fae447c37ee0a9b83a6d2f68d0120c4.jpg",
+        "text": "$$\n\\mathcal { L } ( S , \\theta , \\phi ) = \\sum _ { k = 1 } ^ { B } \\sum _ { t = 0 } ^ { T _ { k } - 1 } C ( s _ { t : t + d } ^ { ( k ) } , \\theta , \\phi ) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We perform gradient descent on $\\theta$ and $\\phi$ to minimize this loss. In practice, we have found that it is beneficial to learn the parameter $\\phi$ at least as fast as $\\theta$ , and accordingly, given a mini-batch of episodes we perform a single gradient update on $\\theta$ and possibly multiple gradient updates on $\\phi$ (see Appendix for details). ",
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+    {
+        "type": "text",
+        "text": "In principle, the mini-batch $S$ may be taken from either on-policy or off-policy trajectories. In our implementation, we utilized a replay buffer prioritized by recency. As episodes (or sub-episodes) are sampled from the environment they are placed in a replay buffer and a priority $p { \\big ( } s _ { 0 : T } )$ is given to a trajectory $s _ { 0 : T }$ equivalent to the current training step. Then, to sample a batch for training, $B$ episodes are sampled from the replay buffer proportional to exponentiated priority $\\exp \\{ \\beta p ( s _ { 0 : T } ) \\}$ for some hyperparameter $\\beta \\geq 0$ . ",
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+        "type": "text",
+        "text": "For the prior policy $\\pi _ { \\tilde { \\theta } }$ , we use a lagged geometric mean of the parameters. At each training step, we update $\\tilde { \\theta }  \\alpha \\tilde { \\theta } + ( 1 - \\alpha ) \\theta$ . Thus on average our training scheme attempts to maximize entropy regularized expected reward while penalizing divergence from a policy roughly $1 / ( 1 - \\alpha )$ training steps in the past. ",
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+    {
+        "type": "text",
+        "text": "4.3 AUTOMATIC TUNING OF THE LAGRANGE MULTIPLIER $\\lambda$ ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The use of a relative entropy regularizer as a penalty rather than a constraint introduces several difficulties. The hyperparameter $\\lambda$ must necessarily adapt to the distribution of rewards. Thus, $\\lambda$ must be tuned not only to each environment but also during training on a single environment, since the observed reward distribution changes as the agent’s behavior policy improves. Using a constraint form of the regularizer is more desirable, and others have advocated its use in practice (Schulman et al., 2015) specifically to robustly allow larger updates during training. ",
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+        "type": "text",
+        "text": "To this end, we propose to redirect the hyperparameter tuning from $\\lambda$ to $\\epsilon$ . Specifically, we present a method which, given a desired hard constraint on the relative entropy defined by $\\epsilon$ , approximates the equivalent penalty coefficient $\\lambda ( \\epsilon )$ . This is a key novelty of our work and is distinct from previous attempts at automatically tuning a regularizing coefficient, which iteratively increase and decrease the coefficient based on observed training behavior (Schulman et al., 2017b; Heess et al., 2017). ",
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+        "type": "text",
+        "text": "We restrict our analysis to the undiscounted setting $\\gamma = 1$ with entropy regularizer $\\tau = 0$ . Additionally, we assume deterministic, finite-horizon environment dynamics. An additional assumption we make is that the expected KL-divergence over states is well-approximated by the KL-divergence starting from the unique initial state $s _ { 0 }$ . Although in our experiments these restrictive assumptions are not met, we still found our method to perform well for adapting $\\lambda$ during training. ",
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+        "type": "text",
+        "text": "In this setting the optimal policy of (14) is proportional to exponentiated scaled reward. Specifically, for a full episode $s _ { 0 : T } = ( s _ { 0 } , a _ { 0 } , r _ { 0 } , \\ldots , s _ { T - 1 } , a _ { T - 1 } , r _ { T - 1 } , s _ { T } )$ , we have ",
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+    {
+        "type": "equation",
+        "img_path": "images/b8aa17d0a3de5c196658b2f48ebe22f3c24268d08679676a10c0be61cca8ff50.jpg",
