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+# REPRESENTATION LEARNING VIA INVARIANT CAUSAL MECHANISMS
+
+Jovana Mitrovic Brian McWilliams Jacob Walker Lars Buesing Charles Blundell   
+DeepMind   
+London, UK   
+{mitrovic, bmcw, jcwalker, lbuesing, cblundell}@google.com
+
+# ABSTRACT
+
+Self-supervised learning has emerged as a strategy to reduce the reliance on costly supervised signals by pretraining representations only using unlabeled data. These methods combine heuristic proxy classification tasks with data augmentations and have achieved significant success, but our theoretical understanding of this success remains limited. In this paper we analyze self-supervised representation learning using a causal framework. We show how data augmentations can be more effectively utilized through explicit invariance constraints on the proxy classifiers employed during pretraining. Based on this, we propose a novel selfsupervised objective, Representation Learning via Invariant Causal Mechanisms (RELIC), that enforces invariant prediction of proxy targets across augmentations through an invariance regularizer which yields improved generalization guarantees. Further, using causality we generalize contrastive learning, a particular kind of self-supervised method, and provide an alternative theoretical explanation for the success of these methods. Empirically, RELIC significantly outperforms competing methods in terms of robustness and out-of-distribution generalization on ImageNet, while also significantly outperforming these methods on Atari achieving above human-level performance on 51 out of 57 games.
+
+# 1 INTRODUCTION
+
+Training deep networks often relies heavily on large amounts of useful supervisory signal, such as labels for supervised learning or rewards for reinforcement learning. These training signals can be costly or otherwise impractical to acquire. On the other hand, unsupervised data is often abundantly available. Therefore, pretraining representations for unknown downstream tasks without the need for labels or extrinsic reward holds great promise for reducing the cost of applying machine learning models. To pretrain representations, self-supervised learning makes use of proxy tasks defined on unsupervised data. Recently, self-supervised methods using contrastive objectives have emerged as one of the most successful strategies for unsupervised representation learning (Oord et al., 2018; Hjelm et al., 2018; Chen et al., 2020a). These methods learn a representation by classifying every datapoint against all others datapoints (negative examples). Under assumptions on how the negative examples are sampled, minimizing the resulting contrastive loss has been justified as maximizing a lower bound on the mutual information (MI) between representations (Poole et al., 2019). However, (Tschannen et al., 2019) has shown that performance on downstream tasks may be more tightly correlated with the choice of encoder architecture than the achieved MI bound, highlighting issues with the MI theory of contrastive learning. Further, contrastive approaches compare different views of the data (usually under different data augmentations) to calculate similarity scores. This approach to computing scores has been empirically observed as a key success factor of contrastive methods, but has yet to be theoretically justified. This lack of a solid theoretical explanation for the effectiveness of contrastive methods hinders their further development.
+
+To remedy the theoretical shortcomings, we analyze the problem of self-supervised representation learning through a causal lens. We formalize intuitions about the data generating process using a causal graph and leverage causal tools to derive properties of the optimal representation. We show that a representation should be an invariant predictor of proxy targets under interventions on features that are only correlated, but not causally related to the downstream targets of interest.
+
+Since neither causally nor purely correlationally related features are observed and thus performing actual interventions on them is not feasible, for learning representation with this property we use data augmentations to simulate a subset of possible interventions. Based on our causal interpretation, we propose a regularizer which enforces that the prediction of the proxy targets is invariant across data augmentations. We propose a novel objective for self-supervised representation learning called REpresentation Learning with Invariant Causal mechanisms (RELIC). We show how this explicit invariance regularization leverages augmentations more effectively than previous self-supervised methods and that representations learned using RELIC are guaranteed to generalize well to downstream tasks under weaker assumptions than those required by previous work (Saunshi et al., 2019).
+
+Next we generalize contrastive learning and provide an alternative theoretical explanation to MI for the success of these methods. We generalize the proxy task of instance discrimination commonly used in contrastive learning using the causal concept of refinements (Chalupka et al., 2014). Intuitively, a refinement of a task can be understood as a more fine-grained variant of the original problem. For example, a refinement for classifying cats against dogs would be the task of classifying individual cat and dog breeds. The instance discrimination task results from the most fine-grained refinement, e.g. discriminating individual cats and dogs from one another. We show that using refinements as proxy tasks enables us to learn useful representations for downstream tasks. Specifically, using causal tools, we show that learning a representation on refinements such that it is an invariant predictor of proxy targets across augmentations is a sufficient condition for these representations to generalize to downstream tasks (cf. Theorem 1). In summary, we provide theoretical support both for the general form of the contrastive objective as well as for the use of data augmentations. Thus, we provide an alternative explanation to mutual information for the success of recent contrastive approaches namely that of causal refinements of downstream tasks.
+
+We test RELIC on a variety of prediction and reinforcement learning problems. First, we evaluate the quality of representations pretrained on ImageNet with a special focus on robustness and out-ofdistribution generalization. RELIC performs competitively with current state-of-the-art methods on ImageNet, while significantly outperforming competing methods on robustness and out-of-distribution generalization of the learned representations when tested on corrupted ImageNet (ImageNet-C (Hendrycks & Dietterich, 2019)) and a version of ImageNet that consist of different renditions of the same classes (ImageNet-R (Hendrycks et al., 2020)). In terms of robustness, RELIC also significantly outperforms the supervised baseline with an absolute reduction of $4 . 9 \%$ in error. Unlike much prior work that specifically focuses on computer vision tasks, we test RELIC for representation learning in the context of reinforcement learning on the Atari suite (Bellemare et al., 2013). There we find that RELIC significantly outperforms competing methods and achieves above human-level performance on 51 out of 57 games.
+
+# Contributions.
+
+• We formalize problem of self-supervised representation learning using causality and propose to more effectively leverage data augmentations through invariant prediction.   
+• We propose a new self-supervised objective, REpresentation Learning with Invariance Causal mechanisms (RELIC), that enforces invariant prediction through an explicit regularizer and show improved generalization guarantees. We generalize contrastive learning using refinements and show that learning on refinements is a sufficient condition for learning useful representations; this provides an alternative explanation to MI for the success of contrastive methods.
+
+# 2 REPRESENTATION LEARNING VIA INVARIANT CAUSAL MECHANISMS
+
+Problem setting. Let $X$ denote the unlabelled observed data and $\mathcal { V } = \{ Y _ { t } \} _ { t = 1 } ^ { T }$ be a set of unknown tasks with $Y _ { t }$ denoting the targets for task $t$ . The tasks $\{ Y _ { t } \} _ { t = 1 } ^ { T }$ can represent both a multi-environment as well as a multi-task setup. Our goal is to pretrain with unsupervised data a representation $f ( X )$ that will be useful for solving the downstream tasks $\mathcal { V }$ .
+
+Causal interpretation. To effectively leverage common assumptions and intuitions about data generation of the unknown downstream tasks for the learning algorithm, we propose to formalize them using a causal graph. We start from the following assumptions: a) the data is generated from content and style variables, with b) only content (and not style) being relevant for the unknown downstream tasks and c) content and style are independent, i.e. style changes are content-preserving. For example, when classifying dogs against giraffes from images, different parts of the animals constitute content, while style could be, for example, background, lighting conditions and camera lens characteristics. By assumption, content is a good representation of the data for downstream tasks and we therefore cast the goal of representation learning as estimating content. In the following, we compactly formalize these assumptions with a causal graph1, see Figure 1a.
+
+![](images/7d6b18a05d7010061f63fdf863493b0e8d6f15c64a17ca006f14b072fe70c129.jpg)  
+Figure 1: (a) Causal graph formalizing assumptions about content and style of the data and the relationship between targets and proxy tasks. The dashed arrows are not causal and represent learning, while the dashdotted lines denote that $Y ^ { R }$ is a refinement $Y _ { t }$ ’s. All other arrows are causal. (b) RELIC objective. KL refers to the Kullback-Leibler divergence, while $\mathbf { X }$ -entropy denotes cross entropy.
+
+Let $C$ and $S$ be the latent variables describing content and style. In Figure 1a, the directed arrows from $C$ and $S$ to the observed data $X$ (e.g. images) indicate that $X$ is generated based on content and style. The directed arrow from $C$ to the target $Y _ { t }$ (e.g. class labels) encodes the assumption that content directly influences the target tasks, while the absence of any directed arrow from $S$ to $Y _ { t }$ indicates that style does not. Thus, content $C$ has all the necessary information to predict $Y _ { t }$ . The absence of any directed path between $C$ and $S$ in Figure 1a encodes the intuition that these variables are independent, i.e. $C \perp \perp S$ .
+
+Using the independence of mechanisms (Peters et al., 2017), we can conclude that under this causal model performing interventions on $S$ does not change the conditional distribution $P ( Y _ { t } | C )$ , i.e. manipulating the value of $S$ does not influence this conditional distribution. Thus, $P ( Y _ { t } | C )$ is invariant under changes in style $S$ . We call $C$ an invariant representation for $Y _ { t }$ under $S$ , i.e.
+
+$$
+p ^ { d o ( S = s _ { i } ) } ( Y _ { t } \mid C ) = p ^ { d o ( S = s _ { j } ) } ( Y _ { t } \mid C ) \quad \forall s _ { i } , s _ { j } \in \mathcal { S } ,
+$$
+
+where $p ^ { d o ( S = s ) }$ denotes the distribution arising from assigning $S$ the value $s$ with $s$ the domain of $S$ (Pearl, 2009). Specifically, using $C$ as a representation allows for us to predict targets stably across perturbations, i.e. content $C$ is both a useful and robust representation for tasks $\mathcal { V }$ .
+
+Since the targets $Y _ { t }$ are unknown, we will construct a proxy task $Y ^ { R }$ in order to learn representations from unlabeled data $X$ only. In order to learn useful representations for $Y _ { t }$ , we will construct proxy tasks that represents more fine-grained problems than $Y _ { t }$ ; for a more formal treatment of proxy tasks please refer to Section 3. Further, to learn invariant representations, such as $C$ , we enforce Equation 1 which requires us to observe data under different style interventions, i.e. we need data that describes the same content under varying style. Since we do not have access to $S$ , to simulate style variability we use content-preserving data augmentations (e.g. rotation, grayscaling, translation, cropping for images). Specifically, we utilize data augmentations as interventions on the style variable $S$ , i.e.
+
+applying data augmentation $a _ { i }$ corresponds to intervening on $S$ and setting it to $s _ { a _ { i } }$ . 2 Although we are not able to generate all possible styles using a fixed set of data augmentations, we will use augmentations that generate large sets of diverse styles as this allows us to learn better representations. Note that the heuristic of estimating similarity based on different views from contrastive learning can be interpreted as an implicit invariance constraint.
+
+RELIC objective. Equation 1 provides a general scheme to estimate content (c.f. Figure 1a). We operationalize this by proposing to learn representations such that prediction of proxy targets from the representation is invariant under data augmentations. The representation $f ( X )$ must fulfill the following invariant prediction criteria
+
+(Invarian
+
+$$
+\begin{array} { r l r } { \prime p r e d i c t i o n ) } & { } & { p ^ { \mathrm { d o } ( a _ { i } ) } ( Y ^ { R } | f ( X ) ) = p ^ { \mathrm { d o } ( a _ { j } ) } ( Y ^ { R } | f ( X ) ) \quad \forall a _ { i } , a _ { j } \in \mathcal { A } . } \end{array}
+$$
+
+$\mathcal { A } = \{ a _ { 1 } , \ldots , a _ { m } \}$ is the set of data augmentations which simulate interventions on the style variables and $p ^ { d o ( a ) }$ denotes $p ^ { d o ( S = s _ { a } ) }$ .
+
+To achieve invariant prediction, we propose to explicitly enforce invariance under augmentations through a regularizer. This gives rise to an objective for self-supervised learning we call Representation Learning via Invariant Causal Mechanisms (RELIC). We write this objective as
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { \stackrel { X \warrow p ( X ) } { \sim \sim p ( X ) } } \underset { \sim \mathcal { A } \times \mathcal { A } } { \mathbb { E } } \quad \underset { b \in \{ a _ { l k } , a _ { q t } \} } { \sum } \mathcal { L } _ { b } ( Y ^ { R } , f ( X ) ) } \\ { s . t . \quad K L \left( p ^ { d o ( a _ { l k } ) } ( Y ^ { R } | f ( X ) ) , p ^ { d o ( a _ { q t } ) } ( Y ^ { R } | f ( X ) ) \right) \leq \rho } \end{array}
+$$
+
+where $\mathcal { L }$ is the proxy task loss and $K L$ is the Kullback-Leibler (KL) divergence. Note that any distance measure on distributions can be used in place of the KL divergence. We explain the remaining terms in detail below.
+
+Concretely, as proxy task we associate to every datapoint $x _ { i }$ the label $y _ { i } ^ { R } = i$ . This corresponds to the instance discrimination task, commonly used in contrastive learning (Hadsell et al., 2006). We take pairs of points $( x _ { i } , x _ { j } )$ to compute similarity scores and use pairs of augmentations $a _ { l k } = ( a _ { l } , a _ { k } ) \in$ $A \times A$ to perform a style intervention. Given a batch of samples $\{ x _ { i } \} _ { i = 1 } ^ { N } \sim \mathcal { D }$ , we use
+
+$$
+p ^ { d o ( a _ { l k } ) } ( Y ^ { R } = j | f ( x _ { i } ) ) \propto \exp \left( \phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { j } ^ { a _ { k } } ) ) / \tau \right) .
+$$
+
+with $x ^ { a }$ data augmented with $a$ and $\tau$ a softmax temperature parameter. We encode $f$ using a neural network and choose $h$ to be related to $f$ , e.g. $h = f$ or as a network with an exponential moving average of the weights of $f$ (e.g. target networks similar to (Grill et al., 2020)). To compare representations we use the function $\bar { \phi } ( f ( \bar { x _ { i } } ) , h ( x _ { j } ) ) \ : = \ : \langle g ( f ( x _ { i } ) ) , g ( h ( x _ { j } ) ) \rangle$ where $g$ is a fullyconnected neural network often called the critic.
+
+Combining these pieces, we learn representations by minimizing the following objective over the full set of data $x _ { i } \in \mathcal { D }$ and augmentations $a _ { l k } \in \mathcal { A } \times \mathcal { A }$
+
+$$
+\operatorname* { m i n } - \sum _ { i = 1 } ^ { N } \sum _ { a _ { l k } } \log \frac { \exp { ( \phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { i } ^ { a _ { k } } ) ) / \tau ) } } { \sum _ { m = 1 } ^ { M } \exp { ( \phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { m } ^ { a _ { k } } ) ) / \tau ) } } + \alpha \sum _ { a _ { l k } , a _ { q t } } K L ( p ^ { d o ( a _ { l k } ) } , p ^ { d o ( a _ { q t } ) } )
+$$
+
+with $M$ the number of points we use to construct the contrast set and $\alpha$ the weighting of the invariance penalty. We used the shorthand $p ^ { d o ( a ) }$ for $p ^ { d o ( a ) } ( Y ^ { R } = j \mid f ( x _ { i } ) )$ . With appropriate choices for $\phi$ , $g$ , $f$ and $h$ above, Equation 3 recovers many recent state-of-the-art methods (c.f. Table 5 in Section A). Figure 1b presents a schematic of the RELIC objective.
+
+The explicit invariance penalty encourages the within-class distances (for a downstream task of interest) of the representations learned by RELIC to be tightly concentrated. We show this empirically in Figure 2 and theoretically in Appendix B. In the following section we provide theoretical justification for using an instance discrimination-based contrastive loss using a causal perspective. We also show (cf. Theorem 1 below) that minimizing the contrastive loss alone (i.e. $\alpha = 0$ ) does not guarantee generalization. Instead, invariance across augmentations must be explicitly enforced.
+
+# 3 GENERALIZING CONTRASTIVE LEARNING
+
+Learning with refinements. In contrastive learning, the task of instance discrimination, i.e. classifying the dataset $\{ ( x _ { i } , y _ { i } ^ { R } = i ) | x _ { i } \in \mathcal { D } \}$ , is used as the proxy task. To better understand contrastive learning and motivate this proxy task, we generalize instance discrimination using the causal concept of refinements (Chalupka et al., 2014). Intuitively, a refinement of one problem is another more fine-grained problem. If task $Y _ { t }$ is to classify cats against dogs, then a refinement of $Y _ { t }$ is the task of classifying cats and dogs into their individual breeds. See Figure 4 for a further visual example. For any set of tasks, there exist many different refinements. However, the most fine-grained refinement corresponds exactly to classifying the dataset $\{ ( x _ { i } , y _ { i } ^ { R } = i ) | x _ { i } \in \mathcal { D } \}$ . Thus, the instance discrimination task used in contrastive learning is a specific type of refinement. For a definition and formal treatment of refinements please refer to Appendix D.
+
+Let $Y ^ { R }$ be targets of a proxy task that is a refinement for all tasks in $\mathcal { V }$ . Leveraging causal tools, we connect learning on refinements to learning on downstream tasks. Specifically, we provide a theoretical justification for exchanging unknown downstream tasks with these specially constructed proxy tasks. We show that if $f ( X )$ is an invariant representation for $Y ^ { R }$ under changes in style $S$ , then $f ( X )$ is also an invariant representation for tasks in $\mathcal { V }$ under changes in style $S$ . Thus by enforcing invariance under style interventions on a refinement, we learn representations that generalize to downstream tasks.3 This is summarized in the following theorem.
+
+Theorem 1. Let $\mathcal { V } = \{ Y _ { t } \} _ { t = 1 } ^ { T }$ be a family of downstream tasks. Let $Y ^ { R }$ be a refinement for all tasks in $\mathcal { V }$ . If $f ( X )$ is an invariant representation for $Y ^ { R }$ under style interventions $S$ , then $f ( X )$ is an invariant representation for all tasks in $\mathcal { V }$ under style interventions $S$ , i.e.
+
+$$
+\begin{array} { r l r } { p ^ { d o ( s _ { i } ) } ( Y ^ { R } \mid f ( X ) ) = p ^ { d o ( s _ { j } ) } ( Y ^ { R } \mid f ( X ) ) } & { { } \Rightarrow } & { p ^ { d o ( s _ { i } ) } ( Y _ { t } \mid f ( X ) ) = p ^ { d o ( s _ { j } ) } ( Y _ { t } \mid f ( X ) ) } \end{array}
+$$
+
+for all $s _ { i } , s _ { j } \in S$ with $p ^ { d o ( s _ { i } ) } = p ^ { d o ( S = s _ { i } ) }$ . Thus, $f ( X )$ is a representation that generalizes to $\mathcal { V }$
+
+Theorem 1 states that if $Y ^ { R }$ is a refinement of $\mathcal { V }$ then learning a representation on $Y ^ { R }$ is a sufficient condition for this representation to be useful on $\mathcal { V }$ . For a formal exposition of these points and accompanying proofs, please refer to Appendix D. Recall that the instance discrimination proxy task is the most fine-grained refinement, and so the right hand side of 4 is satisfied for any downstream task satisfying the stated assumptions of the theorem.
+
+We generalize contrastive learning through refinements and connect representations learned on refinements and downstream tasks in Theorem 1. Thus, using causality we provide an alternative explanation to mutual information for the success of contrastive learning. Note that our methodology of refinements is not limited to instance discrimination tasks and is thus more general than currently used contrastive losses. Real world data often includes rich sources of metadata which can be used to guide the construction of refinements by grouping the data according to any available meta-data. Note that the coarser we can create a refinement, the more data efficient we can expect to be when learning representations for downstream tasks. Further, we can also expect to require less supervised data to finetune the representation.
+
+# 4 RELATED WORK
+
+Contrastive objectives and mutual information maximization. Many recent approaches to selfsupervised learning are rooted in the well-established idea of maximizing mutual information (MI), e.g. Contrastive Predictive Coding (CPC) (Oord et al., 2018; Henaff et al., 2019), Deep InfoMax ´ (DIM) (Hjelm et al., 2018) and Augmented Multiscale DIM (AMDIM) (Bachman et al., 2019). These methods are based on noise contrastive estimation (NCE) (Gutmann & Hyvarinen, 2010) which, ¨ under specific conditions, can be viewed as a bound on MI (Poole et al., 2019). The resulting objective functions are commonly referred to as InfoNCE.
+
+The precise role played by mutual information maximization in self-supervised learning is subject to some debate. (Tschannen et al., 2019) argue that the performance on downstream tasks is not correlated with the achieved bound on MI, but may be more tightly correlated with encoder architecture and capacity. Importantly, InfoNCE objectives require custom architectures to ensure the network does not converge to non-informative solutions thus precluding the use of standard architectures. Recently, several works (He et al., 2019; Chen et al., 2020a) successfully combined contrastive estimation with a standard ResNet-50 architecture. In particular, SimCLR (Chen et al., 2020a) relies on a set of strong augmentations4, while (He et al., 2019) uses a memory bank. Inspired by target networks in reinforcement learning, (Grill et al., 2020) proposed BYOL: an algorithm for self-supervised learning which remarkably does not use a contrastive objective. Although theoretical explanation for the good performance of BYOL is presently missing, interestingly the objective, an $\ell _ { 2 }$ distance between two different embeddings of the input data resembles the $\ell _ { 2 }$ form of our regularizer proposed in Equation 5 in Appendix B.
+
+Recently, (Saunshi et al., 2019) proposed a learning theoretic framework to analyze the performance of contrastive objectives. However, without strong assumptions on intra-class concentration they note that contrastive objectives are fundamentally limited in the representations they are able to learn. RELIC explicitly enforces intra-class concentration via the invariance regularizer, ensuring that it generalizes under weaker assumptions. Unlike (Saunshi et al., 2019) who do not discuss augmentations, we incorporate augmentations into our theoretical explanation of contrastive methods.
+
+The reasons for the improvement in performance from AMDIM through to SimCLR and BYOL are not easily explained by either the MI maximization or the learning theoretic viewpoint. Further, it is not clear why relatively minor architectural differences between the methods result in significant differences in performance nor is it obvious how current state-of-the-art can be improved. In contrast to prior art, the performance of RELIC is explained by connections to causal theory. As such it gives a clear path for improving results by devising problem appropriate refinements, interventions and invariance penalties. Furthermore, the use of invariance penalties in RELIC as dictated by causal theory yields significantly more robust representations that generalize better than those learned with SimCLR or BYOL.
+
+![](images/b9bdc1c7a4b16d60eb71a077be525dc6413a7049dc4f90a27f4eeb110e4847d9.jpg)  
+Figure 2: Distribution of the linear discriminant ratio $F _ { \mathrm { L D A } }$ , see text) of $f$ for RELIC, SimCLR and AMDIM $y$ -axis clipped to aid visualization).
+
+Causality and invariance. Recently, the notion of invariant prediction has emerged as an important operational concept in causal inference (Peters et al., 2016). This idea has been used to learn classifiers which are robust against domain shifts (Gong et al., 2016). Notably, (Heinze-Deml & Meinshausen, 2017) propose to use group structure to delineate between different environments where the aim is to minimize the classification loss while also ensuring that the conditional variance of the prediction function within each group remains small. Unlike (Heinze-Deml & Meinshausen, 2017) who use supervised data and rely on having a grouping in the training data, our approach does not rely on ground-truth targets and can flexibly create groupings of the training data if none are present. Further, we enforce invariant prediction within the group by constraining the distance between distributions resulting from contrasting data across groups.
+
+The relationship between causal inference and semi-supervised learning has been explored in (Scholkopf et al., 2012). In particular, in order for ¨ unlabeled data to be helpful for learning, the relationship between the predictors and targets must be anti-causal. Our setting is somewhat different as we assume a latent decomposition of the observed variables and learn representations based on proxy targets which are refinements of the true targets.
+
+Our invariance penalty is similar in practise to consistency regularizers which have recently gained popularity in semi-supervised learning (Sajjadi et al., 2016; Xie et al., 2019; Sohn et al., 2020). These typically supplement the supervised training signal with a term which enforces similarity between the function’s output on two views of data. Although recently proposed consistency methods are heuristic, they have deeper roots in the more rigorous co-training approach to semi-supervised learning from multiple views (Blum & Mitchell, 1998; Kakade & Foster, 2007; Sridharan & Kakade, 2008; McWilliams & Montana, 2012; McWilliams et al., 2013). The main conceptual difference between our approach and consistency-based methods is that we do not have access to a supervised training signal and that the RELIC penalty is motivated from the underlying causal graph.
+
+# 5 EXPERIMENTS
+
+We first visualize the influence of the explicit invariance constraint in RELIC on the linear separability of the learned representations. We then evaluate RELIC on a number of prediction and reinforcement learning tasks for usefulness and robustness. For the prediction tasks, we test RELIC after pretraining the representation in a self-supervised way on the training set of the ImageNet ILSVRC-2012 dataset (Russakovsky et al., 2015). We evaluate RELIC in the linear evaluation setup on ImageNet and test its robustness and out-of-distribution generalization on datasets related to ImageNet. Unlike much prior work in contrastive learning which focuses specifically on computer vision tasks, we test RELIC also in the context of learning representations for reinforcement learning. Specifically, we test RELIC on the suite of Atari games (Bellemare et al., 2013) which consists of 57 diverse games of varying difficulty.
+
+Linear evaluation. In order to understand how representations learned by RELIC differ from other methods, we compare it against those learned by AMDIM and SimCLR in terms of Fischer’s linear discriminant ratio (Friedman et al., 2009): $\begin{array} { r } { \dot { F _ { \mathrm { L D A } } } { } = \| \mu _ { k } - \mu _ { k ^ { \prime } } \| ^ { 2 } / \sum _ { i , j \in \mathcal { C } _ { k } } \| f ( x _ { i } ) - f ( x _ { j } ) \| ^ { 2 } } \end{array}$ where $\begin{array} { r } { \mu _ { k } = \frac { 1 } { | { \mathcal C } _ { k } | } \sum _ { i \in { \mathcal C } _ { k } } f ( x _ { i } ) } \end{array}$ is the mean of the representations of class $k$ and $\mathcal { C } _ { k }$ is the index set of that class. A larger $F _ { \mathrm { L D A } }$ implies that classes are more easily separated with a linear classifier. This can be achieved by either increasing distances between classes (numerator) or shrinking within-class variance (denominator).
+
+Figure 2 shows the distribution of $F _ { \mathrm { L D A } }$ for RELIC, SimCLR and AMDIM after training as measured on the (downsampled) ImageNet validation set. The distance between medians of RELIC and SimCLR is 162. AMDIM is tightly concentrated close to 20. The invariance penalty ensures that— even though labels are $a$ -priori unknown—for RELIC within-class variability of $f$ is concentrated leading to better linear separability between classes in the downstream task of interest.This is reflected in the rightward shift of the distribution of $F _ { \mathrm { L D A } }$ in Figure 2 for RELIC compared with SimCLR and AMDIM which do not impose such a constraint.
+
+Next we evaluate RELIC’s representation by training a linear classifier of top of the fixed encoder following the procedure in (Kolesnikov et al., 2019; Chen et al., 2020a) and Appendix E.4. In Table 1, we report top-1 and top-5 accuracy on the ImageNet test set. Methods denoted with \* use SimCLR augmentations (Chen et al., 2020a), while methods denoted $\dagger$ use custom, stronger augmentations. Comparing methods which use SimCLR augmentations, RELIC outperforms competing approaches on both ResNet-50 and ResNet-50 with target network. For completeness, we report results for SwAV (Caron et al., 2020) and InfoMin (Tian et al., 2020), but note that these methods use stronger augmentations which alone have been shown to boost performance by over $5 \%$ A fair comparison between different objectives can only be achieved under the same architecture and the same set of augmentations.
+
+Table 1: Accuracy $( \mathrm { i n } \% )$ ) under linear evaluation on ImageNet for different self-supervised representation learning methods. Methods with \* use SimCLR augmentations. Methods with $\dagger$ use custom, stronger augmentations.   
+
+<table><tr><td>Method</td><td>Top-1</td><td>Top-5</td></tr><tr><td>ResNet-50architecture</td><td></td><td></td></tr><tr><td>PIRL</td><td>63.6</td><td></td></tr><tr><td>CPC v2</td><td>63.8</td><td>85.3</td></tr><tr><td>CMC</td><td>66.2</td><td>87.0</td></tr><tr><td>SimCLR(Chen et al.,2020a)</td><td>* 69.3</td><td>89.0</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td>* 70.1</td><td>=</td></tr><tr><td>RELIC (ours)</td><td>* 70.3</td><td>89.5</td></tr><tr><td>InfoMin Aug.(Tian et al.,2020)</td><td>+ 73.0</td><td>91.1</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td>+ 75.3</td><td>=</td></tr><tr><td>ResNet-50 with target network</td><td></td><td></td></tr><tr><td>MoCo v2 (Chen et al., 2020b)</td><td>71.1</td><td></td></tr><tr><td>BYOL (Grill et al., 2020)</td><td>* 74.3</td><td>91.6</td></tr><tr><td>RELIC (ours)</td><td>* 74.8</td><td>92.2</td></tr></table>
+
+Robustness and generalization. We evaluate robustness and out-of-distribution generalization of RELIC’s representation on datasets Imagenet-C (Hendrycks & Dietterich, 2019) and ImageNet-R (Hendrycks et al., 2020), respectively. To evaluate RELIC’s representation, we train a linear classifier on top of the frozen representation following the procedure described in (Chen et al., 2020a) and appendix E.5.2. For Imagenet-C we report the mean Corruption Error (mCE) and Corruption Errors for Noise corruptions in Table 3. RELIC has significantly lower mCE than both the supervised ResNet-50 baseline and the unsupervised methods SimCLR and BYOL. Also, it has the lowest Corruption Error on 14 out of 15 corruptions when compared to SimCLR and BYOL. Thus, we see that RELIC learns the most robust representation. RELIC also outperforms SimCLR and BYOL on ImageNet-R showing its superior out-of-distribution generalization ability; see Table 2. For further details and results please consult E.5.
+
+Table 2: Top-1 error rates for different self-supervised representation learning methods on ImageNetR. All models are trained only on clean ImageNet images and $\mathrm { R E L I C } _ { T }$ refers to RELIC using a ResNet-50 with target network as in BYOL (Grill et al., 2020).   
+
+<table><tr><td>Method</td><td>Supervised</td><td>SimCLR</td><td>RELIC</td><td>BYOL</td><td>RELICT</td></tr><tr><td>Top-1 Error (%)</td><td>63.9</td><td>81.7</td><td>77.4</td><td>77.0</td><td>76.2</td></tr></table>
+
+Table 3: Mean Corruption Error (mCE), mean relative Corruption Error (mrCE) and Corruption Errors for the “Noise” class of corruptions (Gaussian, Shot, Impulse) on ImageNet-C. The mCE value is the average across 75 different corruptions. Methods are trained only on clean ImageNet images.   
+
+<table><tr><td>Method</td><td>mCE</td><td>mrCE</td><td>Gaussian</td><td>Shot</td><td>Impulse</td></tr><tr><td>Supervised</td><td>76.7</td><td>105.0</td><td>80.0</td><td>82.0</td><td>83.0</td></tr><tr><td>ResNet-50 architecture:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SimCLR</td><td>87.5</td><td>111.9</td><td>79.4</td><td>81.9</td><td>89.6</td></tr><tr><td>ReLIC</td><td>76.4</td><td>87.7</td><td>67.8</td><td>70.7</td><td>77.0</td></tr><tr><td>ResNet-50 with target network:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BYOL</td><td>72.3</td><td>90.0</td><td>65.9</td><td>68.4</td><td>73.7</td></tr><tr><td>ReLIC</td><td>70.8</td><td>88.4</td><td>63.6</td><td>65.7</td><td>69.2</td></tr></table>
+
+Reinforcement Learning. Much prior work in contrastive learning has focused specifically on computer vision tasks. In order to compare these approaches in a different domain, we investigate representation learning in the context of reinforcement learning. We compare RELIC as an auxiliary loss against other state of the art self-supervised losses on an agent trained on 57 Atari games. Using human normalized scores as a metric, we use the original architecture and hyperparameters of the R2D2 agent (Kapturowski et al., 2019) and supplement it with a second encoder trained with a given representation learning loss. When auxiliary losses are present, the Q-Network takes the output of the second encoder as an input. The Q-Network and the encoder are trained with separate optimizers. For the augmentation baseline, the Q-Network takes two identical encoders trained end-to-end. Table 4 shows a comparison between RELIC, SimCLR, BYOL, CURL (Srinivas et al., 2020), and feeding augmented observations directly to the agent (Kostrikov et al., 2020). We find that RELIC has a significant advantage over competing self-supervised methods, performing best in 25 out of 57 games. The next best performing method, CURL performs best in 11 games. Note that none of these methods outperform R2D2 (Kapturowski et al., 2019) which achieves superhuman performance in 52 games. Although previously published work shows the auxiliary tasks help on Atari, this is in the very low-data regime. Here we show that ReLIC is able to close the gap in the more common Atari set up. We hypothesize that the performance penalty resulting from adding auxiliary self-supervised losses stems from the naive combining of the outputs of the two encoders; we will explore alternative options for combining encoder outputs in future. Full details are presented in Section E.6.
+
+# 6 CONCLUSION
+
+In this work we have analyzed self-supervised learning using a causal framework. Using a causal graph, we have formalized the problem of self-supervised representation learning and derived properties of the optimal representation. We have shown that representations need to be invariant predictors of proxy targets under interventions on features that are only correlated, but not causally related to the downstream tasks. We have leveraged data augmentations to simulate these interventions and have proposed to explicitly enforce this invariance constraint. Based on this, we have proposed a new self-supervised objective, Representation Learning via Invariant Causal Mechanisms (RELIC), that enforces invariant prediction of proxy targets across augmentations using an invariance regularizer. Further, we have generalized contrastive methods using the concept of refinements and have shown that learning a representation on refinements using the principle of invariant prediction is a sufficient condition for these representations to generalize to downstream tasks. With this, we have provided an alternative explanation to mutual information for the success of contrastive methods. Empirically we have compared RELIC against recent self-supervised methods on a variety of prediction and reinforcement learning tasks. Specifically, RELIC significantly outperforms competing methods in terms of robustness and out-of-distribution generalization of the representations it learns on ImageNet. RELIC also significantly outperforms related self-supervised methods on the Atari suite achieving superhuman performance on 51 out of 57 games. We aim to investigate the construction of more coarse-grained refinements and the empirical evaluation of different kinds of refinements in future work.
+
+Table 4: Human Normalized Scores of Auxiliary Methods over 57 Atari Games.   
+
+<table><tr><td>AtariPerformance</td><td>RELIC</td><td>SimCLR</td><td>CURL</td><td>BYOL</td><td>Augmentation</td></tr><tr><td>Capped mean</td><td>91.46</td><td>88.76</td><td>90.72</td><td>89.43</td><td>80.60</td></tr><tr><td>Number of superhuman games</td><td>51</td><td>49</td><td>49</td><td>49</td><td>34</td></tr><tr><td>Mean</td><td>3003.73</td><td>2086.16</td><td>2413.12</td><td>1769.43</td><td>503.15</td></tr><tr><td>Median</td><td>832.50</td><td>592.83</td><td>819.56</td><td>483.39</td><td>132.17</td></tr><tr><td>40% Percentile</td><td>356.27</td><td>266.07</td><td>409.46</td><td>224.80</td><td>94.35</td></tr><tr><td>30% Percentile</td><td>202.49</td><td>174.19</td><td>190.96</td><td>150.21</td><td>80.04</td></tr><tr><td>20% Percentile</td><td>133.93</td><td>120.84</td><td>126.10</td><td>118.36</td><td>57.95</td></tr><tr><td>10% Percentile</td><td>83.79</td><td>37.19</td><td>59.09</td><td>44.14</td><td>32.74</td></tr><tr><td>5% Percentile</td><td>20.87</td><td>12.74</td><td>20.56</td><td>7.75</td><td>2.85</td></tr></table>
+
+Acknowledgements. We thank David Balduzzi, Melanie Rey, Christina Heinze-Deml, Ilja Kuzborskij, Ali Eslami, Jeffrey de Fauw and Josip Djolonga for invaluable discussions.
