Add files using upload-large-folder tool

parse/train/H1cWzoxA-/H1cWzoxA-.md

@@ -0,0 +1,490 @@
+# BI-DIRECTIONAL BLOCK SELF-ATTENTION FOR FASTAND MEMORY-EFFICIENT SEQUENCE MODELING
+
+Tao Shen†, Tianyi Zhou‡, Guodong Long†, Jing Jiang†& Chengqi Zhang† †Centre for Artificial Intelligence, School of Software, University of Technology Sydney ‡Paul G. Allen School of Computer Science & Engineering, University of Washington tao.shen@student.uts.edu.au,tianyizh@uw.edu {guodong.long,jing.jiang,chengqi.zhang}@uts.edu.au
+
+# ABSTRACT
+
+Recurrent neural networks (RNN), convolutional neural networks (CNN) and selfattention networks (SAN) are commonly used to produce context-aware representations. RNN can capture long-range dependency but is hard to parallelize and not time-efficient. CNN focuses on local dependency but does not perform well on some tasks. SAN can model both such dependencies via highly parallelizable computation, but memory requirement grows rapidly in line with sequence length. In this paper, we propose a model, called “bi-directional block self-attention network (Bi-BloSAN)”, for RNN/CNN-free sequence encoding. It requires as little memory as RNN but with all the merits of SAN. Bi-BloSAN splits the entire sequence into blocks, and applies an intra-block SAN to each block for modeling local context, then applies an inter-block SAN to the outputs for all blocks to capture long-range dependency. Thus, each SAN only needs to process a short sequence, and only a small amount of memory is required. Additionally, we use feature-level attention to handle the variation of contexts around the same word, and use forward/backward masks to encode temporal order information. On nine benchmark datasets for different NLP tasks, Bi-BloSAN achieves or improves upon state-of-the-art accuracy, and shows better efficiency-memory trade-off than existing RNN/CNN/SAN.
+
+# 1 INTRODUCTION
+
+Context dependency provides critical information for most natural language processing (NLP) tasks. In deep neural networks (DNN), context dependency is usually modeled by a context fusion module, whose goal is to learn a context-aware representation for each token from the input sequence. Recurrent neural networks (RNN), convolutional neural networks (CNN) and self-attention networks (SAN) are commonly used as context fusion modules. However, each has its own merits and defects, so which network to use is an open problem and mainly depends on the specific task.
+
+RNN is broadly used given its capability in capturing long-range dependency through recurrent computation. It has been applied to various NLP tasks, e.g., question answering (Wang et al., 2017), neural machine translation (Bahdanau et al., 2015), sentiment analysis (Qian et al., 2017), natural language inference (Liu et al., 2016), etc. However, training the basic RNN encounters the gradient dispersion problem, and is difficult to parallelize. Long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) effectively avoids the vanishing gradient. Gated recurrent unit (GRU) (Chung et al., 2014) and simple recurrent unit (SRU) (Lei & Zhang, 2017) improve the efficiency by reducing parameters and removing partial temporal-dependency, respectively. However, they still suffer from expensive time cost, especially when applied to long sequences.
+
+CNN becomes popular recently on some NLP tasks because of its the highly parallelizable convolution computation (Dong et al., 2017). Unlike RNN, CNN can simultaneously apply convolutions defined by different kernels to multiple chunks of a sequence (Kim, 2014). It is mainly used for sentence-encoding tasks (Lei et al., 2015; Kalchbrenner et al., 2014). Recently, hierarchical CNNs, e.g. ByteNet (Kalchbrenner et al., 2016), and ConvS2S (Gehring et al., 2017), are proposed to capture relatively long-range dependencies by using stacking CNNs to increase the number of input elements represented in a state. Nonetheless, as mentioned by Vaswani et al. (2017), the number of CNNs required to relate signals from two arbitrary input grows in the distance between positions, linearly for ConvS2S and logarithmically for ByteNet. This makes it difficult to learn dependencies between distant positions.