+        "text": "$$\n\\pi ^ { * } ( s _ { 0 : T } ) \\propto \\tilde { \\pi } ( s _ { 0 : T } ) \\exp \\left\\{ \\frac { R ( s _ { 0 : T } ) } { \\lambda } \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\pi ( s _ { 0 : T } ) = \\prod _ { i = 0 } ^ { T - 1 } \\pi ( a _ { i } | s _ { i } ) } \\end{array}$ and $\\begin{array} { r } { R ( s _ { 0 : T } ) = \\sum _ { i = 0 } ^ { T - 1 } r _ { i } } \\end{array}$ . The normalization factor of $\\pi ^ { * }$ ",
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+        "type": "equation",
+        "img_path": "images/f4eedf06c204c8457dc51d2bd64f08449907ce0932cefd50877a3333f0d0c7c0.jpg",
+        "text": "$$\nZ = \\mathbb { E } _ { s _ { 0 : T } \\sim \\tilde { \\pi } } \\left[ \\exp \\left\\{ \\frac { R ( s _ { 0 : T } ) } { \\lambda } \\right\\} \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We would like to approximate the trajectory-wide KL-divergence between $\\pi ^ { * }$ and $\\tilde { \\pi }$ . We may express the KL-divergence analytically: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/44239c5241fef6e1f0e176e02f53d9ab4c46543d6ef3c0a4696d73db9a041cab.jpg",
+        "text": "$$\n\\begin{array} { r l } & { K L ( \\pi ^ { * } | | \\tilde { \\pi } ) = \\mathbb { E } _ { s _ { 0 : T } \\sim \\pi ^ { * } } \\left[ \\log \\left( \\frac { \\pi ^ { * } \\left( s _ { 0 : T } \\right) } { \\tilde { \\pi } \\left( s _ { 0 : T } \\right) } \\right) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { s _ { 0 : T } \\sim \\pi ^ { * } } \\left[ \\frac { R \\left( s _ { 0 : T } \\right) } { \\lambda } - \\log Z \\right] } \\\\ & { \\qquad = - \\log Z + \\mathbb { E } _ { s _ { 0 : T } \\sim \\tilde { \\pi } } \\left[ \\frac { R \\left( s _ { 0 : T } \\right) } { \\lambda } \\cdot \\frac { \\pi ^ { * } \\left( s _ { 0 : T } \\right) } { \\tilde { \\pi } \\left( s _ { 0 : T } \\right) } \\right] } \\\\ & { \\qquad = - \\log Z + \\mathbb { E } _ { s _ { 0 : T } \\sim \\tilde { \\pi } } \\left[ \\frac { R \\left( s _ { 0 : T } \\right) } { \\lambda } \\exp \\{ R ( s _ { 0 : T } ) / \\lambda - \\log Z \\} \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since all expectations are with respect to $\\tilde { \\pi }$ , this quantity is tractable to approximate given episodes sampled from $\\tilde { \\pi }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Therefore, in Trust-PCL, given a set of episodes sampled from the prior policy $\\pi _ { \\tilde { \\theta } }$ and a desired maximum divergence $\\epsilon$ , we can perform a simple line search to find a suitable $\\lambda ( \\epsilon )$ which yields $K L ( \\pi ^ { * } | | \\pi _ { \\tilde { \\theta } } )$ as close as possible to $\\epsilon$ . ",
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+    {
+        "type": "text",
+        "text": "The preceding analysis provided a method to determine $\\lambda ( \\epsilon )$ given a desired maximum divergence $\\epsilon$ . However, there is still a question of whether $\\epsilon$ should change during training. Indeed, as episodes may possibly increase in length, $K L ( \\pi ^ { * } | | \\tilde { \\pi } )$ naturally increases when compared to the average perstate $K L ( \\pi ^ { * } ( - | s ) | | \\tilde { \\pi } ( - | s ) )$ , and vice versa for decreasing length. Thus, in practice, given an $\\epsilon$ and a set of sampled episodes $\\dot { S } = \\{ s _ { 0 : T _ { k } } ^ { ( k ) } \\} _ { k = 1 } ^ { N }$ , we approximate the best $\\lambda$ which yields a maximum divergence of $\\begin{array} { r } { \\frac { \\epsilon } { N } \\sum _ { k = 1 } ^ { N } T _ { k } } \\end{array}$ . This makes it so that $\\epsilon$ corresponds more to a constraint on the lengthaveraged KL-divergence. ",
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+    {
+        "type": "text",
+        "text": "To avoid incurring a prohibitively large number of interactions with the environment for each parameter update, in practice we use the last 100 episodes as the set of sampled episodes $S$ . While this is not exactly the same as sampling episodes from $\\pi _ { \\tilde { \\theta } }$ , it is not too far off since $\\pi _ { \\tilde { \\theta } }$ is a lagged version of the online policy $\\pi _ { \\theta }$ . Moreover, we observed this protocol to work well in practice. A more sophisticated and accurate protocol may be derived by weighting the episodes according to the importance weights corresponding to their true sampling distribution. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate Trust-PCL against TRPO on a number of benchmark tasks. We choose TRPO as a baseline since it is a standard algorithm known to achieve state-of-the-art performance on the continuous control tasks we consider (see e.g., leaderboard results on the OpenAI Gym website (Brockman et al., 2016)). We find that Trust-PCL can match or improve upon TRPO’s performance in terms of both average reward and sample efficiency. ",