+
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+        "text": "REPRESENTATION LEARNING VIA INVARIANT CAUSAL MECHANISMS ",
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+        "type": "text",
+        "text": "Jovana Mitrovic Brian McWilliams Jacob Walker Lars Buesing Charles Blundell   \nDeepMind   \nLondon, UK   \n{mitrovic, bmcw, jcwalker, lbuesing, cblundell}@google.com ",
+        "bbox": [
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            277
+        ],
+        "page_idx": 0
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+        "text": "Self-supervised learning has emerged as a strategy to reduce the reliance on costly supervised signals by pretraining representations only using unlabeled data. These methods combine heuristic proxy classification tasks with data augmentations and have achieved significant success, but our theoretical understanding of this success remains limited. In this paper we analyze self-supervised representation learning using a causal framework. We show how data augmentations can be more effectively utilized through explicit invariance constraints on the proxy classifiers employed during pretraining. Based on this, we propose a novel selfsupervised objective, Representation Learning via Invariant Causal Mechanisms (RELIC), that enforces invariant prediction of proxy targets across augmentations through an invariance regularizer which yields improved generalization guarantees. Further, using causality we generalize contrastive learning, a particular kind of self-supervised method, and provide an alternative theoretical explanation for the success of these methods. Empirically, RELIC significantly outperforms competing methods in terms of robustness and out-of-distribution generalization on ImageNet, while also significantly outperforming these methods on Atari achieving above human-level performance on 51 out of 57 games. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Training deep networks often relies heavily on large amounts of useful supervisory signal, such as labels for supervised learning or rewards for reinforcement learning. These training signals can be costly or otherwise impractical to acquire. On the other hand, unsupervised data is often abundantly available. Therefore, pretraining representations for unknown downstream tasks without the need for labels or extrinsic reward holds great promise for reducing the cost of applying machine learning models. To pretrain representations, self-supervised learning makes use of proxy tasks defined on unsupervised data. Recently, self-supervised methods using contrastive objectives have emerged as one of the most successful strategies for unsupervised representation learning (Oord et al., 2018; Hjelm et al., 2018; Chen et al., 2020a). These methods learn a representation by classifying every datapoint against all others datapoints (negative examples). Under assumptions on how the negative examples are sampled, minimizing the resulting contrastive loss has been justified as maximizing a lower bound on the mutual information (MI) between representations (Poole et al., 2019). However, (Tschannen et al., 2019) has shown that performance on downstream tasks may be more tightly correlated with the choice of encoder architecture than the achieved MI bound, highlighting issues with the MI theory of contrastive learning. Further, contrastive approaches compare different views of the data (usually under different data augmentations) to calculate similarity scores. This approach to computing scores has been empirically observed as a key success factor of contrastive methods, but has yet to be theoretically justified. This lack of a solid theoretical explanation for the effectiveness of contrastive methods hinders their further development. ",
+        "bbox": [
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+        "type": "text",
+        "text": "To remedy the theoretical shortcomings, we analyze the problem of self-supervised representation learning through a causal lens. We formalize intuitions about the data generating process using a causal graph and leverage causal tools to derive properties of the optimal representation. We show that a representation should be an invariant predictor of proxy targets under interventions on features that are only correlated, but not causally related to the downstream targets of interest. ",
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+        "text": "Since neither causally nor purely correlationally related features are observed and thus performing actual interventions on them is not feasible, for learning representation with this property we use data augmentations to simulate a subset of possible interventions. Based on our causal interpretation, we propose a regularizer which enforces that the prediction of the proxy targets is invariant across data augmentations. We propose a novel objective for self-supervised representation learning called REpresentation Learning with Invariant Causal mechanisms (RELIC). We show how this explicit invariance regularization leverages augmentations more effectively than previous self-supervised methods and that representations learned using RELIC are guaranteed to generalize well to downstream tasks under weaker assumptions than those required by previous work (Saunshi et al., 2019). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Next we generalize contrastive learning and provide an alternative theoretical explanation to MI for the success of these methods. We generalize the proxy task of instance discrimination commonly used in contrastive learning using the causal concept of refinements (Chalupka et al., 2014). Intuitively, a refinement of a task can be understood as a more fine-grained variant of the original problem. For example, a refinement for classifying cats against dogs would be the task of classifying individual cat and dog breeds. The instance discrimination task results from the most fine-grained refinement, e.g. discriminating individual cats and dogs from one another. We show that using refinements as proxy tasks enables us to learn useful representations for downstream tasks. Specifically, using causal tools, we show that learning a representation on refinements such that it is an invariant predictor of proxy targets across augmentations is a sufficient condition for these representations to generalize to downstream tasks (cf. Theorem 1). In summary, we provide theoretical support both for the general form of the contrastive objective as well as for the use of data augmentations. Thus, we provide an alternative explanation to mutual information for the success of recent contrastive approaches namely that of causal refinements of downstream tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We test RELIC on a variety of prediction and reinforcement learning problems. First, we evaluate the quality of representations pretrained on ImageNet with a special focus on robustness and out-ofdistribution generalization. RELIC performs competitively with current state-of-the-art methods on ImageNet, while significantly outperforming competing methods on robustness and out-of-distribution generalization of the learned representations when tested on corrupted ImageNet (ImageNet-C (Hendrycks & Dietterich, 2019)) and a version of ImageNet that consist of different renditions of the same classes (ImageNet-R (Hendrycks et al., 2020)). In terms of robustness, RELIC also significantly outperforms the supervised baseline with an absolute reduction of $4 . 9 \\%$ in error. Unlike much prior work that specifically focuses on computer vision tasks, we test RELIC for representation learning in the context of reinforcement learning on the Atari suite (Bellemare et al., 2013). There we find that RELIC significantly outperforms competing methods and achieves above human-level performance on 51 out of 57 games. ",
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+    {
+        "type": "text",
+        "text": "Contributions. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "• We formalize problem of self-supervised representation learning using causality and propose to more effectively leverage data augmentations through invariant prediction.   \n• We propose a new self-supervised objective, REpresentation Learning with Invariance Causal mechanisms (RELIC), that enforces invariant prediction through an explicit regularizer and show improved generalization guarantees. We generalize contrastive learning using refinements and show that learning on refinements is a sufficient condition for learning useful representations; this provides an alternative explanation to MI for the success of contrastive methods. ",
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+    {
+        "type": "text",
+        "text": "2 REPRESENTATION LEARNING VIA INVARIANT CAUSAL MECHANISMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Problem setting. Let $X$ denote the unlabelled observed data and $\\mathcal { V } = \\{ Y _ { t } \\} _ { t = 1 } ^ { T }$ be a set of unknown tasks with $Y _ { t }$ denoting the targets for task $t$ . The tasks $\\{ Y _ { t } \\} _ { t = 1 } ^ { T }$ can represent both a multi-environment as well as a multi-task setup. Our goal is to pretrain with unsupervised data a representation $f ( X )$ that will be useful for solving the downstream tasks $\\mathcal { V }$ . ",
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+        "type": "text",
+        "text": "Causal interpretation. To effectively leverage common assumptions and intuitions about data generation of the unknown downstream tasks for the learning algorithm, we propose to formalize them using a causal graph. We start from the following assumptions: a) the data is generated from content and style variables, with b) only content (and not style) being relevant for the unknown downstream tasks and c) content and style are independent, i.e. style changes are content-preserving. For example, when classifying dogs against giraffes from images, different parts of the animals constitute content, while style could be, for example, background, lighting conditions and camera lens characteristics. By assumption, content is a good representation of the data for downstream tasks and we therefore cast the goal of representation learning as estimating content. In the following, we compactly formalize these assumptions with a causal graph1, see Figure 1a. ",
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+        "image_caption": [
+            "Figure 1: (a) Causal graph formalizing assumptions about content and style of the data and the relationship between targets and proxy tasks. The dashed arrows are not causal and represent learning, while the dashdotted lines denote that $Y ^ { R }$ is a refinement $Y _ { t }$ ’s. All other arrows are causal. (b) RELIC objective. KL refers to the Kullback-Leibler divergence, while $\\mathbf { X }$ -entropy denotes cross entropy. "
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+        "type": "text",
+        "text": "Let $C$ and $S$ be the latent variables describing content and style. In Figure 1a, the directed arrows from $C$ and $S$ to the observed data $X$ (e.g. images) indicate that $X$ is generated based on content and style. The directed arrow from $C$ to the target $Y _ { t }$ (e.g. class labels) encodes the assumption that content directly influences the target tasks, while the absence of any directed arrow from $S$ to $Y _ { t }$ indicates that style does not. Thus, content $C$ has all the necessary information to predict $Y _ { t }$ . The absence of any directed path between $C$ and $S$ in Figure 1a encodes the intuition that these variables are independent, i.e. $C \\perp \\perp S$ . ",
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+        "text": "Using the independence of mechanisms (Peters et al., 2017), we can conclude that under this causal model performing interventions on $S$ does not change the conditional distribution $P ( Y _ { t } | C )$ , i.e. manipulating the value of $S$ does not influence this conditional distribution. Thus, $P ( Y _ { t } | C )$ is invariant under changes in style $S$ . We call $C$ an invariant representation for $Y _ { t }$ under $S$ , i.e. ",
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+        "img_path": "images/66f824ef365d301ad151866230f0527ae3ee6c0c19102f2c93d00c20a9efd277.jpg",
+        "text": "$$\np ^ { d o ( S = s _ { i } ) } ( Y _ { t } \\mid C ) = p ^ { d o ( S = s _ { j } ) } ( Y _ { t } \\mid C ) \\quad \\forall s _ { i } , s _ { j } \\in \\mathcal { S } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p ^ { d o ( S = s ) }$ denotes the distribution arising from assigning $S$ the value $s$ with $s$ the domain of $S$ (Pearl, 2009). Specifically, using $C$ as a representation allows for us to predict targets stably across perturbations, i.e. content $C$ is both a useful and robust representation for tasks $\\mathcal { V }$ . ",
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+        "type": "text",
+        "text": "Since the targets $Y _ { t }$ are unknown, we will construct a proxy task $Y ^ { R }$ in order to learn representations from unlabeled data $X$ only. In order to learn useful representations for $Y _ { t }$ , we will construct proxy tasks that represents more fine-grained problems than $Y _ { t }$ ; for a more formal treatment of proxy tasks please refer to Section 3. Further, to learn invariant representations, such as $C$ , we enforce Equation 1 which requires us to observe data under different style interventions, i.e. we need data that describes the same content under varying style. Since we do not have access to $S$ , to simulate style variability we use content-preserving data augmentations (e.g. rotation, grayscaling, translation, cropping for images). Specifically, we utilize data augmentations as interventions on the style variable $S$ , i.e. ",
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+        "type": "text",
+        "text": "applying data augmentation $a _ { i }$ corresponds to intervening on $S$ and setting it to $s _ { a _ { i } }$ . 2 Although we are not able to generate all possible styles using a fixed set of data augmentations, we will use augmentations that generate large sets of diverse styles as this allows us to learn better representations. Note that the heuristic of estimating similarity based on different views from contrastive learning can be interpreted as an implicit invariance constraint. ",
+        "bbox": [
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+        "type": "text",
+        "text": "RELIC objective. Equation 1 provides a general scheme to estimate content (c.f. Figure 1a). We operationalize this by proposing to learn representations such that prediction of proxy targets from the representation is invariant under data augmentations. The representation $f ( X )$ must fulfill the following invariant prediction criteria ",
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+        "type": "text",
+        "text": "(Invarian ",
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+    {
+        "type": "equation",
+        "img_path": "images/467cf2caee2fe756762941d48676ea4a8f5f45cd709369ff35cb1fc05acb054d.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\prime p r e d i c t i o n ) } & { } & { p ^ { \\mathrm { d o } ( a _ { i } ) } ( Y ^ { R } | f ( X ) ) = p ^ { \\mathrm { d o } ( a _ { j } ) } ( Y ^ { R } | f ( X ) ) \\quad \\forall a _ { i } , a _ { j } \\in \\mathcal { A } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "$\\mathcal { A } = \\{ a _ { 1 } , \\ldots , a _ { m } \\}$ is the set of data augmentations which simulate interventions on the style variables and $p ^ { d o ( a ) }$ denotes $p ^ { d o ( S = s _ { a } ) }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To achieve invariant prediction, we propose to explicitly enforce invariance under augmentations through a regularizer. This gives rise to an objective for self-supervised learning we call Representation Learning via Invariant Causal Mechanisms (RELIC). We write this objective as ",
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+        "type": "equation",
+        "img_path": "images/5490b04aef1959168dd2138793ab9d3720fd7ba171772014d647be03ba8bae8f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatorname* { m i n } _ { \\stackrel { X \\warrow p ( X ) } { \\sim \\sim p ( X ) } } \\underset { \\sim \\mathcal { A } \\times \\mathcal { A } } { \\mathbb { E } } \\quad \\underset { b \\in \\{ a _ { l k } , a _ { q t } \\} } { \\sum } \\mathcal { L } _ { b } ( Y ^ { R } , f ( X ) ) } \\\\ { s . t . \\quad K L \\left( p ^ { d o ( a _ { l k } ) } ( Y ^ { R } | f ( X ) ) , p ^ { d o ( a _ { q t } ) } ( Y ^ { R } | f ( X ) ) \\right) \\leq \\rho } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the proxy task loss and $K L$ is the Kullback-Leibler (KL) divergence. Note that any distance measure on distributions can be used in place of the KL divergence. We explain the remaining terms in detail below. ",
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+        "type": "text",
+        "text": "Concretely, as proxy task we associate to every datapoint $x _ { i }$ the label $y _ { i } ^ { R } = i$ . This corresponds to the instance discrimination task, commonly used in contrastive learning (Hadsell et al., 2006). We take pairs of points $( x _ { i } , x _ { j } )$ to compute similarity scores and use pairs of augmentations $a _ { l k } = ( a _ { l } , a _ { k } ) \\in$ $A \\times A$ to perform a style intervention. Given a batch of samples $\\{ x _ { i } \\} _ { i = 1 } ^ { N } \\sim \\mathcal { D }$ , we use ",
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+        "type": "equation",
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+        "text": "$$\np ^ { d o ( a _ { l k } ) } ( Y ^ { R } = j | f ( x _ { i } ) ) \\propto \\exp \\left( \\phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { j } ^ { a _ { k } } ) ) / \\tau \\right) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "with $x ^ { a }$ data augmented with $a$ and $\\tau$ a softmax temperature parameter. We encode $f$ using a neural network and choose $h$ to be related to $f$ , e.g. $h = f$ or as a network with an exponential moving average of the weights of $f$ (e.g. target networks similar to (Grill et al., 2020)). To compare representations we use the function $\\bar { \\phi } ( f ( \\bar { x _ { i } } ) , h ( x _ { j } ) ) \\ : = \\ : \\langle g ( f ( x _ { i } ) ) , g ( h ( x _ { j } ) ) \\rangle$ where $g$ is a fullyconnected neural network often called the critic. ",
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+        "type": "text",
+        "text": "Combining these pieces, we learn representations by minimizing the following objective over the full set of data $x _ { i } \\in \\mathcal { D }$ and augmentations $a _ { l k } \\in \\mathcal { A } \\times \\mathcal { A }$ ",
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+        "text": "$$\n\\operatorname* { m i n } - \\sum _ { i = 1 } ^ { N } \\sum _ { a _ { l k } } \\log \\frac { \\exp { ( \\phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { i } ^ { a _ { k } } ) ) / \\tau ) } } { \\sum _ { m = 1 } ^ { M } \\exp { ( \\phi ( f ( x _ { i } ^ { a _ { l } } ) , h ( x _ { m } ^ { a _ { k } } ) ) / \\tau ) } } + \\alpha \\sum _ { a _ { l k } , a _ { q t } } K L ( p ^ { d o ( a _ { l k } ) } , p ^ { d o ( a _ { q t } ) } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with $M$ the number of points we use to construct the contrast set and $\\alpha$ the weighting of the invariance penalty. We used the shorthand $p ^ { d o ( a ) }$ for $p ^ { d o ( a ) } ( Y ^ { R } = j \\mid f ( x _ { i } ) )$ . With appropriate choices for $\\phi$ , $g$ , $f$ and $h$ above, Equation 3 recovers many recent state-of-the-art methods (c.f. Table 5 in Section A). Figure 1b presents a schematic of the RELIC objective. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The explicit invariance penalty encourages the within-class distances (for a downstream task of interest) of the representations learned by RELIC to be tightly concentrated. We show this empirically in Figure 2 and theoretically in Appendix B. In the following section we provide theoretical justification for using an instance discrimination-based contrastive loss using a causal perspective. We also show (cf. Theorem 1 below) that minimizing the contrastive loss alone (i.e. $\\alpha = 0$ ) does not guarantee generalization. Instead, invariance across augmentations must be explicitly enforced. ",
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+    {
+        "type": "text",
+        "text": "3 GENERALIZING CONTRASTIVE LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Learning with refinements. In contrastive learning, the task of instance discrimination, i.e. classifying the dataset $\\{ ( x _ { i } , y _ { i } ^ { R } = i ) | x _ { i } \\in \\mathcal { D } \\}$ , is used as the proxy task. To better understand contrastive learning and motivate this proxy task, we generalize instance discrimination using the causal concept of refinements (Chalupka et al., 2014). Intuitively, a refinement of one problem is another more fine-grained problem. If task $Y _ { t }$ is to classify cats against dogs, then a refinement of $Y _ { t }$ is the task of classifying cats and dogs into their individual breeds. See Figure 4 for a further visual example. For any set of tasks, there exist many different refinements. However, the most fine-grained refinement corresponds exactly to classifying the dataset $\\{ ( x _ { i } , y _ { i } ^ { R } = i ) | x _ { i } \\in \\mathcal { D } \\}$ . Thus, the instance discrimination task used in contrastive learning is a specific type of refinement. For a definition and formal treatment of refinements please refer to Appendix D. ",
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+        "type": "text",
+        "text": "Let $Y ^ { R }$ be targets of a proxy task that is a refinement for all tasks in $\\mathcal { V }$ . Leveraging causal tools, we connect learning on refinements to learning on downstream tasks. Specifically, we provide a theoretical justification for exchanging unknown downstream tasks with these specially constructed proxy tasks. We show that if $f ( X )$ is an invariant representation for $Y ^ { R }$ under changes in style $S$ , then $f ( X )$ is also an invariant representation for tasks in $\\mathcal { V }$ under changes in style $S$ . Thus by enforcing invariance under style interventions on a refinement, we learn representations that generalize to downstream tasks.3 This is summarized in the following theorem. ",
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+        "type": "text",
+        "text": "Theorem 1. Let $\\mathcal { V } = \\{ Y _ { t } \\} _ { t = 1 } ^ { T }$ be a family of downstream tasks. Let $Y ^ { R }$ be a refinement for all tasks in $\\mathcal { V }$ . If $f ( X )$ is an invariant representation for $Y ^ { R }$ under style interventions $S$ , then $f ( X )$ is an invariant representation for all tasks in $\\mathcal { V }$ under style interventions $S$ , i.e. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/73b842ff62039898712fc7503cfc46cf67bb8bc9f53ade767245e57ea8c4dea6.jpg",
+        "text": "$$\n\\begin{array} { r l r } { p ^ { d o ( s _ { i } ) } ( Y ^ { R } \\mid f ( X ) ) = p ^ { d o ( s _ { j } ) } ( Y ^ { R } \\mid f ( X ) ) } & { { } \\Rightarrow } & { p ^ { d o ( s _ { i } ) } ( Y _ { t } \\mid f ( X ) ) = p ^ { d o ( s _ { j } ) } ( Y _ { t } \\mid f ( X ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "for all $s _ { i } , s _ { j } \\in S$ with $p ^ { d o ( s _ { i } ) } = p ^ { d o ( S = s _ { i } ) }$ . Thus, $f ( X )$ is a representation that generalizes to $\\mathcal { V }$ ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Theorem 1 states that if $Y ^ { R }$ is a refinement of $\\mathcal { V }$ then learning a representation on $Y ^ { R }$ is a sufficient condition for this representation to be useful on $\\mathcal { V }$ . For a formal exposition of these points and accompanying proofs, please refer to Appendix D. Recall that the instance discrimination proxy task is the most fine-grained refinement, and so the right hand side of 4 is satisfied for any downstream task satisfying the stated assumptions of the theorem. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We generalize contrastive learning through refinements and connect representations learned on refinements and downstream tasks in Theorem 1. Thus, using causality we provide an alternative explanation to mutual information for the success of contrastive learning. Note that our methodology of refinements is not limited to instance discrimination tasks and is thus more general than currently used contrastive losses. Real world data often includes rich sources of metadata which can be used to guide the construction of refinements by grouping the data according to any available meta-data. Note that the coarser we can create a refinement, the more data efficient we can expect to be when learning representations for downstream tasks. Further, we can also expect to require less supervised data to finetune the representation. ",
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+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Contrastive objectives and mutual information maximization. Many recent approaches to selfsupervised learning are rooted in the well-established idea of maximizing mutual information (MI), e.g. Contrastive Predictive Coding (CPC) (Oord et al., 2018; Henaff et al., 2019), Deep InfoMax ´ (DIM) (Hjelm et al., 2018) and Augmented Multiscale DIM (AMDIM) (Bachman et al., 2019). These methods are based on noise contrastive estimation (NCE) (Gutmann & Hyvarinen, 2010) which, ¨ under specific conditions, can be viewed as a bound on MI (Poole et al., 2019). The resulting objective functions are commonly referred to as InfoNCE. ",
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+    {
+        "type": "text",
+        "text": "The precise role played by mutual information maximization in self-supervised learning is subject to some debate. (Tschannen et al., 2019) argue that the performance on downstream tasks is not correlated with the achieved bound on MI, but may be more tightly correlated with encoder architecture and capacity. Importantly, InfoNCE objectives require custom architectures to ensure the network does not converge to non-informative solutions thus precluding the use of standard architectures. Recently, several works (He et al., 2019; Chen et al., 2020a) successfully combined contrastive estimation with a standard ResNet-50 architecture. In particular, SimCLR (Chen et al., 2020a) relies on a set of strong augmentations4, while (He et al., 2019) uses a memory bank. Inspired by target networks in reinforcement learning, (Grill et al., 2020) proposed BYOL: an algorithm for self-supervised learning which remarkably does not use a contrastive objective. Although theoretical explanation for the good performance of BYOL is presently missing, interestingly the objective, an $\\ell _ { 2 }$ distance between two different embeddings of the input data resembles the $\\ell _ { 2 }$ form of our regularizer proposed in Equation 5 in Appendix B. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Recently, (Saunshi et al., 2019) proposed a learning theoretic framework to analyze the performance of contrastive objectives. However, without strong assumptions on intra-class concentration they note that contrastive objectives are fundamentally limited in the representations they are able to learn. RELIC explicitly enforces intra-class concentration via the invariance regularizer, ensuring that it generalizes under weaker assumptions. Unlike (Saunshi et al., 2019) who do not discuss augmentations, we incorporate augmentations into our theoretical explanation of contrastive methods. ",
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+    {
+        "type": "text",
+        "text": "The reasons for the improvement in performance from AMDIM through to SimCLR and BYOL are not easily explained by either the MI maximization or the learning theoretic viewpoint. Further, it is not clear why relatively minor architectural differences between the methods result in significant differences in performance nor is it obvious how current state-of-the-art can be improved. In contrast to prior art, the performance of RELIC is explained by connections to causal theory. As such it gives a clear path for improving results by devising problem appropriate refinements, interventions and invariance penalties. Furthermore, the use of invariance penalties in RELIC as dictated by causal theory yields significantly more robust representations that generalize better than those learned with SimCLR or BYOL. ",
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+        "img_path": "images/b9bdc1c7a4b16d60eb71a077be525dc6413a7049dc4f90a27f4eeb110e4847d9.jpg",
+        "image_caption": [
+            "Figure 2: Distribution of the linear discriminant ratio $F _ { \\mathrm { L D A } }$ , see text) of $f$ for RELIC, SimCLR and AMDIM $y$ -axis clipped to aid visualization). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Causality and invariance. Recently, the notion of invariant prediction has emerged as an important operational concept in causal inference (Peters et al., 2016). This idea has been used to learn classifiers which are robust against domain shifts (Gong et al., 2016). Notably, (Heinze-Deml & Meinshausen, 2017) propose to use group structure to delineate between different environments where the aim is to minimize the classification loss while also ensuring that the conditional variance of the prediction function within each group remains small. Unlike (Heinze-Deml & Meinshausen, 2017) who use supervised data and rely on having a grouping in the training data, our approach does not rely on ground-truth targets and can flexibly create groupings of the training data if none are present. Further, we enforce invariant prediction within the group by constraining the distance between distributions resulting from contrasting data across groups. ",
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+        "type": "text",
+        "text": "The relationship between causal inference and semi-supervised learning has been explored in (Scholkopf et al., 2012). In particular, in order for ¨ unlabeled data to be helpful for learning, the relationship between the predictors and targets must be anti-causal. Our setting is somewhat different as we assume a latent decomposition of the observed variables and learn representations based on proxy targets which are refinements of the true targets. ",
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+    },
+    {
+        "type": "text",
+        "text": "Our invariance penalty is similar in practise to consistency regularizers which have recently gained popularity in semi-supervised learning (Sajjadi et al., 2016; Xie et al., 2019; Sohn et al., 2020). These typically supplement the supervised training signal with a term which enforces similarity between the function’s output on two views of data. Although recently proposed consistency methods are heuristic, they have deeper roots in the more rigorous co-training approach to semi-supervised learning from multiple views (Blum & Mitchell, 1998; Kakade & Foster, 2007; Sridharan & Kakade, 2008; McWilliams & Montana, 2012; McWilliams et al., 2013). The main conceptual difference between our approach and consistency-based methods is that we do not have access to a supervised training signal and that the RELIC penalty is motivated from the underlying causal graph. ",
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+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We first visualize the influence of the explicit invariance constraint in RELIC on the linear separability of the learned representations. We then evaluate RELIC on a number of prediction and reinforcement learning tasks for usefulness and robustness. For the prediction tasks, we test RELIC after pretraining the representation in a self-supervised way on the training set of the ImageNet ILSVRC-2012 dataset (Russakovsky et al., 2015). We evaluate RELIC in the linear evaluation setup on ImageNet and test its robustness and out-of-distribution generalization on datasets related to ImageNet. Unlike much prior work in contrastive learning which focuses specifically on computer vision tasks, we test RELIC also in the context of learning representations for reinforcement learning. Specifically, we test RELIC on the suite of Atari games (Bellemare et al., 2013) which consists of 57 diverse games of varying difficulty. ",
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+        "text": "Linear evaluation. In order to understand how representations learned by RELIC differ from other methods, we compare it against those learned by AMDIM and SimCLR in terms of Fischer’s linear discriminant ratio (Friedman et al., 2009): $\\begin{array} { r } { \\dot { F _ { \\mathrm { L D A } } } { } = \\| \\mu _ { k } - \\mu _ { k ^ { \\prime } } \\| ^ { 2 } / \\sum _ { i , j \\in \\mathcal { C } _ { k } } \\| f ( x _ { i } ) - f ( x _ { j } ) \\| ^ { 2 } } \\end{array}$ where $\\begin{array} { r } { \\mu _ { k } = \\frac { 1 } { | { \\mathcal C } _ { k } | } \\sum _ { i \\in { \\mathcal C } _ { k } } f ( x _ { i } ) } \\end{array}$ is the mean of the representations of class $k$ and $\\mathcal { C } _ { k }$ is the index set of that class. A larger $F _ { \\mathrm { L D A } }$ implies that classes are more easily separated with a linear classifier. This can be achieved by either increasing distances between classes (numerator) or shrinking within-class variance (denominator). ",
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+        "text": "Figure 2 shows the distribution of $F _ { \\mathrm { L D A } }$ for RELIC, SimCLR and AMDIM after training as measured on the (downsampled) ImageNet validation set. The distance between medians of RELIC and SimCLR is 162. AMDIM is tightly concentrated close to 20. The invariance penalty ensures that— even though labels are $a$ -priori unknown—for RELIC within-class variability of $f$ is concentrated leading to better linear separability between classes in the downstream task of interest.This is reflected in the rightward shift of the distribution of $F _ { \\mathrm { L D A } }$ in Figure 2 for RELIC compared with SimCLR and AMDIM which do not impose such a constraint. ",
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+        "text": "Next we evaluate RELIC’s representation by training a linear classifier of top of the fixed encoder following the procedure in (Kolesnikov et al., 2019; Chen et al., 2020a) and Appendix E.4. In Table 1, we report top-1 and top-5 accuracy on the ImageNet test set. Methods denoted with \\* use SimCLR augmentations (Chen et al., 2020a), while methods denoted $\\dagger$ use custom, stronger augmentations. Comparing methods which use SimCLR augmentations, RELIC outperforms competing approaches on both ResNet-50 and ResNet-50 with target network. For completeness, we report results for SwAV (Caron et al., 2020) and InfoMin (Tian et al., 2020), but note that these methods use stronger augmentations which alone have been shown to boost performance by over $5 \\%$ A fair comparison between different objectives can only be achieved under the same architecture and the same set of augmentations. ",
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+        "table_caption": [
+            "Table 1: Accuracy $( \\mathrm { i n } \\% )$ ) under linear evaluation on ImageNet for different self-supervised representation learning methods. Methods with \\* use SimCLR augmentations. Methods with $\\dagger$ use custom, stronger augmentations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Top-1</td><td>Top-5</td></tr><tr><td>ResNet-50architecture</td><td></td><td></td></tr><tr><td>PIRL</td><td>63.6</td><td></td></tr><tr><td>CPC v2</td><td>63.8</td><td>85.3</td></tr><tr><td>CMC</td><td>66.2</td><td>87.0</td></tr><tr><td>SimCLR(Chen et al.,2020a)</td><td>* 69.3</td><td>89.0</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td>* 70.1</td><td>=</td></tr><tr><td>RELIC (ours)</td><td>* 70.3</td><td>89.5</td></tr><tr><td>InfoMin Aug.(Tian et al.,2020)</td><td>+ 73.0</td><td>91.1</td></tr><tr><td>SwAV (Caron et al., 2020)</td><td>+ 75.3</td><td>=</td></tr><tr><td>ResNet-50 with target network</td><td></td><td></td></tr><tr><td>MoCo v2 (Chen et al., 2020b)</td><td>71.1</td><td></td></tr><tr><td>BYOL (Grill et al., 2020)</td><td>* 74.3</td><td>91.6</td></tr><tr><td>RELIC (ours)</td><td>* 74.8</td><td>92.2</td></tr></table>",
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+        "type": "text",
+        "text": "Robustness and generalization. We evaluate robustness and out-of-distribution generalization of RELIC’s representation on datasets Imagenet-C (Hendrycks & Dietterich, 2019) and ImageNet-R (Hendrycks et al., 2020), respectively. To evaluate RELIC’s representation, we train a linear classifier on top of the frozen representation following the procedure described in (Chen et al., 2020a) and appendix E.5.2. For Imagenet-C we report the mean Corruption Error (mCE) and Corruption Errors for Noise corruptions in Table 3. RELIC has significantly lower mCE than both the supervised ResNet-50 baseline and the unsupervised methods SimCLR and BYOL. Also, it has the lowest Corruption Error on 14 out of 15 corruptions when compared to SimCLR and BYOL. Thus, we see that RELIC learns the most robust representation. RELIC also outperforms SimCLR and BYOL on ImageNet-R showing its superior out-of-distribution generalization ability; see Table 2. For further details and results please consult E.5. ",
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+        "img_path": "images/8b3562db7db3c8315ab9ff25104eec59d016f21a882abfa42ebe472852862010.jpg",
+        "table_caption": [
+            "Table 2: Top-1 error rates for different self-supervised representation learning methods on ImageNetR. All models are trained only on clean ImageNet images and $\\mathrm { R E L I C } _ { T }$ refers to RELIC using a ResNet-50 with target network as in BYOL (Grill et al., 2020). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Supervised</td><td>SimCLR</td><td>RELIC</td><td>BYOL</td><td>RELICT</td></tr><tr><td>Top-1 Error (%)</td><td>63.9</td><td>81.7</td><td>77.4</td><td>77.0</td><td>76.2</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: Mean Corruption Error (mCE), mean relative Corruption Error (mrCE) and Corruption Errors for the “Noise” class of corruptions (Gaussian, Shot, Impulse) on ImageNet-C. The mCE value is the average across 75 different corruptions. Methods are trained only on clean ImageNet images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>mCE</td><td>mrCE</td><td>Gaussian</td><td>Shot</td><td>Impulse</td></tr><tr><td>Supervised</td><td>76.7</td><td>105.0</td><td>80.0</td><td>82.0</td><td>83.0</td></tr><tr><td>ResNet-50 architecture:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SimCLR</td><td>87.5</td><td>111.9</td><td>79.4</td><td>81.9</td><td>89.6</td></tr><tr><td>ReLIC</td><td>76.4</td><td>87.7</td><td>67.8</td><td>70.7</td><td>77.0</td></tr><tr><td>ResNet-50 with target network:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BYOL</td><td>72.3</td><td>90.0</td><td>65.9</td><td>68.4</td><td>73.7</td></tr><tr><td>ReLIC</td><td>70.8</td><td>88.4</td><td>63.6</td><td>65.7</td><td>69.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Reinforcement Learning. Much prior work in contrastive learning has focused specifically on computer vision tasks. In order to compare these approaches in a different domain, we investigate representation learning in the context of reinforcement learning. We compare RELIC as an auxiliary loss against other state of the art self-supervised losses on an agent trained on 57 Atari games. Using human normalized scores as a metric, we use the original architecture and hyperparameters of the R2D2 agent (Kapturowski et al., 2019) and supplement it with a second encoder trained with a given representation learning loss. When auxiliary losses are present, the Q-Network takes the output of the second encoder as an input. The Q-Network and the encoder are trained with separate optimizers. For the augmentation baseline, the Q-Network takes two identical encoders trained end-to-end. Table 4 shows a comparison between RELIC, SimCLR, BYOL, CURL (Srinivas et al., 2020), and feeding augmented observations directly to the agent (Kostrikov et al., 2020). We find that RELIC has a significant advantage over competing self-supervised methods, performing best in 25 out of 57 games. The next best performing method, CURL performs best in 11 games. Note that none of these methods outperform R2D2 (Kapturowski et al., 2019) which achieves superhuman performance in 52 games. Although previously published work shows the auxiliary tasks help on Atari, this is in the very low-data regime. Here we show that ReLIC is able to close the gap in the more common Atari set up. We hypothesize that the performance penalty resulting from adding auxiliary self-supervised losses stems from the naive combining of the outputs of the two encoders; we will explore alternative options for combining encoder outputs in future. Full details are presented in Section E.6. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work we have analyzed self-supervised learning using a causal framework. Using a causal graph, we have formalized the problem of self-supervised representation learning and derived properties of the optimal representation. We have shown that representations need to be invariant predictors of proxy targets under interventions on features that are only correlated, but not causally related to the downstream tasks. We have leveraged data augmentations to simulate these interventions and have proposed to explicitly enforce this invariance constraint. Based on this, we have proposed a new self-supervised objective, Representation Learning via Invariant Causal Mechanisms (RELIC), that enforces invariant prediction of proxy targets across augmentations using an invariance regularizer. Further, we have generalized contrastive methods using the concept of refinements and have shown that learning a representation on refinements using the principle of invariant prediction is a sufficient condition for these representations to generalize to downstream tasks. With this, we have provided an alternative explanation to mutual information for the success of contrastive methods. Empirically we have compared RELIC against recent self-supervised methods on a variety of prediction and reinforcement learning tasks. Specifically, RELIC significantly outperforms competing methods in terms of robustness and out-of-distribution generalization of the representations it learns on ImageNet. RELIC also significantly outperforms related self-supervised methods on the Atari suite achieving superhuman performance on 51 out of 57 games. We aim to investigate the construction of more coarse-grained refinements and the empirical evaluation of different kinds of refinements in future work. ",
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+    {
+        "type": "table",
+        "img_path": "images/14b00ad19e70f8d9397920afb6bfd9568509d4cbceea7299b455e4f0506bfab5.jpg",
+        "table_caption": [
+            "Table 4: Human Normalized Scores of Auxiliary Methods over 57 Atari Games. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>AtariPerformance</td><td>RELIC</td><td>SimCLR</td><td>CURL</td><td>BYOL</td><td>Augmentation</td></tr><tr><td>Capped mean</td><td>91.46</td><td>88.76</td><td>90.72</td><td>89.43</td><td>80.60</td></tr><tr><td>Number of superhuman games</td><td>51</td><td>49</td><td>49</td><td>49</td><td>34</td></tr><tr><td>Mean</td><td>3003.73</td><td>2086.16</td><td>2413.12</td><td>1769.43</td><td>503.15</td></tr><tr><td>Median</td><td>832.50</td><td>592.83</td><td>819.56</td><td>483.39</td><td>132.17</td></tr><tr><td>40% Percentile</td><td>356.27</td><td>266.07</td><td>409.46</td><td>224.80</td><td>94.35</td></tr><tr><td>30% Percentile</td><td>202.49</td><td>174.19</td><td>190.96</td><td>150.21</td><td>80.04</td></tr><tr><td>20% Percentile</td><td>133.93</td><td>120.84</td><td>126.10</td><td>118.36</td><td>57.95</td></tr><tr><td>10% Percentile</td><td>83.79</td><td>37.19</td><td>59.09</td><td>44.14</td><td>32.74</td></tr><tr><td>5% Percentile</td><td>20.87</td><td>12.74</td><td>20.56</td><td>7.75</td><td>2.85</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Acknowledgements. We thank David Balduzzi, Melanie Rey, Christina Heinze-Deml, Ilja Kuzborskij, Ali Eslami, Jeffrey de Fauw and Josip Djolonga for invaluable discussions. ",
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parse/train/Ja-hVQrfeGZ/Ja-hVQrfeGZ.md

@@ -0,0 +1,258 @@
+# Rebooting ACGAN: Auxiliary Classifier GANs with Stable Training
+
+Minguk Kang Woohyeon Shim Minsu Cho Jaesik Park
+
+Pohang University of Science and Technology (POSTECH), South Korea {mgkang, wh.shim, mscho, jaesik.park}@postech.ac.kr
+
+# Abstract
+
+Conditional Generative Adversarial Networks (cGAN) generate realistic images by incorporating class information into GAN. While one of the most popular cGANs is an auxiliary classifier GAN with softmax cross-entropy loss (ACGAN), it is widely known that training ACGAN is challenging as the number of classes in the dataset increases. ACGAN also tends to generate easily classifiable samples with a lack of diversity. In this paper, we introduce two cures for ACGAN. First, we identify that gradient exploding in the classifier can cause an undesirable collapse in early training, and projecting input vectors onto a unit hypersphere can resolve the problem. Second, we propose the Data-to-Data Cross-Entropy loss (D2D-CE) to exploit relational information in the class-labeled dataset. On this foundation, we propose the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results show that ReACGAN achieves state-of-the-art generation results on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet datasets. We also verify that ReACGAN benefits from differentiable augmentations and that D2D-CE harmonizes with StyleGAN2 architecture. Model weights and a software package that provides implementations of representative cGANs and all experiments in our paper are available at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN.
+
+# 1 Introduction
+
+Generative Adversarial Networks (GAN) [1] are known for the forefront approach to generating high-fidelity images of diverse categories [2, 3, 4, 5, 6, 7, 8, 9]. Behind the sensational generation ability of GANs, there has been tremendous effort to develop adversarial objectives free from the vanishing gradient problem [10, 11, 12], regularizations for stabilizing adversarial training [11, 13, 14, 3, 15, 16, 17], and conditioning techniques to support the adversarial training using category information of the dataset [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Subsequently, the conditioning techniques have become the de facto standard for high-quality image generation. The models with the conditioning methods are called conditional Generative Adversarial Networks (cGAN), and cGANs can be divided into two groups depending on the discriminator’s conditioning way: classifier-based GANs [18, 20, 23, 25, 27] and projection-based GANs [19, 3, 4, 28].
+
+The classifier-based GANs facilitate an auxiliary classifier to generate class-specific images by penalizing the generator if the synthesized images are not consistent with the conditioned labels. ACGAN [18] has been one of the widely used classifier-based GANs for its simple design and satisfactory generation performance. While ACGAN can exploit class information by pushing and pulling classifier’s weights (proxies) against image embeddings [23], it is well known that ACGAN training is prone to collapsing at the early stage of training as the number of classes increases [19, 23, 25, 27]. In addition, the generator of ACGAN tends to generate easily classifiable images at the cost of reduced diversity [18, 19, 27]. Projection-based GANs, on the other hand, have shown cutting-edge generation results on datasets with a large number of categories. SNGAN [3],
+
+![](images/a77f02b277b4a9b1554bdeb714cab48b71615b6d03f8135fb468d91f2cc6deb5.jpg)  
+Figure 1: Schematics that depict how cGANs perform conditioning. The red color means positive samples/proxy, and the blue color indicates negative samples/proxies. Arrows represent push-pull forces based on the reference sample. The length of an arrow indicates the magnitude of the force.
+
+SAGAN [29], and BigGAN [4] are representatives in this family and can generate realistic images on CIFAR10 [30] and ImageNet [31] datasets. However, projection-based GANs only consider a pairwise relationship between an image and its label proxy (data-to-class relationships). As the result, the projection-based GANs can miss an additional opportunity to consider relation information between data instances (data-to-data relationships) as discovered by [23].
+
+In this paper, we analyze why ACGAN training becomes unstable as the number of classes increases and propose remedies for (1) the instability and (2) the relatively poor generation performance of ACGAN compared with the projection-based models. First, we begin by analytically deriving the gradient of the softmax cross-entropy loss used in ACGAN. By examining the exact values of analytic gradients, we discover that the unboundedness of input feature vectors and poor classification performance in the early training stage can cause an undesirable gradient exploding problem. Second, with alleviating the instability, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) using the Data-to-Data Cross-Entropy loss (D2D-CE). ReACGAN projects image embeddings and proxies onto a unit hypersphere and computes similarities for data-to-data and data-to-class consideration. Additionally, we introduce two margin values for intra-class variations and inter-class separability. In this way, ReACGAN overcomes the training instability and can exploit additional supervisory signals by explicitly considering data-to-class and data-to-data relationships, and also by implicitly looking at class-to-class relationships in the same mini-batch.
+
+To validate our model, we conduct image generation experiments on CIFAR10 [30], TinyImageNet [32], CUB200 [33], and ImageNet [31] datasets. Through extensive experiments, we demonstrate that ReACGAN beats both the classifier-based and projection-based GANs, improving over the state of the art by $2 . 5 \%$ , $1 5 . 8 \%$ , $5 . 1 \%$ , and $1 4 . 5 \%$ in terms of Fréchet Inception Distance (FID) [34] on the four datasets, respectively. We also verify that ReACGAN benefits from consistency regularization [16] and differentiable augmentations [9, 8] for limited data training. Finally, we confirm that D2D-CE harmonizes with the StyleGAN2 architecture [7].
+
+# 2 Background: Generative Adversarial Networks
+
+Generative Adversarial Network (GAN) [1] is an implicit generative model that aims to generate a sample indistinguishable from the real. GAN consists of two networks: a Generator $G : { \mathcal { Z } } \longrightarrow { \mathcal { X } }$ that tries to map a latent variable $\mathbf { z } \sim p ( \mathbf { z } )$ into the real data space $\mathcal { X }$ and a Discriminator $D : \mathcal { X } \longrightarrow$ [0, 1] that strives to discriminate whether a given sample $\mathbf { x }$ is from the real data distribution $p _ { \mathrm { r e a l } } ( \mathbf { x } )$ or from the implicit distribution $p _ { \mathrm { g e n } } ( G ( \mathbf { z } ) )$ derived from the generator $G ( \mathbf { z } )$ . The objective of a vanilla GAN [1] can be expressed as follows:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } [ \log ( D ( \mathbf { x } ) ) ] + \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } [ \log ( 1 - D ( G ( \mathbf { z } ) ) ) ] .
+$$
+
+While GANs have shown impressive results in the image generation task [2, 35, 11, 13], training GANs often ends up encountering a mode-collapse problem [36, 11, 12]. As one of the prescriptions for stabilizing and reinforcing GANs, training GANs with categorical information, named conditional Generative Adversarial Networks (cGAN), is suggested [37, 18, 19]. Depending on the presence of explicit classification losses, cGAN can be divided into two groups: Classifier-based GANs [18, 20, 23, 25, 27] and Projection-based GANs [19, 3, 4, 28]. One of the widely used classifier-based GANs is ACGAN [18], and ACGAN utilizes softmax cross-entropy loss to perform classification task with adversarial training. Although ACGAN has shown satisfactory generation results, training ACGAN becomes unstable as the number of classes in the training dataset increases [19, 23, 25, 27]. Besides, ACGAN tends to generate easily classifiable images at the cost of limited diversity [18, 19, 27]. To alleviate those problems, Zhou et al. [38] have proposed performing adversarial training on the classifier. Gong et al. [20] have introduced an additional classifier to eliminate a conditional entropy minimization process in the adversarial training. However, ACGAN training still suffers from the early-training collapse issue and the reduced diversity problem when trained on datasets with a large number of class categories, such as Tiny-ImageNet [32] and ImageNet [31].
+
+In these circumstances, Miyato et al. [19] have proposed a projection discriminator for cGANs and have shown significant improvement in generating the ImageNet dataset. Motivated by the promising result of the projection discriminator, many projection-based GANs [3, 29, 4, 16, 6, 17] have been proposed and become the standard for conditional image generation. In this paper, we revisit ACGAN and unveil why ACGAN training is so unstable. Coping with the instability, we propose the Rebooted Auxiliary Classifier GANs (ReACGAN) for high-quality and diverse image generation.
+
+# 3 Rebooting Auxiliary Classifier GANs
+
+# 3.1 Feature Normalization
+
+To uncover a nuisance that can cause the instability of ACGAN, we start by analytically deriving the gradients of weight vectors in the softmax classifier. Let the part of the discriminator before the fully connected layer be a Feature extractor $F : \mathcal { X } \longrightarrow \mathcal { F } \in \bar { \mathbb { R } ^ { d } }$ and let Classifier $C : { \mathcal { F } } \longrightarrow \mathbb { R } ^ { c }$ be a single fully connected layer parameterized by $\pmb { W } = [ \pmb { w } _ { 1 } , . . . , \pmb { w } _ { c } ] \in \mathbb { R } ^ { d \times c }$ , where $c$ denotes the number of classess. We sample training images $\pmb { X } = \{ \pmb { x } _ { 1 } , . . . , \pmb { x } _ { N } \}$ and integer labels ${ \textbf { 3 } } =$ $\{ y _ { 1 } , . . . , y _ { N } \}$ from the joint distribution $p ( \mathbf { x } , \mathbf { y } )$ . Using the notations above, we can express the empirical cross-entropy loss used in ACGAN [18] as follows:
+
+$$
+\mathcal { L } _ { \mathrm { C E } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Big ( \frac { \exp \left( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { y _ { i } } \right) } { \sum _ { j = 1 } ^ { c } \exp \left( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { j } \right) } \Big ) .
+$$
+
+Based on Eq. (3), we can derive the derivative of the cross-entropy loss, w.r.t ${ \pmb w } _ { k \in \{ 1 , \dots , c \} }$ as follows:
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { C E } } } { \partial \boldsymbol { w } _ { k } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \bigg \{ F ( \boldsymbol { x } _ { i } ) \bigg ( \mathbf { 1 } _ { y _ { i } = k } - p _ { i , k } \bigg ) \bigg \} ,
+$$
+
+where $\mathbf { 1 } _ { \mathit { y } _ { i } = k }$ is an indicator function that will output 1 if $y _ { i } = k$ is satisfied, and $p _ { i , k }$ is a class probability that represents the probability that $i$ -th sample belongs to class $k$ , mathematically $\frac { \exp { ( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { k } ) } } { \sum _ { j = 1 } ^ { c } \exp { ( F ( \pmb { x } _ { i } ) ^ { \top } \pmb { w } _ { j } ) } }$ . The equation above implies that the norm of the gradient of the softmax cross-entropy loss is coupled with the norms and directions of each input feature map $F ( { \pmb x } _ { i } )$ and the class probabilities. In the early training stage, the classifier is prone to making incorrect predictions, resulting in low probabilities. This phenomenon occurs more frequently as the number of categories in the dataset increases. As the result, the norm of the gradient $| \frac { \partial \hat { \mathcal { L } } _ { \mathrm { C E } } } { \partial w _ { k } } |$ begins to explode as the vector $F ( { \pmb x } _ { i } )$ stretches out to ${ \pmb w } _ { k }$ direction but being located close to the the other vectors ${ \pmb w } _ { j \in \{ 1 , . . . , c \} \backslash \{ k \} }$ . This often breaks the balance between adversarial learning and classifier training, leading to an early-training collapse. Once the early-training collapse occurs, ACGAN training concentrates on classifying categories of images instead of discriminating the authenticity of given samples. We experimentally demonstrate that the average norm of ACGAN’s input feature maps increases as the training progresses (Fig. 2a). Accordingly, the average norm of the gradients increases sharply at the early training stage and decreases with the high class probabilities of the classifier (Fig. 2b, Fig. A3 in Appendix). While the average norm of gradients decreases at some point, the FID value of ACGAN does not decrease, implying the collapse of ACGAN training (Fig. 2c).