+
+Recently, self-attention networks (SAN) have been successfully applied to several NLP tasks. It produces context-aware representation by applying attention to each pair of tokens from the input sequence. Compared to RNN/CNN, SAN is flexible in modeling both long-range and local dependencies. The major computation in SAN is the highly parallelizable matrix multiplication without any temporal iteration, which can be easily accelerated by existing tools. Unlike most works that attach SAN to RNN/CNN as an additional module, two recent works show that SAN independent of any RNN/CNN module can achieve state-of-the-art performance on several NLP tasks. The first, multi-head attention (Vaswani et al., 2017), is a major component of a seq2seq model “Transformer” that outperforms previous methods in neural machine translation. It projects the input sequence into multiple subspaces, applies a SAN to the representation in each subspace, and concatenates the outputs. The second, directional self-attention network (DiSAN) (Shen et al., 2017), computes alignment scores at feature level, rather than at token level, and applies forward/backward masks to the alignment score matrix to encode temporal order information. DiSAN achieves the best or state-ofthe-art test accuracy on several NLP tasks by using less computational time and fewer parameters. More related works can be found in Appendix D.
+
+However, one drawback of SAN is its large memory requirement to store the alignment scores of all the token pairs; the number grows quadratically with the sequence length. By contrast, RNN/CNN demand far less memory. The goal of this paper is to develop a novel SAN for RNN/CNN-free sequence encoding, which requires as little memory as RNN but inherits all the advantages of SAN, i.e., highly parallelizable computation, the capability/flexibility in modeling both long-range/local dependencies, and state-of-the-art performance on multiple NLP tasks.
+
+We propose an attention mechanism, called “bidirectional block self-attention (Bi-BloSA)”, for fast and memory-efficient context fusion. The basic idea is to split a sequence into several length-equal blocks (with padding if necessary), and apply an intra-block SAN to each block independently. The outputs for all the blocks are then processed by an inter-block SAN. The intra-block SAN captures the local dependency within each block, while the inter-block SAN captures the long-range/global dependency. Hence, every SAN only needs to process a short sequence. Compared to a single SAN applied to the whole sequence, such two-layer stacked SAN saves a significant amount of memory. A feature fusion gate combines the outputs of intra-block and inter-block SAN with the original input, to produce the final contextaware representations of all the tokens. Similar to directional self-attention (DiSA) (Shen et al., 2017), BiBloSA uses forward/backward masks to encode the temporal order information, and feature-level attention to handle the variation of contexts around the same word. Further, a RNN/CNN-free sequence encoding model we build based on Bi-BloSA, called “bi-directional block self-attention network (Bi-BloSAN)”, uses an attention mechanism to compress the output of Bi-BloSA into a vector representation.
+
+![](images/4b9c89c715ce4fe3c14628e6de76cd75e581850cd269a07c916426f45e476dfd.jpg)  
+Figure 1: A comparison of Bi-BloSAN and other RNN/CNN/SAN in terms of training time, training memory consumption and test accuracy on SNLI (Bowman et al., 2015). The details of all the models are provided in Section 4.
+
+In experiments1, we implement Bi-BloSAN and popular sequence encoding models on several NLP tasks, e.g., language inference, sentiment analysis, semantic relatedness, reading comprehension, question-type classification, etc. The baseline models include Bi-LSTM, Bi-GRU, Bi-SRU, CNNs, multi-head attention and DiSAN. A thorough comparison on nine benchmark datasets demonstrates the advantages of Bi-BloSAN in terms of training speed, inference accuracy and memory consumption. Figure 1 shows that Bi-BloSAN obtains the best accuracy by costing similar training time to DiSAN, and as little memory as Bi-LSTM, Bi-GRU and multi-head attention. This shows that Bi-BloSAN achieves a better efficiency-memory trade-off than existing RNN/CNN/SAN models.
+
+Our notations follow these conventions: 1) lowercase denotes a vector; 2) bold lowercase denotes a sequence of vectors (stored as a matrix); and 3) uppercase denotes a matrix or a tensor.