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+    {
+        "type": "text",
+        "text": "5.1 SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We chose a number of control tasks available from OpenAI Gym (Brockman et al., 2016). The first task, Acrobot, is a discrete-control task, while the remaining tasks (HalfCheetah, Swimmer, Hopper, Walker2d, and Ant) are well-known continuous-control tasks utilizing the MuJoCo environment (Todorov et al., 2012). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For TRPO we trained using batches of $Q \\ = \\ 2 5 , 0 0 0$ steps (12, 500 for Acrobot), which is the approximate batch size used by other implementations (Duan et al., 2016; Schulman, 2017). Thus, at each training iteration, TRPO samples 25, 000 steps using the policy $\\pi _ { \\tilde { \\theta } }$ and then takes a single step within a KL-ball to yield a new $\\pi _ { \\theta }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Trust-PCL is off-policy, so to evaluate its performance we alternate between collecting experience and training on batches of experience sampled from the replay buffer. Specifically, we alternate between collecting $P = 1 0$ steps from the environment and performing a single gradient step based on a batch of size $Q = 6 4$ sub-episodes of length $P$ from the replay buffer, with a recency weight of $\\beta = 0 . 0 0 1$ on the sampling distribution of the replay buffer. To maintain stability we use $\\alpha = 0 . 9 9$ and we modified the loss from squared loss to Huber loss on the consistency error. Since our policy is parameterized by a unimodal Gaussian, it is impossible for it to satisfy all path consistencies, and so we found this crucial for stability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For each of the variants and for each environment, we performed a hyperparameter search to find the best hyperparameters. The plots presented here show the reward achieved during training on the best hyperparameters averaged over the best 4 seeds of 5 randomly seeded training runs. Note that this reward is based on greedy actions (rather than random sampling). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiments were performed using Tensorflow (Abadi et al., 2016). Although each training step of Trust-PCL (a simple gradient step) is considerably faster than TRPO, we found that this does not have an overall effect on the run time of our implementation, due to a combination of the fact that each environment step is used in multiple training steps of Trust-PCL and that a majority of the run time is spent interacting with the environment. A detailed description of our implementation and hyperparameter search is available in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "5.2 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present the reward over training of Trust-PCL and TRPO in Figure 1. We find that Trust-PCL can match or beat the performance of TRPO across all environments in terms of both final reward and sample efficiency. These results are especially significant on the harder tasks (Walker2d and Ant). We additionally present our results compared to other published results in Table 1. We find that even when comparing across different implementations, Trust-PCL can match or beat the state-of-the-art. ",
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+    {
+        "type": "text",
+        "text": "5.2.1 HYPERPARAMETER ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The most important hyperparameter in our method is $\\epsilon$ , which determines the size of the trust region and thus has a critical role in the stability of the algorithm. To showcase this effect, we present the reward during training for several different values of $\\epsilon$ in Figure 2. As $\\epsilon$ increases, instability increases as well, eventually having an adverse effect on the agent’s ability to achieve optimal reward. ",
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+        "type": "image",
+        "img_path": "images/2819973078574959af663e0a5bd2f625284a6841ecbb7974799c6d7e0e859717.jpg",
+        "image_caption": [