+
+![](images/41f752bd054c08034409d7f5b84c86a0197793ecc9cb9dcf2b22365737c55349.jpg)  
+Figure 2: Merits of integrating feature normalization and data-to-data relationship consideration. All experiments are conducted on Tiny-ImageNet [32] dataset. (a) Average norms of input feature maps $\bar { F } ( \cdot )$ , (b) average norms of gradients of classification losses, and (c) trends of FID scores. Compared with ACGAN [18], the proposed ReACGAN does not experience the training collapse problem caused by excessively large norms of feature maps and gradients at the early training stage. In addition, ReACGAN can converge to better equilibrium by considering data-to-data relationships with easy positive and negative sample suppression.
+
+As one of the remedies for the gradient exploding problem, we find that simply normalizing the feature embeddings onto a unit hypersphere $\frac { F ( { \pmb x } _ { i } ) } { | | F ( { \pmb x } _ { i } ) | | }$ effectively resolves ACGAN’s early-training collapse (see Fig. 2). The motivation is that normalizing the features onto the hypersphere makes the norms of feature maps equal to 1.0. Thus, the discriminator does not experience the gradient exploding problem. From the next section, we will deploy a linear projection layer $P$ on the feature extractor $F$ . And, we will normalize both the embeddings from the projection layer and the weight vectors $( \pmb { w } _ { 1 } , . . . , \pmb { w } _ { c } )$ in the classifier since normalizing both the embeddings and the weight vectors does not degrade image generation performance. We denote the normalized embedding P (F (xi))||P (F (xi))|| as $f _ { i }$ and the normalized weight vector $\frac { { \pmb w } _ { y _ { i } } } { | | { \pmb w } _ { y _ { i } } | | }$ as vyi .
+
+# 3.2 Data-to-Data Cross-Entropy Loss (D2D-CE)
+
+We expand the feature normalized softmax cross-entropy loss described in Sec. 3.1 to the Data-to-Data Cross-Entropy (D2D-CE) loss. The motivations are summarized into two points: (1) replacing datato-class similarities in the denominator of Eq. (3) with data-to-data similarities and (2) introducing two margin values to the modified softmax cross-entropy loss. We expect that point (1) will encourage the feature extractor to consider data-to-class as well as data-to-data relationships, and that point (2) will guarantee inter-class separability and intra-class variations in the feature space while preventing ineffective gradient updates induced by easy negative and positive samples. To develop the feature normalized cross-entropy loss into D2D-CE, we replace the similarities between a sample embedding and all proxies except for the positive one in the denominator, $\begin{array} { r } { \sum _ { j \in \{ 1 , \dots , c \} \backslash \{ y _ { i } \} } \exp \left( \mathbf { { f } } _ { i } ^ { \top } \mathbf { { v } } _ { j } \right) } \end{array}$ with similarities between negative samples in the same mini-batch. The modified cross-entropy loss can be expressed as follows:
+
+$$
+\mathcal { L } _ { \mathrm { C E } } ^ { ' } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Bigg ( \frac { \exp { ( f _ { i } ^ { \top } v _ { y _ { i } } / \tau ) } } { \exp { ( f _ { i } ^ { \top } v _ { y _ { i } } / \tau ) } + \sum _ { j \in N ( i ) } \exp { ( f _ { i } ^ { \top } f _ { j } / \tau ) } } \Bigg ) ,
+$$
+
+where $\tau$ is a temperature, and $\mathcal { N } ( i )$ is the set of indices that point locations of the negative samples whose labels are different from the reference label $\pmb { v } _ { y _ { i } }$ in the mini-batch. The self-similarity matrix of samples in the mini-batch is used to calculate the similarities between negative samples $\pmb { f } _ { i } ^ { \top } \pmb { f } _ { j \in \mathcal { N } ( i ) }$ with a false negative mask (see Fig. 3). Thus, Eq. (5) enables the discriminator to contrastively compare visual differences between multiple images and can supply more informative supervision for image conditioning. Finally, we introduce two margin hyperparameters to $\mathcal { L } _ { \mathrm { C E } } ^ { ' }$ and name it Data-to-Data Cross-Entropy loss (D2D-CE). The proposed D2D-CE can be expressed as follows:
+
+$$
+\mathcal { L } _ { \mathrm { D 2 D - C E } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \Bigg ( \frac { \exp \left( [ f _ { i } ^ { \top } v _ { y _ { i } } - m _ { p } ] _ { - } / \tau \right) } { \exp \left( [ f _ { i } ^ { \top } v _ { y _ { i } } - m _ { p } ] _ { - } / \tau \right) + \sum _ { j \in N ( i ) } \exp \left( [ f _ { i } ^ { \top } f _ { j } - m _ { n } ] _ { + } / \tau \right) } \Bigg ) ,
+$$
+
+![](images/ee7bad162e1754a63a5d7f7b5a7a8d9d8e3e367c1fcd2cc101b8a2534e6e145d.jpg)  
+Figure 3: Overview of ReACGAN. ReACGAN performs adversarial training using the loss ${ \mathcal { L } } _ { \mathrm { A d v } }$ . At the same time, ReACGAN tries to minimize data-to-data cross-entropy loss (D2D-CE) on the linear head $( P )$ to exploit relational information in the labeled dataset. $^ { \circ }$ means element-wise product. Note that False negatives mask operates on the similarity matrix between two batches of sample embeddings to compute the similarities between negative samples in the denominator of Eq. (6).
+
+where $m _ { p }$ is a margin for suppressing a high similarity value between a reference sample and its corresponding proxy (easy positive), $m _ { n }$ is a margin for suppressing low similarity values between negatives samples (easy negatives). $[ \cdot ] _ { - }$ and $[ \cdot ] _ { + }$ denote $\operatorname* { m i n } ( \cdot , 0 )$ and $\operatorname* { m a x } ( \cdot , 0 )$ functions, respectively.
+
+# 3.3 Useful Properties of Data-to-Data Cross-Entropy Loss
+
+In this subsection, we explain four useful properties of D2D-CE. Let $s _ { q }$ be a similarity between the normalized embedding $f _ { q }$ and the corresponding normalized proxy ${ \pmb v } _ { y _ { q } }$ , $s _ { q , r }$ be a similarity between $f _ { q }$ and $f _ { r }$ , and $a$ and $\mathit { b }$ be arbitrary indices of negative samples, i.e., $a , b \in \mathcal { N } ( q )$ . Then the properties of D2D-CE can be summarized as follows:
+
+![](images/284973472b7cc1ba4ea65614768295afd00d54dfd0ba532f646e48451266f16e.jpg)  
+Figure 4: Graph showing regions where hard negative mining, easy positive and negative suppression, and no gradient update occur.
+
+Property 1. Hard negative mining. If the value of $s _ { q , a }$ is greater than $s _ { q , b }$ , the derivative of LD2D-CE w.r.t $s _ { q , a }$ is greater than or equal to the derivative $w . r . t \ s _ { q , b }$ ; that is $\begin{array} { r } { \frac { \partial \mathcal { L } _ { D 2 D - C E } } { \partial s _ { q , a } } \geq \frac { \partial \mathcal { L } _ { D 2 D - C E } } { \partial s _ { q , b } } \geq 0 . } \end{array}$
+
+Property 2. Easy positive suppression. If $s _ { q } \mathrm { ~ - ~ }$ $m _ { p } \geq 0$ , the derivative of $\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q }$ is 0.
+
+Property 3. Easy negative suppression. If $s _ { q , r } - m _ { n } \leq 0 ,$ , the derivative of $\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q , r }$ is 0. Property 4. If $s _ { q } - m _ { p } \ge 0$ and $s _ { q , r } - m _ { n } \leq 0$ are satisfied, $\mathcal { L } _ { D 2 D - C E }$ has the global minima of $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \left( 1 + \bar { | \mathcal { N } ( i ) | } \right) } \end{array}$ .
+
+We put proofs of the above properties in Appendix D. Property 1 indicates that D2D-CE implicitly conducts hard negative mining and benefits from comparing samples with each other. Also, Properties 2 and 3 imply that samples will not affect gradient updates if the samples are trained sufficiently. Consequently, the classifier concentrates on pushing and pulling hard negative and hard positive examples without being dominated by easy negative and positive samples.
+
+# 3.4 Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN)
+
+With the proposed D2D-CE objective, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). As ACGAN does, ReACGAN jointly optimizes an adversarial loss and the classification objective (D2D-CE). Specifically, the discriminator, which consists of the feature extractor, adversarial head, and linear head, of ReACGAN tries to discriminate whether a given image is sampled from the real distribution or not. At the same time, the discriminator tries to maximize similarities between the reference samples and corresponding proxies while minimizing similarities between negative samples using real images and D2D-CE loss. After updating the discriminator a predetermined number of times, the generator strives to deceive the discriminator by generating well-conditioned images that will output a low D2D-CE value. By alternatively training the discriminator and generator until convergence, ReACGAN generates high-quality images of diverse categories without early-training collapse. We attach the algorithm table in Appendix A.
+
+Differences between ReACGAN and ContraGAN. The authors of ContraGAN [23] propose a conditional contrastive loss (2C loss) to cover data-to-data relationships when training cGANs. The main differences between 2C loss and D2D-CE objective are summed up into three points: (1) while 2C loss is derived from NT-Xent loss [39], which is popularly used in the field of the self-supervised learning, our D2D-CE is developed to resolve the early-training collapsing problem and the poor generation results of ACGAN, (2) 2C loss holds false-negative samples in the denominator, and they can cause unexpected positive repulsion forces, and (3) 2C loss contains the similarities between all positive samples in the numerator, and they give rise to large gradients on pulling easy positive samples. Consequently, GANs with 2C loss tend to synthesize images of unintended classes as reported in the author’s software document [40]. However, D2D-CE does not contain the false negatives in the denominator and considers only the similarity between a sample and its proxy in the numerator. Therefore, ReACGAN is free from the undesirable repelling forces and does not conduct unnecessary easy positive mining. More detailed explanations are attached in Appendix F.
+
+Consistency Regularization and D2D-CE Loss. Zhang et al. [16] propose a consistency regularization to force the discriminator to make consistent predictions if given two images are visually close to each other. They create a visually similar image pair by augmenting a reference image with pre-defined augmentations. After that, they let the discriminator minimize L2 distance between the logit of the reference image and the logit of the augmented counterpart. While ReACGAN locates an image embedding nearby its corresponding proxy but far apart multiple image embeddings of different classes, consistency regularization only pulls a reference and the augmented image towards each other. Since consistency regularization and D2D-CE can be applied together, we will show that ReACGAN benefits from consistency regularization in the experiments section (Table 1).
+
+# 4 Experiments
+
+# 4.1 Datasets and Evaluation Metrics
+
+To verify the effectiveness of ReACGAN, we conduct conditional image generation experiments using five datasets: CIFAR10 [30], Tiny-ImageNet [32], CUB200 [33], ImageNet [31], and AFHQ [41] datasets and four evaluation metrics: Inception Score (IS) [42], Fréchet Inception Distance (FID) [34], and $\mathrm { F _ { 0 . 1 2 5 } }$ (Precision) and $\mathrm { F } _ { 8 }$ (Recall) [43]. The details on the training datasets are in Appendix C.1.
+
+Inception Score (IS) [42] and Fréchet Inception Distance (FID) [34] are widely used metrics for evaluating generative models. We utilize IS and FID together because some studies [4, 6, 25] have shown that IS has a tendency to measure the fidelity of images better while FID tends to weight capturing the diversity of images.
+
+Precision $\bf ( F _ { 0 . 1 2 5 } )$ and Recall $\bf ( F _ { 8 } )$ [43] are metrics for estimating precision and recall of the approximated distribution $p ( G ( \mathbf { z } ) )$ against the true data distribution $p ( \mathbf { x } )$ . Instead of evaluating generative models using one-dimensional scores, such as IS and FID, Sajjadi et al. [43] have suggested using a two-dimensional score $\mathrm { F _ { 0 . 1 2 5 } }$ and $\mathrm { F } _ { 8 }$ that can quantify how precisely the generated images are and how well the generated images cover the reference distribution.
+
+# 4.2 Experimental Details
+
+The implementation of ReACGAN basically follows details of PyTorch-StudioGAN library [40]1 that supports various experimental setups from ACGAN [18] to StyleGAN2 [7] $^ +$ ADA [8] with different scales of datasets [30, 32, 33, 31, 41] and architectures [2, 13, 4, 7]. For a fair comparison, we use the same backbones over all baselines, except otherwise noted, for both the discriminator and generator. For stable training, we apply spectral normalization (SN) [3] to the generator and discriminator except for experiments using SNGAN (in this case, we apply SN to the discriminator only) and StyleGAN2 [7]. We also use the same conditioning method for generators with conditional batch normalization (cBN) [44, 45, 19]. This allows us to investigate solely the conditioning methods of the discriminator, which are the main interest of our paper. If not specified, we use hinge loss [46] as a default for the adversarial loss.
+
+Table 1: Comparison with classifier-based GANs [18, 23] and projection-based GANs [3, 4, 29] on CIFAR10 [30], Tiny-ImageNet [32], and CUB200 [33] datasets using IS [42], FID [34], $\mathrm { F } _ { 0 . 1 2 5 }$ , and $\mathrm { F } _ { 8 }$ [43] metrics. For baselines, both the numbers from the cited paper (denoted as \* in method) and from our experiments using StudioGAN library [40] are reported. The numbers in bold-faced denote the best performance and in underline indicate the values are in one standard deviation from the best.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">CIFAR10 [30]</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td></tr><tr><td>SNGAN*[3]</td><td>8.22</td><td>21.7</td><td>=</td><td>-</td><td>1</td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>=</td><td>-</td></tr><tr><td>BigGAN*[4]</td><td>9.22</td><td>14.73</td><td>=</td><td>-</td><td>=</td><td>-</td><td></td><td>=</td><td>-</td><td></td><td>-</td><td>1</td></tr><tr><td>ContraGAN* [23]</td><td>1</td><td>10.60</td><td></td><td></td><td></td><td>29.49</td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>ACGAN[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>SNGAN [3]</td><td>8.67</td><td>13.33</td><td>0.985</td><td>0.976</td><td>8.71</td><td>51.15</td><td>0.900</td><td>0.702</td><td>5.41</td><td>47.75</td><td>0.754</td><td>0.912</td></tr><tr><td>SAGAN [29]</td><td>8.66</td><td>14.31</td><td>0.983</td><td>0.973</td><td>8.74</td><td>49.90</td><td>0.872</td><td>0.712</td><td>5.48</td><td>54.29</td><td>0.728</td><td>0.882</td></tr><tr><td>BigGAN[4]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>ContraGAN [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>ReACGAN</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td></td><td>14.06 27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr><tr><td>BigGAN +CR*[16]</td><td>-</td><td>11.48</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BigGAN [4]+ CR[16]</td><td>9.97</td><td>7.18</td><td>0.995</td><td>0.993</td><td></td><td>15.94 19.96</td><td>0.972</td><td>0.950 5.14</td><td></td><td>11.97</td><td>0.978</td><td>0.981</td></tr><tr><td>ContraGAN [23] + CR[16]</td><td>9.59</td><td>8.55</td><td>0.992</td><td>0.972</td><td>15.81</td><td>19.21</td><td>0.983</td><td>0.941</td><td>4.90</td><td>11.08</td><td>0.984</td><td>0.967</td></tr><tr><td>ReACGAN + CR [16]</td><td>10.11</td><td>7.20</td><td>0.996</td><td>0.994 16.56</td><td></td><td>19.69</td><td>0.984</td><td>0.940</td><td>4.87</td><td>10.72</td><td>0.985</td><td>0.971</td></tr><tr><td>BigGAN + DiffAug*[9]</td><td>9.17</td><td>8.49</td><td></td><td></td><td>=</td><td></td><td>=</td><td>=</td><td>=</td><td></td><td></td><td></td></tr><tr><td>BigGAN [4] + DiffAug [9]</td><td>9.94</td><td>7.17</td><td>0.995</td><td>0.992</td><td></td><td>18.0815.70</td><td>0.980</td><td>0.972</td><td>5.53</td><td>12.15</td><td>0.967</td><td>0.981</td></tr><tr><td>ContraGAN [23] + DiffAug [9]</td><td>9.95</td><td>7.27</td><td>0.995</td><td>0.992</td><td>18.20</td><td>15.40</td><td>0.986</td><td>0.963</td><td>5.39</td><td>11.02</td><td>0.978</td><td>0.970</td></tr><tr><td>ReACGAN + DiffAug [9]</td><td>10.22</td><td>6.79</td><td>0.996</td><td>0.993</td><td>20.60</td><td>14.25</td><td>0.988</td><td>0.972</td><td>5.22</td><td>9.27</td><td>0.985</td><td>0.983</td></tr><tr><td>Real Data</td><td colspan="2">11.54</td><td colspan="2">34.11</td><td colspan="2"></td><td colspan="2"></td><td colspan="2">5.49</td><td colspan="2"></td></tr></table>
+
+Before conducting main experiments, we perform hyperparameter search with candidates of a temperature $\tau \in \{ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \}$ and a positive margin $m _ { p } \in \ \{ 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 8 , \bar { 1 . 0 } \}$ . We set a negative margin $m _ { n }$ as $1 \ : - \ : m _ { p }$ to reduce search time. Through extensive experiments with 3 runs per each setting, we select $\tau$ with $\{ 0 . 5 , 0 . 7 5 , 0 . 2 5 , 0 . 5 , 0 . 2 5 \}$ and $m _ { p }$ with $\{ 0 . 9 8 , 1 . 0 , 0 . 9 5 , 0 . 9 \bar { 8 } , 0 . 9 0 \}$ for CIFAR10, Tiny-ImageNet, CUB200, ImageNet 256 B.S., and ImageNet 2048 B.S. experiments, respectively. A low temperature seems to work well on fine-grained image generation tasks, but generally, ReACGAN is robust to the choice of hyperparameters. The results of the hyperparameter search and other hyperparameter setups are provided in Appendix C.2.
+
+We evaluate all methods through the same protocol of [9, 16, 17], which uses the same amounts of generated images from the reference split specialized for each dataset.2 Besides, we run all the experiments three times with random seeds and report the averaged best performances for reliable evaluation with the lone exception of ImageNet and StyleGAN2 related experiments. Please refer to Appendix C.2 for other experimental details. The numbers in bold-faced denote the best performance and in underline indicate that the values are in one standard deviation from the best.
+
+# 4.3 Evaluation Results
+
+Comparison with Other cGANs. We compare ReACGAN with previous state-of-the-art cGANs in Tables 1 and 2. We employ the implementations of GANs in PyTorch-StudioGAN library as it provides improved results on standard benchmark datasets [30, 31]. For a fair comparison, we provide results from each original paper (denoted as \* in method) as well as those from StudioGAN library [40]. We also conduct experiments with popular augmentation-based methods: consistency regularization (CR) [16] and differentiable augmentation (DiffAug) [9].
+
+Table 2: Experiments using ImageNet [31] dataset. B.S. means batch size for training. (Left): Comparisons with previous cGAN approaches. (Right): Learning curves of BigGAN [4], ContraGAN [23], and ReACGAN (ours) which are trained using the batch size of 256.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">ImageNet [31]</td></tr><tr><td>IS↑</td><td>FID↓</td><td></td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>ACGAN*</td><td>[20]</td><td>7.26</td><td>184.41</td><td></td><td></td></tr><tr><td>SNGAN*[3]</td><td></td><td>36.80</td><td>27.62</td><td></td><td></td></tr><tr><td>SAGAN*[29]</td><td></td><td>52.52</td><td>18.28</td><td></td><td></td></tr><tr><td>BigGAN*[20]</td><td></td><td>38.05</td><td>22.77</td><td></td><td></td></tr><tr><td></td><td>TAC-GAN*[20]</td><td>28.86</td><td>23.75</td><td></td><td></td></tr><tr><td>ContraGAN*[23]</td><td></td><td>31.10</td><td>19.69</td><td>0.951</td><td>0.927</td></tr><tr><td></td><td>ACGAN[18]</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>35=58</td><td>SNGAN [3]</td><td>32.25</td><td>26.79</td><td>0.938</td><td>0.913</td></tr><tr><td></td><td>SAGAN [29]</td><td>29.85</td><td>34.73</td><td>0.849</td><td>0.914</td></tr><tr><td></td><td>BigGAN [4]</td><td>43.97</td><td>16.36</td><td>0.964</td><td>0.955</td></tr><tr><td></td><td>ContraGAN [23]</td><td>25.25</td><td>25.16</td><td>0.947</td><td>0.855</td></tr><tr><td>ReACGAN</td><td></td><td>68.27</td><td>13.98</td><td>0.976</td><td>0.977</td></tr><tr><td></td><td>BigGAN [4] + DiffAug [9]</td><td>36.97</td><td>18.57</td><td>0.956</td><td>0.941</td></tr><tr><td></td><td>ReACGAN +DiffAug[9]</td><td>69.74</td><td>11.95</td><td>0.977</td><td>0.975</td></tr><tr><td></td><td>BigGAN*[4]</td><td>99.31</td><td>8.51</td><td>-</td><td>-</td></tr><tr><td>P55=44</td><td>BigGAN*[25]</td><td>104.57</td><td>9.18</td><td></td><td>■</td></tr><tr><td></td><td>BigGAN[4]</td><td>99.71</td><td>7.89</td><td>0.985</td><td>0.989</td></tr><tr><td></td><td>ReACGAN</td><td>92.74</td><td>8.23</td><td>0.991</td><td>0.990</td></tr><tr><td></td><td>Real Data</td><td>173.33</td><td></td><td></td><td></td></tr></table>
+
+![](images/53ae9768944e305563820e0c40eca80c05a3af3fed3352bd392c4954a6225bf3.jpg)
+
+Compared with other cGANs, ReACGAN performs the best on most benchmarks, surpassing the previous methods by $2 . 5 \%$ , $5 . 1 \%$ , $1 5 . 8 \%$ , and $1 4 . 5 \%$ in FID on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet (256 B.S.), respectively. ReACGAN also harmonizes with augmentation-based regularizations, bringing incremental improvements on all the metrics. For the ImageNet experiments using a batch size of 256, ReACGAN reaches higher IS and lower FID with fewer iterations than other models in comparison. Finally, we demonstrate that ReACGAN can learn with a larger batch size on ImageNet. While some recent methods [20, 38, 27] have been built on ACGAN to improve the generation performance of ACGAN, large-scale image generation experiments with the batch size of 2048 have never been reported. Table 2 shows that our ReACGAN reaches FID score of 8.23 on ImageNet, being comparable with the value of 7.89 from BigGAN implementation in PyTorch-StudioGAN library. ReACGAN, however, provides better synthesis results than other implementations of BigGAN [4, 25]. Note that our result on ImageNet is obtained in only two runs while the training setup and architecture of BigGAN have been extensively searched and finely tuned.
+
+Comparison with Other Conditioning Losses. We investigate how the generation qualities vary with different conditioning losses while keeping the other configurations fixed. We compare D2D-CE loss with cross-entropy loss of ACGAN (AC) [18], loss used in the projection discriminator (PD) [19], multi-hinge loss (MH) [26], and conditional contrastive loss (2C) [23]. The results are shown in Table 3. AC- and MH-based models present decent results on CIFAR10, but undergo early-training collapse on Tiny-ImageNet and CUB200 datasets. Replacing them with PD, 2C, and D2D-CE losses produce satisfactory performances across all datasets, where PD loss makes the best $\mathrm { F } _ { 8 }$ (recall) on CUB200 dataset while giving third $\mathrm { F _ { 0 . 1 2 5 } }$ (precision) value. The noticeable point is that the proposed D2D-CE shows consistent results across all datasets, showing the lowest FID and the highest $\mathrm { F _ { 8 } }$ and $\mathrm { F _ { 0 . 1 2 5 } }$ values in most cases. This means ReACGAN can generate high-fidelity images and is relatively free from the precision and recall trade-off [43] than the others.
+
+Consistent Performance of ReACGAN on Adversarial Loss Selection. We validate the consistent performance of ReACGAN on four adversarial losses: non-saturation loss [1], least square loss [10], Wasserstein loss with gradient penalty regularization (W-GP) [13], and hinge loss [46] on CIFAR10 and Tiny-ImageNet datasets in Table 4. The experimental results show that $\mathrm { B i g G A N } + \mathrm { D } 2 \mathrm { D } { \cdot } \mathrm { C E }$ (ReACGAN) consistently outperforms the projection discriminator (PD) and conditional contrastive loss (2C) counterparts over three adversarial losses [1, 13, 46]. However, for the experiments using the least square loss [10], ReACGAN exhibits inferior generation performances to the projection discriminator. We speculate that minimizing the least square distance between an adversarial logit and the target scalar (1 or 0) might affect the norms of feature maps and spoil the classifier training performed by D2D-CE loss.
+
+Table 3: Experiments on the effectiveness of D2D-CE loss compared with other conditioning losses.   
+
+<table><tr><td rowspan="2">Conditioning Method</td><td colspan="4">CIFAR10 [30]</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>BigGAN w/o Condition [4] (Abbreviated to Big)</td><td>9.46</td><td>12.21</td><td>0.987</td><td>0.982</td><td>7.38</td><td>76.15</td><td>0.804</td><td>0.576</td><td>5.16</td><td>35.17</td><td>0.852</td><td>0.936</td></tr><tr><td>Big + AC[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>Big + PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>Big + MH[26]</td><td>10.05</td><td>7.94</td><td>0.994</td><td>0.990</td><td>4.37</td><td>140.74</td><td>0.282</td><td>0.156</td><td>5.18</td><td>245.69</td><td>0.625</td><td>0.832</td></tr><tr><td>Big + 2C[23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>Big + D2D-CE (ReACGAN)</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr></table>
+
+Table 4: Experiments to identify the consistent performance of D2D-CE on adversarial loss selection.   
+
+<table><tr><td rowspan="2">Adversarial Loss</td><td rowspan="2">Conditioning Method</td><td colspan="5">CIFAR10 [30]</td><td colspan="5">Tiny-ImageNet [32]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>Better?</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td><td>Better?</td></tr><tr><td rowspan="3">Non-saturation [1]</td><td>PD[19]</td><td>9.75</td><td>8.29</td><td>0.993</td><td>0.991</td><td>√</td><td>8.27</td><td>58.85</td><td>0.816</td><td>0.713</td><td></td></tr><tr><td>2C [23]</td><td>9.30</td><td>10.47</td><td>0.990</td><td>0.959</td><td></td><td>6.57</td><td>84.27</td><td>0.745</td><td>0.556</td><td></td></tr><tr><td>D2D-CE</td><td>9.79</td><td>8.27</td><td>0.993</td><td>0.991</td><td>√</td><td>11.76</td><td>39.32</td><td>0.942</td><td>0.852</td><td>√</td></tr><tr><td rowspan="3">Least square [10]</td><td>PD[19]</td><td>9.94</td><td>8.26</td><td>0.993</td><td>0.992</td><td>√</td><td>12.74</td><td>37.14</td><td>0.920</td><td>0.900</td><td>√</td></tr><tr><td>2C[23]</td><td>8.66</td><td>12.18</td><td>0.986</td><td>0.941</td><td></td><td>9.58</td><td>53.10</td><td>0.916</td><td>0.706</td><td></td></tr><tr><td>D2D-CE</td><td>9.70</td><td>9.56</td><td>0.991</td><td>0.987</td><td></td><td>9.50</td><td>57.67</td><td>0.848</td><td>0.692</td><td></td></tr><tr><td rowspan="3">W-GP [13]</td><td>PD[19]</td><td>5.71</td><td>64.75</td><td>0.792</td><td>0.652</td><td></td><td>6.67</td><td>84.16</td><td>0.696</td><td>0.498</td><td></td></tr><tr><td>2C [23]</td><td>5.93</td><td>55.99</td><td>0.842</td><td>0.709</td><td></td><td>6.89</td><td>74.45</td><td>0.812</td><td>0.536</td><td></td></tr><tr><td>D2D-CE</td><td>7.30</td><td>35.94</td><td>0.942</td><td>0.847</td><td>√</td><td>8.92</td><td> 52.74</td><td>0.856</td><td>0.689</td><td>√</td></tr><tr><td rowspan="3">Hinge [46]</td><td>PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td></td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td></td></tr><tr><td>2C [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td></td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td></td></tr><tr><td>D2D-CE</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>√</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>√</td></tr></table>
+
+Table 5: FID [34] values of conditional StyleGAN2 (cStyleGAN2) [7] and StyleGAN2 $[ 7 ] + \mathrm { D } 2 \mathrm { D } { \cdot } \mathrm { C E }$ on CIFAR10 [30] and AFHQ [41] datasets. ADA means the adaptive discriminator augmentation [8]. FID is exceptionally computed using the training split for calculating the reference moments since FID value of StyleGAN2 is often calculated using the moments of the training dataset.   
+
+<table><tr><td>Conditioning method</td><td>CIFAR10 [30]</td><td>AFHQ [41]</td></tr><tr><td>cStyleGAN2 [7]</td><td>3.88</td><td>=</td></tr><tr><td>StyleGAN2 [7] + D2D-CE</td><td> 3.34</td><td></td></tr><tr><td>cStyleGAN2 [7] + ADA [8]</td><td>2.43</td><td>4.99</td></tr><tr><td>StyleGAN2 [7] + ADA [8] + D2D-CE</td><td>2.38</td><td>4.95</td></tr><tr><td> StyleGAN2 [7] + DiffAug [9] + D2D-CE + Tuning</td><td>2.26</td><td>-</td></tr></table>
+
+Effect of D2D-CE for Different GAN Architectures. We study the effect of D2D-CE loss with different GAN architectures. In Table 5, we validate that D2D-CE is effective for StyleGAN2 [7] backbone and also fits well with the adaptive discriminator augmentation (ADA) [8]. StyleGAN2 with D2D-CE loss produces $1 3 . 9 \%$ better generation result than the conditional version of StyleGAN2 (cStyleGAN2) on CIFAR10. Moreover, StyleGAN2 with D2D-CE can be reinforced with ADA or DiffAug when train StyleGAN2 $^ +$ D2D-CE under the limited data situation. Among GANs, ${ \mathrm { S t y l e G A N } } 2 + { \mathrm { D i f f A u g } } + { \mathrm { D } } 2 { \mathrm { D } } { \mathrm { - C E } } + { \mathrm { 1 } }$ Tuning achieves the best performance on CIFAR10, even outperforming some diffusion-based methods [47, 48]. Additional results with other architectures, i.e., a deep convolutional network [2] and a resnet style backbone [13], are provided in Appendix E.
+
+# 4.4 Ablation Study
+
+We study how each component of ReACGAN affects ACGAN training. By adding or ablating each part of ReACGAN, as shown in Table 6, we identify four major observations. (1) Feature and weight normalization greatly stabilize ACGAN training and improve generation performances on Tiny-ImageNet and CUB200 datasets. (2) D2D-CE enhances the generation performance by considering data-to-data relationships and by performing easy sample suppression (3th and 4th rows). (3) the suppression technique does not work well on the feature normalized cross-entropy loss (5th row). (4) While ACGAN shows high Inception score on ImageNet experiment, it shows relatively poor FID, $\mathrm { F _ { 0 . 1 2 5 } }$ , and $\mathrm { F _ { 8 } }$ values compared with the model trained with normalization, data-to-data consideration, and the suppression technique. This result is consistent with the qualitative results on ImageNet, where the images from ACGAN are easily classifiable, but the images from ReACGAN are high quality and diverse. We attribute the improvement to the discriminator that successfully leverages informative data-to-data and data-to-class relationships with easy sample suppression.
+
+![](images/a59ce57281fb29c29f8a0af958d8e87aa81a5b050192d0c14ce2973864977a69.jpg)  
+Figure 5: Curated images generated by the proposed ReACGAN. More qualitative results on ReACGAN, BigGAN [4], ContraGAN [23], and ACGAN [18] are attached in Appendix J.
+
+Table 6: Ablation study on feature map normalization, data-to-data consideration, and easy positive and negative sample suppression.   
+
+<table><tr><td rowspan="2">Ablation</td><td colspan="4">Tiny-ImageNet [32]</td><td colspan="4">CUB200 [33]</td><td colspan="4">ImageNet [49]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td></tr><tr><td>ACGAN [18]</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>+ Normalization</td><td>13.46</td><td>30.33</td><td>0.955</td><td>0.889</td><td>4.78</td><td>25.54</td><td>0.883</td><td>0.952</td><td>18.16</td><td>36.40</td><td>0.879</td><td>0.787</td></tr><tr><td>+ Data-to-data (Eq. (5))</td><td>12.96</td><td>28.71</td><td>0.967</td><td>0.863</td><td>5.08</td><td>25.12</td><td>0.894</td><td>0.946</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td> + Suppression (Eq. (6))</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td><td>63.16 14.59</td><td></td><td></td><td>0.9740.974</td></tr><tr><td>- Data-to-data</td><td>12.96</td><td>30.79</td><td>0.960</td><td>0.857</td><td>5.39</td><td>30.36</td><td>0.863</td><td>0.947</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we have analyzed why training ACGAN becomes unstable as the number of classes in the dataset increases. By deriving the analytic form of gradient in the classifier and numerically checking the gradient values, we have discovered that the unstable training comes from a gradient exploding problem caused by the unboundedness of input feature vectors and poor classification ability of the classifier in the early training stage. To alleviate the instability and reinforce ACGAN, we have proposed the Data-to-Data Cross-Entropy loss (D2D-CE) and the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results verify the superiority of ReACGAN compared with the existing classifier- and projection-based GANs on five benchmark datasets. Moreover, exhaustive analyses on ReACGAN prove that ReACGAN is robust to hyperparameter selection and harmonizes with various architectures and differentiable augmentations.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the IITP grants (No.2019-0-01906: AI Graduate School Program - POSTECH, No.2021-0-00537: Visual Common Sense, No.2021-0-02068: AI Innovation Hub) funded by Ministry of Science and ICT, Korea.
+
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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] The main contributions of our paper can be summarized into two points: (1) analyzing why ACGAN training becomes unstable as the number of classes in the training dataset increases, (2) developing a new cGAN model: the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). Both claims are well reflected in the abstract and introduction.   
+(b) Did you describe the limitations of your work? [Yes] The proposed ReACGAN generates high-quality images using informative datato-data and data-to-class relationships in the dataset. However, it requires additional hyperparameter tuning for a temperature and margins values. Also, we analyze the weakness of ReACGAN in Appendix F.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Generative Adversarial Networks can be used to synthesize indistinguishable images from the real. This might cause fake news, sexual harnesses, and hacking machine vision applications. Although the potential negative social impacts are not discussed in our main paper, we deal with this issue in Appendix G.   
+(d) Have you read the ethics review and ensured that your paper conforms to them? [Yes] We carefully read the ethics review guidelines and comply with them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all assumptions that arise from the theoretical results, i.e., Property 1-4 in Sec. 3.3 in the main paper.   
+(b) Did you include complete proofs of all theoretical results? [Yes] We provide the complete proofs in Appendix D.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We release model weights and the software package used in our paper at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We state details for training models in Sec. 4.2 in the main paper, which include the evaluation procedure, hyperparameter selection, and controlled environment for a fair comparison. In addition, we add how we pre-process training datasets in the Appendix C.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For reliable evaluation, we run all the experiments three times with random seeds. We provide standard deviations of Tables 1, 3, 4, 6 in Appendix I.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We include the total amount of compute and the type of resources in Appendix H.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We cite papers and GitHub repository that we used for our experiments in Sec. 4.2.   
+(b) Did you mention the license of the assets? [No] We utilize five datasets to validate our method: CIFAR10, Tiny-ImageNet, CUB200, ImageNet, and AFHQ. These are widely used public datasets, and we cite them in the paper. However, we could not find any license information for CIFAR10, Tiny-ImageNet, and CUB200. The ImageNet and AFHQ datasets can be used for noncommercial purposes only. In addition, we mention the license of PyTorch-StudioGAN in the main paper, which is used for our experiments.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide the URL of our source code in the abstract.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] The CIFAR10, Tiny-ImageNet, CUB200, ImageNet, and AFHQ datasets are properly curated, and they are popular datasets for research purpose. To our best knowledge, there is no inappropriate contents in the datasets.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] The benchmark datasets we used are widely adopted by others and do not contain any harmful or troublesome contents.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] Our research has no relevance to the crowdsourcing or human subjects.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] Our research has no relevance to the crowdsourcing or human subjects.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Our research has no relevance to the crowdsourcing or human subjects.