+
+# 2 BACKGROUND
+
+# 2.1 WORD EMBEDDING
+
+Word embedding is the basic processing unit in most DNN for sequence modeling. It transfers each discrete token into a representation vector of real values. Given a sequence of tokens (e.g., words or characters) $\pmb { w } = [ w _ { 1 } , w _ { 2 } , \ldots , w _ { n } ] \in \mathbb { R } ^ { N \times n }$ , where $w _ { i }$ is a one-hot vector, $N$ is the vocabulary size and $n$ is the sequence length. A pre-trained token embedding (e.g. word2vec (Mikolov et al., 2013b)) is applied to $\textbf { \em w }$ , which outputs a sequence of low dimensional vectors $\pmb { x } = [ x _ { 1 } , x _ { 2 } , \ldots , x _ { n } ] \in \mathbb { R } ^ { d _ { e } \times n }$ . This process can be formally written as $\mathbf { \Delta } \mathbf { x } ~ = ~ W ^ { \left( e \right) } \mathbf { \Delta } w$ , where $W ^ { ( e ) } \in \mathbb { R } ^ { d _ { e } \times N }$ is the embedding weight matrix that can be fine-tuned during the training phase.
+
+# 2.2 VANILLA ATTENTION AND MULTI-DIMENSIONAL ATTENTION
+
+Vanilla Attention: Given an input sequence $\pmb { x } = [ x _ { 1 } , x _ { 2 } , \dots , x _ { n } ]$ composed of token embeddings and a vector representation of a query $\mathbf { \bar { \boldsymbol { q } } } \in \mathbb { R } ^ { d _ { q } }$ , vanilla attention (Bahdanau et al., 2015) computes the alignment score between $q$ and each token $x _ { i }$ (reflecting the attention of $q$ to $x _ { i }$ ) using a compatibility function $f ( x _ { i } , q )$ . A softmax function then transforms the alignment scores $a \in \mathbb { R } ^ { n }$ to a probability distribution $p ( z | \mathbf { \boldsymbol { x } } , q )$ , where $z$ is an indicator of which token is important to $q$ . A large $\bar { p } ( z = i | \mathbf { x } , q )$ means that $x _ { i }$ contributes important information to $q$ . This process can be written as
+
+$$
+\begin{array} { r l } & { a = [ f ( x _ { i } , q ) ] _ { i = 1 } ^ { n } , } \\ & { p ( z | \mathbf { x } , q ) = \mathrm { s o f t m a x } ( a ) . } \end{array}
+$$
+
+The output $s$ is the expectation of sampling a token according to its importance, i.e.,
+
+$$
+s = \sum _ { i = 1 } ^ { n } p ( z = i | \pmb { x } , q ) x _ { i } = \mathbb { E } _ { i \sim p ( z | \pmb { x } , q ) } ( x _ { i } ) .
+$$
+
+Multiplicative attention (or dot-product attention) (Vaswani et al., 2017; Sukhbaatar et al., 2015; Rush et al., 2015) and additive attention (or multi-layer perceptron attention) (Bahdanau et al., 2015; Shang et al., 2015) are two commonly used attention mechanisms. They differ in the choice of compatibility function $f ( x _ { i } , q )$ . Multiplicative attention uses the cosine similarity for $f ( x _ { i } , q )$ , i.e.,
+
+$$
+f ( x _ { i } , q ) = \left. W ^ { ( 1 ) } x _ { i } , W ^ { ( 2 ) } q \right. ,
+$$
+
+where $W ^ { ( 1 ) } \in \mathbb { R } ^ { d _ { h } \times d _ { e } } , W ^ { ( 2 ) } \in \mathbb { R } ^ { d _ { h } \times d _ { q } }$ are the learnable parameters. Additive attention is defined as
+
+$$
+f ( x _ { i } , q ) = w ^ { T } \sigma ( W ^ { ( 1 ) } x _ { i } + W ^ { ( 2 ) } q + b ^ { ( 1 ) } ) + b ,
+$$
+
+where $w \in \mathbb { R } ^ { d _ { h } }$ , $b ^ { ( 1 ) }$ and $b$ are the biases, and $\sigma ( \cdot )$ is an activation function. Additive attention usually achieves better empirical performance than multiplicative attention, but is expensive in time cost and memory consumption.
+
+Multi-dimensional Attention: Unlike vanilla attention, in multi-dimensional (multi-dim) attention (Shen et al., 2017), the alignment score is computed for each feature, i.e., the score of a token pair is a vector rather than a scalar, so the score might be large for some features but small for others. Therefore, it is more expressive than vanilla attention, especially for the words whose meaning varies in different contexts.