+            "Figure 1: The results of Trust-PCL against a TRPO baseline. Each plot shows average greedy reward with single standard deviation error intervals capped at the min and max across 4 best of 5 randomly seeded training runs after choosing best hyperparameters. The $\\mathbf { X }$ -axis shows millions of environment steps. We observe that Trust-PCL is consistently able to match and, in many cases, beat TRPO’s performance both in terms of reward and sample efficiency. "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/34d09bb3531e483b9be242882f2de4c101ffb45016e355dc9b7df44bf18184cb.jpg",
+        "image_caption": [
+            "Figure 2: The results of Trust-PCL across several values of $\\epsilon$ , defining the size of the trust region. Each plot shows average greedy reward across 4 best of 5 randomly seeded training runs after choosing best hyperparameters. The $\\mathbf { X }$ -axis shows millions of environment steps. We observe that instability increases with $\\epsilon$ , thus concluding that the use of trust region is crucial. "
+        ],
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+    {
+        "type": "text",
+        "text": "Note that standard PCL (Nachum et al., 2017) corresponds to $\\epsilon  \\infty$ (that is, $\\lambda = 0$ ). Therefore, standard PCL would fail in these environments, and the use of trust region is crucial. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main advantage of Trust-PCL over existing trust region methods for continuous control is its ability to learn in an off-policy manner. The degree to which Trust-PCL is off-policy is determined by a combination of the hyparparameters $\\alpha , \\beta$ , and $P$ . To evaluate the importance of training off-policy, we evaluate Trust-PCL with a hyperparameter setting that is more on-policy. We set $\\alpha = 0 . 9 5$ , $\\beta = 0 . 1$ , and $P = 1 , 0 0 0$ . In this setting, we also use large batches of $Q = 2 5$ episodes of length $P$ (a total of 25, 000 environment steps per batch). Figure 3 shows the results of Trust-PCL with our original parameters and this new setting. We note a dramatic advantage in sample efficiency when using off-policy training. Although Trust-PCL (on-policy) can achieve state-of-the-art reward performance, it requires an exorbitant amount of experience. On the other hand, Trust-PCL (off",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b0d97e378156ad5edbbab6dfd5281f228dce1b8dcd57a11bb07f63cb417ae41b.jpg",
+        "image_caption": [
+            "Figure 3: The results of Trust-PCL varying the degree of on/off-policy. We see that Trust-PCL (on-policy) has a behavior similar to TRPO, achieving good final reward but requiring an exorbitant number of experience collection. When collecting less experience per training step in Trust-PCL (off-policy), we are able to improve sample efficiency while still achieving a competitive final reward. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/53c954ae74a9a7c6efff8f9a07081ba4f869841cea5ec8cbc039211bbe15b587.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Domain</td><td>TRPO-GAE</td><td>TRPO (rllab)</td><td>TRPO (ours)</td><td>Trust-PCL</td><td>IPG</td></tr><tr><td>HalfCheetah</td><td>4871.36</td><td>2889</td><td>4343.6</td><td>7057.1</td><td>4767</td></tr><tr><td>Swimmer</td><td>137.25</td><td>1</td><td>288.1</td><td>297.0</td><td></td></tr><tr><td>Hopper</td><td>3765.78</td><td>1</td><td>3516.7</td><td>3804.9</td><td>一</td></tr><tr><td>Walker2d</td><td>6028.73</td><td>1487</td><td>2838.4</td><td>5027.2</td><td>3047</td></tr><tr><td>Ant</td><td>2918.25</td><td>1520</td><td>4347.5</td><td>6104.2</td><td>4415</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Results for best average reward in the first 10M steps of training for our implementations (TRPO (ours) and Trust-PCL) and external implementations. TRPO-GAE are results of Schulman (2017) available on the OpenAI Gym website. TRPO (rllab) and IPG are taken from Gu et al. (2017b). These results are each on different setups with different hyperparameter searches and in some cases different evaluation protocols (e.g.,TRPO (rllab) and IPG were run with a simple linear value network instead of the two-hidden layer network we use). Thus, it is not possible to make any definitive claims based on this data. However, we do conclude that our results are overall competitive with state-of-the-art external implementations. ",
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+    {
+        "type": "text",