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+        "text": "Conditional Generative Adversarial Networks (cGAN) generate realistic images by incorporating class information into GAN. While one of the most popular cGANs is an auxiliary classifier GAN with softmax cross-entropy loss (ACGAN), it is widely known that training ACGAN is challenging as the number of classes in the dataset increases. ACGAN also tends to generate easily classifiable samples with a lack of diversity. In this paper, we introduce two cures for ACGAN. First, we identify that gradient exploding in the classifier can cause an undesirable collapse in early training, and projecting input vectors onto a unit hypersphere can resolve the problem. Second, we propose the Data-to-Data Cross-Entropy loss (D2D-CE) to exploit relational information in the class-labeled dataset. On this foundation, we propose the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results show that ReACGAN achieves state-of-the-art generation results on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet datasets. We also verify that ReACGAN benefits from differentiable augmentations and that D2D-CE harmonizes with StyleGAN2 architecture. Model weights and a software package that provides implementations of representative cGANs and all experiments in our paper are available at https://github.com/POSTECH-CVLab/PyTorch-StudioGAN. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Generative Adversarial Networks (GAN) [1] are known for the forefront approach to generating high-fidelity images of diverse categories [2, 3, 4, 5, 6, 7, 8, 9]. Behind the sensational generation ability of GANs, there has been tremendous effort to develop adversarial objectives free from the vanishing gradient problem [10, 11, 12], regularizations for stabilizing adversarial training [11, 13, 14, 3, 15, 16, 17], and conditioning techniques to support the adversarial training using category information of the dataset [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Subsequently, the conditioning techniques have become the de facto standard for high-quality image generation. The models with the conditioning methods are called conditional Generative Adversarial Networks (cGAN), and cGANs can be divided into two groups depending on the discriminator’s conditioning way: classifier-based GANs [18, 20, 23, 25, 27] and projection-based GANs [19, 3, 4, 28]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The classifier-based GANs facilitate an auxiliary classifier to generate class-specific images by penalizing the generator if the synthesized images are not consistent with the conditioned labels. ACGAN [18] has been one of the widely used classifier-based GANs for its simple design and satisfactory generation performance. While ACGAN can exploit class information by pushing and pulling classifier’s weights (proxies) against image embeddings [23], it is well known that ACGAN training is prone to collapsing at the early stage of training as the number of classes increases [19, 23, 25, 27]. In addition, the generator of ACGAN tends to generate easily classifiable images at the cost of reduced diversity [18, 19, 27]. Projection-based GANs, on the other hand, have shown cutting-edge generation results on datasets with a large number of categories. SNGAN [3], ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a77f02b277b4a9b1554bdeb714cab48b71615b6d03f8135fb468d91f2cc6deb5.jpg",
+        "image_caption": [
+            "Figure 1: Schematics that depict how cGANs perform conditioning. The red color means positive samples/proxy, and the blue color indicates negative samples/proxies. Arrows represent push-pull forces based on the reference sample. The length of an arrow indicates the magnitude of the force. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "SAGAN [29], and BigGAN [4] are representatives in this family and can generate realistic images on CIFAR10 [30] and ImageNet [31] datasets. However, projection-based GANs only consider a pairwise relationship between an image and its label proxy (data-to-class relationships). As the result, the projection-based GANs can miss an additional opportunity to consider relation information between data instances (data-to-data relationships) as discovered by [23]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we analyze why ACGAN training becomes unstable as the number of classes increases and propose remedies for (1) the instability and (2) the relatively poor generation performance of ACGAN compared with the projection-based models. First, we begin by analytically deriving the gradient of the softmax cross-entropy loss used in ACGAN. By examining the exact values of analytic gradients, we discover that the unboundedness of input feature vectors and poor classification performance in the early training stage can cause an undesirable gradient exploding problem. Second, with alleviating the instability, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) using the Data-to-Data Cross-Entropy loss (D2D-CE). ReACGAN projects image embeddings and proxies onto a unit hypersphere and computes similarities for data-to-data and data-to-class consideration. Additionally, we introduce two margin values for intra-class variations and inter-class separability. In this way, ReACGAN overcomes the training instability and can exploit additional supervisory signals by explicitly considering data-to-class and data-to-data relationships, and also by implicitly looking at class-to-class relationships in the same mini-batch. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To validate our model, we conduct image generation experiments on CIFAR10 [30], TinyImageNet [32], CUB200 [33], and ImageNet [31] datasets. Through extensive experiments, we demonstrate that ReACGAN beats both the classifier-based and projection-based GANs, improving over the state of the art by $2 . 5 \\%$ , $1 5 . 8 \\%$ , $5 . 1 \\%$ , and $1 4 . 5 \\%$ in terms of Fréchet Inception Distance (FID) [34] on the four datasets, respectively. We also verify that ReACGAN benefits from consistency regularization [16] and differentiable augmentations [9, 8] for limited data training. Finally, we confirm that D2D-CE harmonizes with the StyleGAN2 architecture [7]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Background: Generative Adversarial Networks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Network (GAN) [1] is an implicit generative model that aims to generate a sample indistinguishable from the real. GAN consists of two networks: a Generator $G : { \\mathcal { Z } } \\longrightarrow { \\mathcal { X } }$ that tries to map a latent variable $\\mathbf { z } \\sim p ( \\mathbf { z } )$ into the real data space $\\mathcal { X }$ and a Discriminator $D : \\mathcal { X } \\longrightarrow$ [0, 1] that strives to discriminate whether a given sample $\\mathbf { x }$ is from the real data distribution $p _ { \\mathrm { r e a l } } ( \\mathbf { x } )$ or from the implicit distribution $p _ { \\mathrm { g e n } } ( G ( \\mathbf { z } ) )$ derived from the generator $G ( \\mathbf { z } )$ . The objective of a vanilla GAN [1] can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af21913ce94f55d589da37b72bcd0d6981cac1e7074fd7e7c6991e20448adb0b.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { \\mathbf { x } \\sim p ( \\mathbf { x } ) } [ \\log ( D ( \\mathbf { x } ) ) ] + \\mathbb { E } _ { \\mathbf { z } \\sim p ( \\mathbf { z } ) } [ \\log ( 1 - D ( G ( \\mathbf { z } ) ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While GANs have shown impressive results in the image generation task [2, 35, 11, 13], training GANs often ends up encountering a mode-collapse problem [36, 11, 12]. As one of the prescriptions for stabilizing and reinforcing GANs, training GANs with categorical information, named conditional Generative Adversarial Networks (cGAN), is suggested [37, 18, 19]. Depending on the presence of explicit classification losses, cGAN can be divided into two groups: Classifier-based GANs [18, 20, 23, 25, 27] and Projection-based GANs [19, 3, 4, 28]. One of the widely used classifier-based GANs is ACGAN [18], and ACGAN utilizes softmax cross-entropy loss to perform classification task with adversarial training. Although ACGAN has shown satisfactory generation results, training ACGAN becomes unstable as the number of classes in the training dataset increases [19, 23, 25, 27]. Besides, ACGAN tends to generate easily classifiable images at the cost of limited diversity [18, 19, 27]. To alleviate those problems, Zhou et al. [38] have proposed performing adversarial training on the classifier. Gong et al. [20] have introduced an additional classifier to eliminate a conditional entropy minimization process in the adversarial training. However, ACGAN training still suffers from the early-training collapse issue and the reduced diversity problem when trained on datasets with a large number of class categories, such as Tiny-ImageNet [32] and ImageNet [31]. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In these circumstances, Miyato et al. [19] have proposed a projection discriminator for cGANs and have shown significant improvement in generating the ImageNet dataset. Motivated by the promising result of the projection discriminator, many projection-based GANs [3, 29, 4, 16, 6, 17] have been proposed and become the standard for conditional image generation. In this paper, we revisit ACGAN and unveil why ACGAN training is so unstable. Coping with the instability, we propose the Rebooted Auxiliary Classifier GANs (ReACGAN) for high-quality and diverse image generation. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Rebooting Auxiliary Classifier GANs ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Feature Normalization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To uncover a nuisance that can cause the instability of ACGAN, we start by analytically deriving the gradients of weight vectors in the softmax classifier. Let the part of the discriminator before the fully connected layer be a Feature extractor $F : \\mathcal { X } \\longrightarrow \\mathcal { F } \\in \\bar { \\mathbb { R } ^ { d } }$ and let Classifier $C : { \\mathcal { F } } \\longrightarrow \\mathbb { R } ^ { c }$ be a single fully connected layer parameterized by $\\pmb { W } = [ \\pmb { w } _ { 1 } , . . . , \\pmb { w } _ { c } ] \\in \\mathbb { R } ^ { d \\times c }$ , where $c$ denotes the number of classess. We sample training images $\\pmb { X } = \\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { N } \\}$ and integer labels ${ \\textbf { 3 } } =$ $\\{ y _ { 1 } , . . . , y _ { N } \\}$ from the joint distribution $p ( \\mathbf { x } , \\mathbf { y } )$ . Using the notations above, we can express the empirical cross-entropy loss used in ACGAN [18] as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3cde793fe61034f79c6982cfce3d15f0a2d0f599275318cc5332121eac04663.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C E } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Big ( \\frac { \\exp \\left( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { y _ { i } } \\right) } { \\sum _ { j = 1 } ^ { c } \\exp \\left( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { j } \\right) } \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on Eq. (3), we can derive the derivative of the cross-entropy loss, w.r.t ${ \\pmb w } _ { k \\in \\{ 1 , \\dots , c \\} }$ as follows: ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/634f34fe1b9cb8058559e57be9cafdd69cbbcbcc35cc37fe0b2346817ebaa3ee.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { L } _ { \\mathrm { C E } } } { \\partial \\boldsymbol { w } _ { k } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\bigg \\{ F ( \\boldsymbol { x } _ { i } ) \\bigg ( \\mathbf { 1 } _ { y _ { i } = k } - p _ { i , k } \\bigg ) \\bigg \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            630,
+            650,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { 1 } _ { \\mathit { y } _ { i } = k }$ is an indicator function that will output 1 if $y _ { i } = k$ is satisfied, and $p _ { i , k }$ is a class probability that represents the probability that $i$ -th sample belongs to class $k$ , mathematically $\\frac { \\exp { ( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { k } ) } } { \\sum _ { j = 1 } ^ { c } \\exp { ( F ( \\pmb { x } _ { i } ) ^ { \\top } \\pmb { w } _ { j } ) } }$ . The equation above implies that the norm of the gradient of the softmax cross-entropy loss is coupled with the norms and directions of each input feature map $F ( { \\pmb x } _ { i } )$ and the class probabilities. In the early training stage, the classifier is prone to making incorrect predictions, resulting in low probabilities. This phenomenon occurs more frequently as the number of categories in the dataset increases. As the result, the norm of the gradient $| \\frac { \\partial \\hat { \\mathcal { L } } _ { \\mathrm { C E } } } { \\partial w _ { k } } |$ begins to explode as the vector $F ( { \\pmb x } _ { i } )$ stretches out to ${ \\pmb w } _ { k }$ direction but being located close to the the other vectors ${ \\pmb w } _ { j \\in \\{ 1 , . . . , c \\} \\backslash \\{ k \\} }$ . This often breaks the balance between adversarial learning and classifier training, leading to an early-training collapse. Once the early-training collapse occurs, ACGAN training concentrates on classifying categories of images instead of discriminating the authenticity of given samples. We experimentally demonstrate that the average norm of ACGAN’s input feature maps increases as the training progresses (Fig. 2a). Accordingly, the average norm of the gradients increases sharply at the early training stage and decreases with the high class probabilities of the classifier (Fig. 2b, Fig. A3 in Appendix). While the average norm of gradients decreases at some point, the FID value of ACGAN does not decrease, implying the collapse of ACGAN training (Fig. 2c). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/41f752bd054c08034409d7f5b84c86a0197793ecc9cb9dcf2b22365737c55349.jpg",
+        "image_caption": [
+            "Figure 2: Merits of integrating feature normalization and data-to-data relationship consideration. All experiments are conducted on Tiny-ImageNet [32] dataset. (a) Average norms of input feature maps $\\bar { F } ( \\cdot )$ , (b) average norms of gradients of classification losses, and (c) trends of FID scores. Compared with ACGAN [18], the proposed ReACGAN does not experience the training collapse problem caused by excessively large norms of feature maps and gradients at the early training stage. In addition, ReACGAN can converge to better equilibrium by considering data-to-data relationships with easy positive and negative sample suppression. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As one of the remedies for the gradient exploding problem, we find that simply normalizing the feature embeddings onto a unit hypersphere $\\frac { F ( { \\pmb x } _ { i } ) } { | | F ( { \\pmb x } _ { i } ) | | }$ effectively resolves ACGAN’s early-training collapse (see Fig. 2). The motivation is that normalizing the features onto the hypersphere makes the norms of feature maps equal to 1.0. Thus, the discriminator does not experience the gradient exploding problem. From the next section, we will deploy a linear projection layer $P$ on the feature extractor $F$ . And, we will normalize both the embeddings from the projection layer and the weight vectors $( \\pmb { w } _ { 1 } , . . . , \\pmb { w } _ { c } )$ in the classifier since normalizing both the embeddings and the weight vectors does not degrade image generation performance. We denote the normalized embedding P (F (xi))||P (F (xi))|| as $f _ { i }$ and the normalized weight vector $\\frac { { \\pmb w } _ { y _ { i } } } { | | { \\pmb w } _ { y _ { i } } | | }$ as vyi . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Data-to-Data Cross-Entropy Loss (D2D-CE) ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We expand the feature normalized softmax cross-entropy loss described in Sec. 3.1 to the Data-to-Data Cross-Entropy (D2D-CE) loss. The motivations are summarized into two points: (1) replacing datato-class similarities in the denominator of Eq. (3) with data-to-data similarities and (2) introducing two margin values to the modified softmax cross-entropy loss. We expect that point (1) will encourage the feature extractor to consider data-to-class as well as data-to-data relationships, and that point (2) will guarantee inter-class separability and intra-class variations in the feature space while preventing ineffective gradient updates induced by easy negative and positive samples. To develop the feature normalized cross-entropy loss into D2D-CE, we replace the similarities between a sample embedding and all proxies except for the positive one in the denominator, $\\begin{array} { r } { \\sum _ { j \\in \\{ 1 , \\dots , c \\} \\backslash \\{ y _ { i } \\} } \\exp \\left( \\mathbf { { f } } _ { i } ^ { \\top } \\mathbf { { v } } _ { j } \\right) } \\end{array}$ with similarities between negative samples in the same mini-batch. The modified cross-entropy loss can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2083ef980138ec9937eb51f9829dbe3f337cc0b83827c7b723f912acedff8847.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C E } } ^ { ' } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Bigg ( \\frac { \\exp { ( f _ { i } ^ { \\top } v _ { y _ { i } } / \\tau ) } } { \\exp { ( f _ { i } ^ { \\top } v _ { y _ { i } } / \\tau ) } + \\sum _ { j \\in N ( i ) } \\exp { ( f _ { i } ^ { \\top } f _ { j } / \\tau ) } } \\Bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            702,
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+            746
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a temperature, and $\\mathcal { N } ( i )$ is the set of indices that point locations of the negative samples whose labels are different from the reference label $\\pmb { v } _ { y _ { i } }$ in the mini-batch. The self-similarity matrix of samples in the mini-batch is used to calculate the similarities between negative samples $\\pmb { f } _ { i } ^ { \\top } \\pmb { f } _ { j \\in \\mathcal { N } ( i ) }$ with a false negative mask (see Fig. 3). Thus, Eq. (5) enables the discriminator to contrastively compare visual differences between multiple images and can supply more informative supervision for image conditioning. Finally, we introduce two margin hyperparameters to $\\mathcal { L } _ { \\mathrm { C E } } ^ { ' }$ and name it Data-to-Data Cross-Entropy loss (D2D-CE). The proposed D2D-CE can be expressed as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f95d96f99ec92483130927216fa5f895d898e210867b75a033db7d5a3213348.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { D 2 D - C E } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\Bigg ( \\frac { \\exp \\left( [ f _ { i } ^ { \\top } v _ { y _ { i } } - m _ { p } ] _ { - } / \\tau \\right) } { \\exp \\left( [ f _ { i } ^ { \\top } v _ { y _ { i } } - m _ { p } ] _ { - } / \\tau \\right) + \\sum _ { j \\in N ( i ) } \\exp \\left( [ f _ { i } ^ { \\top } f _ { j } - m _ { n } ] _ { + } / \\tau \\right) } \\Bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/ee7bad162e1754a63a5d7f7b5a7a8d9d8e3e367c1fcd2cc101b8a2534e6e145d.jpg",
+        "image_caption": [
+            "Figure 3: Overview of ReACGAN. ReACGAN performs adversarial training using the loss ${ \\mathcal { L } } _ { \\mathrm { A d v } }$ . At the same time, ReACGAN tries to minimize data-to-data cross-entropy loss (D2D-CE) on the linear head $( P )$ to exploit relational information in the labeled dataset. $^ { \\circ }$ means element-wise product. Note that False negatives mask operates on the similarity matrix between two batches of sample embeddings to compute the similarities between negative samples in the denominator of Eq. (6). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            93,
+            812,
+            332
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $m _ { p }$ is a margin for suppressing a high similarity value between a reference sample and its corresponding proxy (easy positive), $m _ { n }$ is a margin for suppressing low similarity values between negatives samples (easy negatives). $[ \\cdot ] _ { - }$ and $[ \\cdot ] _ { + }$ denote $\\operatorname* { m i n } ( \\cdot , 0 )$ and $\\operatorname* { m a x } ( \\cdot , 0 )$ functions, respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Useful Properties of Data-to-Data Cross-Entropy Loss ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            492,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we explain four useful properties of D2D-CE. Let $s _ { q }$ be a similarity between the normalized embedding $f _ { q }$ and the corresponding normalized proxy ${ \\pmb v } _ { y _ { q } }$ , $s _ { q , r }$ be a similarity between $f _ { q }$ and $f _ { r }$ , and $a$ and $\\mathit { b }$ be arbitrary indices of negative samples, i.e., $a , b \\in \\mathcal { N } ( q )$ . Then the properties of D2D-CE can be summarized as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/284973472b7cc1ba4ea65614768295afd00d54dfd0ba532f646e48451266f16e.jpg",
+        "image_caption": [
+            "Figure 4: Graph showing regions where hard negative mining, easy positive and negative suppression, and no gradient update occur. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Property 1. Hard negative mining. If the value of $s _ { q , a }$ is greater than $s _ { q , b }$ , the derivative of LD2D-CE w.r.t $s _ { q , a }$ is greater than or equal to the derivative $w . r . t \\ s _ { q , b }$ ; that is $\\begin{array} { r } { \\frac { \\partial \\mathcal { L } _ { D 2 D - C E } } { \\partial s _ { q , a } } \\geq \\frac { \\partial \\mathcal { L } _ { D 2 D - C E } } { \\partial s _ { q , b } } \\geq 0 . } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "Property 2. Easy positive suppression. If $s _ { q } \\mathrm { ~ - ~ }$ $m _ { p } \\geq 0$ , the derivative of $\\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q }$ is 0. ",
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+    {
+        "type": "text",
+        "text": "Property 3. Easy negative suppression. If $s _ { q , r } - m _ { n } \\leq 0 ,$ , the derivative of $\\mathcal { L } _ { D 2 D - C E } w . r . t s _ { q , r }$ is 0. Property 4. If $s _ { q } - m _ { p } \\ge 0$ and $s _ { q , r } - m _ { n } \\leq 0$ are satisfied, $\\mathcal { L } _ { D 2 D - C E }$ has the global minima of $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\left( 1 + \\bar { | \\mathcal { N } ( i ) | } \\right) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "We put proofs of the above properties in Appendix D. Property 1 indicates that D2D-CE implicitly conducts hard negative mining and benefits from comparing samples with each other. Also, Properties 2 and 3 imply that samples will not affect gradient updates if the samples are trained sufficiently. Consequently, the classifier concentrates on pushing and pulling hard negative and hard positive examples without being dominated by easy negative and positive samples. ",
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+    {
+        "type": "text",
+        "text": "3.4 Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the proposed D2D-CE objective, we propose the Rebooted Auxiliary Classifier Generative Adversarial Networks (ReACGAN). As ACGAN does, ReACGAN jointly optimizes an adversarial loss and the classification objective (D2D-CE). Specifically, the discriminator, which consists of the feature extractor, adversarial head, and linear head, of ReACGAN tries to discriminate whether a given image is sampled from the real distribution or not. At the same time, the discriminator tries to maximize similarities between the reference samples and corresponding proxies while minimizing similarities between negative samples using real images and D2D-CE loss. After updating the discriminator a predetermined number of times, the generator strives to deceive the discriminator by generating well-conditioned images that will output a low D2D-CE value. By alternatively training the discriminator and generator until convergence, ReACGAN generates high-quality images of diverse categories without early-training collapse. We attach the algorithm table in Appendix A. ",
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+        "type": "text",
+        "text": "Differences between ReACGAN and ContraGAN. The authors of ContraGAN [23] propose a conditional contrastive loss (2C loss) to cover data-to-data relationships when training cGANs. The main differences between 2C loss and D2D-CE objective are summed up into three points: (1) while 2C loss is derived from NT-Xent loss [39], which is popularly used in the field of the self-supervised learning, our D2D-CE is developed to resolve the early-training collapsing problem and the poor generation results of ACGAN, (2) 2C loss holds false-negative samples in the denominator, and they can cause unexpected positive repulsion forces, and (3) 2C loss contains the similarities between all positive samples in the numerator, and they give rise to large gradients on pulling easy positive samples. Consequently, GANs with 2C loss tend to synthesize images of unintended classes as reported in the author’s software document [40]. However, D2D-CE does not contain the false negatives in the denominator and considers only the similarity between a sample and its proxy in the numerator. Therefore, ReACGAN is free from the undesirable repelling forces and does not conduct unnecessary easy positive mining. More detailed explanations are attached in Appendix F. ",
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+        "type": "text",
+        "text": "Consistency Regularization and D2D-CE Loss. Zhang et al. [16] propose a consistency regularization to force the discriminator to make consistent predictions if given two images are visually close to each other. They create a visually similar image pair by augmenting a reference image with pre-defined augmentations. After that, they let the discriminator minimize L2 distance between the logit of the reference image and the logit of the augmented counterpart. While ReACGAN locates an image embedding nearby its corresponding proxy but far apart multiple image embeddings of different classes, consistency regularization only pulls a reference and the augmented image towards each other. Since consistency regularization and D2D-CE can be applied together, we will show that ReACGAN benefits from consistency regularization in the experiments section (Table 1). ",
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+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Datasets and Evaluation Metrics ",
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+        "text": "To verify the effectiveness of ReACGAN, we conduct conditional image generation experiments using five datasets: CIFAR10 [30], Tiny-ImageNet [32], CUB200 [33], ImageNet [31], and AFHQ [41] datasets and four evaluation metrics: Inception Score (IS) [42], Fréchet Inception Distance (FID) [34], and $\\mathrm { F _ { 0 . 1 2 5 } }$ (Precision) and $\\mathrm { F } _ { 8 }$ (Recall) [43]. The details on the training datasets are in Appendix C.1. ",
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+        "type": "text",
+        "text": "Inception Score (IS) [42] and Fréchet Inception Distance (FID) [34] are widely used metrics for evaluating generative models. We utilize IS and FID together because some studies [4, 6, 25] have shown that IS has a tendency to measure the fidelity of images better while FID tends to weight capturing the diversity of images. ",
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+        "type": "text",
+        "text": "Precision $\\bf ( F _ { 0 . 1 2 5 } )$ and Recall $\\bf ( F _ { 8 } )$ [43] are metrics for estimating precision and recall of the approximated distribution $p ( G ( \\mathbf { z } ) )$ against the true data distribution $p ( \\mathbf { x } )$ . Instead of evaluating generative models using one-dimensional scores, such as IS and FID, Sajjadi et al. [43] have suggested using a two-dimensional score $\\mathrm { F _ { 0 . 1 2 5 } }$ and $\\mathrm { F } _ { 8 }$ that can quantify how precisely the generated images are and how well the generated images cover the reference distribution. ",
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+    {
+        "type": "text",
+        "text": "4.2 Experimental Details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The implementation of ReACGAN basically follows details of PyTorch-StudioGAN library [40]1 that supports various experimental setups from ACGAN [18] to StyleGAN2 [7] $^ +$ ADA [8] with different scales of datasets [30, 32, 33, 31, 41] and architectures [2, 13, 4, 7]. For a fair comparison, we use the same backbones over all baselines, except otherwise noted, for both the discriminator and generator. For stable training, we apply spectral normalization (SN) [3] to the generator and discriminator except for experiments using SNGAN (in this case, we apply SN to the discriminator only) and StyleGAN2 [7]. We also use the same conditioning method for generators with conditional batch normalization (cBN) [44, 45, 19]. This allows us to investigate solely the conditioning methods of the discriminator, which are the main interest of our paper. If not specified, we use hinge loss [46] as a default for the adversarial loss. ",
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+    {
+        "type": "table",
+        "img_path": "images/4b04bdeeb55f3430ef72f3f3db9f9e61e72c6ba6199d770fd52b912b38354bce.jpg",
+        "table_caption": [
+            "Table 1: Comparison with classifier-based GANs [18, 23] and projection-based GANs [3, 4, 29] on CIFAR10 [30], Tiny-ImageNet [32], and CUB200 [33] datasets using IS [42], FID [34], $\\mathrm { F } _ { 0 . 1 2 5 }$ , and $\\mathrm { F } _ { 8 }$ [43] metrics. For baselines, both the numbers from the cited paper (denoted as \\* in method) and from our experiments using StudioGAN library [40] are reported. The numbers in bold-faced denote the best performance and in underline indicate the values are in one standard deviation from the best. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">CIFAR10 [30]</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td><td></td><td></td><td>IS↑FID↓F0.125↑F8↑</td><td></td></tr><tr><td>SNGAN*[3]</td><td>8.22</td><td>21.7</td><td>=</td><td>-</td><td>1</td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>=</td><td>-</td></tr><tr><td>BigGAN*[4]</td><td>9.22</td><td>14.73</td><td>=</td><td>-</td><td>=</td><td>-</td><td></td><td>=</td><td>-</td><td></td><td>-</td><td>1</td></tr><tr><td>ContraGAN* [23]</td><td>1</td><td>10.60</td><td></td><td></td><td></td><td>29.49</td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>ACGAN[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>SNGAN [3]</td><td>8.67</td><td>13.33</td><td>0.985</td><td>0.976</td><td>8.71</td><td>51.15</td><td>0.900</td><td>0.702</td><td>5.41</td><td>47.75</td><td>0.754</td><td>0.912</td></tr><tr><td>SAGAN [29]</td><td>8.66</td><td>14.31</td><td>0.983</td><td>0.973</td><td>8.74</td><td>49.90</td><td>0.872</td><td>0.712</td><td>5.48</td><td>54.29</td><td>0.728</td><td>0.882</td></tr><tr><td>BigGAN[4]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>ContraGAN [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>ReACGAN</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td></td><td>14.06 27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr><tr><td>BigGAN +CR*[16]</td><td>-</td><td>11.48</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BigGAN [4]+ CR[16]</td><td>9.97</td><td>7.18</td><td>0.995</td><td>0.993</td><td></td><td>15.94 19.96</td><td>0.972</td><td>0.950 5.14</td><td></td><td>11.97</td><td>0.978</td><td>0.981</td></tr><tr><td>ContraGAN [23] + CR[16]</td><td>9.59</td><td>8.55</td><td>0.992</td><td>0.972</td><td>15.81</td><td>19.21</td><td>0.983</td><td>0.941</td><td>4.90</td><td>11.08</td><td>0.984</td><td>0.967</td></tr><tr><td>ReACGAN + CR [16]</td><td>10.11</td><td>7.20</td><td>0.996</td><td>0.994 16.56</td><td></td><td>19.69</td><td>0.984</td><td>0.940</td><td>4.87</td><td>10.72</td><td>0.985</td><td>0.971</td></tr><tr><td>BigGAN + DiffAug*[9]</td><td>9.17</td><td>8.49</td><td></td><td></td><td>=</td><td></td><td>=</td><td>=</td><td>=</td><td></td><td></td><td></td></tr><tr><td>BigGAN [4] + DiffAug [9]</td><td>9.94</td><td>7.17</td><td>0.995</td><td>0.992</td><td></td><td>18.0815.70</td><td>0.980</td><td>0.972</td><td>5.53</td><td>12.15</td><td>0.967</td><td>0.981</td></tr><tr><td>ContraGAN [23] + DiffAug [9]</td><td>9.95</td><td>7.27</td><td>0.995</td><td>0.992</td><td>18.20</td><td>15.40</td><td>0.986</td><td>0.963</td><td>5.39</td><td>11.02</td><td>0.978</td><td>0.970</td></tr><tr><td>ReACGAN + DiffAug [9]</td><td>10.22</td><td>6.79</td><td>0.996</td><td>0.993</td><td>20.60</td><td>14.25</td><td>0.988</td><td>0.972</td><td>5.22</td><td>9.27</td><td>0.985</td><td>0.983</td></tr><tr><td>Real Data</td><td colspan=\"2\">11.54</td><td colspan=\"2\">34.11</td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\">5.49</td><td colspan=\"2\"></td></tr></table>",
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+        "type": "text",
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+        "type": "text",
+        "text": "Before conducting main experiments, we perform hyperparameter search with candidates of a temperature $\\tau \\in \\{ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \\}$ and a positive margin $m _ { p } \\in \\ \\{ 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 8 , \\bar { 1 . 0 } \\}$ . We set a negative margin $m _ { n }$ as $1 \\ : - \\ : m _ { p }$ to reduce search time. Through extensive experiments with 3 runs per each setting, we select $\\tau$ with $\\{ 0 . 5 , 0 . 7 5 , 0 . 2 5 , 0 . 5 , 0 . 2 5 \\}$ and $m _ { p }$ with $\\{ 0 . 9 8 , 1 . 0 , 0 . 9 5 , 0 . 9 \\bar { 8 } , 0 . 9 0 \\}$ for CIFAR10, Tiny-ImageNet, CUB200, ImageNet 256 B.S., and ImageNet 2048 B.S. experiments, respectively. A low temperature seems to work well on fine-grained image generation tasks, but generally, ReACGAN is robust to the choice of hyperparameters. The results of the hyperparameter search and other hyperparameter setups are provided in Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": "We evaluate all methods through the same protocol of [9, 16, 17], which uses the same amounts of generated images from the reference split specialized for each dataset.2 Besides, we run all the experiments three times with random seeds and report the averaged best performances for reliable evaluation with the lone exception of ImageNet and StyleGAN2 related experiments. Please refer to Appendix C.2 for other experimental details. The numbers in bold-faced denote the best performance and in underline indicate that the values are in one standard deviation from the best. ",
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+    {
+        "type": "text",
+        "text": "4.3 Evaluation Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Comparison with Other cGANs. We compare ReACGAN with previous state-of-the-art cGANs in Tables 1 and 2. We employ the implementations of GANs in PyTorch-StudioGAN library as it provides improved results on standard benchmark datasets [30, 31]. For a fair comparison, we provide results from each original paper (denoted as \\* in method) as well as those from StudioGAN library [40]. We also conduct experiments with popular augmentation-based methods: consistency regularization (CR) [16] and differentiable augmentation (DiffAug) [9]. ",
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+    {
+        "type": "table",
+        "img_path": "images/d5bbe5ee1dbe2195a327ca572b5f43ba744dd66b42d23636d28e4a2d3dd61fd8.jpg",
+        "table_caption": [
+            "Table 2: Experiments using ImageNet [31] dataset. B.S. means batch size for training. (Left): Comparisons with previous cGAN approaches. (Right): Learning curves of BigGAN [4], ContraGAN [23], and ReACGAN (ours) which are trained using the batch size of 256. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"5\">ImageNet [31]</td></tr><tr><td>IS↑</td><td>FID↓</td><td></td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>ACGAN*</td><td>[20]</td><td>7.26</td><td>184.41</td><td></td><td></td></tr><tr><td>SNGAN*[3]</td><td></td><td>36.80</td><td>27.62</td><td></td><td></td></tr><tr><td>SAGAN*[29]</td><td></td><td>52.52</td><td>18.28</td><td></td><td></td></tr><tr><td>BigGAN*[20]</td><td></td><td>38.05</td><td>22.77</td><td></td><td></td></tr><tr><td></td><td>TAC-GAN*[20]</td><td>28.86</td><td>23.75</td><td></td><td></td></tr><tr><td>ContraGAN*[23]</td><td></td><td>31.10</td><td>19.69</td><td>0.951</td><td>0.927</td></tr><tr><td></td><td>ACGAN[18]</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>35=58</td><td>SNGAN [3]</td><td>32.25</td><td>26.79</td><td>0.938</td><td>0.913</td></tr><tr><td></td><td>SAGAN [29]</td><td>29.85</td><td>34.73</td><td>0.849</td><td>0.914</td></tr><tr><td></td><td>BigGAN [4]</td><td>43.97</td><td>16.36</td><td>0.964</td><td>0.955</td></tr><tr><td></td><td>ContraGAN [23]</td><td>25.25</td><td>25.16</td><td>0.947</td><td>0.855</td></tr><tr><td>ReACGAN</td><td></td><td>68.27</td><td>13.98</td><td>0.976</td><td>0.977</td></tr><tr><td></td><td>BigGAN [4] + DiffAug [9]</td><td>36.97</td><td>18.57</td><td>0.956</td><td>0.941</td></tr><tr><td></td><td>ReACGAN +DiffAug[9]</td><td>69.74</td><td>11.95</td><td>0.977</td><td>0.975</td></tr><tr><td></td><td>BigGAN*[4]</td><td>99.31</td><td>8.51</td><td>-</td><td>-</td></tr><tr><td>P55=44</td><td>BigGAN*[25]</td><td>104.57</td><td>9.18</td><td></td><td>■</td></tr><tr><td></td><td>BigGAN[4]</td><td>99.71</td><td>7.89</td><td>0.985</td><td>0.989</td></tr><tr><td></td><td>ReACGAN</td><td>92.74</td><td>8.23</td><td>0.991</td><td>0.990</td></tr><tr><td></td><td>Real Data</td><td>173.33</td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "image",
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+        "text": "Compared with other cGANs, ReACGAN performs the best on most benchmarks, surpassing the previous methods by $2 . 5 \\%$ , $5 . 1 \\%$ , $1 5 . 8 \\%$ , and $1 4 . 5 \\%$ in FID on CIFAR10, Tiny-ImageNet, CUB200, and ImageNet (256 B.S.), respectively. ReACGAN also harmonizes with augmentation-based regularizations, bringing incremental improvements on all the metrics. For the ImageNet experiments using a batch size of 256, ReACGAN reaches higher IS and lower FID with fewer iterations than other models in comparison. Finally, we demonstrate that ReACGAN can learn with a larger batch size on ImageNet. While some recent methods [20, 38, 27] have been built on ACGAN to improve the generation performance of ACGAN, large-scale image generation experiments with the batch size of 2048 have never been reported. Table 2 shows that our ReACGAN reaches FID score of 8.23 on ImageNet, being comparable with the value of 7.89 from BigGAN implementation in PyTorch-StudioGAN library. ReACGAN, however, provides better synthesis results than other implementations of BigGAN [4, 25]. Note that our result on ImageNet is obtained in only two runs while the training setup and architecture of BigGAN have been extensively searched and finely tuned. ",
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+        "type": "text",
+        "text": "Comparison with Other Conditioning Losses. We investigate how the generation qualities vary with different conditioning losses while keeping the other configurations fixed. We compare D2D-CE loss with cross-entropy loss of ACGAN (AC) [18], loss used in the projection discriminator (PD) [19], multi-hinge loss (MH) [26], and conditional contrastive loss (2C) [23]. The results are shown in Table 3. AC- and MH-based models present decent results on CIFAR10, but undergo early-training collapse on Tiny-ImageNet and CUB200 datasets. Replacing them with PD, 2C, and D2D-CE losses produce satisfactory performances across all datasets, where PD loss makes the best $\\mathrm { F } _ { 8 }$ (recall) on CUB200 dataset while giving third $\\mathrm { F _ { 0 . 1 2 5 } }$ (precision) value. The noticeable point is that the proposed D2D-CE shows consistent results across all datasets, showing the lowest FID and the highest $\\mathrm { F _ { 8 } }$ and $\\mathrm { F _ { 0 . 1 2 5 } }$ values in most cases. This means ReACGAN can generate high-fidelity images and is relatively free from the precision and recall trade-off [43] than the others. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Consistent Performance of ReACGAN on Adversarial Loss Selection. We validate the consistent performance of ReACGAN on four adversarial losses: non-saturation loss [1], least square loss [10], Wasserstein loss with gradient penalty regularization (W-GP) [13], and hinge loss [46] on CIFAR10 and Tiny-ImageNet datasets in Table 4. The experimental results show that $\\mathrm { B i g G A N } + \\mathrm { D } 2 \\mathrm { D } { \\cdot } \\mathrm { C E }$ (ReACGAN) consistently outperforms the projection discriminator (PD) and conditional contrastive loss (2C) counterparts over three adversarial losses [1, 13, 46]. However, for the experiments using the least square loss [10], ReACGAN exhibits inferior generation performances to the projection discriminator. We speculate that minimizing the least square distance between an adversarial logit and the target scalar (1 or 0) might affect the norms of feature maps and spoil the classifier training performed by D2D-CE loss. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5f9662b078f6d463fde84887bf9dcf884f89b25fc053625a9e84dc7b9f48ef56.jpg",
+        "table_caption": [
+            "Table 3: Experiments on the effectiveness of D2D-CE loss compared with other conditioning losses. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Conditioning Method</td><td colspan=\"4\">CIFAR10 [30]</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td></tr><tr><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS↑</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td></tr><tr><td>BigGAN w/o Condition [4] (Abbreviated to Big)</td><td>9.46</td><td>12.21</td><td>0.987</td><td>0.982</td><td>7.38</td><td>76.15</td><td>0.804</td><td>0.576</td><td>5.16</td><td>35.17</td><td>0.852</td><td>0.936</td></tr><tr><td>Big + AC[18]</td><td>9.84</td><td>8.45</td><td>0.993</td><td>0.992</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td></tr><tr><td>Big + PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td>4.98</td><td>18.30</td><td>0.924</td><td>0.967</td></tr><tr><td>Big + MH[26]</td><td>10.05</td><td>7.94</td><td>0.994</td><td>0.990</td><td>4.37</td><td>140.74</td><td>0.282</td><td>0.156</td><td>5.18</td><td>245.69</td><td>0.625</td><td>0.832</td></tr><tr><td>Big + 2C[23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td>5.34</td><td>21.16</td><td>0.935</td><td>0.942</td></tr><tr><td>Big + D2D-CE (ReACGAN)</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8543ec2bce142bf9ba668eff880dd37e8f3be1d789b055b0f875b5f0e5b0a8a8.jpg",
+        "table_caption": [
+            "Table 4: Experiments to identify the consistent performance of D2D-CE on adversarial loss selection. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Adversarial Loss</td><td rowspan=\"2\">Conditioning Method</td><td colspan=\"5\">CIFAR10 [30]</td><td colspan=\"5\">Tiny-ImageNet [32]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>Better?</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td><td>Better?</td></tr><tr><td rowspan=\"3\">Non-saturation [1]</td><td>PD[19]</td><td>9.75</td><td>8.29</td><td>0.993</td><td>0.991</td><td>√</td><td>8.27</td><td>58.85</td><td>0.816</td><td>0.713</td><td></td></tr><tr><td>2C [23]</td><td>9.30</td><td>10.47</td><td>0.990</td><td>0.959</td><td></td><td>6.57</td><td>84.27</td><td>0.745</td><td>0.556</td><td></td></tr><tr><td>D2D-CE</td><td>9.79</td><td>8.27</td><td>0.993</td><td>0.991</td><td>√</td><td>11.76</td><td>39.32</td><td>0.942</td><td>0.852</td><td>√</td></tr><tr><td rowspan=\"3\">Least square [10]</td><td>PD[19]</td><td>9.94</td><td>8.26</td><td>0.993</td><td>0.992</td><td>√</td><td>12.74</td><td>37.14</td><td>0.920</td><td>0.900</td><td>√</td></tr><tr><td>2C[23]</td><td>8.66</td><td>12.18</td><td>0.986</td><td>0.941</td><td></td><td>9.58</td><td>53.10</td><td>0.916</td><td>0.706</td><td></td></tr><tr><td>D2D-CE</td><td>9.70</td><td>9.56</td><td>0.991</td><td>0.987</td><td></td><td>9.50</td><td>57.67</td><td>0.848</td><td>0.692</td><td></td></tr><tr><td rowspan=\"3\">W-GP [13]</td><td>PD[19]</td><td>5.71</td><td>64.75</td><td>0.792</td><td>0.652</td><td></td><td>6.67</td><td>84.16</td><td>0.696</td><td>0.498</td><td></td></tr><tr><td>2C [23]</td><td>5.93</td><td>55.99</td><td>0.842</td><td>0.709</td><td></td><td>6.89</td><td>74.45</td><td>0.812</td><td>0.536</td><td></td></tr><tr><td>D2D-CE</td><td>7.30</td><td>35.94</td><td>0.942</td><td>0.847</td><td>√</td><td>8.92</td><td> 52.74</td><td>0.856</td><td>0.689</td><td>√</td></tr><tr><td rowspan=\"3\">Hinge [46]</td><td>PD[19]</td><td>9.81</td><td>8.08</td><td>0.993</td><td>0.992</td><td></td><td>12.78</td><td>32.03</td><td>0.948</td><td>0.868</td><td></td></tr><tr><td>2C [23]</td><td>9.70</td><td>8.22</td><td>0.993</td><td>0.991</td><td></td><td>13.46</td><td>28.55</td><td>0.974</td><td>0.881</td><td></td></tr><tr><td>D2D-CE</td><td>9.89</td><td>7.88</td><td>0.994</td><td>0.992</td><td>√</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>√</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/ae88335aa74ef3f4e06044ae71e86fd59505e41e675a879a0dae35e5dc54b2d4.jpg",
+        "table_caption": [
+            "Table 5: FID [34] values of conditional StyleGAN2 (cStyleGAN2) [7] and StyleGAN2 $[ 7 ] + \\mathrm { D } 2 \\mathrm { D } { \\cdot } \\mathrm { C E }$ on CIFAR10 [30] and AFHQ [41] datasets. ADA means the adaptive discriminator augmentation [8]. FID is exceptionally computed using the training split for calculating the reference moments since FID value of StyleGAN2 is often calculated using the moments of the training dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Conditioning method</td><td>CIFAR10 [30]</td><td>AFHQ [41]</td></tr><tr><td>cStyleGAN2 [7]</td><td>3.88</td><td>=</td></tr><tr><td>StyleGAN2 [7] + D2D-CE</td><td> 3.34</td><td></td></tr><tr><td>cStyleGAN2 [7] + ADA [8]</td><td>2.43</td><td>4.99</td></tr><tr><td>StyleGAN2 [7] + ADA [8] + D2D-CE</td><td>2.38</td><td>4.95</td></tr><tr><td> StyleGAN2 [7] + DiffAug [9] + D2D-CE + Tuning</td><td>2.26</td><td>-</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Effect of D2D-CE for Different GAN Architectures. We study the effect of D2D-CE loss with different GAN architectures. In Table 5, we validate that D2D-CE is effective for StyleGAN2 [7] backbone and also fits well with the adaptive discriminator augmentation (ADA) [8]. StyleGAN2 with D2D-CE loss produces $1 3 . 9 \\%$ better generation result than the conditional version of StyleGAN2 (cStyleGAN2) on CIFAR10. Moreover, StyleGAN2 with D2D-CE can be reinforced with ADA or DiffAug when train StyleGAN2 $^ +$ D2D-CE under the limited data situation. Among GANs, ${ \\mathrm { S t y l e G A N } } 2 + { \\mathrm { D i f f A u g } } + { \\mathrm { D } } 2 { \\mathrm { D } } { \\mathrm { - C E } } + { \\mathrm { 1 } }$ Tuning achieves the best performance on CIFAR10, even outperforming some diffusion-based methods [47, 48]. Additional results with other architectures, i.e., a deep convolutional network [2] and a resnet style backbone [13], are provided in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4.4 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We study how each component of ReACGAN affects ACGAN training. By adding or ablating each part of ReACGAN, as shown in Table 6, we identify four major observations. (1) Feature and weight normalization greatly stabilize ACGAN training and improve generation performances on Tiny-ImageNet and CUB200 datasets. (2) D2D-CE enhances the generation performance by considering data-to-data relationships and by performing easy sample suppression (3th and 4th rows). (3) the suppression technique does not work well on the feature normalized cross-entropy loss (5th row). (4) While ACGAN shows high Inception score on ImageNet experiment, it shows relatively poor FID, $\\mathrm { F _ { 0 . 1 2 5 } }$ , and $\\mathrm { F _ { 8 } }$ values compared with the model trained with normalization, data-to-data consideration, and the suppression technique. This result is consistent with the qualitative results on ImageNet, where the images from ACGAN are easily classifiable, but the images from ReACGAN are high quality and diverse. We attribute the improvement to the discriminator that successfully leverages informative data-to-data and data-to-class relationships with easy sample suppression. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/a59ce57281fb29c29f8a0af958d8e87aa81a5b050192d0c14ce2973864977a69.jpg",
+        "image_caption": [
+            "Figure 5: Curated images generated by the proposed ReACGAN. More qualitative results on ReACGAN, BigGAN [4], ContraGAN [23], and ACGAN [18] are attached in Appendix J. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            130,
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d7edc5dfca5366716959d4814eb9bf165b368a9f9e4aa544c93ae94e2b174a7.jpg",
+        "table_caption": [
+            "Table 6: Ablation study on feature map normalization, data-to-data consideration, and easy positive and negative sample suppression. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ablation</td><td colspan=\"4\">Tiny-ImageNet [32]</td><td colspan=\"4\">CUB200 [33]</td><td colspan=\"4\">ImageNet [49]</td></tr><tr><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8↑</td><td>IS个</td><td>FID↓</td><td>F0.125↑</td><td>F8个</td></tr><tr><td>ACGAN [18]</td><td>6.00</td><td>96.04</td><td>0.656</td><td>0.368</td><td>6.09</td><td>60.73</td><td>0.726</td><td>0.891</td><td>62.99</td><td>26.35</td><td>0.935</td><td>0.963</td></tr><tr><td>+ Normalization</td><td>13.46</td><td>30.33</td><td>0.955</td><td>0.889</td><td>4.78</td><td>25.54</td><td>0.883</td><td>0.952</td><td>18.16</td><td>36.40</td><td>0.879</td><td>0.787</td></tr><tr><td>+ Data-to-data (Eq. (5))</td><td>12.96</td><td>28.71</td><td>0.967</td><td>0.863</td><td>5.08</td><td>25.12</td><td>0.894</td><td>0.946</td><td>-</td><td>=</td><td>-</td><td></td></tr><tr><td> + Suppression (Eq. (6))</td><td>14.06</td><td>27.10</td><td>0.970</td><td>0.894</td><td>4.91</td><td>15.40</td><td>0.970</td><td>0.954</td><td>63.16 14.59</td><td></td><td></td><td>0.9740.974</td></tr><tr><td>- Data-to-data</td><td>12.96</td><td>30.79</td><td>0.960</td><td>0.857</td><td>5.39</td><td>30.36</td><td>0.863</td><td>0.947</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we have analyzed why training ACGAN becomes unstable as the number of classes in the dataset increases. By deriving the analytic form of gradient in the classifier and numerically checking the gradient values, we have discovered that the unstable training comes from a gradient exploding problem caused by the unboundedness of input feature vectors and poor classification ability of the classifier in the early training stage. To alleviate the instability and reinforce ACGAN, we have proposed the Data-to-Data Cross-Entropy loss (D2D-CE) and the Rebooted Auxiliary Classifier Generative Adversarial Network (ReACGAN). The experimental results verify the superiority of ReACGAN compared with the existing classifier- and projection-based GANs on five benchmark datasets. Moreover, exhaustive analyses on ReACGAN prove that ReACGAN is robust to hyperparameter selection and harmonizes with various architectures and differentiable augmentations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the IITP grants (No.2019-0-01906: AI Graduate School Program - POSTECH, No.2021-0-00537: Visual Common Sense, No.2021-0-02068: AI Innovation Hub) funded by Ministry of Science and ICT, Korea. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems (NeurIPS), pages 2672–2680, 2014.   \n[2] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. arXiv preprint arXiv 1511.06434, 2016.   \n[3] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   \n[4] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large Scale GAN Training for High Fidelity Natural Image Synthesis. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.   \n[5] Tero Karras, Samuli Laine, and Timo Aila. 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Technical report, California Institute of Technology, 2010.   \n[34] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In Advances in Neural Information Processing Systems (NeurIPS), pages 6626–6637, 2017.   \n[35] Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization. In Advances in Neural Information Processing Systems (NeurIPS), pages 271–279, 2016.   \n[36] Akash Srivastava, Lazar Valkov, Chris Russell, Michael U Gutmann, and Charles A Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In Advances in Neural Information Processing Systems (NeurIPS), 2017.   \n[37] Mehdi Mirza and Simon Osindero. Conditional Generative Adversarial Nets. arXiv preprint arXiv 1411.1784, 2014.   \n[38] Zhiming Zhou, Han Cai, Shu Rong, Yuxuan Song, Kan Ren, Weinan Zhang, Jun Wang, and Yong Yu. Activation Maximization Generative Adversarial Nets. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.   \n[39] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey E. Hinton. A Simple Framework for Contrastive Learning of Visual Representations. arXiv preprint arXiv 2002.05709, 2020.   \n[40] Minguk Kang and Jaesik Park. Pytorch-studiogan. https://github.com/POSTECH-CVLab/ PyTorch-StudioGAN, 2020.   \n[41] Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. StarGAN v2: Diverse Image Synthesis for Multiple Domains. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   \n[42] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, and Xi Chen. Improved Techniques for Training GANs. In Advances in Neural Information Processing Systems (NeurIPS), pages 2234–2242, 2016.   \n[43] Mehdi SM Sajjadi, Olivier Bachem, Mario Lucic, Olivier Bousquet, and Sylvain Gelly. Assessing generative models via precision and recall. In Advances in Neural Information Processing Systems (NeurIPS), 2018.   \n[44] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur. A Learned Representation For Artistic Style. In Proceedings of the International Conference on Learning Representations (ICLR), 2017.   \n[45] Harm de Vries, Florian Strub, Jeremie Mary, Hugo Larochelle, Olivier Pietquin, and Aaron C Courville. Modulating early visual processing by language. In Advances in Neural Information Processing Systems (NeurIPS), pages 6594–6604, 2017.   \n[46] Jae Hyun Lim and Jong Chul Ye. Geometric GAN. arXiv preprint arXiv 1705.02894, 2017.   \n[47] Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Proceedings of the International Conference on Machine Learning (ICML), 2021.   \n[48] Dongjun Kim, Seungjae Shin, Kyungwoo Song, Wanmo Kang, and Il-Chul Moon. Score Matching Model for Unbounded Data Score. arXiv preprint arXiv:2106.05527, 2021.   \n[49] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou. Arcface: Additive angular margin loss for deep face recognition. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2019. ",
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parse/train/r1Usiwcex/r1Usiwcex.md

@@ -0,0 +1,267 @@
+# COUNTERPOINT BY CONVOLUTION
+
+Cheng-Zhi Anna Huang∗ MILA, Universite de Montr ´ eal ´ chengzhiannahuang@gmail.com
+
+Tim Cooijmans† MILA, Universite de Montr ´ eal ´ tim.cooijmans@umontreal.ca
+
+Adam Roberts   
+Google Brain   
+adarob@google.com
+
+Aaron Courville MILA, Universite de Montr´ eal´ aaron.courville@umontreal.ca
+
+Douglas Eck   
+Google Brain   
+deck@google.com
+
+# ABSTRACT
+
+Machine learning models of music typically break down the task of composition into a chronological process, composing a piece of music in a single pass from beginning to end. On the contrary, human composers write music in a nonlinear fashion, scribbling motifs here and there, often revisiting choices previously made. We explore the use of blocked Gibbs sampling as an analogue to the human approach, and introduce COCONET, a convolutional neural network in the NADE family of generative models (Uria et al., 2016). Despite ostensibly sampling from the same distribution as the NADE ancestral sampling procedure, we find that a blocked Gibbs approach significantly improves sample quality. We provide evidence that this is due to some conditional distributions being poorly modeled. Moreover, we show that even the cheap approximate blocked Gibbs procedure from Yao et al. (2014) yields better samples than ancestral sampling. We demonstrate the versatility of our method on unconditioned polyphonic music generation.