+
+Multi-dim attention has $d _ { e }$ indicators $z _ { 1 } , \ldots , z _ { d _ { e } }$ for $d _ { e }$ features. Each indicator has a probability distribution that is generated by applying softmax to the $n$ alignment scores of the corresponding feature. Hence, for each feature $k$ in each token $i$ , we have $P _ { k i } \triangleq p ( z _ { k } = i | \pmb { x } , q )$ where $P \in \mathbb { R } ^ { d _ { e } \times n }$ .
+
+A large $P _ { k i }$ means that the feature $k$ in token $i$ is important to $q$ . The output of multi-dim attention is written as
+
+$$
+s = \left[ \sum _ { i = 1 } ^ { n } P _ { k i } \pmb { x } _ { k i } \right] _ { k = 1 } ^ { d _ { e } } = \left[ \mathbb { E } _ { i \sim p ( z _ { k } | \pmb { x } , q ) } ( \pmb { x } _ { k i } ) \right] _ { k = 1 } ^ { d _ { e } } .
+$$
+
+For simplicity, we ignore the subscript $k$ where no confusion is caused. Then, Eq.(6) can be rewritten as an element-wise product, i.e., $\begin{array} { r } { s \stackrel { - } { = } \sum _ { i = 1 } ^ { n } P _ { \cdot i } \odot x _ { i } } \end{array}$ . Here, $P _ { \cdot i }$ is computed by the additive attention in Eq.(5) where $w ^ { T }$ is replaced with a weight matrix $W \in \mathbb { R } ^ { d _ { h } \times d _ { e } }$ , which leads to a score vector for each token pair.
+
+# 2.3 TWO TYPES OF SELF-ATTENTION
+
+token2token self-attention ( $\mathrm { H u }$ et al., 2017; Vaswani et al., 2017; Shen et al., 2017) produces context-aware representations by exploring the dependency between two tokens $x _ { i }$ and $x _ { j }$ from the same sequence $_ { \textbf { \em x } }$ . In particular, $q$ in Eq.(5) is replaced with $x _ { j }$ , i.e.,
+
+$$
+f ( x _ { i } , x _ { j } ) = W ^ { T } \sigma ( W ^ { ( 1 ) } x _ { i } + W ^ { ( 2 ) } x _ { j } + b ^ { ( 1 ) } ) + b .
+$$
+
+Similar to the $P$ in multi-dim attention, each input token $x _ { j }$ is associated with a probability matrix $P ^ { j }$ such that $P _ { k i } ^ { j } \triangleq p ( z _ { k } = i | \pmb { x } , x _ { j } )$ . The output representation for $x _ { j }$ is $\begin{array} { r } { s _ { j } = \sum _ { i = 1 } ^ { n } P _ { \cdot i } ^ { j } \odot x _ { i } } \end{array}$ and the final output of token2token self-attention is $\pmb { s } = \left[ s _ { 1 } , s _ { 2 } , \ldots , s _ { n } \right]$ .
+
+source2token self-attention (Lin et al., 2017; Shen et al., 2017; Liu et al., 2016) explores the importance of each token to the entire sentence given a specific task. In particular, $q$ is removed from Eq.(5), and the following equation is used as the compatibility function.
+
+$$
+\begin{array} { r } { f ( x _ { i } ) = W ^ { T } \sigma ( W ^ { ( 1 ) } x _ { i } + b ^ { ( 1 ) } ) + b . } \end{array}
+$$
+
+The probability matrix $P$ is defined as $P _ { k i } \triangleq p ( z _ { k } = i | \pmb { x } )$ . The final output of source2token selfki  k  attention has the same form as multi-dim attention, i.e., $\begin{array} { r } { \dot { s } = \sum _ { i = 1 } ^ { n } P _ { \cdot i } \odot \dot { x _ { i } } } \end{array}$
+
+# 2.4 MASKED SELF-ATTENTION
+
+Temporal order information is difficult to encode in token2token self-attention introduced above because the alignment score between two tokens is symmetric. Masked self-attention (Shen et al., 2017) applies a mask $M \in \mathbb { R } ^ { n \times n }$ to the alignment score matrix (or tensor due to feature-level score) computed by Eq.(7), so it allows one-way attention from one token to another. Specifically, the bias $b$ in Eq.(7) is replaced with a constant vector $M _ { i j } { \bf 1 }$ , where the 1 is an all-one vector. In addition, $W$ is fixed to a scalar $c$ and $\operatorname { t a n h } ( \cdot / c )$ is used as the activation function $\sigma ( \cdot )$ , i.e.,
+
+![](images/12fad078d040654ecac6b98a91435c605945ff1804df1076d398ab5ecfbbb190.jpg)  
+Figure 2: Masked self-attention mechanism. $f _ { i j }$ denotes $f ( x _ { i } , x _ { j } )$ in Eq.(9).