+        "text": "policy) can be competitive in terms of reward while providing a significant improvement in sample efficiency. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "One last hyperparameter is $\\tau$ , determining the degree of exploration. Anecdotally, we found $\\tau$ to not be of high importance for the tasks we evaluated. Indeed many of our best results use $\\tau =$ 0. Including $\\tau > 0$ had a marginal effect, at best. The reason for this is likely due to the tasks themselves. Indeed, other works which focus on exploration in continuous control have found the need to propose exploration-advanageous variants of these standard benchmarks (Haarnoja et al., 2017; Houthooft et al., 2016). ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented Trust-PCL, an off-policy algorithm employing a relative-entropy penalty to impose a trust region on a maximum reward objective. We found that Trust-PCL can perform well on a set of standard control tasks, improving upon TRPO both in terms of average reward and sample efficiency. Our best results on Trust-PCL are able to maintain the stability and solution quality of TRPO while approaching the sample-efficiency of value-based methods (see e.g., Metz et al. (2017)). This gives hope that the goal of achieving both stability and sample-efficiency without trading-off one for the other is attainable in a single unifying RL algorithm. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGMENT ",
+        "text_level": 1,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank Matthew Johnson, Luke Metz, Shane Gu, and the Google Brain team for insightful comments and discussions. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "type": "text",
+        "text": "A IMPLEMENTATION BENEFITS OF TRUST-PCL ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We have already highlighted the ability of Trust-PCL to use off-policy data to stably train both a parameterized policy and value estimate, which sets it apart from previous methods. We have also noted the ease with which exploration can be incorporated through the entropy regularizer. We elaborate on several additional benefits of Trust-PCL. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Compared to TRPO, Trust-PCL is much easier to implement. Standard TRPO implementations perform second-order gradient calculations on the KL-divergence to construct a Fisher information matrix (more specifically a vector product with the inverse Fisher information matrix). This yields a vector direction for which a line search is subsequently employed to find the optimal step. Compare this to Trust-PCL which employs simple gradient descent. This makes implementation much more straightforward and easily realizable within standard deep learning frameworks. ",
+        "bbox": [
+            174,
+            198,
+            825,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Even if one replaces the constraint on the average KL-divergence of TRPO with a simple regularization penalty (as in proximal policy gradient methods (Schulman et al., 2017b; Wang et al., 2016)), optimizing the resulting objective requires computing the gradient of the KL-divergence. In Trust-PCL, there is no such necessity. The per-state KL-divergence need not have an analytically computable gradient. In fact, the KL-divergence need not have a closed form at all. The only requirement of Trust-PCL is that the log-density be analytically computable. This opens up the possible policy parameterizations to a much wider class of functions. While continuous control has traditionally used policies parameterized by unimodal Gaussians, with Trust-PCL the policy can be replaced with something much more expressive—for example, mixtures of Gaussians or autoregressive policies as in Metz et al. (2017). ",
+        "bbox": [
+            174,
+            287,
+            825,
+            426
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We have yet to fully explore these additional benefits in this work, but we hope that future investigations can exploit the flexibility and ease of implementation of Trust-PCL to further the progress of RL in continuous control environments. ",
+        "bbox": [
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+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            498,
+            401,
+            513
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We describe in detail the experimental setup regarding implementation and hyperparameter search. ",
+        "bbox": [
+            174,
+            531,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 ENVIRONMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
+            331,
+            579