+
+![](images/76c6a45fb4238b7ac19558e4f5a13b990679ecf9101a0c589cf902dafaaa8e6e.jpg)  
+Figure 1: Ancestral inpainting of a corrupted Bach chorale by COCONET. Colors are used to distinguish the four voices. Grayscale heatmaps show predictions $p ( \mathbf { x } _ { i } \mid \mathbf { x } _ { C } )$ . The pitch sampled in the current step is indicated by a rectangular outline. The original Bach chorale is shown in the bottom right. Step 0 shows the corrupted Bach chorale. Step 64 shows the result.
+
+# 1 INTRODUCTION
+
+Machine learning can be used to create compelling art. This was shown recently by DeepDream (Mordvintsev et al., 2015), an optimization process that created psychedelic transformations of images. A similar idea underlies a variety of style transfer algorithms (Gatys et al., 2015), which impose textures and colors from one image onto another. More recently, the multistyle pastiche generator (Dumoulin et al., 2016) exposes adjustable knobs that allow users of the system fine-grained control over style transfers. Neural doodle (Champandard, 2016) further closes the feedback loop between algorithm and artist.
+
+We wish to bring similar artistic tools to the domain of music. Whereas previous work in music has relied mainly on sequence models such as Hidden Markov Models (HMMs, Baum & Petrie (1966)) and Recurrent Neural Networks (RNNs, Rumelhart et al. (1988)), we instead employ convolutional neural networks due to their emphasis on capturing local structure and their invariance properties. Moreover, convolutional neural networks have shown to be extremely versatile once trained, as shown by a variety of creative uses in the literature (Mordvintsev et al., 2015; Gatys et al., 2015; Almahairi et al., 2015; Lamb et al., 2016).
+
+We introduce COCONET, a deep convolutional model trained to reconstruct partial scores. Once trained, COCONET provides direct access to all conditionals of the form $p ( \mathbf { x } _ { i } \mid \mathbf { x } _ { C } )$ where $\mathbf { x } _ { C }$ is a fragment of a musical score $\mathbf { x }$ and $i \not \in C$ is in its complement. Figure 1 shows an example of such conditionals used in completing a partial score.
+
+COCONET is an instance of deep orderless NADE Uria et al. (2014), and thus learns an ensemble of factorizations of the joint $p ( \mathbf { x } )$ . However, the sampling procedure for orderless NADE is not orderless. Sampling from an orderless NADE involves (randomly) choosing an ordering, and sampling ancestrally according to the chosen ordering. We have found that this produces poor results for the highly structured and complex domain of musical counterpoint.
+
+Instead, we propose to use blocked-Gibbs sampling, essentially improving sample quality through rewriting. An instance of this was previously explored by Yao et al. (2014) who employed a NADE in the transition operator for a Markov Chain, yielding a Generative Stochastic Network (GSN). The transition consists of a corruption process that masks out a subset $\mathbf { x } _ { \lnot C }$ of variables, followed by a process that independently resamples variables $\mathbf { x } _ { i } , i \notin C$ according to the distribution $p _ { \theta } ( \mathbf { x } _ { i } \mid \mathbf { x } _ { C } )$ emitted by the NADE. Crucially, the effects of independent sampling are amortized by annealing the probability with which variables are masked out. Whereas Yao et al. (2014) treat their procedure as a cheap approximation to ancestral sampling, we find that it produces superior samples.
+
+We show the versatility of our method on unconditioned polyphonic music generation.
+
+Section 2 discusses previous work in the area of automatic musical composition. The details of our model and training procedure are laid out in Section 3. In Section 4 we show that our approach is equivalent to that of deep and orderless NADE Uria et al. (2014). We discuss sampling from our model in Section 5. Results of quantitative and qualitative evaluations are reported in Section 6. Finally, Section 7 concludes.
+
+# 2 RELATED WORK
+
+Sequence models such as HMMs and RNNs are a natural choice for modeling music. However, one of the challenges in adapting such models to music is that music generally consists of multiple interdependent streams of events. This can be most clearly seen in the notion of counterpoint, which refers to the relationships between the movement of individual instruments in a musical work. Compare this to typical sequence domains such as speech and language, which involve modeling a single stream of events: a single speaker or a single stream of words.
+
+Successful application of sequence models to music hence requires serializing or otherwise rerepresenting the music to fit the sequence paradigm. For instance, Liang (2016) serialize four-part Bach chorales by interleaving the parts, while Allan & Williams (2005) construct a chord vocabulary. Boulanger-Lewandowski et al. (2012) adopt a piano roll representation, which is a binary matrix $\mathbf { x }$ such that $\mathbf { x } _ { i t }$ is hot if some instrument is playing pitch $i$ at time $t$ . To model the joint probability distribution of the multi-hot pitch vector $\mathbf { x } _ { t }$ , they employ a Restricted Boltzmann Machine (RBM (Smolensky, 1986; Hinton et al., 2006)) or Neural Autoregressive Distribution Estimator (Uria et al., 2016) at each time step.
+
+Moreover, the behavior of human composers does not fit the chronological mold assumed by previous authors. A human composer might start his work with a coarse chord progression and iteratively refine it, revisiting choices previously made. Sampling according to $x _ { t } \sim p ( x _ { t } | \boldsymbol { x } _ { < t } )$ , as is common, cannot account for the kinds of timeless dependencies that composers employ. Hadjeres et al. (2016) sidestep the choice of causal factorization and instead employ an undirected Markov model to learn pairwise relationships between neighboring notes up to a specified number of steps away in a score. Sampling involves Markov Chain Monte Carlo (MCMC) using the model as a Metropolis-Hastings (MH) objective. The model permits constraints on the state space to support tasks such as melody harmonization. However, the Markov assumption severely limits the expressivity of the model.
+
+We opt instead for a convolutional approach that avoids many of these issues and naturally captures both relationships across time and interactions between instruments.
+
+# 3 COUNTERPOINT BY CONVOLUTION
+
+We approach the task of music composition with a deep convolutional neural network (Krizhevsky et al., 2012). This choice is motivated by the locality of contrapuntal rules and their near-invariance to translation, both in time and in the frequency spectrum.
+
+We represent the music as a stack of piano rolls encoded in a binary three-tensor $\mathbf { x } \in \{ 0 , 1 \} ^ { I \times T \times P }$ . Here $I$ denotes the number of instruments, $T$ the number of time steps, $P$ the number of pitches, and $\mathbf { x } _ { i , t , p } = 1$ iff the ith instrument plays pitch $p$ at time $t$ . We will assume each instrument plays exactly one pitch at a time, that is, $\begin{array} { r } { \sum _ { p } \mathbf { x } _ { i , t , p } = 1 } \end{array}$ for all $i , t$ .
+
+For the present work we will restrict ourselves to the study of four-part Bach chorales as used in prior work (Allan & Williams, 2005; Boulanger-Lewandowski et al., 2012; Goel et al., 2014; Liang, 2016; Hadjeres et al., 2016). Hence we assume $I = 4$ throughout. We discretize pitch according to equal temperament, but constrain ourselves to only the range that appears in our training data (MIDI pitches 36 through 88). Time is discretized at the level of 16th notes for similar reasons. To curb memory requirements, we enforce $T = 6 4$ by randomly cropping the training examples.
+
+Given a training example $\mathbf { x } \sim p ( \mathbf { x } )$ , we present the model with the values of only a strict subset of its elements $\mathbf { x } _ { C } = \{ \mathbf { x } _ { ( i , t ) } \mid ( i , t ) \in C \}$ and ask it to reconstruct its complement $\mathbf { x } _ { \lnot C }$ . The input to the model is obtained by masking the piano rolls $\mathbf { x }$ to obtain the context $\mathbf { x } _ { C }$ and concatenating this with the corresponding mask:
+
+$$
+\begin{array} { r } { \mathbf { h } _ { i , t , p } ^ { 0 } = \mathbb { 1 } _ { ( i , t ) \in C } \mathbf { x } _ { i , t , p } } \\ { \mathbf { h } _ { I + i , t , p } ^ { 0 } = \mathbb { 1 } _ { ( i , t ) \in C } } \end{array}
+$$
+
+where the first dimension ranges over channels and the time and pitch dimensions are convolved over.
+
+$$
+\begin{array} { l l } { { \displaystyle { \bf { a } } ^ { l } = { \mathrm { B N } } ( { \bf { W } } ^ { l } * { \bf { h } } ^ { l - 1 } ; \gamma ^ { l } , \beta ^ { l } ) } } \\ { { \displaystyle { \bf { h } } ^ { l } = { \mathrm { R e L U } } ( { \bf { a } } ^ { l } + { \bf { h } } ^ { l - 2 } ) } } & { { \qquad \mathrm { f o r ~ } 3 < l < L - 1 \mathrm { ~ a n d ~ } l = 0 \mathrm { ~ m o d ~ } 2 } } \\ { { \displaystyle { \bf { h } } ^ { L } = { \bf { a } } ^ { L } } } \end{array}
+$$
+
+With the exception of the first and final layers, all of our convolutions preserve the size of the input. That is, we use “same” padding throughout and all activations $h ^ { l } , 1 \overline { { < } } \ l < L$ have 128 channels. The network consists of 64 layers with $3 \times 3$ filters on each layer. After each convolution we apply batch normalization Ioffe & Szegedy (2015) (denoted by $\mathrm { B N } ( \cdot ) ,$ ) with statistics tied across time and pitch. After every second convolution, we introduce a skip connection from the hidden state two levels below to reap the benefits of residual learning He et al. (2015).
+
+Finally, we obtain predictions for the pitch at each instrument/time pair:
+
+$$
+p _ { \theta } ( \mathbf { x } _ { i , t , p } \mid \mathbf { x } _ { C } , C ) = \frac { \exp ( h _ { i , t , p } ^ { L } ) } { \sum _ { p } \exp ( h _ { i , t , p } ^ { L } ) }
+$$
+
+The loss function is given by
+
+$$
+\begin{array} { l } { { \mathcal { L } } ( \mathbf { x } ; C , \theta ) = - \displaystyle \sum _ { ( i , t ) \notin C } \log p _ { \theta } ( \mathbf { x } _ { i , t } \mid \mathbf { x } _ { C } , C ) } \\ { = - \displaystyle \sum _ { ( i , t ) \notin C } \sum _ { p } \mathbf { x } _ { i , t , p } \log p _ { \theta } ( \mathbf { x } _ { i , t , p } \mid \mathbf { x } _ { C } , C ) } \end{array}
+$$
+
+where $p _ { \theta }$ denotes the probability under the model with parameters $\theta \ : \ : = \ : \ : \mathbf { W } ^ { 1 } , \gamma ^ { 1 } , \beta ^ { 1 } , \dots ,$ , $\mathbf { W } ^ { L - 1 } , \gamma ^ { L - 1 } , \beta ^ { L - 1 }$ . To minimize the expected loss
+
+$$
+\mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } \mathbb { E } _ { C \sim p ( C ) } \mathcal { L } ( \mathbf { x } ; C , \boldsymbol { \theta } )
+$$
+
+with respect to $\theta$ , we sample piano rolls $\mathbf { x }$ from the training set and contexts $C \sim p ( C )$ and optimize by stochastic gradient descent with step size determined by Adam (Kingma & Ba, 2014).
+
+# 4 RELATIONSHIP TO NADE
+
+Our approach is an instance of orderless and deep Neural Autoregressive Distribution Estimators (Uria et al., 2016). NADE models a $d$ -variate distribution $p ( \mathbf { x } )$ through a factorization
+
+$$
+p _ { \theta } ( \mathbf { x } ) = \prod _ { d } p _ { \theta } ( \mathbf { x } _ { o _ { d } } \mid \mathbf { x } _ { o _ { < d } } )
+$$
+
+where $o$ is a permutation, and the parameters $\theta$ are shared among the conditionals. NADE can be trained for all orderings $o$ simultaneously using the orderless NADE (Uria et al., 2014) training procedure. This procedure relies on the observation that, thanks to parameter sharing, computing $\bar { p } _ { \theta } ( \mathbf { x } _ { o _ { d ^ { \prime } } } \mid \mathbf { x } _ { o _ { < d } } )$ for all $d ^ { \prime } \geq d$ is no more expensive than computing it only for $d ^ { \prime } = d$ . Hence for a given $o$ and $d$ we can simultaneously obtain partial losses for all orderings that agree with $o$ up to $d$ :
+
+$$
+\mathcal { L } _ { \mathrm { N A D E } } ( \mathbf { x } ; o _ { < d } , \theta ) = - \sum _ { o _ { d } } \log p _ { \theta } ( \mathbf { x } _ { o _ { d } } \mid \mathbf { x } _ { o _ { < d } } , o _ { < d } , o _ { d } )
+$$
+
+Letting $o _ { < d } = C$ , we obtain our loss from Equation 7
+
+$$
+\mathcal { L } _ { \mathrm { C o c o n g r } } ( \mathbf { x } ; C , \theta ) = - \sum _ { ( i , t ) \notin C } \log p _ { \theta } ( \mathbf { x } _ { i , t } \mid \mathbf { x } _ { C } , C )
+$$
+
+For any one sample $( \mathbf { x } , C )$ , this loss consists of $| \neg C |$ terms of the form $\log p _ { \theta } ( \mathbf { x } _ { i , t } \mid \mathbf { x } _ { C } , C )$ . We let $p ( C )$ be uniform in the size of the mask $| C |$ and reweight the sample losses according to
+
+$$
+\tilde { \mathcal { L } } ( \mathbf { x } ; C , \theta ) = \frac { 1 } { | \neg C | } \mathcal { L } ( \mathbf { x } ; C , \theta ) .
+$$
+
+This correction, due to Uria et al. (2014), ensures consistent estimation of the negative log-likelihood of the joint $p _ { \theta } ( \mathbf { x } )$ .
+
+However, we might wish to increase the difficulty by choosing $p ( C )$ so as to frequently mask out large contiguous regions, as otherwise the model might learn only superficial local relationships. This is discussed in Pathak et al. (2016) for the case of images, where a model might learn only that pixels are similar to their neighbors. Similar low-level relationships hold in our case, as our piano roll representation is binary and very sparse. For instance, if we mask out only a single sixteenth step in the middle of a long-held note, reconstructing the masked out step does not require any deep understanding of music. To this end we also consider choosing the context $C$ by independent Bernoulli samples, such that each variable has a low probability of being included in the context.
+
+# 5 SAMPLING
+
+We can sample from the model using the NADE ancestral ordering procedure. However, we find that this yields poor samples, and we propose instead to use Gibbs sampling.
+
+# 5.1 NADE SAMPLING
+
+To sample according to NADE, we start with an empty (zero everywhere) piano roll $\mathbf { x } ^ { 0 }$ and context $C ^ { 0 }$ and populate them iteratively by the following process. We feed the piano roll $\mathbf { x } ^ { s }$ and context $C ^ { s }$ into the model to obtain a set of categorical distributions $p _ { \theta } ( \mathbf { x } _ { i , t } | \mathbf { x } _ { C ^ { s } } ^ { s } , C ^ { s } )$ for $( i , t ) \notin C ^ { s }$ . As the $\mathbf { x } _ { i , t }$ are not conditionally independent, we cannot simply sample from these distributions independently. However, if we sample from one of thfor the others. Hence we randomly choose one $( i , t ) ^ { s + 1 }$ an compute new conditio to sample from, and let $\mathbf { x } _ { i , t } ^ { s + 1 }$ stributionsequal the one-hot realization. Augment the context with $C ^ { s + 1 } = C ^ { s } \cup ( i , t )$ and repeat until the piano roll is populated. This procedure is easily generalized to tasks such as melody harmonization and partial score completion by starting with a nonempty piano roll.
+
+Unfortunately, samples thus generated are of low quality, which we surmise is due to accumulation of errors. While the model provides conditionals $p _ { \theta } ( \mathbf { x } _ { i , t } | \mathbf { x } _ { C } , C )$ for all $( i , t ) \notin C$ , some of these conditionals may be better modeled than others. We suspect in particular those conditionals used early on in the procedure, for which the context $C$ consists of very few variables. Moreover, although the model is trained to be order-agnostic, different orderings invoke different distributions, which is another indication that some conditionals are poorly learned. We test this hypothesis in Section 6.2.
+
+# 5.2 GIBBS SAMPLING
+
+To remedy this, we allow the model to revisit its choices: we repeatedly mask out some part of the piano roll and then repopulate it. This is a form of blocked Gibbs sampling (Liu, 1994). Blocked sampling is crucial for mixing, as the high temporal resolution of our representation causes strong correlations between consecutive notes. For instance, without blocked sampling, it would take many steps to snap out of a long-held note. Similar observations hold for the Ising model from statistical mechanics, leading to the development of the Swendsen-Wang algorithm (Swendsen & Wang, 1987) in which large clusters of variables are resampled at once.
+
+We consider two strategies for resampling a given block of variables: ancestral sampling and independent sampling. Ancestral sampling invokes the orderless NADE sampling procedure described in Section 5.1 on the masked-out portion of the piano roll. Independent sampling simply treats the masked-out variables $\mathbf { x } { \lrcorner } C$ as independent given the context $\mathbf { x } _ { C }$ .
+
+Using independent blocked Gibbs to sample from a NADE model has been studied by Yao et al. (2014), who propose to use an annealed masking probability given by
+
+$$
+\alpha _ { n } = \operatorname* { m a x } \Bigg ( \alpha _ { \operatorname* { m i n } } , \alpha _ { \operatorname* { m a x } } - \frac { n } { \eta N } ( \alpha _ { \operatorname* { m a x } } - \alpha _ { \operatorname* { m i n } } ) \Bigg )
+$$
+
+for some minimum and maximum probabilities $\alpha _ { \mathrm { m i n } } , \alpha _ { \mathrm { m a x } }$ , number of Gibbs steps $N$ and fraction $\eta$ of time spent before settling onto the minimum probability $\alpha _ { \mathrm { m i n } }$ . This scheme ensures the Gibbs process with independent resampling produces samples from the model distribution $p _ { \theta } ( \mathbf { x } )$ . Initially, when the masking probability is high, the chain mixes fast but samples are poor due to independent sampling. As the masking probability reduces, fewer variables are sampled at a time, until finally variables are sampled one at a time and conditioned on all the others.
+
+Table 1: Negative log-likelihood on the test set for the Bach corpus. As discussed in the text, our numbers are not directly comparable to those of other authors due to the use of different splits. Results from Boulanger-Lewandowski et al. (2012) were based on an eighth-note temporal resolution (our resolution is sixteenth notes). Please note that our results are preliminary validation likelihoods.   
+
+<table><tr><td>Model</td><td>Notewise NLL</td><td>Framewise NLL</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Bachbot (Liang, 2016)</td><td>0.477</td><td></td></tr><tr><td>NADE (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>7.19</td></tr><tr><td>RNN-RBM (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>6.27</td></tr><tr><td>RNN-NADE (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>5.56</td></tr><tr><td>CoCONET, i.i.d Bernoulli(0.50)</td><td>0.924</td><td>8</td></tr><tr><td>CoCONET, i.i.d Bernoulli(0.25)</td><td>0.655</td><td>4.48</td></tr><tr><td>COCONET, i.i.d Bernoulli(0.10)</td><td>0.812</td><td>4.66</td></tr><tr><td>CocONET,importance sampling</td><td>0.569</td><td>3.73</td></tr></table>
+
+Yao et al. (2014) treat independent blocked Gibbs as a cheap approximation to ancestral sampling. Indeed, per Gibbs step, independent sampling requires only a single model evaluation, whereas ancestral sampling requires as many model evaluations as there are variables to sample. Moreover, we find that independent blocked Gibbs sampling in fact yields better samples than the NADE procedure from Section 5.1. Samples can be heard here: https:// soundcloud.com/czhuang/sets/coconet-nade and https://soundcloud.com/ czhuang/sets/coconet-independent-gibbs.
+
+# 6 EVALUATION
+
+We evaluate our approach on a corpus of four-part Bach chorales. The literature features many variants of this dataset (Allan & Williams, 2005; Boulanger-Lewandowski et al., 2012; Liang, 2016; Hadjeres et al., 2016), and we follow the unfortunate tradition of introducing our own adaptation. Although this complicates comparisons against earlier work, we feel justified in doing so as our approach requires instruments to be separated, and other authors’ eighth-note temporal resolution is too coarse to accurately convey counterpoint.
+
+We rebuilt our dataset from the Bach chorale musicXML scores readily available through (Cuthbert & Ariza, 2010), which was also the basis for the dataset used in (Liang, 2016). The scores included 357 four-part Bach chorales. We excluded scores that included note durations less than sixteenth notes, resulting in 354 pieces. These pieces were split into train/valid/test in $6 0 / 2 0 / 2 0 \%$ ratios.
+
+We compare with Liang (2016) based on note-level likelihood and Boulanger-Lewandowski et al. (2012) based on frame-level likelihood. Note that train/valid/test differs among both prior work and also with our work, and that Liang (2016) uses a $8 0 / 1 0 / 1 0 \%$ split instead.
+
+However, evaluation of generative models is hard (Theis et al., 2015). The gold standard for evaluation is qualitative comparison by humans, and we therefore report human evaluation results.
+
+# 6.1 EVALUATING LOG-LIKELIHOOD
+
+To estimate the log-likelihood of a datapoint $\mathbf { x }$ , we follow the orderless NADE approach. That is, we sample a random ordering $( i _ { 1 } , t _ { 1 } ) , ( i _ { 2 } , t _ { 2 } ) , \dots ( i _ { I T } , t _ { I T } )$ , and compute the notewise log-likelihood
+
+$$
+\log \hat { p } _ { \boldsymbol { \theta } } ( \mathbf { x } ) = \frac { 1 } { I T } \sum _ { d = 1 } ^ { I T } \log p _ { \boldsymbol { \theta } } ( \mathbf { x } _ { i _ { d } , t _ { d } } \mid \mathbf { x } _ { C _ { d - 1 } } , C _ { d - 1 } )
+$$
+
+where $\begin{array} { r } { C _ { d } = \bigcup _ { c = 1 } ^ { d } \{ ( i _ { c } , t _ { c } ) \} } \end{array}$ . Note that we randomly crop each datapoint to be $T$ time steps long before processing it, as this facilitates batch processing.
+
+We repeat this procedure $k$ times and average across all point estimates. The numbers for our models in Table 1 were obtained with $k = 5$ .
+
+The process for computing the notewise log-likelihood is akin to teacher-forcing, where at each step of the way the model observes the ground truth for all its previous predictions. To compute the framewise log-likelihood, we instead let the model run free within each frame $t$ . This results in a more representative measure of the model’s quality as it is sensitive to accumulation of error.
+
+Table 1 lists notewise and framewise likelihoods of the validation data under variants of our model, as well as comparable results from other authors. We include four variants of COCONET that differ in the choice of the distribution $p ( C )$ over contexts during training. By importance sampling we refer to the orderless NADE strategy discussed in Section 4, in which $p ( C )$ is uniform over $| C |$ and the sampled losses are reweighted by $1 / | \neg C |$ . We also evaluate three variants where the contexts are chosen by biased coin flips, that is, $\operatorname* { P r } ( ( i , t ) \in C ) = \rho$ , for $\rho \in { 0 . 5 , 0 . 2 5 , 0 . 1 }$ . The framewise log-likelihood for $\rho = 0 . 5$ is listed as $\infty$ as its estimation repeatedly overflowed.
+
+Overall, COCONET seems to underperform in terms of notewise likelihood, yet perform well in terms of framewise likelihood. Estimating the loss by importance sampling appears to work significantly better than determining the context using independent Bernoulli variables, as one might expect. However, the choice of Bernoulli probability $\rho$ strongly affects the resulting loss, which suggests that some of the conditionals benefit from more training.
+
+Table 2: Mean $( \pm \mathrm { S E M } )$ negative log-likelihood under the model of unconditioned samples generated from the model by various procedures.   
+
+<table><tr><td>Sampling scheme</td><td>Notewise NLL</td><td>Framewise NLL</td></tr><tr><td>Ancestral Gibbs,ρ = O.0O (NADE)</td><td>0.565 ± 0.011</td><td>3.872 ± 0.052</td></tr><tr><td>Ancestral Gibbs,ρ= 0.01</td><td>0.560 ± 0.010</td><td>3.824 ± 0.052</td></tr><tr><td>Ancestral Gibbs,ρ= 0.25</td><td>0.444 ± 0.008</td><td>3.276 ± 0.036</td></tr><tr><td>Ancestral Gibbs,ρ = 0.50</td><td>0.438 ± 0.007</td><td>3.332 ± 0.040</td></tr><tr><td>Contiguous Gibbs, ρ = 0.50</td><td>0.447 ± 0.008</td><td>3.476 ± 0.048</td></tr><tr><td>Independent Gibbs (Yao et al., 2014)</td><td>0.440 ± 0.008</td><td>3.348 ± 0.040</td></tr></table>
+
+# 6.2 SAMPLE QUALITY
+
+In Section 5 we conjectured that the low quality of NADE samples is due to poorly modeled conditionals $p _ { \theta } ( \mathbf { x } _ { i , t } \mid \mathbf { x } _ { C } , C )$ where $C$ is small. We test this hypothesis by evaluating the likelihood under the model of samples generated by the ancestral blocked Gibbs procedure with $C$ chosen according to independent Bernoulli variables. When we set the inclusion probability $\rho$ to 0, we obtain NADE. Increasing $\rho$ increases the expected context size $| C |$ , which should yield better samples if our hypothesis is true. The results shown in Table 2 confirm that this is the case. For these experiments, we used sample length $T = 3 2$ time steps and number of Gibbs steps $N = 1 0 0$ .
+
+Figure 2 shows the convergence behavior of the various Gibbs procedures, averaged over 100 runs. We see that for low values of $\rho$ (small $C$ ), the chains hardly make progress beyond NADE in terms of likelihood. Higher values of $\rho$ (large $C$ ) enable the model to bootstrap and reach significantly better likelihood. However, high values of $\rho$ cause the chain to mix slowly, as can be seen in the case where $\rho ~ = ~ 0 . 5 0$ . For comparison, we included a variant, Contiguous(0.50), that always masks out in contiguous chunks of at least four sixteenth notes. This variant converges much more rapidly than Bernoulli(0.50) despite masking out equally many variables on average. Note that whereas ancestral sampling (NADE) requires ${ \cal O } ( I T )$ model evaluations and ancestral Gibbs requires ${ \cal O } ( I T N )$ model evaluations, independent Gibbs requires only $O ( N )$ model evaluations, with typically $N < I T$ .
+
+![](images/36e9ce8ae5d09b5df6de91e8d0188fee78bc2bfacf3592f33b5b94a137775065.jpg)  
+Figure 2: Likelihood under the model of Gibbs samples obtained with various context distributions $p ( C )$ . NADE (equivalent to Bernoulli(0.00)) is included for reference.
+
+# 6.3 HUMAN EVALUATIONS
+
+We carried out a listening test on Amazon’s Mechanical Turk (MTurk) to compare quality of samples from different sources (sampling schemes and Bach). The sampling schemes under study are ancestral Gibbs with Bernoulli(0.00) masking (NADE), independent Gibbs (Yao et al., 2014), and ancestral Gibbs with Contiguous(0.50). For each scheme, we generate four unconditioned samples from empty piano rolls. For Bach, we randomly crop four fragments from the chorale validation set. We thus obtain four sets of four sounds each. All fragments are two measures long, and last twelve seconds after synthesis.
+
+![](images/c6e8aa922a515dfbdc710a2f43e84aa1496664fd38d1ba314bf4cb0648e7a622.jpg)  
+Figure 3: Human evaluations from MTurk on comparing sampling schemes.
+
+For each MTurk hit, users are asked to rate on a Likert scale which of two random samples they perceive as more musical. The study resulted in 192 ratings, where each source was involved in 92 pairwise comparisons. Figure 6.3 reports for each source the number of times it was rated as more musical. We see that although ancestral sampling on NADE performs poorly compared to Bach, both ancestral and independent Gibbs Yao et al. (2014) were considered at least as musical as fragments from Bach, with independent Gibbs Yao et al. (2014) outperforming ancestral sampling (NADE) by a large margin. Pairwise comparisons are listed in Appendix A.
+
+# 7 CONCLUSION
+
+We introduced a convolutional approach to modeling musical scores based on the NADE (Uria et al., 2016) framework. Our experiments show that the NADE ancestral sampling procedure yields poor samples for our domain, which we have argued is because some conditionals are not captured well by the model. We have shown that sample quality improves significantly when we use blocked Gibbs sampling to iteratively rewrite parts of the score. Moreover, annealed independent blocked Gibbs sampling as proposed by Yao et al. (2014) is not only faster but in fact produces better samples.
+
+# ACKNOWLEDGMENTS
+
+We thank Kyle Kastner and Guillaume Alain, Curtis (Fjord) Hawthorne, the Google Brain Magenta team, as well as Jason Freidenfelds for helpful feedback, discussions, suggestions and support.
+
+# REFERENCES
+
+Moray Allan and Christopher KI Williams. Harmonising chorales by probabilistic inference. Advances in neural information processing systems, 17:25–32, 2005.
+
+Amjad Almahairi, Nicolas Ballas, Tim Cooijmans, Yin Zheng, Hugo Larochelle, and Aaron Courville. Dynamic capacity networks. arXiv preprint arXiv:1511.07838, 2015.
+
+Leonard E Baum and Ted Petrie. Statistical inference for probabilistic functions of finite state markov chains. The annals of mathematical statistics, 37(6):1554–1563, 1966.
+
+Nicolas Boulanger-Lewandowski, Yoshua Bengio, and Pascal Vincent. Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription. International Conference on Machine Learning, 2012.
+
+Alex J. Champandard. Neural doodle, 2016. URL https://github.com/alexjc/ neural-doodle.
+
+Michael Scott Cuthbert and Christopher Ariza. music21: A toolkit for computer-aided musicology and symbolic music data. 2010.
+
+Vincent Dumoulin, Johnathon Shlens, and Manjunath Kudlur. A learned representation for artistic style. arXiv preprint arXiv:1610.07629, 2016.
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+Leon A Gatys, Alexander S Ecker, and Matthias Bethge. A neural algorithm of artistic style. arXiv preprint arXiv:1508.06576, 2015.
+
+Kratarth Goel, Raunaq Vohra, and JK Sahoo. Polyphonic music generation by modeling temporal dependencies using a rnn-dbn. In International Conference on Artificial Neural Networks, pp. 217–224. Springer, 2014.
+
+Gaetan Hadjeres, Jason Sakellariou, and Franc¸ois Pachet. Style imitation and chord invention in ¨ polyphonic music with exponential families. arXiv preprint arXiv:1609.05152, 2016.
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+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015.
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+Geoffrey E Hinton, Simon Osindero, and Yee-Whye Teh. A fast learning algorithm for deep belief nets. Neural computation, 18(7):1527–1554, 2006.
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+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
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+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
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+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Alex Lamb, Vincent Dumoulin, and Aaron Courville. Discriminative regularization for generative models. arXiv preprint arXiv:1602.03220, 2016.
+
+Feynman Liang. Bachbot: Automatic composition in style of bach chorales. Masters thesis, University of Cambridge, 2016.
+
+Jun S Liu. The collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem. Journal of the American Statistical Association, 89(427):958–966, 1994.
+
+MIDI. Midi tuning standard. https://en.wikipedia.org/wiki/MIDI_Tuning_ Standard. Accessed: 2016-11-12.   
+Alexander Mordvintsev, Christopher Olah, and Mike Tyka. Inceptionism: Going deeper into neural networks, 2015. URL https://research.googleblog.com/2015/06/ inceptionism-going-deeper-into-neural.html.   
+Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. arXiv preprint arXiv:1604.07379, 2016.   
+David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by backpropagating errors. Cognitive modeling, 5(3):1, 1988.   
+Paul Smolensky. Information processing in dynamical systems: Foundations of harmony theory. Technical report, DTIC Document, 1986.   
+Robert H Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in monte carlo simulations. Physical review letters, 58(2):86, 1987.   
+Lucas Theis, Aaron van den Oord, and Matthias Bethge. A note on the evaluation of generative ¨ models. arXiv preprint arXiv:1511.01844, 2015.   
+Benigno Uria, Iain Murray, and Hugo Larochelle. A deep and tractable density estimator. In ICML, pp. 467–475, 2014.   
+Benigno Uria, Marc-Alexandre Cotˆ e, Karol Gregor, Iain Murray, and Hugo Larochelle. Neural ´ autoregressive distribution estimation. arXiv preprint arXiv:1605.02226, 2016.   
+Li Yao, Sherjil Ozair, Kyunghyun Cho, and Yoshua Bengio. On the equivalence between deep nade and generative stochastic networks. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 322–336. Springer, 2014.
+
+# A PAIRWISE HUMAN EVALUATION RESULTS
+
+This appendix supplements Section 6.3 on the evaluation of samples by human subjects. Figure 3 lists the number of wins, ties and losses for each sample source against each other sample source. All pairs of sources were compared 32 times.
+
+$$
+\begin{array} { r } { \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c } { \mathrm { ~ I } } & { \mathrm { ~ C } } & { \mathrm { ~ N } } & { \mathrm { ~ B } } & & & & & { \mathrm { ~ I } } & { \mathrm { ~ C } } & { \mathrm { ~ N } } & { \mathrm { ~ B } } & & & & { \mathrm { ~ I } } & { \mathrm { ~ C } } & { \mathrm { ~ N } } & { \mathrm { ~ B } } \\ { \mathrm { ~ I } } & { \mathrm { ~ I ~ } } & { 2 0 } & { 2 0 } & & & { \mathrm { ~ I } } & & { 6 } & { 7 } & { 2 } & & & { \mathrm { ~ I } } & { 1 5 } & { 5 } & { 1 0 } \\ { \mathrm { ~ C } } & { 1 5 } & & & { 1 2 } & { 1 6 } & { \mathrm { ~ C } } & { 6 } & & & { 6 } & { 6 } & { \mathrm { ~ C } } & { 1 1 } & & & { 1 4 } & { 1 0 } \\ { \mathrm { ~ N } } & { 5 } & { 1 4 } & & { 2 } & & { 7 } & & { \mathrm { ~ N } } & { 7 } & { 6 } & & & { 3 } & & { 3 } & & { \mathrm { ~ N } } & { 2 0 } & { 1 2 } & { 1 2 } \\ { \mathrm { ~ B } } & { 1 0 } & { 1 0 } & { 2 2 } & & & { \mathrm { ~ B } } & { 2 } & { 6 } & { 3 } & & & { \mathrm { ~ B } } & { 2 0 } & { 1 6 } & { 7 } & & \\ { \mathrm { ~ ( a ) ~ W i n s } } & & & & & { \mathrm { ~ ( b ) ~ T i e s } } & & & & { \mathrm { ~ ( c ) ~ L o s s e s } } & & & & & { } \end{array} } \end{array}
+$$
+
+Figure 3: Pairwise human evaluation results. Each element of Table 3(a) shows the number of times the source corresponding to the row was preferred over the source corresponding to the column. Table 3(b) shows the number of ties. Table 3(c) shows the number of losses and is the transpose of Table 3(a). Source legend: I denotes Independent Gibbs (Yao et al., 2014), C denotes Contiguous Gibbs, N denotes NADE and B denotes Bach.