+
+$$
+f ( x _ { i } , x _ { j } ) = c \cdot \operatorname { t a n h } \left( [ W ^ { ( 1 ) } x _ { i } + W ^ { ( 2 ) } x _ { j } + b ^ { ( 1 ) } ] / c \right) + M _ { i j } \mathbf { 1 } ,
+$$
+
+where $W ^ { ( 1 ) } \in \mathbb { R } ^ { d _ { e } \times d _ { e } } , W ^ { ( 2 ) } \in \mathbb { R } ^ { d _ { e } \times d _ { q } }$ . The procedures to calculate the attention output from $f ( x _ { i } , x _ { j } )$ are identical to those in token2token self-attention. We use $\pmb { s } = g ^ { m } ( \pmb { x } , M )$ to denote the complete process of masked self-attention with $\pmb { s } = [ s _ { 1 } , s _ { 2 } , . . . , s _ { n } ]$ as the output sequence. An illustration of masked self-attention is given in Figure 2.
+
+In order to model bi-directional order information, forward mask $M ^ { f w }$ and backward mask $M ^ { b w }$ are respectively substituted into Eq.(9), which results in forward and backward self-attentions. These two masks are defined as
+
+$$
+M _ { i j } ^ { f w } = \left\{ \begin{array} { l l } { 0 , } & { i < j } \\ { - \infty , } & { \mathrm { o t h e r w i s e } } \end{array} \right. \quad M _ { i j } ^ { b w } = \left\{ \begin{array} { l l } { 0 , } & { i > j } \\ { - \infty , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+The outputs of forward and backward self-attentions are denoted by $\begin{array} { r } { s ^ { f w } \ = \ g ^ { m } ( \pmb { x } , M ^ { f w } ) } \end{array}$ and $\begin{array} { r } { s ^ { b w } = \dot { g } ^ { m } ( { \pmb x } , M ^ { b w } ) } \end{array}$ , respectively.
+
+# 3 PROPOSED MODEL
+
+In this section, we first introduce the “masked block self-attention (mBloSA)” (Section 3.1) as a fundamental self-attention module. Then, we present the “bi-directional block self-attention network (Bi-BloSAN)” (Section 3.2) for sequence encoding, which uses the “bi-directional block selfattention (Bi-BloSA)” (mBloSA with forward and backward masks) as its context fusion module.
+
+# 3.1 MASKED BLOCK SELF-ATTENTION
+
+As shown in Figure 3, masked block self-attention (mBloSA) has three parts from its bottom to top, i.e., 1) intra-block self-attention, 2) inter-block self-attention, and 3) the context fusion.
+
+![](images/9fc74bf1887d76a783aeb4a2f2918d964204be160c2c116de6e59441d6f49758.jpg)  
+Figure 3: Masked block self-attention (mBloSA) mechanism.
+
+Intra-block self-attention: We firstly split the input sequence of token/word embeddings into $m$ blocks of equal length $r$ , i.e., $[ { \pmb x } ^ { l } ] _ { l = 1 } ^ { m } \ = \ [ { \pmb x } ^ { 1 } , { \pmb x } ^ { 2 } , . . . , \bar { { \pmb x } } ^ { m } ]$ where $\pmb { x } ^ { 1 } = [ x _ { 1 } , x _ { 2 } , \ldots , { \bar { x _ { r } } } ] , \pmb { x } ^ { 2 } =$ $[ x _ { r + 1 } , x _ { r + 2 } , \ldots , x _ { 2 r } ]$ and $\pmb { x } ^ { m } = [ x _ { n - r + 1 } , x _ { n - r + 2 } , . ~ . ~ , x _ { n } ]$ . Padding can be applied to the last block if necessary. Intra-block self-attention applies the masked self-attentions $g ^ { m } ( \cdot , M )$ with shared parameters to all the blocks , i.e.,
+
+$$
+\pmb { h } ^ { l } = g ^ { m } ( \pmb { x } ^ { l } , M ) , l = 1 , 2 , . . . , m .