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Acrobot, episodes were cut-off at step 500. For the remaining environments, episodes were cutoff at step $1 , 0 0 0$ . ",
+        "bbox": [
+            173,
+            592,
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+            619
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Acrobot, HalfCheetah, and Swimmer are all non-terminating environments. Thus, for these environments, each episode had equal length and each batch contained the same number of episodes. Hopper, Walker2d, and Ant are environments that can terminate the agent. Thus, for these environments, the batch size throughout training remained constant in terms of steps but not in terms of episodes. ",
+        "bbox": [
+            174,
+            626,
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+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "There exists an additional common MuJoCo task called Humanoid. We found that neither our implementation of TRPO nor Trust-PCL could make more than negligible headway on this task, and so omit it from the results. We are aware that TRPO with the addition of GAE and enough finetuning can be made to achieve good results on Humanoid (Schulman et al., 2016). We decided to not pursue a GAE implementation to keep a fair comparison between variants. Trust-PCL can also be made to incorporate an analogue to GAE (by maintaining consistencies at varying time scales), but we leave this to future work. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            819,
+            408,
+            833
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use fully-connected feed-forward neural networks to represent both policy and value. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The policy $\\pi _ { \\theta }$ is represented by a neural network with two hidden layers of dimension 64 with tanh activations. At time step $t$ , the network is given the observation $s _ { t }$ . It produces a vector $\\mu _ { t }$ , which is combined with a learnable (but $t$ -agnostic) parameter $\\xi$ to parametrize a unimodal Gaussian with mean $\\mu _ { t }$ and standard deviation $\\exp ( \\xi )$ . The next action $a _ { t }$ is sampled randomly from this Gaussian. ",
+        "bbox": [
+            176,
+            867,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The value network $V _ { \\phi }$ is represented by a neural network with two hidden layers of dimension 64 with tanh activations. At time step $t$ the network is given the observation $s _ { t }$ and the component-wise squared observation $s _ { t } \\odot s _ { t }$ . It produces a single scalar value. ",
+        "bbox": [
+            174,
+            103,
+            823,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2.1 TRPO LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            161,
+            354,
+            175
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "At each training iteration, both the policy and value parameters are updated. The policy is trained by performing a trust region step according to the procedure described in Schulman et al. (2015). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The value parameters at each step are solved using an LBFGS optimizer. To avoid instability, the value parameters are solved to fit a mixture of the empirical values and the expected values. That is, we determine $\\phi$ to minimize $\\begin{array} { r } { \\sum _ { s \\in \\mathrm { b a t c h } } ( V _ { \\phi } ( s ) - \\kappa V _ { \\tilde { \\phi } } ( s ) - ( 1 - \\kappa ) \\hat { V } _ { \\tilde { \\phi } } ( s ) ) ^ { 2 } } \\end{array}$ , where again $\\tilde { \\phi }$ is the previous value parameterization. We use $\\kappa = 0 . 9$ . This method for training $\\phi$ is according to that used in Schulman (2017). ",
+        "bbox": [
+            174,
+            219,
+            825,
+            295
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2.2 TRUST-PCL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            309,
+            395,
+            324
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "At each training iteration, both the policy and value parameters are updated. The specific updates are slightly different between Trust-PCL (on-policy) and Trust-PCL (off-policy). ",
+        "bbox": [
+            176,
+            334,
+            821,
+            362