parse/train/r1Usiwcex/r1Usiwcex_content_list.json

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+    {
+        "type": "text",
+        "text": "COUNTERPOINT BY CONVOLUTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            598,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Cheng-Zhi Anna Huang∗ MILA, Universite de Montr ´ eal ´ chengzhiannahuang@gmail.com ",
+        "bbox": [
+            183,
+            145,
+            450,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tim Cooijmans† MILA, Universite de Montr ´ eal ´ tim.cooijmans@umontreal.ca ",
+        "bbox": [
+            557,
+            145,
+            813,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adam Roberts   \nGoogle Brain   \nadarob@google.com ",
+        "bbox": [
+            183,
+            208,
+            352,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aaron Courville MILA, Universite de Montr´ eal´ aaron.courville@umontreal.ca ",
+        "bbox": [
+            370,
+            208,
+            645,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Douglas Eck   \nGoogle Brain   \ndeck@google.com ",
+        "bbox": [
+            663,
+            208,
+            813,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            287,
+            544,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning models of music typically break down the task of composition into a chronological process, composing a piece of music in a single pass from beginning to end. On the contrary, human composers write music in a nonlinear fashion, scribbling motifs here and there, often revisiting choices previously made. We explore the use of blocked Gibbs sampling as an analogue to the human approach, and introduce COCONET, a convolutional neural network in the NADE family of generative models (Uria et al., 2016). Despite ostensibly sampling from the same distribution as the NADE ancestral sampling procedure, we find that a blocked Gibbs approach significantly improves sample quality. We provide evidence that this is due to some conditional distributions being poorly modeled. Moreover, we show that even the cheap approximate blocked Gibbs procedure from Yao et al. (2014) yields better samples than ancestral sampling. We demonstrate the versatility of our method on unconditioned polyphonic music generation. ",
+        "bbox": [
+            232,
+            325,
+            764,
+            520
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/76c6a45fb4238b7ac19558e4f5a13b990679ecf9101a0c589cf902dafaaa8e6e.jpg",
+        "image_caption": [
+            "Figure 1: Ancestral inpainting of a corrupted Bach chorale by COCONET. Colors are used to distinguish the four voices. Grayscale heatmaps show predictions $p ( \\mathbf { x } _ { i } \\mid \\mathbf { x } _ { C } )$ . The pitch sampled in the current step is indicated by a rectangular outline. The original Bach chorale is shown in the bottom right. Step 0 shows the corrupted Bach chorale. Step 64 shows the result. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            336,
+            117
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Machine learning can be used to create compelling art. This was shown recently by DeepDream (Mordvintsev et al., 2015), an optimization process that created psychedelic transformations of images. A similar idea underlies a variety of style transfer algorithms (Gatys et al., 2015), which impose textures and colors from one image onto another. More recently, the multistyle pastiche generator (Dumoulin et al., 2016) exposes adjustable knobs that allow users of the system fine-grained control over style transfers. Neural doodle (Champandard, 2016) further closes the feedback loop between algorithm and artist. ",
+        "bbox": [
+            174,
+            136,
+            825,
+            232
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We wish to bring similar artistic tools to the domain of music. Whereas previous work in music has relied mainly on sequence models such as Hidden Markov Models (HMMs, Baum & Petrie (1966)) and Recurrent Neural Networks (RNNs, Rumelhart et al. (1988)), we instead employ convolutional neural networks due to their emphasis on capturing local structure and their invariance properties. Moreover, convolutional neural networks have shown to be extremely versatile once trained, as shown by a variety of creative uses in the literature (Mordvintsev et al., 2015; Gatys et al., 2015; Almahairi et al., 2015; Lamb et al., 2016). ",
+        "bbox": [
+            174,
+            239,
+            825,
+            337
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We introduce COCONET, a deep convolutional model trained to reconstruct partial scores. Once trained, COCONET provides direct access to all conditionals of the form $p ( \\mathbf { x } _ { i } \\mid \\mathbf { x } _ { C } )$ where $\\mathbf { x } _ { C }$ is a fragment of a musical score $\\mathbf { x }$ and $i \\not \\in C$ is in its complement. Figure 1 shows an example of such conditionals used in completing a partial score. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "COCONET is an instance of deep orderless NADE Uria et al. (2014), and thus learns an ensemble of factorizations of the joint $p ( \\mathbf { x } )$ . However, the sampling procedure for orderless NADE is not orderless. Sampling from an orderless NADE involves (randomly) choosing an ordering, and sampling ancestrally according to the chosen ordering. We have found that this produces poor results for the highly structured and complex domain of musical counterpoint. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Instead, we propose to use blocked-Gibbs sampling, essentially improving sample quality through rewriting. An instance of this was previously explored by Yao et al. (2014) who employed a NADE in the transition operator for a Markov Chain, yielding a Generative Stochastic Network (GSN). The transition consists of a corruption process that masks out a subset $\\mathbf { x } _ { \\lnot C }$ of variables, followed by a process that independently resamples variables $\\mathbf { x } _ { i } , i \\notin C$ according to the distribution $p _ { \\theta } ( \\mathbf { x } _ { i } \\mid \\mathbf { x } _ { C } )$ emitted by the NADE. Crucially, the effects of independent sampling are amortized by annealing the probability with which variables are masked out. Whereas Yao et al. (2014) treat their procedure as a cheap approximation to ancestral sampling, we find that it produces superior samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show the versatility of our method on unconditioned polyphonic music generation. ",
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+        "type": "text",
+        "text": "Section 2 discusses previous work in the area of automatic musical composition. The details of our model and training procedure are laid out in Section 3. In Section 4 we show that our approach is equivalent to that of deep and orderless NADE Uria et al. (2014). We discuss sampling from our model in Section 5. Results of quantitative and qualitative evaluations are reported in Section 6. Finally, Section 7 concludes. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Sequence models such as HMMs and RNNs are a natural choice for modeling music. However, one of the challenges in adapting such models to music is that music generally consists of multiple interdependent streams of events. This can be most clearly seen in the notion of counterpoint, which refers to the relationships between the movement of individual instruments in a musical work. Compare this to typical sequence domains such as speech and language, which involve modeling a single stream of events: a single speaker or a single stream of words. ",
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+    {
+        "type": "text",
+        "text": "Successful application of sequence models to music hence requires serializing or otherwise rerepresenting the music to fit the sequence paradigm. For instance, Liang (2016) serialize four-part Bach chorales by interleaving the parts, while Allan & Williams (2005) construct a chord vocabulary. Boulanger-Lewandowski et al. (2012) adopt a piano roll representation, which is a binary matrix $\\mathbf { x }$ such that $\\mathbf { x } _ { i t }$ is hot if some instrument is playing pitch $i$ at time $t$ . To model the joint probability distribution of the multi-hot pitch vector $\\mathbf { x } _ { t }$ , they employ a Restricted Boltzmann Machine (RBM (Smolensky, 1986; Hinton et al., 2006)) or Neural Autoregressive Distribution Estimator (Uria et al., 2016) at each time step. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "text",
+        "text": "Moreover, the behavior of human composers does not fit the chronological mold assumed by previous authors. A human composer might start his work with a coarse chord progression and iteratively refine it, revisiting choices previously made. Sampling according to $x _ { t } \\sim p ( x _ { t } | \\boldsymbol { x } _ { < t } )$ , as is common, cannot account for the kinds of timeless dependencies that composers employ. Hadjeres et al. (2016) sidestep the choice of causal factorization and instead employ an undirected Markov model to learn pairwise relationships between neighboring notes up to a specified number of steps away in a score. Sampling involves Markov Chain Monte Carlo (MCMC) using the model as a Metropolis-Hastings (MH) objective. The model permits constraints on the state space to support tasks such as melody harmonization. However, the Markov assumption severely limits the expressivity of the model. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We opt instead for a convolutional approach that avoids many of these issues and naturally captures both relationships across time and interactions between instruments. ",
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+        "type": "text",
+        "text": "3 COUNTERPOINT BY CONVOLUTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We approach the task of music composition with a deep convolutional neural network (Krizhevsky et al., 2012). This choice is motivated by the locality of contrapuntal rules and their near-invariance to translation, both in time and in the frequency spectrum. ",
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+        "type": "text",
+        "text": "We represent the music as a stack of piano rolls encoded in a binary three-tensor $\\mathbf { x } \\in \\{ 0 , 1 \\} ^ { I \\times T \\times P }$ . Here $I$ denotes the number of instruments, $T$ the number of time steps, $P$ the number of pitches, and $\\mathbf { x } _ { i , t , p } = 1$ iff the ith instrument plays pitch $p$ at time $t$ . We will assume each instrument plays exactly one pitch at a time, that is, $\\begin{array} { r } { \\sum _ { p } \\mathbf { x } _ { i , t , p } = 1 } \\end{array}$ for all $i , t$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the present work we will restrict ourselves to the study of four-part Bach chorales as used in prior work (Allan & Williams, 2005; Boulanger-Lewandowski et al., 2012; Goel et al., 2014; Liang, 2016; Hadjeres et al., 2016). Hence we assume $I = 4$ throughout. We discretize pitch according to equal temperament, but constrain ourselves to only the range that appears in our training data (MIDI pitches 36 through 88). Time is discretized at the level of 16th notes for similar reasons. To curb memory requirements, we enforce $T = 6 4$ by randomly cropping the training examples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given a training example $\\mathbf { x } \\sim p ( \\mathbf { x } )$ , we present the model with the values of only a strict subset of its elements $\\mathbf { x } _ { C } = \\{ \\mathbf { x } _ { ( i , t ) } \\mid ( i , t ) \\in C \\}$ and ask it to reconstruct its complement $\\mathbf { x } _ { \\lnot C }$ . The input to the model is obtained by masking the piano rolls $\\mathbf { x }$ to obtain the context $\\mathbf { x } _ { C }$ and concatenating this with the corresponding mask: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/839a62beae821091b9ba677be8c422731035f3709d972d4a52a30557f11ae4d6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { h } _ { i , t , p } ^ { 0 } = \\mathbb { 1 } _ { ( i , t ) \\in C } \\mathbf { x } _ { i , t , p } } \\\\ { \\mathbf { h } _ { I + i , t , p } ^ { 0 } = \\mathbb { 1 } _ { ( i , t ) \\in C } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the first dimension ranges over channels and the time and pitch dimensions are convolved over. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/8378e01221cd1a8daf2e50957371787cb4eea513999fdd7a78ae18b60c746186.jpg",
+        "text": "$$\n\\begin{array} { l l } { { \\displaystyle { \\bf { a } } ^ { l } = { \\mathrm { B N } } ( { \\bf { W } } ^ { l } * { \\bf { h } } ^ { l - 1 } ; \\gamma ^ { l } , \\beta ^ { l } ) } } \\\\ { { \\displaystyle { \\bf { h } } ^ { l } = { \\mathrm { R e L U } } ( { \\bf { a } } ^ { l } + { \\bf { h } } ^ { l - 2 } ) } } & { { \\qquad \\mathrm { f o r ~ } 3 < l < L - 1 \\mathrm { ~ a n d ~ } l = 0 \\mathrm { ~ m o d ~ } 2 } } \\\\ { { \\displaystyle { \\bf { h } } ^ { L } = { \\bf { a } } ^ { L } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "With the exception of the first and final layers, all of our convolutions preserve the size of the input. That is, we use “same” padding throughout and all activations $h ^ { l } , 1 \\overline { { < } } \\ l < L$ have 128 channels. The network consists of 64 layers with $3 \\times 3$ filters on each layer. After each convolution we apply batch normalization Ioffe & Szegedy (2015) (denoted by $\\mathrm { B N } ( \\cdot ) ,$ ) with statistics tied across time and pitch. After every second convolution, we introduce a skip connection from the hidden state two levels below to reap the benefits of residual learning He et al. (2015). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Finally, we obtain predictions for the pitch at each instrument/time pair: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/670d3de8602c894ec57f3ff7bf4eef8fa50e27371dfe937a24a3475fdf33d088.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { x } _ { i , t , p } \\mid \\mathbf { x } _ { C } , C ) = \\frac { \\exp ( h _ { i , t , p } ^ { L } ) } { \\sum _ { p } \\exp ( h _ { i , t , p } ^ { L } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            123,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The loss function is given by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe2939ac92b857547018c3f1771cdd6b0551b5d7e429bfb6db1afdfa72795013.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\mathcal { L } } ( \\mathbf { x } ; C , \\theta ) = - \\displaystyle \\sum _ { ( i , t ) \\notin C } \\log p _ { \\theta } ( \\mathbf { x } _ { i , t } \\mid \\mathbf { x } _ { C } , C ) } \\\\ { = - \\displaystyle \\sum _ { ( i , t ) \\notin C } \\sum _ { p } \\mathbf { x } _ { i , t , p } \\log p _ { \\theta } ( \\mathbf { x } _ { i , t , p } \\mid \\mathbf { x } _ { C } , C ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            215,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { \\theta }$ denotes the probability under the model with parameters $\\theta \\ : \\ : = \\ : \\ : \\mathbf { W } ^ { 1 } , \\gamma ^ { 1 } , \\beta ^ { 1 } , \\dots ,$ , $\\mathbf { W } ^ { L - 1 } , \\gamma ^ { L - 1 } , \\beta ^ { L - 1 }$ . To minimize the expected loss ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08ba1fe0538f9adba46872a1e612d0926a438457603571991ea2d2f10643946a.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { x } \\sim p ( \\mathbf { x } ) } \\mathbb { E } _ { C \\sim p ( C ) } \\mathcal { L } ( \\mathbf { x } ; C , \\boldsymbol { \\theta } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with respect to $\\theta$ , we sample piano rolls $\\mathbf { x }$ from the training set and contexts $C \\sim p ( C )$ and optimize by stochastic gradient descent with step size determined by Adam (Kingma & Ba, 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 RELATIONSHIP TO NADE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our approach is an instance of orderless and deep Neural Autoregressive Distribution Estimators (Uria et al., 2016). NADE models a $d$ -variate distribution $p ( \\mathbf { x } )$ through a factorization ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c41c24238ddaf420d269ab0e8eff685e0cca40643d7533b3265210da5541f07.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { x } ) = \\prod _ { d } p _ { \\theta } ( \\mathbf { x } _ { o _ { d } } \\mid \\mathbf { x } _ { o _ { < d } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $o$ is a permutation, and the parameters $\\theta$ are shared among the conditionals. NADE can be trained for all orderings $o$ simultaneously using the orderless NADE (Uria et al., 2014) training procedure. This procedure relies on the observation that, thanks to parameter sharing, computing $\\bar { p } _ { \\theta } ( \\mathbf { x } _ { o _ { d ^ { \\prime } } } \\mid \\mathbf { x } _ { o _ { < d } } )$ for all $d ^ { \\prime } \\geq d$ is no more expensive than computing it only for $d ^ { \\prime } = d$ . Hence for a given $o$ and $d$ we can simultaneously obtain partial losses for all orderings that agree with $o$ up to $d$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fb429f6c9de79044bb2900f191549b41697be4b890f40082c846d05c2523653.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { N A D E } } ( \\mathbf { x } ; o _ { < d } , \\theta ) = - \\sum _ { o _ { d } } \\log p _ { \\theta } ( \\mathbf { x } _ { o _ { d } } \\mid \\mathbf { x } _ { o _ { < d } } , o _ { < d } , o _ { d } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Letting $o _ { < d } = C$ , we obtain our loss from Equation 7 ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24bbd2343df0226a539e1f8eecae405a18bbf1125c3b924d1f3711506df9afd6.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C o c o n g r } } ( \\mathbf { x } ; C , \\theta ) = - \\sum _ { ( i , t ) \\notin C } \\log p _ { \\theta } ( \\mathbf { x } _ { i , t } \\mid \\mathbf { x } _ { C } , C )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For any one sample $( \\mathbf { x } , C )$ , this loss consists of $| \\neg C |$ terms of the form $\\log p _ { \\theta } ( \\mathbf { x } _ { i , t } \\mid \\mathbf { x } _ { C } , C )$ . We let $p ( C )$ be uniform in the size of the mask $| C |$ and reweight the sample losses according to ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf473aa26dc22960f3213c75a535614141f09086091a4ec64607e272a5e36a87.jpg",
+        "text": "$$\n\\tilde { \\mathcal { L } } ( \\mathbf { x } ; C , \\theta ) = \\frac { 1 } { | \\neg C | } \\mathcal { L } ( \\mathbf { x } ; C , \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This correction, due to Uria et al. (2014), ensures consistent estimation of the negative log-likelihood of the joint $p _ { \\theta } ( \\mathbf { x } )$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, we might wish to increase the difficulty by choosing $p ( C )$ so as to frequently mask out large contiguous regions, as otherwise the model might learn only superficial local relationships. This is discussed in Pathak et al. (2016) for the case of images, where a model might learn only that pixels are similar to their neighbors. Similar low-level relationships hold in our case, as our piano roll representation is binary and very sparse. For instance, if we mask out only a single sixteenth step in the middle of a long-held note, reconstructing the masked out step does not require any deep understanding of music. To this end we also consider choosing the context $C$ by independent Bernoulli samples, such that each variable has a low probability of being included in the context. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can sample from the model using the NADE ancestral ordering procedure. However, we find that this yields poor samples, and we propose instead to use Gibbs sampling. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 NADE SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To sample according to NADE, we start with an empty (zero everywhere) piano roll $\\mathbf { x } ^ { 0 }$ and context $C ^ { 0 }$ and populate them iteratively by the following process. We feed the piano roll $\\mathbf { x } ^ { s }$ and context $C ^ { s }$ into the model to obtain a set of categorical distributions $p _ { \\theta } ( \\mathbf { x } _ { i , t } | \\mathbf { x } _ { C ^ { s } } ^ { s } , C ^ { s } )$ for $( i , t ) \\notin C ^ { s }$ . As the $\\mathbf { x } _ { i , t }$ are not conditionally independent, we cannot simply sample from these distributions independently. However, if we sample from one of thfor the others. Hence we randomly choose one $( i , t ) ^ { s + 1 }$ an compute new conditio to sample from, and let $\\mathbf { x } _ { i , t } ^ { s + 1 }$ stributionsequal the one-hot realization. Augment the context with $C ^ { s + 1 } = C ^ { s } \\cup ( i , t )$ and repeat until the piano roll is populated. This procedure is easily generalized to tasks such as melody harmonization and partial score completion by starting with a nonempty piano roll. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, samples thus generated are of low quality, which we surmise is due to accumulation of errors. While the model provides conditionals $p _ { \\theta } ( \\mathbf { x } _ { i , t } | \\mathbf { x } _ { C } , C )$ for all $( i , t ) \\notin C$ , some of these conditionals may be better modeled than others. We suspect in particular those conditionals used early on in the procedure, for which the context $C$ consists of very few variables. Moreover, although the model is trained to be order-agnostic, different orderings invoke different distributions, which is another indication that some conditionals are poorly learned. We test this hypothesis in Section 6.2. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.2 GIBBS SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To remedy this, we allow the model to revisit its choices: we repeatedly mask out some part of the piano roll and then repopulate it. This is a form of blocked Gibbs sampling (Liu, 1994). Blocked sampling is crucial for mixing, as the high temporal resolution of our representation causes strong correlations between consecutive notes. For instance, without blocked sampling, it would take many steps to snap out of a long-held note. Similar observations hold for the Ising model from statistical mechanics, leading to the development of the Swendsen-Wang algorithm (Swendsen & Wang, 1987) in which large clusters of variables are resampled at once. ",
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+    {
+        "type": "text",
+        "text": "We consider two strategies for resampling a given block of variables: ancestral sampling and independent sampling. Ancestral sampling invokes the orderless NADE sampling procedure described in Section 5.1 on the masked-out portion of the piano roll. Independent sampling simply treats the masked-out variables $\\mathbf { x } { \\lrcorner } C$ as independent given the context $\\mathbf { x } _ { C }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "Using independent blocked Gibbs to sample from a NADE model has been studied by Yao et al. (2014), who propose to use an annealed masking probability given by ",
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+        "text": "$$\n\\alpha _ { n } = \\operatorname* { m a x } \\Bigg ( \\alpha _ { \\operatorname* { m i n } } , \\alpha _ { \\operatorname* { m a x } } - \\frac { n } { \\eta N } ( \\alpha _ { \\operatorname* { m a x } } - \\alpha _ { \\operatorname* { m i n } } ) \\Bigg )\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "for some minimum and maximum probabilities $\\alpha _ { \\mathrm { m i n } } , \\alpha _ { \\mathrm { m a x } }$ , number of Gibbs steps $N$ and fraction $\\eta$ of time spent before settling onto the minimum probability $\\alpha _ { \\mathrm { m i n } }$ . This scheme ensures the Gibbs process with independent resampling produces samples from the model distribution $p _ { \\theta } ( \\mathbf { x } )$ . Initially, when the masking probability is high, the chain mixes fast but samples are poor due to independent sampling. As the masking probability reduces, fewer variables are sampled at a time, until finally variables are sampled one at a time and conditioned on all the others. ",
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+        "img_path": "images/df0976f61118c39cdd809376b37133bde031f9b635ee6db4f818a36949cd113c.jpg",
+        "table_caption": [
+            "Table 1: Negative log-likelihood on the test set for the Bach corpus. As discussed in the text, our numbers are not directly comparable to those of other authors due to the use of different splits. Results from Boulanger-Lewandowski et al. (2012) were based on an eighth-note temporal resolution (our resolution is sixteenth notes). Please note that our results are preliminary validation likelihoods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Notewise NLL</td><td>Framewise NLL</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Bachbot (Liang, 2016)</td><td>0.477</td><td></td></tr><tr><td>NADE (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>7.19</td></tr><tr><td>RNN-RBM (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>6.27</td></tr><tr><td>RNN-NADE (Boulanger-Lewandowski et al., 2012)</td><td>1</td><td>5.56</td></tr><tr><td>CoCONET, i.i.d Bernoulli(0.50)</td><td>0.924</td><td>8</td></tr><tr><td>CoCONET, i.i.d Bernoulli(0.25)</td><td>0.655</td><td>4.48</td></tr><tr><td>COCONET, i.i.d Bernoulli(0.10)</td><td>0.812</td><td>4.66</td></tr><tr><td>CocONET,importance sampling</td><td>0.569</td><td>3.73</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Yao et al. (2014) treat independent blocked Gibbs as a cheap approximation to ancestral sampling. Indeed, per Gibbs step, independent sampling requires only a single model evaluation, whereas ancestral sampling requires as many model evaluations as there are variables to sample. Moreover, we find that independent blocked Gibbs sampling in fact yields better samples than the NADE procedure from Section 5.1. Samples can be heard here: https:// soundcloud.com/czhuang/sets/coconet-nade and https://soundcloud.com/ czhuang/sets/coconet-independent-gibbs. ",
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+    {
+        "type": "text",
+        "text": "6 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate our approach on a corpus of four-part Bach chorales. The literature features many variants of this dataset (Allan & Williams, 2005; Boulanger-Lewandowski et al., 2012; Liang, 2016; Hadjeres et al., 2016), and we follow the unfortunate tradition of introducing our own adaptation. Although this complicates comparisons against earlier work, we feel justified in doing so as our approach requires instruments to be separated, and other authors’ eighth-note temporal resolution is too coarse to accurately convey counterpoint. ",
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+    {
+        "type": "text",
+        "text": "We rebuilt our dataset from the Bach chorale musicXML scores readily available through (Cuthbert & Ariza, 2010), which was also the basis for the dataset used in (Liang, 2016). The scores included 357 four-part Bach chorales. We excluded scores that included note durations less than sixteenth notes, resulting in 354 pieces. These pieces were split into train/valid/test in $6 0 / 2 0 / 2 0 \\%$ ratios. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare with Liang (2016) based on note-level likelihood and Boulanger-Lewandowski et al. (2012) based on frame-level likelihood. Note that train/valid/test differs among both prior work and also with our work, and that Liang (2016) uses a $8 0 / 1 0 / 1 0 \\%$ split instead. ",
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+    {
+        "type": "text",
+        "text": "However, evaluation of generative models is hard (Theis et al., 2015). The gold standard for evaluation is qualitative comparison by humans, and we therefore report human evaluation results. ",
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+    {
+        "type": "text",
+        "text": "6.1 EVALUATING LOG-LIKELIHOOD ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "To estimate the log-likelihood of a datapoint $\\mathbf { x }$ , we follow the orderless NADE approach. That is, we sample a random ordering $( i _ { 1 } , t _ { 1 } ) , ( i _ { 2 } , t _ { 2 } ) , \\dots ( i _ { I T } , t _ { I T } )$ , and compute the notewise log-likelihood ",
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+        "img_path": "images/88e98161459b9d3933b86edbc5fa5f1e8d2a0569aeeae895c04bb24a11cdb139.jpg",
+        "text": "$$\n\\log \\hat { p } _ { \\boldsymbol { \\theta } } ( \\mathbf { x } ) = \\frac { 1 } { I T } \\sum _ { d = 1 } ^ { I T } \\log p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i _ { d } , t _ { d } } \\mid \\mathbf { x } _ { C _ { d - 1 } } , C _ { d - 1 } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { C _ { d } = \\bigcup _ { c = 1 } ^ { d } \\{ ( i _ { c } , t _ { c } ) \\} } \\end{array}$ . Note that we randomly crop each datapoint to be $T$ time steps long before processing it, as this facilitates batch processing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We repeat this procedure $k$ times and average across all point estimates. The numbers for our models in Table 1 were obtained with $k = 5$ . ",
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+    {
+        "type": "text",
+        "text": "The process for computing the notewise log-likelihood is akin to teacher-forcing, where at each step of the way the model observes the ground truth for all its previous predictions. To compute the framewise log-likelihood, we instead let the model run free within each frame $t$ . This results in a more representative measure of the model’s quality as it is sensitive to accumulation of error. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1 lists notewise and framewise likelihoods of the validation data under variants of our model, as well as comparable results from other authors. We include four variants of COCONET that differ in the choice of the distribution $p ( C )$ over contexts during training. By importance sampling we refer to the orderless NADE strategy discussed in Section 4, in which $p ( C )$ is uniform over $| C |$ and the sampled losses are reweighted by $1 / | \\neg C |$ . We also evaluate three variants where the contexts are chosen by biased coin flips, that is, $\\operatorname* { P r } ( ( i , t ) \\in C ) = \\rho$ , for $\\rho \\in { 0 . 5 , 0 . 2 5 , 0 . 1 }$ . The framewise log-likelihood for $\\rho = 0 . 5$ is listed as $\\infty$ as its estimation repeatedly overflowed. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall, COCONET seems to underperform in terms of notewise likelihood, yet perform well in terms of framewise likelihood. Estimating the loss by importance sampling appears to work significantly better than determining the context using independent Bernoulli variables, as one might expect. However, the choice of Bernoulli probability $\\rho$ strongly affects the resulting loss, which suggests that some of the conditionals benefit from more training. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/d1af8beaa20b1d252c992f1ef1b7929a42cb4c0d0caaa7e8e88232b9ba0ae2eb.jpg",
+        "table_caption": [
+            "Table 2: Mean $( \\pm \\mathrm { S E M } )$ negative log-likelihood under the model of unconditioned samples generated from the model by various procedures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Sampling scheme</td><td>Notewise NLL</td><td>Framewise NLL</td></tr><tr><td>Ancestral Gibbs,ρ = O.0O (NADE)</td><td>0.565 ± 0.011</td><td>3.872 ± 0.052</td></tr><tr><td>Ancestral Gibbs,ρ= 0.01</td><td>0.560 ± 0.010</td><td>3.824 ± 0.052</td></tr><tr><td>Ancestral Gibbs,ρ= 0.25</td><td>0.444 ± 0.008</td><td>3.276 ± 0.036</td></tr><tr><td>Ancestral Gibbs,ρ = 0.50</td><td>0.438 ± 0.007</td><td>3.332 ± 0.040</td></tr><tr><td>Contiguous Gibbs, ρ = 0.50</td><td>0.447 ± 0.008</td><td>3.476 ± 0.048</td></tr><tr><td>Independent Gibbs (Yao et al., 2014)</td><td>0.440 ± 0.008</td><td>3.348 ± 0.040</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6.2 SAMPLE QUALITY",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Section 5 we conjectured that the low quality of NADE samples is due to poorly modeled conditionals $p _ { \\theta } ( \\mathbf { x } _ { i , t } \\mid \\mathbf { x } _ { C } , C )$ where $C$ is small. We test this hypothesis by evaluating the likelihood under the model of samples generated by the ancestral blocked Gibbs procedure with $C$ chosen according to independent Bernoulli variables. When we set the inclusion probability $\\rho$ to 0, we obtain NADE. Increasing $\\rho$ increases the expected context size $| C |$ , which should yield better samples if our hypothesis is true. The results shown in Table 2 confirm that this is the case. For these experiments, we used sample length $T = 3 2$ time steps and number of Gibbs steps $N = 1 0 0$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 2 shows the convergence behavior of the various Gibbs procedures, averaged over 100 runs. We see that for low values of $\\rho$ (small $C$ ), the chains hardly make progress beyond NADE in terms of likelihood. Higher values of $\\rho$ (large $C$ ) enable the model to bootstrap and reach significantly better likelihood. However, high values of $\\rho$ cause the chain to mix slowly, as can be seen in the case where $\\rho ~ = ~ 0 . 5 0$ . For comparison, we included a variant, Contiguous(0.50), that always masks out in contiguous chunks of at least four sixteenth notes. This variant converges much more rapidly than Bernoulli(0.50) despite masking out equally many variables on average. Note that whereas ancestral sampling (NADE) requires ${ \\cal O } ( I T )$ model evaluations and ancestral Gibbs requires ${ \\cal O } ( I T N )$ model evaluations, independent Gibbs requires only $O ( N )$ model evaluations, with typically $N < I T$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/36e9ce8ae5d09b5df6de91e8d0188fee78bc2bfacf3592f33b5b94a137775065.jpg",
+        "image_caption": [
+            "Figure 2: Likelihood under the model of Gibbs samples obtained with various context distributions $p ( C )$ . NADE (equivalent to Bernoulli(0.00)) is included for reference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.3 HUMAN EVALUATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We carried out a listening test on Amazon’s Mechanical Turk (MTurk) to compare quality of samples from different sources (sampling schemes and Bach). The sampling schemes under study are ancestral Gibbs with Bernoulli(0.00) masking (NADE), independent Gibbs (Yao et al., 2014), and ancestral Gibbs with Contiguous(0.50). For each scheme, we generate four unconditioned samples from empty piano rolls. For Bach, we randomly crop four fragments from the chorale validation set. We thus obtain four sets of four sounds each. All fragments are two measures long, and last twelve seconds after synthesis. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6e8aa922a515dfbdc710a2f43e84aa1496664fd38d1ba314bf4cb0648e7a622.jpg",
+        "image_caption": [
+            "Figure 3: Human evaluations from MTurk on comparing sampling schemes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For each MTurk hit, users are asked to rate on a Likert scale which of two random samples they perceive as more musical. The study resulted in 192 ratings, where each source was involved in 92 pairwise comparisons. Figure 6.3 reports for each source the number of times it was rated as more musical. We see that although ancestral sampling on NADE performs poorly compared to Bach, both ancestral and independent Gibbs Yao et al. (2014) were considered at least as musical as fragments from Bach, with independent Gibbs Yao et al. (2014) outperforming ancestral sampling (NADE) by a large margin. Pairwise comparisons are listed in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We introduced a convolutional approach to modeling musical scores based on the NADE (Uria et al., 2016) framework. Our experiments show that the NADE ancestral sampling procedure yields poor samples for our domain, which we have argued is because some conditionals are not captured well by the model. We have shown that sample quality improves significantly when we use blocked Gibbs sampling to iteratively rewrite parts of the score. Moreover, annealed independent blocked Gibbs sampling as proposed by Yao et al. (2014) is not only faster but in fact produces better samples. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Kyle Kastner and Guillaume Alain, Curtis (Fjord) Hawthorne, the Google Brain Magenta team, as well as Jason Freidenfelds for helpful feedback, discussions, suggestions and support. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    },
+    {
+        "type": "text",
+        "text": "Feynman Liang. Bachbot: Automatic composition in style of bach chorales. Masters thesis, University of Cambridge, 2016. ",
+        "bbox": [
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+        "text": "Jun S Liu. The collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem. Journal of the American Statistical Association, 89(427):958–966, 1994. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "MIDI. Midi tuning standard. https://en.wikipedia.org/wiki/MIDI_Tuning_ Standard. Accessed: 2016-11-12.   \nAlexander Mordvintsev, Christopher Olah, and Mike Tyka. Inceptionism: Going deeper into neural networks, 2015. URL https://research.googleblog.com/2015/06/ inceptionism-going-deeper-into-neural.html.   \nDeepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. arXiv preprint arXiv:1604.07379, 2016.   \nDavid E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by backpropagating errors. Cognitive modeling, 5(3):1, 1988.   \nPaul Smolensky. Information processing in dynamical systems: Foundations of harmony theory. Technical report, DTIC Document, 1986.   \nRobert H Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in monte carlo simulations. Physical review letters, 58(2):86, 1987.   \nLucas Theis, Aaron van den Oord, and Matthias Bethge. A note on the evaluation of generative ¨ models. arXiv preprint arXiv:1511.01844, 2015.   \nBenigno Uria, Iain Murray, and Hugo Larochelle. A deep and tractable density estimator. In ICML, pp. 467–475, 2014.   \nBenigno Uria, Marc-Alexandre Cotˆ e, Karol Gregor, Iain Murray, and Hugo Larochelle. Neural ´ autoregressive distribution estimation. arXiv preprint arXiv:1605.02226, 2016.   \nLi Yao, Sherjil Ozair, Kyunghyun Cho, and Yoshua Bengio. On the equivalence between deep nade and generative stochastic networks. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 322–336. Springer, 2014. ",
+        "bbox": [
+            169,
+            102,
+            828,
+            503
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A PAIRWISE HUMAN EVALUATION RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            526,
+            550,
+            542
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This appendix supplements Section 6.3 on the evaluation of samples by human subjects. Figure 3 lists the number of wins, ties and losses for each sample source against each other sample source. All pairs of sources were compared 32 times. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6dafc7160e6cb47cc0699ec85d33385190e3ccdc9bb50551c9d44432f09d38ce.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c } { \\mathrm { ~ I } } & { \\mathrm { ~ C } } & { \\mathrm { ~ N } } & { \\mathrm { ~ B } } & & & & & { \\mathrm { ~ I } } & { \\mathrm { ~ C } } & { \\mathrm { ~ N } } & { \\mathrm { ~ B } } & & & & { \\mathrm { ~ I } } & { \\mathrm { ~ C } } & { \\mathrm { ~ N } } & { \\mathrm { ~ B } } \\\\ { \\mathrm { ~ I } } & { \\mathrm { ~ I ~ } } & { 2 0 } & { 2 0 } & & & { \\mathrm { ~ I } } & & { 6 } & { 7 } & { 2 } & & & { \\mathrm { ~ I } } & { 1 5 } & { 5 } & { 1 0 } \\\\ { \\mathrm { ~ C } } & { 1 5 } & & & { 1 2 } & { 1 6 } & { \\mathrm { ~ C } } & { 6 } & & & { 6 } & { 6 } & { \\mathrm { ~ C } } & { 1 1 } & & & { 1 4 } & { 1 0 } \\\\ { \\mathrm { ~ N } } & { 5 } & { 1 4 } & & { 2 } & & { 7 } & & { \\mathrm { ~ N } } & { 7 } & { 6 } & & & { 3 } & & { 3 } & & { \\mathrm { ~ N } } & { 2 0 } & { 1 2 } & { 1 2 } \\\\ { \\mathrm { ~ B } } & { 1 0 } & { 1 0 } & { 2 2 } & & & { \\mathrm { ~ B } } & { 2 } & { 6 } & { 3 } & & & { \\mathrm { ~ B } } & { 2 0 } & { 1 6 } & { 7 } & & \\\\ { \\mathrm { ~ ( a ) ~ W i n s } } & & & & & { \\mathrm { ~ ( b ) ~ T i e s } } & & & & { \\mathrm { ~ ( c ) ~ L o s s e s } } & & & & & { } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            616,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "",
+        "image_caption": [
+            "Figure 3: Pairwise human evaluation results. Each element of Table 3(a) shows the number of times the source corresponding to the row was preferred over the source corresponding to the column. Table 3(b) shows the number of ties. Table 3(c) shows the number of losses and is the transpose of Table 3(a). Source legend: I denotes Independent Gibbs (Yao et al., 2014), C denotes Contiguous Gibbs, N denotes NADE and B denotes Bach. "
+        ],
+        "image_footnote": [],
+        "page_idx": 9
+    }
+]

parse/train/r1xCMyBtPS/r1xCMyBtPS.md

@@ -0,0 +1,296 @@
+# MULTILINGUAL ALIGNMENT OF CONTEXTUAL WORD REPRESENTATIONS
+
+Steven Cao, Nikita Kitaev & Dan Klein   
+Computer Science Division   
+University of California, Berkeley   
+{stevencao,kitaev,klein}@berkeley.edu
+
+# ABSTRACT
+
+We propose procedures for evaluating and strengthening contextual embedding alignment and show that they are useful in analyzing and improving multilingual BERT. In particular, after our proposed alignment procedure, BERT exhibits significantly improved zero-shot performance on XNLI compared to the base model, remarkably matching pseudo-fully-supervised translate-train models for Bulgarian and Greek. Further, to measure the degree of alignment, we introduce a contextual version of word retrieval and show that it correlates well with downstream zero-shot transfer. Using this word retrieval task, we also analyze BERT and find that it exhibits systematic deficiencies, e.g. worse alignment for open-class parts-of-speech and word pairs written in different scripts, that are corrected by the alignment procedure. These results support contextual alignment as a useful concept for understanding large multilingual pre-trained models.
+
+# 1 INTRODUCTION
+
+![](images/4c1b569466a38a121d3526249ec6c6f2cc465ee1df5b4c3e76c884e1d6ab2f89.jpg)  
+Figure 1: t-SNE (Maaten & Hinton, 2008) visualization of the embedding space of multilingual BERT for English-German word pairs (left: pre-alignment, right: post-alignment). Each point is a different instance of the word in the Europarl corpus. This figure suggests that BERT begins already somewhat aligned out-of-the-box but becomes much more aligned after our proposed procedure.
+
+Embedding alignment was originally studied for word vectors with the goal of enabling cross-lingual transfer, where the embeddings for two languages are in alignment if word translations, e.g. cat and Katze, have similar representations (Mikolov et al., 2013a; Smith et al., 2017). Recently, large pretrained models have largely subsumed word vectors based on their accuracy on downstream tasks, partly due to the fact that their word representations are context-dependent, allowing them to more richly capture the meaning of a word (Peters et al., 2018; Howard & Ruder, 2018; Radford et al., 2018; Devlin et al., 2018). Therefore, with the same goal of cross-lingual transfer but for these more complex models, we might consider contextual embedding alignment, where we observe whether word pairs within parallel sentences, e.g. cat in “The cat sits” and Katze in “Die Katze sitzt,” have similar representations.
+
+One model relevant to these questions is multilingual BERT, a version of BERT pre-trained on 104 languages that achieves remarkable transfer on downstream tasks. For example, after the model is fine-tuned on the English MultiNLI training set, it achieves $7 4 . 3 \%$ accuracy on the test set in Spanish, which is only $7 . 1 \%$ lower than the English accuracy (Devlin et al., 2018; Conneau et al., 2018b). Furthermore, while the model transfers better to languages similar to English, it still achieves reasonable accuracies even on languages with different scripts.
+
+However, given the way that multilingual BERT was pre-trained, it is unclear why we should expect such high zero-shot performance. Compared to monolingual BERT which exhibits no zero-shot transfer, multilingual BERT differs only in that (1) during pre-training (i.e. masked word prediction), each batch contains sentences from all of the languages, and (2) it uses a single shared vocabulary, formed by WordPiece on the concatenated monolingual corpora (Devlin et al., 2019). Therefore, we might wonder: (1) How can we better understand BERT’s multilingualism? (2) Can we further improve BERT’s cross-lingual transfer?
+
+In this paper, we show that contextual embedding alignment is a useful concept for addressing these questions. First, we propose a contextual version of word retrieval to evaluate the degree of alignment, where a model is presented with two parallel corpora, and given a word within a sentence in one corpus, it must find the correct word and sentence in the other. Using this metric of alignment, we show that multilingual BERT achieves zero-shot transfer because its embeddings are partially aligned, as depicted in Figure 1, with the degree of alignment predicting the degree of downstream transfer.
+
+Next, using between 10K and 250K sentences per language from the Europarl corpus as parallel data (Koehn, 2005), we propose a fine-tuning-based alignment procedure and show that it significantly improves BERT as a multilingual model. Specifically, on zero-shot XNLI, where the model is trained on English MultiNLI and tested on other languages (Conneau et al., 2018b), the aligned model improves accuracies by $2 . 7 8 \%$ on average over the base model, and it remarkably matches translate-train models for Bulgarian and Greek, which approximate the fully-supervised setting.
+
+To put our results in the context of past work, we also use word retrieval to compare our finetuning procedure to two alternatives: (1) fastText augmented with sentence and aligned using rotations (Bojanowski et al., 2017; Ruckl ¨ e et al., 2018; Artetxe et al., 2018), and (2) BERT aligned using ´ rotations (Aldarmaki & Diab, 2019; Schuster et al., 2019; Wang et al., 2019). We find that when there are multiple occurences per word, fine-tuned BERT outperforms fastText, which outperforms rotation-aligned BERT. This result supports the intuition that contextual alignment is more difficult than its non-contextual counterpart, given that a rotation, at least when applied naively, is no longer sufficient to produce strong alignments. In addition, when there is only one occurrence per word, fine-tuned BERT matches the performance of fastText. Given that context disambiguation is no longer necessary, this result suggests that our fine-tuning procedure is able to align BERT at the type level to a degree that matches non-contextual approaches.
+
+Finally, we use the contextual word retrieval task to conduct finer-grained analysis of multilingual BERT, with the goal of better understanding its strengths and shortcomings. Specifically, we find that base BERT has trouble aligning open-class compared to closed-class parts-of-speech, as well as word pairs that have large differences in usage frequency, suggesting insight into the pre-training procedure that we explore in Section 5. Together, these experiments support contextual alignment as an important task that provides useful insight into large multilingual pre-trained models.
+
+# 2 RELATED WORK
+
+Word vector alignment. There has been a long line of works that learn aligned word vectors from varying levels of supervision (Ruder et al., 2019). One popular family of methods starts with word vectors learned independently for each language (using a method like skip-gram with negative sampling (Mikolov et al., 2013b)), and it learns a mapping from source language vectors to target language vectors with a bilingual dictionary as supervision (Mikolov et al., 2013a; Smith et al., 2017; Artetxe et al., 2017). When the mapping is constrained to be an orthogonal linear transformation, the optimal mapping that minimizes distances between word pairs can be solved in closed form (Artetxe et al., 2016; Schonemann, 1966). Alignment is evaluated using bilingual lexicon induction, so these papers also propose ways to mitigate the hubness problem in nearest neighbors, e.g. by using alternate similarity functions like CSLS (Conneau et al., 2018a). A recent set of works has also shown that the mapping can be learned with minimal to no supervision by starting with some minimal seed dictionary and alternating between learning the linear map and inducing the dictionary (Artetxe et al., 2018; Conneau et al., 2018a; Hoshen & Wolf, 2018; Xu et al., 2018; Chen & Cardie, 2018).