+$$
+
+Its goal is to capture the local context dependency inside each block. Similar to $\mathbf { \Delta } _ { x } l$ , the output representations of the tokens in the $l$ -th block are denoted by $\pmb { h } ^ { l } = [ h _ { r ( l - 1 ) + 1 } , h _ { r ( l - 1 ) + 2 } , \ldots , h _ { r \times l } ]$ . Note, the block length $r$ is a hyper-parameter and $m = n / r$ . In Appendix A, we introduce an approach to selecting the optimal $r$ , which results in the maximum memory utility rate in expectation.
+
+Inter-block self-attention: To generate a vector representation $v ^ { l }$ of each block, a source2token self-attention $g ^ { s 2 t } ( \cdot )$ is applied to the output $h ^ { l }$ of the intra-block self-attention on each block, i.e.,
+
+$$
+v ^ { l } = g ^ { s 2 t } ( h ^ { l } ) , l = 1 , 2 , \ldots , m .
+$$
+
+Note we apply the parameter-shared $g ^ { s 2 t } ( \cdot )$ to $h ^ { l }$ for different blocks. This provides us with a sequence $\pmb { v } = [ v _ { 1 } , v _ { 2 } , \dots , v _ { m } ]$ of local-context representations at block level. Inter-block self
+
+attention then applies a masked self-attention to $\textbf {  { v } }$ in order to capture the long-range/global dependency among the blocks, i.e.,
+
+$$
+\begin{array} { r } { \pmb { o } = \pmb { g } ^ { m } ( \pmb { v } , \pmb { M } ) . } \end{array}
+$$
+
+To combine the local and global context features at block level, a gate is used to merge the input and the output of the masked self-attention dynamically. This is similar to the gates in LSTM. The output sequence $\boldsymbol { e } = [ e _ { 1 } , \dots , e _ { m } ]$ of the gate is computed by
+
+$$
+\begin{array} { r l } & { G = \mathrm { s i g m o i d } \left( W ^ { \left( g 1 \right) } o + W ^ { \left( g 2 \right) } v + b ^ { \left( g \right) } \right) , } \\ & { e = G \odot o + \left( 1 - G \right) \odot v } \end{array}
+$$
+
+Context fusion: Given the long-range context representations $\pmb { e } = [ e _ { 1 } , \dots , e _ { m } ] \in \mathbb { R } ^ { d _ { e } \times m }$ at block level, we duplicate $e _ { l }$ for $r$ times to get $e ^ { l } = [ \dot { e _ { l } } , e _ { l } , \dots , e _ { l } ]$ (each token in block $l$ has the global context feature representation $e _ { l }$ ). Let $E \triangleq [ e ^ { l } ] _ { l = 1 } ^ { m } \in \mathbb { R } ^ { d _ { e } \times n }$ . Now, we have the input sequence $_ { \textbf { \em x } }$ of word embeddings, the local context features $^ { h }$ produced by intra-block self-attention, and the long-range/global context features $E$ produced by inter-block self-attention. A feature fusion gate (Gong $\&$ Bowman, 2017) is employed to combine them, and generates the final context-aware representations of all tokens, i.e.,
+
+$$
+\begin{array} { r l } & { F = \sigma \left( W ^ { ( f 1 ) } [ \pmb { x } ; \pmb { h } ; E ] + b ^ { ( f 1 ) } \right) , } \\ & { G = \mathrm { s i g m o i d } \left( W ^ { ( f 2 ) } [ \pmb { x } ; \pmb { h } ; E ] + b ^ { ( f 2 ) } \right) , } \\ & { \pmb { u } = G \odot F + ( 1 - G ) \odot \pmb { x } , } \end{array}
+$$
+
+where $\sigma ( \cdot )$ is an activation function, and $\pmb { \mathscr { u } } = [ u _ { 1 } , u _ { 2 } , \ldots , u _ { n } ] \in \mathbb { R } ^ { d _ { e } \times n }$ is the mBloSA output, which consists of the context-aware representations of the $n$ tokens.