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For Trust-PCL (on-policy), the policy is trained by taking a single gradient step using the Adam optimizer (Kingma & Ba, 2015) with learning rate 0.001. The value network update is inspired by that used in TRPO we perform 5 gradients steps with learning rate 0.001, calculated with regards to a mix between the empirical values and the expected values according to the previous $\\tilde { \\phi }$ . We use $\\kappa = 0 . 9 5$ . ",
+        "bbox": [
+            174,
+            368,
+            823,
+            440
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For Trust-PCL (off-policy), both the policy and value parameters are updated in a single step using the Adam optimizer with learning rate 0.0001. For this variant, we also utilize a target value network (lagged at the same rate as the target policy network) to replace the value estimate at the final state for each path. We do not mix between empirical and expected values. ",
+        "bbox": [
+            173,
+            446,
+            825,
+            503
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.3 HYPERPARAMETER SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            521,
+            413,
+            535
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We found the most crucial hyperparameters for effective learning in both TRPO and TrustPCL to be $\\epsilon$ (the constraint defining the size of the trust region) and $d$ (the rollout determining how to evaluate the empirical value of a state). For TRPO we performed a grid search over $\\epsilon \\in \\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 \\} , d \\in \\{ 1 0 , 5 0 \\}$ . For Trust-PCL we performed a grid search over $\\epsilon \\in \\{ 0 . 0 0 1 , 0 . 0 0 2 , 0 . 0 0 5 , 0 . 0 1 \\} , d \\in \\{ 1 0$ $d \\in \\{ 1 0 , 5 0 \\}$ . For Trust-PCL we also experimented with the value of $\\tau$ , either keeping it at a constant 0 (thus, no exploration) or decaying it from 0.1 to 0.0 by a smoothed exponential rate of 0.1 every 2,500 training iterations. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We fix the discount to $\\gamma = 0 . 9 9 5$ for all environments. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C PSEUDOCODE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            685,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A simplified pseudocode for Trust-PCL is presented in Algorithm 1. ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Trust-PCL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Input: Environment $E N V$ , trust region constraint $\\epsilon$ , learning rates $\\eta _ { \\pi } , \\eta _ { v }$ , discount factor $\\gamma$ , rollout $d$ , batch size $Q$ , collect steps per train step $P$ , number of training steps $N$ , replay buffer $R B$ with exponential lag $\\beta$ , lag on prior policy $\\alpha$ . ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "function Gradients $( \\{ s _ { t : t + P } ^ { ( k ) } \\} _ { k = 1 } ^ { B } )$ $/ / C$ is thpute $\\begin{array} { r } { \\Delta \\theta = \\sum _ { k = 1 } ^ { B } \\sum _ { p = 0 } ^ { P - 1 } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \\theta , \\phi ) \\nabla _ { \\theta } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \\theta , \\phi ) . } \\end{array}$ $\\begin{array} { r } { \\Delta \\phi = \\sum _ { k = 1 } ^ { B } \\sum _ { p = 0 } ^ { P - 1 } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \\theta , \\phi ) \\nabla _ { \\phi } C ( s _ { t + p : t + p + d } ^ { ( k ) } , \\theta , \\phi ) . } \\end{array}$ Return $\\Delta \\theta , \\Delta \\phi$   \nend function   \nInitialize $\\theta , \\phi , \\lambda$ , set $\\tilde { \\theta } = \\theta$ .   \nInitialize empty replay buffer $R B ( \\beta )$ .   \nfor $i = 0$ to $N - 1$ do // Collect Sample $P$ steps $s _ { t : t + P } \\sim \\pi _ { \\theta }$ on $E N V$ . Insert $s _ { t : t + P }$ to $R B$ . // Train   \nSample batch $\\{ s _ { t : t + P } ^ { ( k ) } \\} _ { k = 1 } ^ { B }$ from $R B$ to contain a total of $Q$ transitions $( B \\approx Q / P )$ . $\\Delta \\theta , \\Delta \\phi = \\mathrm { G r a d i e n t s } \\big ( \\{ s _ { t : t + P } ^ { ( k ) } \\} _ { k = 1 } ^ { B } \\big )$ . Update $\\theta  \\theta - \\eta _ { \\pi } \\Delta \\theta$ . Update $\\phi  \\phi - \\eta _ { v } \\Delta \\phi$ . // Update auxiliary variables Update $\\tilde { \\theta } = \\alpha \\tilde { \\theta } + \\mathrm { \\Gamma } ( 1 - \\alpha ) \\theta$ . Update $\\lambda$ in terms of $\\epsilon$ according to Section 4.3.   \nend for ",
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+    }
+]