+
+Incorporating context into alignment. One key challenge in making alignment context aware is that the embeddings are now different across multiple occurrences of the same word. Past papers have handled this issue by removing context and aligning the “average sense” of a word. In one such study, Schuster et al. (2019) learn a rotation to align contextual ELMo embeddings (Peters et al., 2018) with the goal of improving zero-shot multilingual dependency parsing, and they handle context by taking the average embedding for a word in all of its contexts. In another paper, Aldarmaki & Diab (2019) learn a rotation on sentence vectors, produced by taking the average word vector over the sentence, and they show that the resulting alignment also works well for word-level tasks. In a contemporaneous work, Wang et al. (2019) align not only the word but also the context by learning a linear transformation using word-aligned parallel data to align multilingual BERT, with the goal of improving zero-shot dependency parsing numbers. In this paper, we similarly align not only the word but also the context, and we also depart from these past works by using more expressive alignment methods than rotation.
+
+Incorporating parallel texts into pre-training. Instead of performing alignment post-hoc, another line of works proposes contextual pre-training procedures that are more cross-lingually-aware. Wieting et al. (2019) pre-train sentence embeddings using parallel texts by maximizing similarity between sentence pairs while minimizing similarity with negative examples. Lample & Conneau (2019) propose a cross-lingual pre-training objective that incorporates parallel data in addition to monolingual corpora, leading to improved downstream cross-lingual transfer. In contrast, our method uses less parallel data and aligns existing pre-trained models rather than requiring pretraining from scratch.
+
+Analyzing multilingual BERT. Pires et al. (2019) present a series of probing experiments to better understand multilingual BERT, and they find that transfer is possible even between dissimilar languages, but that it works better between languages that are typologically similar. They conclude that BERT is remarkably multilingual but falls short for certain language pairs.
+
+# 3 METHODS
+
+# 3.1 MULTILINGUAL PRE-TRAINING
+
+We first briefly describe multilingual BERT (Devlin et al., 2018). Like monolingual BERT, multilingual BERT is pre-trained on sentences from Wikipedia to perform two tasks: masked word prediction, where it must predict words that are masked within a sentence, and next sentence prediction, where it must predict whether the second sentence follows the first one. The model is trained on 104 languages, with each batch containing training sentences from each language, and it uses a shared vocabulary formed by WordPiece on the 104 Wikipedias concatenated (Wu et al., 2016).
+
+# 3.2 DEFINING AND EVALUATING CONTEXTUAL ALIGNMENT
+
+In the following sections, we describe how to define, evaluate, and improve contextual alignment. Given two languages, a model is in contextual alignment if it has similar representations for word pairs within parallel sentences. More precisely, suppose we have $N$ parallel sentences $C = \{ ( \mathbf { s } ^ { 1 ^ { \prime } } , \mathbf { t } ^ { 1 } ) , . . . , ( \mathbf { s } ^ { N ^ { \prime } } , \mathbf { t } ^ { N } ) \}$ , where $( \mathbf { s } , \mathbf { t } )$ is a source-target sentence pair. Also, let each sentence pair $( \mathbf { s } , \mathbf { t } )$ have word pairs, denoted $a ( \mathbf { s } , \mathbf { t } ) = \{ ( i _ { 1 } , j _ { 1 } ) , . . . , ( i _ { m } , j _ { m } ) \}$ , containing position tuples $( i , j )$ such that the words $\mathbf { s } _ { i }$ and $\mathbf { t } _ { j }$ are translations of each other.1 We will use $f$ to represent a pre-trained model such that $f ( i , { \bf s } )$ is the contextual embedding for the ith word in s.
+
+As an example, we might have the following sentence pair:
+
+$$
+\begin{array} { c } { { { \bf s } = \{ { \stackrel { 0 } { I } } { } _ { a t e t h e a p p l e } ^ { 3 } { } _ { \cdot } \} \quad { \bf t } = \{ I c h { \stackrel { 1 } { h a b e } } d e n A p { \stackrel { 3 } { f } } e l g e g e s s e n { } ^ { 4 } \} } } \\ { { a ( { \bf s } , { \bf t } ) = \{ ( 0 , 0 ) , ( 1 , 4 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 5 ) \} } } \end{array}
+$$
+
+Then, using the parallel corpus $C$ , we can measure the contextual alignment of the model $f$ using its accuracy in contextual word retrieval. In this task, the model is presented with two parallel corpora, and given a word within a sentence in one corpus, it must find the correct word and sentence in the other. Specifically, we can define a nearest neighbor retrieval function
+
+$$
+\mathrm { n e i g h b o r } ( i , \mathbf { s } ; f , C ) = \underset { \mathbf { t } \in C , 0 \leq j \leq \mathrm { l e n } ( \mathbf { t } ) } { \mathrm { a r g m a x } } \ \mathrm { s i m } ( f ( i , \mathbf { s } ) , f ( j , \mathbf { t } ) ) ,
+$$
+
+where $i$ and $j$ denote the position within a sentence and sim is a similarity function. The accuracy is then given by the percentage of exact matches over the entire corpus, or
+
+$$
+A ( f ; C ) = \frac { 1 } { N } \sum _ { ( \mathbf { s } , \mathbf { t } ) \in C } \sum _ { ( i , j ) \in a ( \mathbf { s } , \mathbf { t } ) } \mathbb { I } ( \mathrm { n e i g h b o r } ( i , \mathbf { s } ; f , C ) = ( j , \mathbf { t } ) ) ,
+$$
+
+where $\mathbb { I }$ represents the indicator function. We can perform the same procedure in the other direction, where we pull target words given source words, so we report the average between the two directions. As our similarity function, we use CSLS, a modified version of cosine similarity that mitigates the hubness problem, with neighborhood size 10 (Conneau et al., 2018a). One additional point is that this procedure can be made more or less contextual based on the corpus: a corpus with more occurrences for each word type requires better representations of context. Therefore, we also test non-contextual word retrieval by removing all but the first occurrence of each word type.
+
+Given parallel data, these word pairs can be procured in an unsupervised fashion using standard techniques developed by the machine translation community (Brown et al., 1993). While these methods can be noisy, by running the algorithm in both the source-target and target-source directions and only keeping word pairs in their intersection, we can trade-off coverage for accuracy, producing a reasonably high-precision dataset (Och & Ney, 2003).
+
+# 3.3 ALIGNING PRE-TRAINED CONTEXTUAL EMBEDDINGS
+
+To improve the alignment of the model $f$ with respect to the corpus $C$ , we can encapsulate alignment in the loss function
+
+$$
+L ( f ; C ) = - \sum _ { ( \mathbf { s } , \mathbf { t } ) \in C } \sum _ { ( i , j ) \in a ( \mathbf { s } , \mathbf { t } ) } \sin ( f ( i , \mathbf { s } ) , f ( j , \mathbf { t } ) ) ,
+$$
+
+where we sum the similarities between word pairs. Because the CSLS metric is not easily optimized, we instead use the squared error loss, or $\mathrm { s i m \bar { ( } } f ( i , \mathbf { s ) } , f ( j , \mathbf { t } ) ) = - | | f ( i , \mathbf { s } ) - f ( j , \mathbf { t } ) | | _ { 2 } ^ { 2 }$ .
+
+However, note that this loss function does not account for the informativity of $f$ ; for example, it is zero if $f$ is constant. Therefore, at a high level, we would like to minimize $L ( f ; C )$ while maintaining some aspect of $f$ that makes it useful, e.g. its high accuracy when fine-tuned on downstream tasks. Letting $f _ { 0 }$ denote the initial pre-trained model before alignment, we achieve this goal by defining a regularization term
+
+$$
+R ( f ; C ) = \sum _ { \mathbf { t } \in C } \sum _ { i = 1 } ^ { \mathrm { l e n } ( \mathbf { t } ) } \vert \vert f ( j , \mathbf { t } ) - f _ { 0 } ( j , \mathbf { t } ) \vert \vert _ { 2 } ^ { 2 } ,
+$$
+
+which imposes a penalty if the target language embeddings stray from their initialization. Then, we sample minibatches $B \subset C$ and take gradient steps of the function $L ( f ; B ) + \lambda R ( f ; B )$ directly on the weights of $f$ , which moves the source embeddings toward the target embeddings while preventing the latter from drifting too far. In our experiments, we set $\lambda = 1$ .
+
+In the multilingual case, suppose we have $k$ parallel corpora $C ^ { 1 } , . . . , C ^ { k }$ , where each corpus has a different source language with the target language as English. Then, we sample equal-sized batches $B ^ { i } \subset C ^ { i }$ from each corpus and take gradient steps on $\begin{array} { r } { \sum _ { i = 1 } ^ { k } L ( f ; B ^ { i } ) + \lambda R ( f ; B ^ { i } ) } \end{array}$ , which moves all of the non-English embeddings toward English.
+
+Note that this alignment method departs from prior work, in which each non-English language is rotated to match the English embedding space through individual learned matrices. Specifically, the most widely used post-hoc alignment method learns a rotation $W$ applied to the source vectors to minimize the distance between parallel word pairs, or
+
+$$
+\operatorname* { m i n } _ { W } \sum _ { ( \mathbf { s } , \mathbf { t } ) \in C } \sum _ { ( i , j ) \in a ( \mathbf { s } , \mathbf { t } ) } | | W f ( i , \mathbf { s } ) - f ( j , \mathbf { t } ) | | _ { 2 } ^ { 2 } \quad s . t . \quad W ^ { \top } W = I .
+$$
+
+This problem is known as the Procrustes problem and can be solved in closed form (Schonemann, 1966). This approach has the nice property that the vectors are only rotated, preserving distances and therefore the semantic information captured by the embeddings (Artetxe et al., 2016). However, rotation requires the strong assumption that the embedding spaces are roughly isometric (Søgaard et al., 2018), an assumption that may not hold for contextual pre-trained models because they represent more aspects of a word than just its type, i.e. context and syntax, which are less likely to be isomorphic between languages. Given that past work has also found independent alignment per language pair to be inferior to joint training (Heyman et al., 2019), another advantage of our method is that the alignment for all languages is done simultaneously.
+
+As our dataset, we use the Europarl corpora for English paired with Bulgarian, German, Greek, Spanish, and French, the languages represented in both Europarl and XNLI (Koehn, 2005). After tokenization (Koehn et al., 2007), we produce word pairs using fastAlign and keep the one-to-one pairs in the intersection (Dyer et al., 2013). We use the most recent 1024 sentences as the test set, the previous 1024 sentences as the development set, and the following 250K sentences as the training set. Furthermore, we modify the test set accuracy calculation to only include word pairs not seen in the training set. We also remove any exact matches, e.g. punctuation and numbers, because BERT is already aligned for these pairs due to its shared vocabulary. Given that parallel data may be limited for low-resource language pairs, we also report numbers for 10K and 50K parallel sentences.
+
+# 3.4 SENTENCE-AUGMENTED NON-CONTEXTUAL BASELINE
+
+Given that there has been a long line of work on word vector alignment (Artetxe et al., 2018; Conneau et al., 2018a; Smith et al., 2017, inter alia), we also compare BERT to a sentence-augmented fastText baseline (Bojanowski et al., 2017). Following Artetxe et al. (2018), we first normalize, then mean-center, then normalize the word vectors, and we then learn a rotation with the same parallel data as in the contextual case, as described in Equation 1. We also strengthen this baseline by including sentence information: specifically, during word retrieval, we concatenate each word vector with a vector representing its sentence. Following Ruckl ¨ e et al. (2018), we compute the sentence ´ vector by concatenating the average, maximum, and minimum vector over all of the words in the sentence, a method that was shown to be state-of-the-art for a suite of cross-lingual tasks. We also experimented with other methods, such as first retrieving the sentence and then the word, but found this method resulted in the highest accuracy. As a result, the fastText vectors are 1200-dimensional, while the BERT vectors are 768-dimensional.
+
+# 3.5 TESTING ZERO-SHOT TRANSFER
+
+The next step is to determine whether better alignment improves cross-lingual transfer. As our downstream task, we use the XNLI dataset, where the English MultiNLI development and test sets are human-translated into multiple languages (Conneau et al., 2018b; Williams et al., 2018). Given a pair of sentences, the task is to predict whether the first sentence implies the second, where there are three labels: entailment, neutral, or contradiction. Starting from either the base or aligned multilingual BERT, we train on English and evaluate on Bulgarian, German, Greek, Spanish, and French, the XNLI languages represented in Europarl.
+
+As our architecture, following Devlin et al. (2018), we apply a linear layer followed by softmax on the [CLS] embedding of the sentence pair, producing scores for each of the three labels. The model is trained using cross-entropy loss and selected based on its development set accuracy averaged across all of the languages. As a fully-supervised ceiling, we also compare to models trained and tested on the same language, where for the non-English training data, we use the machine translations of the English MultiNLI training data provided by Conneau et al. (2018b). While the quality of the training data is affected by the quality of the MT system, this comparison nevertheless serves as a good approximation for the fully-supervised setting.
+
+<table><tr><td></td><td>English</td><td>Bulgarian</td><td>German</td><td>Greek</td><td>Spanish</td><td>French</td><td>Average</td></tr><tr><td colspan="8">Translate-Train</td></tr><tr><td>Base BERT</td><td>81.9</td><td>73.6</td><td>75.9</td><td>71.6</td><td>77.8</td><td>76.8</td><td>76.3</td></tr><tr><td colspan="8">Zero-Shota</td></tr><tr><td>Base BERT</td><td>80.4</td><td>68.7</td><td>70.4</td><td>67.0</td><td>74.5</td><td>73.4</td><td>72.4</td></tr><tr><td>Sentence-aligned BERT (rotation)</td><td>81.1</td><td>68.9</td><td>71.2</td><td>66.7</td><td>74.9</td><td>73.5</td><td>72.7</td></tr><tr><td>Word-aligned BERT (rotation)</td><td>78.8</td><td>69.0</td><td>71.3</td><td>67.1</td><td>74.3</td><td>73.0</td><td>72.2</td></tr><tr><td>Word-aligned BERT (fine-tuned)</td><td>80.1</td><td>73.4</td><td>73.1</td><td>71.4</td><td>75.5</td><td>74.5</td><td>74.7</td></tr><tr><td>XLM (MLM + TLM)</td><td>85.0</td><td>77.4</td><td>77.8</td><td>76.6</td><td>78.9</td><td>78.7</td><td>79.1</td></tr></table>
+
+Table 1: Accuracy on the XNLI test set, where we compare to base BERT (Devlin et al., 2018) and two rotation-based methods, sentence alignment (Aldarmaki & Diab, 2019) and word alignment (Wang et al., 2019). We also include the current state-of-the-art zero-shot achieved by XLM (Lample & Conneau, 2019). Rotation-based methods provide small gains on some languages but not others. On the other hand, after fine-tuning-based alignment, Bulgarian and Greek match the translate-train ceiling, while German, Spanish, and French close roughly one-third of the gap.
+
+aNote that the zero-shot Base BERT numbers are slightly different from those reported in Devlin et al. (2019) because we select a single model using the average accuracy across the six languages. This selection method also accounts for the varying English accuracies across the zero-shot methods.   
+Table 2: Zero-shot accuracy on the XNLI test set, where we align BERT with varying amounts of parallel data. The method scales with the amount of data but achieves a large fraction of the gains with 50K sentences per language pair.   
+
+<table><tr><td>Sentences</td><td>English</td><td>Bulgarian</td><td>German</td><td>Greek</td><td>Spanish</td><td>French</td><td>Average</td></tr><tr><td>None</td><td>80.4</td><td>68.7</td><td>70.4</td><td>67.0</td><td>74.5</td><td>73.4</td><td>72.4</td></tr><tr><td>10K</td><td>79.2</td><td>71.0</td><td>71.8</td><td>67.5</td><td>75.3</td><td>74.1</td><td>73.2</td></tr><tr><td>50K</td><td>81.1</td><td>73</td><td>72.6</td><td>69.6</td><td>75</td><td>74.5</td><td>74.3</td></tr><tr><td>250K</td><td>80.1</td><td>73.4</td><td>73.1</td><td>71.4</td><td>75.5</td><td>74.5</td><td>74.7</td></tr></table>
+
+# 4 RESULTS
+
+# 4.1 ZERO-SHOT XNLI TRANSFER
+
+First, we test whether alignment improves multilingual BERT by applying the models to zero-shot XNLI, as displayed in Table 1. We see that our alignment procedure greatly improves accuracies, with all languages seeing a gain of at least $1 \%$ . In particular, the Bulgarian and Greek zero-shot numbers are boosted by almost $5 \%$ each and match the translate-train numbers, suggesting that the alignment procedure is especially effective for languages that are initially difficult for BERT. We also run alignment for more distant language pairs (Chinese, Arabic, Urdu) and find similar results, which we report in the appendix.
+
+Comparing to rotation-based methods (Aldarmaki & Diab, 2019; Wang et al., 2019), we find that a rotation produces small gains for some languages, namely Bulgarian, German, and Spanish, but is sub-optimal overall, providing evidence that the increased expressivity of our proposed procedure is beneficial for contextual alignment. We explore this comparison more in Section 5.1.
+
+# 4.2 ALIGNMENT WITH LESS DATA
+
+Given that our goal is zero-shot transfer, we cannot expect to always have large amounts of parallel data. Therefore, we also characterize the performance of our alignment method with varying amounts of data, as displayed in Table 2. We find that it improves transfer with as little as 10K sentences per language, making it a promising approach for low-resource languages.
+
+<table><tr><td></td><td>bg-en</td><td>de-en</td><td>el-en</td><td>es-en</td><td>fr-en</td><td>Average</td></tr><tr><td colspan="7">Contextual</td></tr><tr><td>Aligned fastText + sentence</td><td>44.0</td><td>46.4</td><td>42.0</td><td>48.6</td><td>44.5</td><td>45.1</td></tr><tr><td>Base BERT</td><td>19.5</td><td>26.1</td><td>13.9</td><td>32.5</td><td>28.3</td><td>24.1</td></tr><tr><td>Word-aligned BERT (rotation)</td><td>29.8</td><td>31.6</td><td>20.8</td><td>36.8</td><td>31.0</td><td>30.0</td></tr><tr><td>Word-aligned BERT (fine-tuned)</td><td>50.7</td><td>51.3</td><td>49.8</td><td>51.0</td><td>48.6</td><td>50.3</td></tr><tr><td colspan="7">Non-Contextual</td></tr><tr><td>Aligned fastText + sentence</td><td>61.3</td><td>65.4</td><td>61.6</td><td>71.1</td><td>64.8</td><td>64.8</td></tr><tr><td>Base BERT</td><td>29.1</td><td>37.0</td><td>22.3</td><td>46.5</td><td>41.8</td><td>35.3</td></tr><tr><td>Word-aligned BERT (rotation)</td><td>39.6</td><td>43.6</td><td>32.4</td><td>51.4</td><td>46.1</td><td>42.6</td></tr><tr><td>Word-aligned BERT (fine-tuned)</td><td>62.8</td><td>64.3</td><td>67.5</td><td>68.4</td><td>66.3</td><td>65.9</td></tr></table>
+
+Table 3: Word retrieval accuracy for the aligned sentence-augmented fastText baseline and BERT pre- and post-alignment. Across languages, base BERT has variable accuracy while fine-tuningaligned BERT is consistently effective. Fine-tuned BERT also matches fastText in a version of the task where context is not necessary, suggesting that our method matches the type-level alignment of fastText while also aligning context.
+
+# 5 ANALYSIS
+
+# 5.1 WORD RETRIEVAL
+
+In the following sections, we present word retrieval results to both compare our method to past work and better understand the strengths and weaknesses of multilingual BERT. Table 3 displays the word retrieval accuracies for the aligned sentence-augmented fastText baseline and BERT pre- and postalignment. First, we find that in contextual retrieval, fine-tuned BERT outperforms fastText, which outperforms rotation-aligned BERT. This result supports the intuition that aligning large pre-trained models is more difficult than aligning word vectors, given that a rotation, at least when applied naively, produces sub-par alignments. In addition, fine-tuned BERT matches the performance of fastText in non-contextual retrieval, suggesting that our alignment procedure overcomes these challenges and achieves type-level alignment that matches non-contextual approaches. In the appendix, we also provide examples of aligned BERT disambiguating between different meanings of a word, giving qualitative evidence of the benefit of context alignment.
+
+We also find that before alignment, BERT’s performance varies greatly between languages, while after alignment it is consistently effective. In particular, Bulgarian and Greek initially have very low accuracies. This phenomenon is also reflected in the XNLI numbers (Table 1), where Bulgarian and Greek receive the largest boosts from alignment. Examining the connection between alignment and zero-shot more closely, we find that the word retrieval accuracies are highly correlated with downstream zero-shot performance (Figure 2), supporting our evaluation measure as predictive of cross-lingual transfer.
+
+The language discrepancies are also consistent with a hypothesis by Pires et al. (2019) to explain BERT’s multilingualism. They posit that due to the shared vocabulary, shared words between languages, e.g. numbers and names, are forced to have the same representation. Then, due to the masked word prediction task, other words that co-occur with these shared words also receive similar representations. If this hypothesis is true, then languages with higher lexical overlap with English are likely to experience higher transfer. As an extreme form of this phenomenon, Bulgarian and Greek have completely different scripts and should experience worse transfer than the common-script languages, an intuition that is confirmed by the word retrieval and XNLI accuracies. The fact that all languages are equally aligned with English post-alignment suggests that the pre-training procedure is suboptimal for these languages.
+
+<table><tr><td>Lexical Overlap</td><td>Numeral</td><td>Punctuation</td><td>Proper Noun</td><td></td><td></td><td>Average</td></tr><tr><td>Base BERT</td><td>0.90</td><td>0.88</td><td>0.80</td><td></td><td></td><td>0.86</td></tr><tr><td>Aligned BERT</td><td>0.97</td><td>0.96</td><td>0.95</td><td></td><td></td><td>0.96</td></tr><tr><td>Closed-Class</td><td>Determiner</td><td>Preposition</td><td>Conjunction</td><td>Pronoun</td><td>Auxiliary</td><td>Average</td></tr><tr><td>Base BERT Aligned BERT</td><td>0.76 0.91</td><td>0.72</td><td>0.71 0.89</td><td>0.70</td><td>0.61 0.84</td><td>0.70 0.88</td></tr><tr><td></td><td></td><td>0.86</td><td></td><td>0.89</td><td></td><td></td></tr><tr><td>Open-Class</td><td>Noun</td><td>Adverb</td><td> Adjective</td><td>Verb</td><td></td><td>Average</td></tr><tr><td>Base BERT</td><td>0.61</td><td>0.57</td><td>0.50</td><td>0.49</td><td></td><td>0.54</td></tr><tr><td>Aligned BERT</td><td>0.90</td><td>0.88</td><td>0.90</td><td>0.89</td><td></td><td>0.89</td></tr></table>
+
+Table 4: Accuracy by part-of-speech tag for non-contextual word retrieval. To achieve better word type coverage, we do not remove word pairs seen in the training set. The tags are grouped into lexically overlapping, closed-class, and open-class groups. The “Particle,” “Symbol,” “Interjection,” and “Other” tags are omitted.
+
+![](images/e1a9b32051eef1a96b60f5d7735160d2364f6bc899a30cce49a764a54d24797e.jpg)  
+Figure 2: XNLI zero-shot versus word retrieval accuracy for base BERT, where each point is a language paired with English. This plot suggests that alignment correlates well with cross-lingual transfer.
+
+![](images/0a321f0f63a8cf4e40a803ea3bc122f0996b9da54be228ec672f00c82584d179.jpg)  
+Figure 3: Contextual word retrieval accuracy plotted against difference in frequency rank between source and target. The accuracy of base BERT plummets for larger differences, suggesting that its alignment depends on word pairs having similar usage statistics.
+
+# 5.2 WORD RETRIEVAL PART-OF-SPEECH ANALYSIS
+
+Next, to gain insight into the multilingual pre-training procedure, we analyze the accuracy broken down by part-of-speech using the Universal Part-of-Speech Tagset (Petrov et al., 2012), annotated using polyglot (Al-Rfou et al., 2013) for Bulgarian and spaCy (Honnibal & Montani, 2017) for the other languages, as displayed in Table 4. Unsurprisingly, multilingual BERT has high alignment out-of-the-box for groups with high lexical overlap, e.g. numerals, punctuation, and proper nouns, due to its shared vocabulary. We further divide the remaining tags into closed-class and open-class, where closed-class parts-of-speech correspond to fixed sets of words serving grammatical functions (e.g. determiner, preposition, conjunction, pronoun, and auxiliary), while open-class parts-of-speech correspond to lexical words (e.g. noun, adverb, adjective, verb). Interestingly, we see that base BERT has consistently lower accuracy for closed-class versus open-class categories (0.70 vs 0.54), but that this discrepancy disappears after alignment (0.88 vs 0.89).
+
+# 5.3 USAGE HYPOTHESIS FOR ALIGNMENT
+
+From this closed-class vs open-class difference, we hypothesize that BERT’s alignment of a particular word pair is influenced by the similarity of their usage statistics. Specifically, given that BERT is trained through masked word prediction, its embeddings are in large part determined by the co-occurrences between words. Therefore, two words that are used in similar contexts should be better aligned. This hypothesis provides an explanation of the closed-class vs open-class difference: closed-class words are typically grammatical, so they are used in similar ways across typologically similar languages. Furthermore, these words cannot be substituted for one another due to their grammatical function. Therefore, their usage statistics are a strong signature that can be used for alignment. On the other hand, open-class words can be substituted for one another: for example, in most sentences, the noun tokens could be replaced by a wide range of semantically dissimilar nouns with the sentence remaining syntactically well-formed. By this effect, many nouns have similar co-occurrences, making them difficult to align through masked word prediction alone.
+
+To further test this hypothesis, we plot the word retrieval accuracy versus the difference between the frequency rank of the target and source word, where this difference measures discrepancies in usage, as depicted in Figure 3. We see that accuracy drops off significantly as the source-target difference increases, supporting our hypothesis. Furthermore, this shortcoming is remedied by alignment, revealing another systematic deficiency of multilingual pre-training.
+
+# 6 CONCLUSION
+
+Given that the degree of alignment is causally predictive of downstream cross-lingual transfer, contextual alignment proves to be a useful concept for understanding and improving multilingual pretrained models. Given small amounts of parallel data, our alignment procedure improves multilingual BERT and corrects many of its systematic deficiencies. Contextual word retrieval also provides useful new insights into the pre-training procedure, opening up new avenues for analysis.
+
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+<table><tr><td></td><td>English</td><td>Bulgarian</td><td>German</td><td>Greek</td><td>Spanish</td><td>French</td><td>Arabic</td><td>Chinese</td><td>Urdu</td><td>Average</td></tr><tr><td>Translate-Train</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Base BERT</td><td>81.9</td><td>73.6</td><td>75.9</td><td>71.6</td><td>77.8</td><td>76.8</td><td>70.7</td><td>76.6</td><td>61.6</td><td>74.1</td></tr><tr><td>Zero-Shot</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Base BERT</td><td>80.4</td><td>68.7</td><td>70.4</td><td>67.0</td><td>74.5</td><td>73.4</td><td>65.6</td><td>70.6</td><td>60.3</td><td>70.1</td></tr><tr><td>Aligned BERT (2OK sent)</td><td>80.8</td><td>71.6</td><td>72.5</td><td>68.1</td><td>74.7</td><td>73.6</td><td>66.3</td><td>71.5</td><td>61.1</td><td>71.1</td></tr></table>
+
+Table 5: Zero-shot accuracy on the XNLI test set with more languages, where we use 20K parallel sentences for each language paired with English. This result confirms that the alignment method works for distant languages and a variety of parallel corpora, including Europarl, MultiUN, and Tanzil, which contains sentences from the Quran (Koehn, 2005; Eisele & Chen, 2010; Tiedemann, 2012).
+
+# A APPENDIX
+
+# A.1 OPTIMIZATION HYPERPARAMETERS
+
+For both alignment and XNLI optimization, we use a learning rate of $5 \times 1 0 ^ { - 5 }$ with Adam hyperparameters $\beta = ( 0 . 9 , 0 . 9 8 )$ , $\epsilon = 1 \bar { 0 } ^ { - 9 }$ and linear learning rate warmup for the first $1 0 \%$ of the training data. For alignment, the model is trained for one epoch, with each batch containing 2 sentence pairs per language. For XNLI, each model is trained for 3 epochs with 32 examples per batch, and $1 0 \%$ dropout is applied to the BERT embeddings.
+
+# A.2 ALIGNMENT OF CHINESE, ARABIC, AND URDU
+
+In Table 5, we report numbers for additional languages, where we align a single BERT model for all eight languages and then fine-tune on XNLI. We use 20K sentences per language, where we use the MultiUN corpus for Arabic and Chinese (Eisele & Chen, 2010), the Tanzil corpus for Urdu (Tiedemann, 2012), and the Europarl corpus for the other five languages (Koehn, 2005). This result confirms that the alignment method works for a variety of languages and corpora. Furthermore, the Tanzil corpus consists of sentences from the Quran, suggesting that the method works even when the parallel corpus and downstream task contain sentences from entirely different domains.
+
+A.3 EXAMPLES OF CONTEXT-AWARE RETRIEVAL
+
+In this section, we qualitatively show that aligned BERT is able to disambiguate between different occurences of a word.
+
+First, we find two meanings of the word “like” occurring in the English-German Europarl test set.   
+Note also that in the second and third example, the two senses of “like” occur in the same sentence.   
+• This empire did not look for colonies far from home or overseas, like most Western European States, but close by. Dieses Reich suchte seine Kolonien nicht weit von zu Hause und in bersee wie die meisten westeuropaischen Staaten, sondern in der unmittelbaren Umgebung. ¨ Like other speakers, I would like to support the call for the arms embargo to remain. Wie andere Sprecher, so mochte auch ich den Aufruf zur Aufrechterhaltung des Waffen-¨ embargos untersttzen.   
+• Like other speakers, I would like to support the call for the arms embargo to remain. Wie andere Sprecher, so mochte ¨ auch ich den Aufruf zur Aufrechterhaltung des Waffenembargos untersttzen.   
+• I would also like, although they are absent, to mention the Commission and the Council. Ich mochte ¨ mir sogar erlauben, die Kommission und den Rat zu nennen, auch wenn sie nicht anwesend sind.
+
+Multiple meanings of “order”:
+
+• Moreover, the national political elite had to make a detour in Ambon in order to reach the civil governor’s residence by warship. In Ambon mußte die politische Spitze des Landes auch noch einen Umweg machen, um mit einem Kriegsschiff die Residenz des Provinzgouverneurs zu erreichen. Although the European Union has an interest in being surrounded by large, stable regions, the tools it has available in order to achieve this are still very limited. Der Europaischen Union ist zwar an großen stabilen Regionen in ihrer Umgebung gelegen, ¨ aber sie verfgt nach wie vor nur ber recht begrenzte Instrumente, um das zu erreichen.   
+• We could reasonably expect the new Indonesian government to take action in three fundamental areas: restoring public order, prosecuting and punishing those who have blood on their hands and entering into a political dialogue with the opposition. Von der neuen indonesischen Regierung darf man mit Fug und Recht drei elementare Maß- nahmen erwarten: die Wiederherstellung der offentlichen ¨ Ordnung, die Verfolgung und Bestrafung derjenigen, an deren Handen Blut klebt, und die Aufnahme des politischen Di- ¨ alogs mit den Gegnern. Firstly, I might mention the fact that the army needs to be reformed, secondly that a stable system of law and order needs to be introduced. Ich nenne hier an erster Stelle die notwendige Reform der Armee, ferner die Einfhrung eines stabilen Systems rechtsstaatlicher Ordnung.
+
+Multiple meanings of “support”:
+
+• Financial support is needed to enable poor countries to take part in these court activities. Arme Lander m ¨ ussen finanziell ¨ unterstutzt ¨ werden, damit auch sie sich an der Arbeit des Gerichtshofs beteiligen konnen. ¨   
+• We must help them and ensure that a proper action plan is implemented to support their work. Es gilt einen wirklichen Aktionsplan auf den Weg zu bringen, um die Arbeit dieser Organisationen zu unterstutzen ¨ .   
+• So I hope that you will all support this resolution condemning the abominable conditions of prisoners and civilians in Djibouti. Ich hoffe daher, daß Sie alle diese Entschließung befurworten ¨ , die die entsetzlichen Bedingungen von Inhaftierten und Zivilpersonen in Dschibuti verurteilt. It would be difficult to support a subsidy scheme that channelled most of the aid to the large farms in the best agricultural regions. Es ware auch problematisch, ein Beihilfesystem zu ¨ befurworten ¨ , das die meisten Beihilfen in die großen Betriebe in den besten landwirtschaftlichen Gebieten lenkt.
+
+Multiple meanings of “close”:
+
+• This empire did not look for colonies far from home or overseas, like most Western European States, but close by. Dieses Reich suchte seine Kolonien nicht weit von zu Hause und in bersee wie die meisten westeuropaischen Staaten, sondern in der unmittelbaren ¨ Umgebung.   
+In addition, if we are to shut down or refuse investment from every company which may have an association with the arms industry, then we would have to close virtually every American and Japanese software company on the island of Ireland with catastrophic consequences. Wenn wir zudem jedes Unternehmen, das auf irgendeine Weise mit der Rstungsindustrie verbunden ist, schließen oder Investitionen dieser Unternehmen unterbinden, dann mßten wir so ziemlich alle amerikanischen und japanischen Softwareunternehmen auf der irischen Insel schließen, was katastrophale Auswirkungen hatte. ¨
+
+• On the other hand, the deployment of resources left over in the Structural Funds from the programme planning period 1994 to 1999 is hardly worth considering as the available funds have already been allocated to specific measures, in this case in close collaboration with the relevant French authorities. Die Verwendung verbliebener Mittel der Strukturfonds aus dem Programmplanungszeitraum 1994 bis 1999 ist dagegen kaum in Erwagung zu ziehen, da die verfgbaren ¨ Mittel bereits bestimmten Maßnahmen zugewiesen sind, und zwar im konkreten Fall im engen Zusammenwirken mit den zustandigen franz ¨ osischen Beh ¨ orden. ¨ This is particularly justified given that, as already stated, many Member States have very close relations with Djibouti. Zumal, wie erwahnt, viele Mitgliedstaaten sehr ¨ enge Beziehungen zu Dschibuti unterhalten. Mr President, it is regrettable that, at the close of the 20th century, a century symbolised so positively by the peaceful women’s revolution, there are still countries, such as Kuwait and Afghanistan, where half the population, women that is, is still denied fundamental human rights. Herr Prasident! Es ist wirklich bedauerlich, daß es am ¨ Ende des 20. Jahrhunderts, eines so positiv von der friedlichen Revolution der Frauen gepragten Jahrhunderts, noch immer ¨ Lander wie Kuwait und Afghanistan gibt, in denen der H ¨ alfte der Bev ¨ olkerung, den Frauen, ¨ die elementaren Menschenrechte verweigert werden.

parse/train/rkg98yBFDr/rkg98yBFDr_content_list.json

@@ -0,0 +1,1770 @@
+[
+    {
+        "type": "text",
+        "text": "REJECT ILLEGAL INPUTS: SCALING GENERATIVE CLASSIFIERS WITH SUPERVISED DEEP INFOMAX ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
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+        "text": "Deep Infomax (DIM) is an unsupervised representation learning framework by maximizing the mutual information between the inputs and the outputs of an encoder, while probabilistic constraints are imposed on the outputs. In this paper, we propose Supervised Deep InfoMax (SDIM), which introduces supervised probabilistic constraints to the encoder outputs. The supervised probabilistic constraints are equivalent to a generative classifier on high-level data representations, where class conditional log-likelihoods of samples can be evaluated. Unlike other works building generative classifiers with conditional generative models, SDIMs scale on complex datasets, and can achieve comparable performance with discriminative counterparts. With SDIM, we could perform classification with rejection. Instead of always reporting a class label, SDIM only makes predictions when test samples’ largest logits surpass some pre-chosen thresholds, otherwise they will be deemed as out of the data distributions, and be rejected. Our experiments show that SDIM with rejection policy can effectively reject illegal inputs including outof-distribution samples and adversarial examples. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
+            336,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Non-robustness of neural network models emerges as a pressing concern since they are observed to be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2014). Many attack methods have been developed to find imperceptible perturbations to fool the target classifiers (Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2017; Brendel et al., 2017). Meanwhile, many defense schemes have also been proposed to improve the robustnesses of the target models (Goodfellow et al., 2014; Tramer et al., 2017; Madry et al., 2017; Samangouei et al., 2018). \\` ",
+        "bbox": [
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+            618
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An important fact about these works is that they focus on discriminative classifiers, which directly model the conditional probabilities of labels given samples. Another promising direction, which is almost neglected so far, is to explore robustness of generative classifiers (Ng & Jordan, 2002). A generative classifier explicitly model conditional distributions of inputs given the class labels. During inference, it evaluates all the class conditional likelihoods of the test input, and outputs the class label corresponding to the maximum. Conditional generative models are powerful and natural choices to model the class conditional distributions, but they suffer from two big problems: (1) it is hard to scale generative classifiers on high-dimensional tasks, like natural images classification, with comparable performance to the discriminative counterparts. Though generative classifiers have shown promising results of adversarial robustness, they hardly achieve acceptable classification performance even on CIFAR10 (Li et al., 2018; Schott et al., 2018; Fetaya et al., 2019). (2) The behaviors of likelihood-based generative models can be counter-intuitive and brittle. They may assign surprisingly higher likelihoods to out-of-distribution (OoD) samples (Nalisnick et al., 2018; Choi & Jang, 2018). Fetaya et al. (2019) discuss the issues of likelihood as a metric for density modeling, which may be the reason of non-robust classification, e.g. OoD samples detection. ",
+        "bbox": [
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+            625,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose supervised deep infomax (SDIM) by introducing supervised statistical constraints into deep infomax (DIM, Hjelm et al. (2018)), an unsupervised learning framework by maximizing the mutual information between representations and data. SDIM is trained by optimizing two objectives: (1) maximizing the mutual information (MI) between the inputs and the high-level data representations from encoder; (2) ensuring that the representations satisfy the supervised statistical constraints. The supervised statistical constraints can be interpreted as a generative classifier on high-level data representations giving up the full generative process. Unlike full generative models making implicit manifold assumptions, the supervised statistical constraints of SDIM serve as explicit enforcement of manifold assumption: data representations (low-dimensional) are trained to form clusters corresponding to their class labels. With SDIM, we could perform classification with rejection (Nalisnick et al., 2019; Geifman & El-Yaniv, 2017). SDIMs reject illegal inputs based on off-manifold conjecture (Samangouei et al., 2018; Gu & Rigazio, 2014), where illegal inputs, e.g. adversarial examples, lie far away from the data manifold. Samples whose class conditionals are smaller than the pre-chosen thresholds will be deemed as off-manifold, and prediction requests on them will be rejected. The contributions of this paper are : ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We propose Supervised Deep Infomax (SDIM), an end-to-end framework whose probabilistic constraints are equivalent to a generative classifier. SDIMs can achieve comparable classification performance with similar discrinimative counterparts at the cost of small over-parameterization. We propose a simple but novel rejection policy based on off-manifold conjecture: SDIM outputs a class label only if the test sample’s largest class conditional surpasses the prechosen class threshold, otherwise outputs rejection. The choice of thresholds relies only on training set, and takes no additional computations. Experiments show that SDIM with rejection policy can effectively reject illegal inputs, including OoD samples and adversarial examples generated by a comprehensive group of adversarial attacks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND: DEEP INFOMAX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            419,
+            475,
+            435
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep InfoMax (DIM, Hjelm et al. (2018)) is an unsupervised representation learning framework by maximizing the mutual information (MI) of the inputs and outputs of an encoder. The computation of MI takes only input-output pairs with the deep neural networks based esimator MINE (Belghazi et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $E _ { \\phi }$ be an encoder parameterized by $\\phi$ , working on the training set $\\mathcal { X } = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ , and generating output set $\\mathcal { V } = \\{ E ( x _ { i } ) \\} _ { i = 1 } ^ { N }$ . DIM is trained to find the set of parameters $\\phi$ such that: (1) the mutual information ${ \\mathcal { T } } ( X , Y )$ is maximized over sample sets $\\mathcal { X }$ and $\\mathcal { V }$ . (2) the representations, depending on the potential downstream tasks, match some prior distribution. Denote $\\mathbb { J }$ and $\\mathbb { M }$ the joint and product of marginals of random variables $X$ , $Y$ respectively. MINE estimates a lower-bound of MI with Donsker-Varadhan (Donsker & Varadhan, 1983) representation of KL-divergence: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35895dce495f9fcf4f271100c4f3721ec12152f961942490676f00533491da00.jpg",
+        "text": "$$\n\\mathcal { Z } ( X , Y ) = D _ { K L } ( \\mathbb { J } | | \\mathbb { M } ) \\geq \\mathbb { E } _ { \\mathbb { J } } [ T _ { \\omega } ( x , y ) ] - \\log \\mathbb { E } _ { \\mathbb { M } } [ e ^ { T _ { \\omega } ( x , y ) } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            599,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $T _ { \\omega } ( x , y ) \\in \\mathbb { R }$ is a family of functions with parameters $\\omega$ represented by a neural network. Since in representation learning we are more interested in maximizing MI, than its exact value, nonKL divergences are also favorable candidates. We can get a family of variational lower-bounds using $f$ -divergence representations (Nguyen et al., 2010): ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dd80b927f9836fe6426cb54577a003c5c4b5f3296f082fc2f03b2c45937b231c.jpg",
+        "text": "$$\n\\mathbb { Z } _ { f } ( X , Y ) \\ge \\mathbb { E } _ { \\mathbb { J } } [ T _ { \\omega } ( x , y ) ] - \\mathbb { E } _ { \\mathbb { M } } [ f ^ { * } \\left( T _ { \\omega } ( x , y ) \\right) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $f ^ { * }$ is the Fenchel conjugate of a specific divergence $f$ . For KL-divergence, $f ^ { * } ( t ) = e ^ { ( t - 1 ) }$ . A full $f ^ { * }$ list is provided in Tab. 6 of Nowozin et al. (2016). Noise-Contrastive Estimation (Gutmann & Hyvarinen, 2010) can also be used as lower-bound of MI in “infoNCE” (Oord et al., 2018) . ¨ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 SUPERVISED DEEP INFOMAX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            758,
+            450,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "All the components of SDIM framework are summurized in Fig. 1. The focus of Supervised Deep InfoMax (SDIM) is on introducing supervision to probabilistic constraints of DIM for (generative) classification. We choose to maximize the local MI, which has shown to be more effective in classification tasks than maximizing global MI (Hjelm et al., 2018). Equivalently, we minimize $\\mathcal { I } _ { \\mathrm { M I } }$ : ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2465f01f66e89c8feca9bf84626fdb2420c00d491c1bbcf1610e772c42db68ab.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { M I } } = - \\frac { 1 } { M ^ { 2 } } \\sum _ { i = 1 } ^ { M ^ { 2 } } \\tilde { \\mathcal { T } } ( L _ { \\phi } ^ { ( i ) } ( \\pmb { x } ) , E _ { \\phi } ( \\pmb { x } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            847,
+            622,
+            892
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $L _ { \\phi } ( { \\pmb x } )$ is a local $M \\times M$ feature map of $_ { x }$ extracted from some intermediate layer of encoder $E$ , and $\\tilde { \\mathcal { T } }$ can be any possible MI lower-bounds. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/4057617de4c52d01f42003f6e7da7d01924abf89f79d7ff02acce1cb109853f9.jpg",
+        "image_caption": [
+            "Figure 1: Components of SDIM framework. (1)The encoder $E _ { \\phi }$ takes input $_ { \\textbf { \\em x } }$ , and produces pairs of local feature maps $L _ { \\phi } ( \\pmb { x } )$ and global representations $E _ { \\phi } ( \\pmb { x } )$ . (2) The MI evaluation network $T _ { \\omega }$ maps every possible positive pairs and negative pairs to MI scores specified by corresponding MI lower-bound. Negative pairs are simply obtained by combine all unpaired local feature maps and global representations within the same mini-batch. (3) Supervised constraints are imposed on the global representations $\\tilde { \\pmb { x } } = E _ { \\phi } ( \\pmb { x } )$ with loss $\\mathcal { T } _ { \\mathrm { N L L } } + \\mathcal { T } _ { \\mathrm { L M } }$ for generative classification. The true class conditionals are maximized, while false class conditionals are minimized. See following parts of this section for details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 EXPLICIT ENFORCEMENT OF MANIFOLD ASSUMPTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            583,
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+            598
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By adopting a generative approach $p ( \\pmb { x } , y ) = p ( y ) p ( \\pmb { x } | y )$ , we assume that the data follows the manifold assumption: the (high-dimensional) data lies on low-dimensional manifolds corresponding to their class labels. Denote $\\tilde { \\pmb x }$ the compact representation generated with encoder $E _ { \\phi } ( \\pmb { x } )$ . In order to explicitly enforce the manifold assumption , we admit the existence of data manifold in the representation space. Assume that $y$ is a discrete random variable representing class labels, and $p ( \\tilde { \\mathbfit { x } } | \\boldsymbol { y } )$ is the real class conditional distribution of the data manifold given $y$ . Let $p _ { \\theta } ( \\tilde { \\pmb x } | y )$ be the class conditionals we model parameterized with $\\theta$ . We approximate $p ( \\tilde { \\mathbfit { x } } | \\boldsymbol { y } )$ by minimizing the KL-divergence between $p ( \\tilde { \\mathbfit { x } } | \\boldsymbol { y } )$ and our model $p _ { \\theta } ( \\tilde { \\pmb x } | y )$ , which is given by: ",
+        "bbox": [
+            173,
+            608,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84d38e6c8d4d57ade11e6797907ebb8c241f21e0049449e5d08e71fa392138f5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { D _ { K L } \\big ( p ( \\tilde { x } | y ) | | p _ { \\theta } ( \\tilde { x } | y ) \\big ) = \\mathbb { E } _ { \\tilde { x } , y \\sim p ( \\tilde { x } , y ) } [ \\log p ( \\tilde { x } | y ) - \\log p _ { \\theta } ( \\tilde { x } | y ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { \\tilde { x } , y \\sim p ( \\tilde { x } , y ) } [ \\log p ( \\tilde { x } | y ) ] - \\mathbb { E } _ { \\tilde { x } , y \\sim p ( \\tilde { x } , y ) } [ \\log p _ { \\theta } ( \\tilde { x } | y ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            724,
+            759,
+            765
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the first item on RHS is a constant independent of the model parameters $\\theta$ . Eq. 4 equals to maximize the expectation $\\mathbb { E } _ { \\tilde { \\mathbf { x } } , y \\sim p ( \\tilde { \\mathbf { x } } , y ) } [ \\log p _ { \\theta } ( \\tilde { \\mathbf { x } } | y ) ]$ . ",
+        "bbox": [
+            171,
+            766,
+            823,
+            797
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In practice, we minimize the following loss $\\mathcal { T } _ { \\mathrm { N L L } }$ , equivalent to empicically maximize the above expectation over $\\{ \\tilde { \\pmb { x } } _ { i } = E _ { \\phi } ( \\pmb { x } _ { i } ) , y _ { i } \\} _ { i = 1 } ^ { N }$ : ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9aa5ad65f48a82ba27d1517e7b56fd93be8b96555521952a5a38be0bc0b3ceee.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { N L L } } = - \\mathbb { E } _ { \\tilde { \\mathbf { x } } , y \\sim p ( \\tilde { \\mathbf { x } } , y ) } [ \\log p _ { \\theta } ( \\tilde { \\mathbf { x } } | y ) ] \\approx - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log p _ { \\theta } ( \\tilde { \\mathbf { x } } _ { i } | y _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Besides the introduction of supervision, SDIM differs from DIM in its way of enforcing the statistical constraints: DIM use adversarial learning (Makhzani et al., 2015) to push the representations to the desired priors, while SDIM directly maximizes the parameterized class conditional probability. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Maximize Likelihood Margins Since a generative classifier, at inference, decides which class a test input $_ { \\textbf { \\em x } }$ belongs to according to its class conditional probability. On one hand, we maximize samples’ true class conditional probabilities (classes they belong to) using $\\mathcal { T } _ { \\mathrm { N L L } }$ ; On the other hand, we also hope that samples’ false class conditional probabilities (classes they do not belong to) can be minimized. This is assured by the following likelihood margin loss $\\mathcal { I } _ { \\mathrm { L M } }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6ed3f7d35972129b8b8633536bc2f31b9d603a10f8d5b38aef984cb02945ceb.jpg",
+        "text": "$$\n\\mathcal { I } _ { \\mathrm { L M } } = \\frac { 1 } { N } \\cdot \\frac { 1 } { C - 1 } \\sum _ { i = 1 } ^ { N } \\sum _ { c = 1 , c \\neq y _ { i } } ^ { C } \\operatorname* { m a x } ( \\log p ( \\tilde { \\mathbf { x } } _ { i } | y = c ) + K - \\log p ( \\tilde { \\mathbf { x } } _ { i } | y = y _ { i } ) , 0 ) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $K$ is a positive constant to control the margin. For each encoder output $\\tilde { \\mathbf { x } } _ { i }$ , the $C - 1$ truefalse class conditional gaps are squared1, which quadratically increases the penalties when the gap becomes large, then are averaged. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Putting all these together, the complete loss function we minimize is: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e67675b111887be06e0a23a5089b2084fd039bac7bd9f583c1edaff61c07108c.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { S D I M } } = \\alpha \\cdot \\mathcal { T } _ { \\mathrm { M I } } + \\beta \\cdot \\mathcal { T } _ { \\mathrm { N L L } } + \\gamma \\cdot \\mathcal { T } _ { \\mathrm { L M } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Parameterization of Class Conditional Probability Each of the class conditional distribution is represented as an isotropic Gaussian. So the generative classifier is simply a embedding layer with $C$ entries, and each entry contains the trainable mean and variance of a Gaussian. This minimized parameterization encourages the encoder to learn simple and stable low-dimensional representations that can be easily explained by even unimodal distributions. Considering that we maximize the true class conditional probability, and minimize the false class conditional probability at the same time, we do not choose conditional normalizing flows, since the parameters are shared across class labels, and the training can be very difficult. In Schott et al. (2018), each class conditional probability is represented with a VAE, thus scaling to complex datasets with huge number of classes, e.g. ImageNet, is almost impossible. ",
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+    {
+        "type": "text",
+        "text": "3.2 DECISION FUNCTION WITH REJECTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A generative approach models the class-conditional distributions $p ( { \\pmb x } | { \\pmb y } )$ , as well as the class priors $p ( y )$ . For classification, we compute the posterior probabilities $p ( y | \\mathbf { \\boldsymbol { x } } )$ through Bayes’ rule: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a041fd13d5dffdb94922185f70e40a21941f88811720a3c44063429458e1899e.jpg",
+        "text": "$$\np ( y | \\pmb { x } ) = \\frac { p ( \\pmb { x } | y ) p ( y ) } { p ( \\pmb { x } ) } \\propto p ( \\pmb { x } | y ) p ( y )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The prior $p ( y )$ can be computed from the training set, or we simply use uniform class prior for all class labels by default. Then the prediction of test sample $\\mathbf { \\boldsymbol { x } } ^ { * }$ from posteriors is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0a1d14ff2dfcf784e584443efbb79150c506d8c4f727895f6b7f5e88107820a.jpg",
+        "text": "$$\ny ^ { * } = \\underset { c = [ 1 \\ldots C ] } { \\arg \\operatorname* { m a x } } \\log p ( \\pmb { x } ^ { * } | y = c ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The drawback of the above decision function is that it always gives a prediction even for illegal inputs. Instead of simply outputting the class label that maximizes class conditional probability of $\\pmb { x } ^ { * }$ , we set a threshold for each class conditional probability, and define our decision function with rejection to be: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8476b5c4fe224595bb8dbd3ed1a7df98fef92414f64a3242b6be76464221b4eb.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { y ^ { * } , } & { \\mathrm { i f } \\log p ( \\pmb { x } ^ { * } | y ^ { * } ) \\geq \\delta _ { y ^ { * } } } \\\\ { R e j e c t i o n , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "The model gives a rejection when $\\log p ( \\boldsymbol { x } ^ { * } | \\boldsymbol { y } ^ { * } )$ is smaller than the threshold $\\delta _ { y ^ { * } }$ . Note that here we can use $p ( x ^ { * } | y ^ { * } )$ and $p ( \\tilde { \\boldsymbol { x } } ^ { * } | \\boldsymbol { y } ^ { * } )$ interchangeably. This is also known as selective classification (Geifman & El-Yaniv, 2017) or classification with reject option (Nalisnick et al., 2019)(See Supp. A) ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Robustness of Likelihood-based Generative Models Though likelihood-based generative models have achieved great success in samples synthesis, the behaviors of their likelihoods can be counter-intuitive. Flow-based models (Kingma & Dhariwal, 2018) and as well as VAEs (Kingma & Welling, 2013), surprisingly assign even higher likelihoods to out-of-distribution samples than the samples in the training set (Nalisnick et al., 2018; Choi & Jang, 2018). Pixel-level statistical analyses in Nalisnick et al. (2018) show that OoD dataset may “sit inside of” the in-distribution dataset (i.e. with roughly the same mean but smaller variance). ",
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+    },
+    {
+        "type": "text",
+        "text": "Off-Manifold Conjecture Grosse et al. (2017) observe that adversarial examples are outside the training distribution via statistical testing. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "DefenseGAN (Samangouei et al., 2018) models real data distribution with the generator $G$ of GAN. At inference, instead of feeding the test input $_ { \\textbf { \\em x } }$ to the target classifier directly, it searches for the “closest” sample $G ( z ^ { \\ast } )$ from generator distribution to $_ { \\textbf { \\em x } }$ as the final input to the classifier. It ensures that the classifier only make predictions on the data manifold represented by the generator, ruling out the potential adversarial perturbations in $_ { \\textbf { \\em x } }$ . PixelDefend (Song et al., 2017) takes a similar approach which uses likelihood-based generative model - PixelCNN to model the data distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Both DefenseGAN and PixelDefend are additionally trained as peripheral defense schemes agnostic to the target classifiers. Training generative models on complex datasets notoriously takes huge amount of computational resources (Brock et al., 2018). In contrast, the training of SDIM is computationally similar to its discriminative counterpart. The verification of whether inputs are offmanifold is a built-in property of the SDIM generative classifier. The class conditionals of SDIM are modeled on low-dimensional data representations with simple Gaussians, which is much easier, and incurs very small computations. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Datasets We evaluate the effectiveness of the rejection policy of SDIM on four image datasets: MNIST, FashionMNIST (both resized to $3 2 \\times 3 2$ from $2 8 \\times 2 8$ ); and CIFAR10, SVHN. See App. B.1 for details of data processing. For out-of-distribution samples detection, we use the dataset pairs on which likelihood-based generative models fail (Nalisnick et al., 2018; Choi & Jang, 2018): FashionMNIST (in)-MNIST (out) and CIFAR10 (in)-SVHN (out). Adversarial examples detection are evaluated on MNIST and CIFAR10. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Choice of thresholds It is natural that choosing thresholds based on what the model knows, i.e. training set, and can reject what the model does not know, i.e. possible illegal inputs. We set one threshold for each class conditional. For each class conditional probability, we choose to evaluate on two different thresholds: 1st and $2 n d$ percentiles of class conditional log-likelihoods of the correctly classified training samples. Compared to the detection methods proposed in Li et al. (2018), our choice of thresholds is much simpler, and takes no additional computations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Models A typical SDIM instance consists of three networks: an encoder, parameterized by $\\phi$ , which outputs a $d$ -dimensional representation; mutual information evaluation networks, i.e. $T _ { \\omega }$ in Eqn. (1) and Eqn. (2); and $C$ -way class conditional embedding layer, parameterized by $\\theta$ , with each entry a $2 d$ -dimensional vector. We set $d = 6 4$ in all our experiments. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For encoder of SDIM, we use ResNet (He et al., 2016) on $3 2 \\times 3 2$ with a stack of $8 n + 2$ layers, and 4 filter sizes $\\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 \\}$ . The architecture is summarized as: ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b989ee5b1c0540cb6f85878f2d0f8ed1930006df4e6b2e9d8c1d810b020f6369.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td> output map size</td><td>32×32</td><td>16 ×16</td><td>8×8</td><td>4×4</td></tr><tr><td>#layers</td><td>1+2n</td><td>2n</td><td>2n</td><td>2n</td></tr><tr><td># filters</td><td>32</td><td>64</td><td>128</td><td>256</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The last layer of encoder is a $d$ -way fully-connected layer. To construct a discriminative counterpart, we simply set the output size of the encoder’s last layer to $C$ for classification. We use ResNet10 $\\overset { \\cdot } { n } = 1$ ) on MNIST, FashionMNIST, and ResNet26 $\\ R = 3$ ) on CIFAR10, SVHN. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 EVALUATION ON CLEAN DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We report the classification accuracies (see Tab. 1) of SDIMs and the discriminative counterparts on clean test sets . Results show that SDIMs achieve the same level of accuracy as the discriminative counterparts with slightly increased number of parameters ( $1 7 \\%$ increase for ResNet10, and $5 \\%$ increase for ResNet26). We are aware of the existence of better results reported on these datasets using more complex models (Huang et al., 2017; Han et al., 2017) or automatically designed architectures (Cai et al., 2018), but pushing the state-of-the-art is not the focus of this paper. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/c9e4173df794c63087eb400d55db1e26e63b1da21e323c876f272405e7114383.jpg",
+        "table_caption": [
+            "Table 1: Clean test accuracies of SDIMs and the discriminative counterparts. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td># Parameters</td><td>MNIST</td><td>FashionMNIST</td><td>CIFAR10</td><td>SVHN</td></tr><tr><td>Disc. (ResNet10,n = 1)</td><td>1.25M</td><td>99.42%</td><td>94.25%</td><td>-</td><td>-</td></tr><tr><td>SDIM (ResNet10, n = 1)</td><td>1.46M (17% 个)</td><td>99.55%</td><td>94.58%</td><td>-</td><td>-</td></tr><tr><td>Disc. (ResNet26,(n = 3))</td><td>4.39M</td><td>-</td><td>1</td><td>92.35%</td><td>95.96%</td></tr><tr><td>SDIM (ResNet26,n = 3)</td><td>4.60M (5% ↑)</td><td>1</td><td>1</td><td>92.53%</td><td>95.74%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Is Fully Generative Model Necessary for Generative Classification? In the evaluations of Li et al. (2018) and Schott et al. (2018), both model class conditional probability with VAE (Kingma & Welling, 2013; Rezende et al., 2014), and achieve acceptable accuracies $\\mathrm { \\dot { ( > ~ 9 8 \\% ) } }$ on MNIST. However, it is hard for fully conditional generative models to achieve satisfactory classification accuracies even on CIFAR10. On CIFAR10, methods in Li et al. (2018) achieve only $< 5 0 \\%$ accuracy. They also point out that the classification accuracy of a conditional $\\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017) is only $7 2 . 4 \\%$ . The test accuracy of ABS in (Schott et al., 2018) is only $5 4 \\%$ . In contrast, SDIM could achieve almost the same performance with similar discriminative classifier by giving up the full generative process, and building generative classifier on high-level representations. Li et al. (2018) improves the accuracy to $9 2 \\%$ by feeding the features learned by powerful discriminative classifier-VGG16 (Simonyan & Zisserman, 2014) to their generative classifiers, which also suggests that modeling likelihood on high-level representation (features) is more favorable for generative classification than pixel-level likelihood of fully generative classifiers. For classification tasks, discovering discriminative features is much more important than reconstructing the all the image pixels. Thus performing generative classification with full generative models may not be the right choice. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Decision with Rejection We also investigate the implications of the proposed decision function with rejection under different thresholds. The results in Tab. 2 show that choosing a higher percentile as threshold will reject more prediction requests. At the same time, the classification accuracies of SDIM on the left test sets become increasingly better. This demonstrate that out rejection policy tend to reject the ones on which SDIMs make wrong predictions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac2f282d7515a48b0c590762d0cf98d4ec87c60c34abd9e4ffa0045285ec4e6a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Original Acc.</td><td colspan=\"2\">1st percentile</td><td colspan=\"2\">2nd percentile</td></tr><tr><td>Acc.Left</td><td>Rej. Rate</td><td>Acc.Left</td><td>Rej. Rate</td></tr><tr><td>MNIST</td><td>99.55%</td><td>99.95%</td><td>3.02%</td><td>99.97%</td><td>4.00%</td></tr><tr><td>FashionMNIST</td><td>94.58%</td><td>96.45%</td><td>4.63%</td><td>96.94%</td><td>6.60%</td></tr><tr><td>CIFAR10</td><td>92.53%</td><td>96.18%</td><td>8.90%</td><td>96.60%</td><td>10.86%</td></tr><tr><td>SVHN</td><td>95.74%</td><td>97.43%</td><td>3.99%</td><td>98.00%</td><td>6.36%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Classification performances of SDIMs using the proposed decision function with rejection.   \nWe report the rejection rates of the test sets and the accuracies on the left test sets for each threshold. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 OUT-OF-DISTRIBUTION SAMPLES DETECTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Class-wise OoD detections are performed, and mean detection rates over all in-distribution classes are reported in Tab. 3. For each in-distribution class $c$ , we evaluate the log-likelihoods of the whole OoD dataset. Samples whose log-likelihoods are lower the class threshold $\\delta _ { c }$ will be detected as OoD samples. Same evaluations are applied on conditional Glows with 10th percentile thresholds, but the results are not good. The results are clear and confirm that SDIMs, generative classifiers on high-level representations, are more effective on classification tasks than fully conditional generative models on raw pixels. Note that fully generative models including VAE used in Li et al. (2018); Schott et al. (2018) fail on OoD detection. The stark difference between SDIM and full generative models (flows or VAEs) is that SDIM models samples’ likelihoods in the high-level representation spaces, while generative models evaluate directly on the raw pixels. See Supp. C for more results about the histograms of the class conditionals of in-out distributions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c585eaa14050dd27b90a775a140c6f3f68913c9dbd1612e86628a7c780d0e5f7.jpg",
+        "table_caption": [
+            "Table 3: Mean detection rates of SDIMs and Glows with different thresholds on OoD detection. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>FashionMNIST(in)-MNIST(out)</td><td>CIFAR10(in)-SVHN(out)</td></tr><tr><td>SDIM(1st Per.)</td><td>99.36%</td><td>94.24 %</td></tr><tr><td>SDIM(2nd Per.)</td><td>99.64%</td><td>95.81%</td></tr><tr><td>Glow(10th Per.)</td><td>3.53%</td><td>0.02%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 ROBUSTNESS AGAINST ADVERSARIAL EXAMPLES AND DETECTION",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We comprehensively evaluate the robustness of SDIMs against various attacks: ",
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+    },
+    {
+        "type": "text",
+        "text": "gradient-based attacks: one-step gradient attack FGSM (Goodfellow et al., 2014), its iterative variant projected gradient descent (PGD, Kurakin et al. (2016); Madry et al. (2017)), CW- $L _ { 2 }$ attack (Carlini & Wagner, 2017), deepfool (Moosavi-Dezfooli et al., 2016). • score-based attacks: local search attack (Narodytska & Kasiviswanathan, 2016). • decision-based attack: boundary attack (Brendel et al., 2017). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Attacks Using Cross-Entropy We find that SDIMs are much more robust to gradient-based attacks using cross-entropy, e.g. FGSM and PGD, since the gradients numerically vanish as a side effect of the likelihood margin loss $\\mathcal { I } _ { \\mathrm { L M } }$ of SDIM. This phenomenon is similar to some defences that try to hinder generations of adversarial examples by masking the gradients on inputs. While full generative classifiers in Li et al. (2018) still suffer from these attacks. See Supp. D.1 for detailed results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Conservative Adversarial Examples Adversarial attacks aim to find the minimal perturbations that sufficiently change the classification labels, i.e. flip other logits to be the largest one. We show case examples on MNIST generated by untargeted attacks and their logits in Tab. 4 (See Supp. D.2 for examples of CIFAR10). Though these attacks successfully flip the logits, they are designed to be conservative to avoid more distortions to the original images. As a result, the largest logits of adversarial examples are still much lower than the thresholds, so they can be detected by our rejection policy. We find that our rejection policy performs perfectly on MNIST, but fails to detect all adversarial examples on CIFAR10 except for Boundary attack (See Tab. 5). It seems to be a well-known observation that models trained on CIFAR10 are more vulnerable than one trained on MNIST. Gilmer et al. (2018) connects this observation to the generalization of models. They found that many test samples, though correctly classified, are close to the misclassfied samples, which implies the existence of adversarial examples. If a model has higher error rate, it would take smaller perturbations to move correctly classified samples to misclassified areas. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b31684597747c3525f50cfd2b9f46996d720a0ec92fab82aeb65f6063cf98f7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Samples</td><td colspan=\"2\">6</td><td colspan=\"2\">回</td><td colspan=\"2\">6</td><td colspan=\"2\">6</td><td colspan=\"2\"></td></tr><tr><td colspan=\"2\">Original</td><td colspan=\"2\">DeepFool</td><td colspan=\"2\">CW-L2</td><td colspan=\"2\">Boundary</td><td colspan=\"2\">LocalSearch</td></tr><tr><td>1st Per. Original</td><td>477.6 482.5</td><td>465.2 -644.5</td><td>481.1 -440.8</td><td>477.7 -378.8</td><td>465.4 -1082.5</td><td>407.2 -409.8</td><td>474.5 -473.8</td><td>470.4 -850.0</td><td>472.3 -415.0</td><td>463.5 -699.2</td></tr><tr><td>DeepFool</td><td>243.4</td><td>-306.4</td><td>-172.4</td><td>-394.1</td><td>-538.3</td><td>-181.0</td><td>243.5</td><td>-944.2</td><td>-107.7</td><td>-524.1</td></tr><tr><td>CW-L2</td><td>175.9</td><td>-244.9</td><td>-486.4</td><td>-287.5</td><td>-500.8</td><td>-257.6</td><td>-409.5</td><td>-233.5</td><td>-174.3</td><td>176.5</td></tr><tr><td>Boundary</td><td>-58.5</td><td>11.5</td><td>205.7</td><td>-149.7</td><td>-415.1</td><td>-308.0</td><td>-356.3</td><td>205.8</td><td>-223.3</td><td>-250.3</td></tr><tr><td>LocalSearch</td><td>180.4</td><td>-225.3</td><td>-481.9</td><td>-281.3</td><td>-498.9</td><td>-223.2</td><td>-378.8</td><td>-257.9</td><td>-143.1</td><td>189.9</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Full logits of the adversarial examples generated with different attacks. The original image is the first sample of class 0 of MNIST test set. The first row gives the 1st percentile thresholds, and the second row shows the logits of the original image. The largest logits are marked in bold. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Adversarial examples with more confidence Based on the observations above, a natural question we should ask is: can we generate adversarial examples with not only successfully flipped logits, ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/57bd6f7dd9fc9506e61f4012c8a3bed1c8e8f769eb6598e22c2dbd18581d683c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Attacks</td><td colspan=\"2\">MNIST</td><td colspan=\"2\">CIFAR10</td></tr><tr><td>1st Per.</td><td>2nd Per.</td><td>1st Per.</td><td>2nd Per.</td></tr><tr><td>DeepFool</td><td>98.00%</td><td>98.60%</td><td>61.10%</td><td>64.30%</td></tr><tr><td>Boundary</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td></tr><tr><td>LocalSearch</td><td>99.90%</td><td>100%</td><td>88.80%</td><td>93.10%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Detection rates of our rejection policies. We perform untargeted adversarial evaluation on the first 1000 images of test sets. CW- $. L _ { 2 }$ is not involved here, but carefully investigated below. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "but also the largest logit larger than some threshold value? Unlike the conservativeness on paying more distortions of other attacks, CW attack allows us to control the gap between largest and second largest logits with some confidence value $\\kappa$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We perform targeted CW attacks with confidences $\\kappa = \\{ 0 , 5 0 0 , 1 0 0 0 \\}$ (Tab. 6). We find that increasing the confidences help increasing the largest logits of adversarial examples to some extent, but may lead to failures of generation. The sensitivity to confidence values is also different given different targets. The success rates of generating adversarial examples monotonically decreases with the confidences increasing (Tab. 7). Note that on discriminative counterparts, $\\mathrm { C W } { \\cdot } L _ { 2 }$ with the same settings easily achieves $1 0 0 \\%$ success rates. This means that explicitly forcing data representations to form clusters with maximum margins between them help increase average distances between normal samples and the nearest misclassified areas, thus increase the hardness of finding minimal adversarial perturbations . In this case, it takes a large enough adversarial perturbation to move a sample from its cluster to the other. Meanwhile, detection rates remain satisfactory on MNIST, but obviously decline on CIFAR10. For victim generative classifiers in (Li et al., 2018) under CW- $L _ { 2 }$ attack, the detection rates of adversarial examples using the proposed detection methods can be $> 9 5 \\%$ on MNIST, but fall $< 5 0 \\%$ on even CIFAR10-binary (their models don’t scale on CIFAR10, and CW- $L _ { 2 }$ with non-zero confidences are also not evaluated). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3de047c92cb082ce505375e1c84b0bae71c627780c0550b81aa5d06f6b17785e.jpg",
+        "image_caption": [],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e9d882b4d907daa5a2f5ef180dc7aed6ff9da5136fe9ddb8c0d3fe523b9afbbf.jpg",
+        "table_caption": [
+            "Table 6: Adversarial examples generated with targeted CW with different confidences. The original image is the fist sample of class 0. The first row gives the 1st percentile thresholds. Below the images are the logits corresponding to the given targets. “-” denotes failure of generation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Attacks</td><td colspan=\"3\">MNIST</td><td colspan=\"3\">CIFAR10</td></tr><tr><td>1st Per.</td><td>2nd Per.</td><td>success rate</td><td>1st Per.</td><td>2nd Per.</td><td>success rate</td></tr><tr><td>CW-L2(K = 0)</td><td>100%</td><td>100%</td><td>84.65%</td><td>93.93%</td><td>94.80%</td><td>70.84%</td></tr><tr><td>CW-L2(K = 500)</td><td>99.78%</td><td>99.78%</td><td>76.61%</td><td>76.55%</td><td>84.07%</td><td>61.26%</td></tr><tr><td>CW-L2(κ = 1000)</td><td>90.24%</td><td>95.98%</td><td>45.56%</td><td>47.86%</td><td>75.13%</td><td>48.86%</td></tr></table>",
+        "bbox": [
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+            734,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Targeted adversarial evaluations results of our rejection policies on the first 1000 test samples. We report the detection rates with different thresholds and success rates of generating adversarial examples. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Discussions on off-manifold conjecture Gilmer et al. (2018) challenges whether the off-manifold conjecture holds in general. They experiment on synthetic dataset-two high-dimensional concentric spheres with theoretical analyses, showing that even for a trained classifier with close to zero test error, there may be a constant fraction of the data manifold misclassified, which indicates the existence of adversarial examples within the manifold. But there are still several concerns to be addressed: First, as also pointed out by the authors, the manifolds in natural datasets can be quite complex than that of simple synthesized dataset. Fetaya et al. (2019) draws similar conclusion from analyses on synthesized data with particular geometry. So the big concern is whether the conclusions in Gilmer et al. (2018); Fetaya et al. (2019) still hold for the manifolds in natural datasets. A practical obstacle to verify this conclusion is that works modeling the full generative processes are based on manifold assumption, but provide no explicit manifolds for analytical analyses like Gilmer et al. (2018); Fetaya et al. (2019). While SDIM enables explicit and customized manifolds on high-level data representations via probabilistic constraints, thus enables analytical analyses. In this paper, samples of different classes are trained to form isotropic Gaussians corresponding to their classes in representation space (other choices are possible). The relation between the adversarial robustness and the forms and dimensionalities of data manifolds is to be explored. Second, in their experiments, all models evaluated are discriminative classifiers. Considering the recent promising results of generative classifiers against adversarial examples, would using generative classifiers lead to different results? One thing making us feel optimistic is that even though the existence of adversarial examples is inevitable, Gilmer et al. (2018) suggest that adversarial robustness can be improved by minimizing the test errors, which is also supported by our experimental differences on MNIST and CIFAR10. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduce supervised probabilistic constraints to DIM. Giving up the full generative process, SDIMs are equivalent to generative classifiers on high-level data representations. Unlike full conditional generative models which achieve poor classification performance even on CIFAR10, SDIMs attain comparable performance as the discriminative counterparts on complex datasets. The training of SDIM is also computationally similar to discriminative classifiers, and does not require prohibitive computational resources. Our proposed rejection policy based on off-manifold conjecture, a built-in property of SDIM, can effectively reject illegal inputs including OoD samples and adversarial examples. We demonstrate that likelihoods modeled on high-level data representations, rather than raw pixel intensities, are more robust on downstream tasks without the requirement of generating real samples. ",
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+            804,
+            823,
+            847
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Pouya Samangouei, Maya Kabkab, and Rama Chellappa. Defense-gan: Protecting classifiers against adversarial attacks using generative models. arXiv preprint arXiv:1805.06605, 2018. ",
+        "bbox": [
+            173,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lukas Schott, Jonas Rauber, Matthias Bethge, and Wieland Brendel. Towards the first adversarially robust neural network model on mnist. arXiv preprint arXiv:1805.09190, 2018. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. ",
+        "bbox": [
+            169,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yang Song, Taesup Kim, Sebastian Nowozin, Stefano Ermon, and Nate Kushman. Pixeldefend: Leveraging generative models to understand and defend against adversarial examples. arXiv preprint arXiv:1710.10766, 2017. ",
+        "bbox": [
+            176,
+            140,
+            823,
+            183
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013. ",
+        "bbox": [
+            171,
+            193,
+            823,
+            222
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Florian Tramer, Alexey Kurakin, Nicolas Papernot, Ian Goodfellow, Dan Boneh, and Patrick Mc-\\` Daniel. Ensemble adversarial training: Attacks and defenses. arXiv preprint arXiv:1705.07204, 2017. ",
+        "bbox": [
+            176,
+            231,
+            823,
+            272
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A CLASSIFICATION WITH REJECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            501,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A very related work to our paper is (Nalisnick et al., 2019), which propose a hybrid model modeling distribution of features $p$ (features) and predictive distribution $p$ (targets|features) at the same time. Normalizing flow is used to learn invertible features as inputs of discriminative model, i.e. predictive distribution, and provides evaluation of features $\\pmb { x } ^ { * }$ . Inputs out of the training data distribution are rejected by setting a threshold for $\\ b { p } ( \\ b { x } ^ { * } )$ . For SDIM, illagel inputs are rejected by setting thresholds for each of the class conditional. The class conditionals are modeled on the data representations from encoder regularized by MI loss $\\mathcal { I } _ { \\mathrm { M I } }$ . The hybrid model in Nalisnick et al. (2019) can successfully distinguish in-distribution dataset and OoD dataset. While SDIM reject illegal inputs including OoD dataset samples and adversarial examples with more fine-grained class conditionals. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            258
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Geifman & El-Yaniv (2017) propose a selection method to perform selective classification with desired risk level given a trained model, but they focus on discriminative models. ",
+        "bbox": [
+            173,
+            266,
+            823,
+            295
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B EVALUATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            314,
+            397,
+            330
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1 DATA PROCESSING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            347,
+            349,
+            361
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• MNIST, FashionMNIST: All images are resized to $3 2 \\times 3 2$ from $2 8 \\times 2 8$ . We scale the images pixels to $[ 0 , 1 ]$ , and normalization is not used for fair comparisons with the baselines in (Li et al., 2018). CIFAR10, SVHN: We follow the simple data augmentation in (He et al., 2016) for training: 4 pixels are padded on each side, and a $3 2 \\times 3 2$ crop is randomly sampled from the padded image or its horizontal flip. For testing, we only evaluate the single view of the original $3 2 \\times 3 2$ image. All image pixels are also scaled to $[ 0 , 1 ]$ . ",
+        "bbox": [
+            215,
+            373,
+            825,
+            478
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "No other data augmentations or processings are used except for the explicitly listed above. ",
+        "bbox": [
+            173,
+            491,
+            764,
+            505
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2 MODELS AND TRAINING SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            522,
+            462,
+            536
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "MI computation The local MI $\\mathcal { I } _ { \\mathrm { M I } }$ is computed between the $4 \\times 4$ feature maps and the encoder outputs. We use lower-bound of Jensen-Shannon divergence to estimate the MI $\\mathcal { I } _ { J S D } ( \\boldsymbol { X } , \\boldsymbol { Y } )$ , and leave other lower-bounds unexplored. In practice, we find using other bounds would take more computational resources. However, we think better results can be expected if using these bounds according to the experimental results reported in Hjelm et al. (2018). ",
+        "bbox": [
+            174,
+            547,
+            825,
+            618
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "MI evaluation network Following Hjelm et al. (2018), we parameterize the MI evaluation network $T _ { \\omega }$ as a $1 \\times 1$ convolutional neural network with architecture: ",
+        "bbox": [
+            174,
+            633,
+            821,
+            662
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/0425e1b7996e163d9752cb006140172406ae3df3168ee66b545e23f0ee80b28a.jpg",
+        "table_caption": [
+            "Table 8: Local MI evaluation concat-and-convolve network architecture. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Operation</td><td>Size</td><td>Activation</td></tr><tr><td>Input -→1 × 1 conv</td><td>256</td><td>ReLU</td></tr><tr><td>1 ×1conv</td><td>256</td><td>ReLU</td></tr><tr><td>1×1</td><td>1</td><td></td></tr></table>",
+        "bbox": [
+            354,
+            675,
+            643,
+            748
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The input of $T _ { \\omega }$ is positive or negative pair of local feature map $L _ { \\phi } ( \\pmb { x } )$ and encoder output $E _ { \\phi } ( \\pmb { x } )$ . The positive pairs, samples from the joint distribution, are constructed by concat the encoder output $E _ { \\phi } ( \\pmb { x } )$ and the corresponding feature map $L _ { \\phi } ( \\pmb { x } )$ of the given batch. The negative pairs, samples from product of marginals, can be constructed by $E _ { \\phi } ( \\pmb { x } )$ and shuffled $L _ { \\phi } ( { \\pmb x } )$ along the batch-axis. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            852
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Optimization We set $\\alpha = \\beta = \\gamma = 1$ in our experiments. The constant $K$ in the likelihood margin loss $\\mathcal { I } _ { \\mathrm { L M } }$ is 10. All models are trained 500 epochs, and we always save the checkpoints reporting the minimum ongoing training losses for evaluation. We use optimizer Adam (Kingma & Ba, 2014) with default learning rate 0.001. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb04944933ee3f018ea1c8af1820eeb37c8beb8db9fcb24e8ff9bb94a6ddc00b.jpg",
+        "image_caption": [
+            "Figure 2: Class-wise distributions of the class conditionals of both in-distribution samples and outof-distribution samples. The in-out distribution pair is FashionMNIST-MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            137,
+            799,
+            303
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a5169fced8d34a83f04758c5dc342d75e8518d6f0a2cf3bb720c909c1ba68ca.jpg",
+        "image_caption": [
+            "Figure 3: Class-wise distributions of the class conditionals of both in-distribution samples and outof-distribution samples. The in-out distribution pair is CIFAR10-SVHN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            364,
+            797,
+            530
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D ADDITIONAL RESULTS OF ADVERSARIAL EVALUATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            604,
+            676,
+            619
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 ROBUSTNESS AGAINST ATTACKS USING CROSS-ENTROPY ",
+        "bbox": [
+            174,
+            636,
+            612,
+            650
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We make comparisons between SDIM and GBZ (Li et al., 2018), which consistently performs best in Deep Bayes. ",
+        "bbox": [
+            174,
+            662,
+            823,
+            690
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "FGSM and PGD- $L _ { \\infty }$ The results in Fig 4 and Fig 5 show that SDIM performs consistently better than the baseline. We find that increasing the distortion factor $\\epsilon$ of FGSM has no influences of SDIM’s accuracy, and the adversarial examples keep the same. Recall that the class conditionals are optimized to keep a considerable margin. Before evaluating the cross entropy loss, softmax is applied on the class conditionals $\\log p ( { \\pmb x } | c )$ to generate a even sharper distribution. So for the samples that are correctly classified, their losses are numerically zeros, and the gradient on inputs $\\nabla J _ { x } ( { \\pmb x } , y )$ are also numerically zeros. The PGD- $L _ { \\infty }$ we use here is the randomized version (Madry et al., $2 0 1 7 ) ^ { 2 }$ , which adds a small random perturbation before the iterative loop. The randomness is originally introduced to generate different adversarial examples for adversarial training, but here it breaks the zero loss so that the gradient on inputs $\\nabla J _ { x } ( { \\pmb x } , { \\bar { y } } )$ will not be zeros in the loop. FGSM can also be randomized (Tramer et al., 2017), which can be seen as a one-step variant of randomized \\` PGD. ",
+        "bbox": [
+            173,
+            707,
+            825,
+            872
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This phenomena is similar to what some defenses using gradient obfuscation want to achieve. Defensive distillation (Carlini & Wagner, 2016) masks the gradients of cross-entropy by increasing the temperature of softmax. But for CW attacks, which do not use cross-entropy, and operate on logits directly, this could be ineffective. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            159
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ee0e3001d7635d2990911b057f4190f0e443ff2925667f782d7c2638e661ac57.jpg",
+        "image_caption": [
+            "Figure 4: The adversarial classification accuracies of SDIM and GBZ on MNIST under FGSM- $. L _ { \\infty }$ and PGD- $L _ { \\infty }$ attacks. On the right are the generated adversarial examples with $\\epsilon$ from 0 to 0.5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            172,
+            820,
+            315
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb67a848cdd3382d1f6b17fdaca62a32856ae217d485d1aebae8b59fa7d91735.jpg",
+        "image_caption": [
+            "Figure 5: The adversarial classification accuracies of SDIM and GBZ on CIFAR10 under FGSM$L _ { \\infty }$ and PGD- $L _ { \\infty }$ attacks. On the right are the generated adversarial examples with $\\epsilon$ from 0 to 0.5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            387,
+            823,
+            529
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.2 ADVERSARIAL EXAMPLES OF CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
+            495,
+            625
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/500dbfba163b7eb6fe5b69b9d9edfb2c5e14276d1bf2da87ae06e486897d6ed2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Samples</td><td colspan=\"2\">国</td><td colspan=\"2\"></td><td colspan=\"2\">国</td><td colspan=\"2\">日</td><td colspan=\"2\">1</td></tr><tr><td colspan=\"2\">original</td><td colspan=\"2\">deepfool</td><td colspan=\"2\">cw-L2</td><td colspan=\"2\">boundary</td><td colspan=\"2\">JSMA</td></tr><tr><td>1st Per. original</td><td>408.6 424.2</td><td>396.6 -386.1</td><td>375.9 -379.2</td><td>378.4 -319.8</td><td>363.3 -370.2</td><td>376.7 -357.2</td><td>409.4 -356.9</td><td>383.0 -259.3</td><td>397.1 -291.9</td><td>412.2 -239.0</td></tr><tr><td>DeepFool</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>153.4</td><td>-391.6</td><td>-344.1</td><td>-262.6</td><td>-376.1</td><td>-345.9</td><td>215.1</td><td>-306.5</td><td>-244.3</td><td>-326.4</td></tr><tr><td>CW-L2</td><td>129.9</td><td>-555.3</td><td>235.4</td><td>-353.1</td><td>-471.6</td><td>-400.3</td><td>-342.7</td><td>-367.2</td><td>-486.4</td><td>-326.4</td></tr><tr><td>Boundary</td><td>213.9</td><td>-417.4</td><td>-458.3</td><td>-548.0</td><td>-587.4</td><td>-236.3</td><td>214.0</td><td>-1246.1</td><td>-171.2</td><td>-555.6</td></tr><tr><td>LocalSearch</td><td>165.2</td><td>-485.7</td><td>190.9</td><td>-325.6</td><td>-439.0</td><td>-379.0</td><td>-318.8</td><td>-327.5</td><td>-357.9</td><td>-272.3</td></tr></table>",
+        "bbox": [
+            173,
+            642,
+            846,
+            796
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 9: Full logits of adversarial examples generated with different attacks. The original image is the fist sample of class 0 of CIFAR10 test set. The first row gives the 1st percentile thresholds, and the second row shows the logits of the original image. The largest logits are marked in bold. ",
+        "bbox": [
+            173,
+            813,
+            826,
+            856
+        ],
+        "page_idx": 14
+    }
+]