+
+# 3.2 BI-DIRECTIONAL BLOCK SELF-ATTENTION NETWORK FOR SEQUENCE ENCODING
+
+We propose a sequence encoding model “Bi-directional block self-attention network (Bi-BloSAN)” with mBloSA as its major components. Its architecture is shown in Figure 4. In Bi-BloSAN, two fully connected layers (with untied parameters) are applied to the input sequence of token embeddings. Their outputs are processed by two mBloSA modules respectively. One uses the forward mask $\Dot { M } ^ { f w }$ and another uses the backward mask Mbw. Their outputs $\mathbf { \nabla } _ { u } f ^ { w }$ and $\mathbf { \Delta } u ^ { b w }$ are concatenated as $\mathbf { \dot { \boldsymbol { u } } } ^ { b i } = [ \mathbf { \boldsymbol { u } } ^ { f w } ; \mathbf { \boldsymbol { u } } ^ { b w } ] \in \mathbb { R } ^ { 2 d _ { e } \times n }$ . The idea of bi-directional attention follows the same spirit as Bi-LSTM and DiSAN. It
+
+![](images/24f82d8f73a072f687078960e1cea1ae995b46653058814ac133fc01d41eb984.jpg)  
+Figure 4: Bi-directional block self-attention network (BiBloSAN) for sequence encoding.
+
+encodes temporal order information lacking in existing SAN models. The context fusion module in Bi-BloSAN, with the input $_ { \textbf { \em x } }$ and the output $\boldsymbol { u } ^ { b i }$ , is called “Bi-BloSA”. In order to obtain a sequence encoding, a source2token self-attention transforms the sequence $\boldsymbol { u } ^ { b i }$ of concatenated token representations into a vector representation $s$ .
+
+# 4 EXPERIMENTS
+
+We conduct the experiments of Bi-BloSAN and several popular RNN/CNN/SAN-based sequence encoding models on nine benchmark datasets for multiple different NLP tasks. Note that, in some baseline models, a source2token self-attention is on the top of the models to generate an encoding for the entire sequence. All the models used for comparisons are listed as follows.
+
+• Bi-LSTM: 600D Bi-directional LSTM (300D forward LST $\Lambda + 3 0 0 \mathrm { D }$ backward LSTM) (Graves et al., 2013).
+
+• Bi-GRU: 600D Bi-directional GRU (Chung et al., 2014).   
+• Bi-SRU: 600D Bi-directional SRU (Lei & Zhang, 2017) (with sped-up recurrence but no CUDA level optimization for fair comparison).   
+• Multi-CNN: 600D CNN sentence embedding model (Kim, 2014) (200D for each of 3, 4, 5-gram).   
+• Hrchy-CNN: 3-layer 300D CNN (Gehring et al., 2017) with kernel length 5, to which gated linear units (Dauphin et al., 2016) and residual connection (He et al., 2016) are applied.   
+Multi-head: 600D Multi-head attention (Vaswani et al., 2017) (8 heads, each has 75 hidden units). The positional encoding method used in Vaswani et al. (2017) is applied to the input sequence to encode temporal order information.   
+• DiSAN: 600D Directional self-attention network (Shen et al., 2017) (300D forward masked self-attention $+ ~ 3 0 0 \mathrm { D }$ backward masked self-attention).
+
+All experimental codes are implemented in Python with Tensorflow and run on a single Nvidia GTX 1080Ti graphic card. Both time cost and memory load data are collected under Tensorflow1.3 with CUDA8 and cuDNN6021. In the rest of this section, we conduct the experiments on natural language inference in Section 4.1, reading comprehension in Section 4.2, semantic relatedness in Section 4.3 and sentence classifications in Section 4.4. Finally, we analyze the time cost and memory load of the different models vs. the sequence length in Section 4.5.
+
+# 4.1 NATURAL LANGUAGE INFERENCE
+
+Natural language inference (NLI) aims to reason the semantic relationship between a pair of sentences, i.e., a premise sentence and a hypothesis sentence. This relationship could be entailment, neutral or contradiction. In the experiment, we compare Bi-BloSAN to other baselines on the Stanford Natural Language Inference (Bowman et al., 2015) (SNLI)2 dataset, which contains standard training/dev/test split of 549,367/9,842/9,824 samples.
+
+Table 1: Experimental results for different methods on SNLI. $| \theta |$ : the number of parameters (excluding word embedding part). Train Accu and Test Accu: the accuracies on training and test sets respectively.   
+
+<table><tr><td>Model</td><td>|0</td><td>Train Accu</td><td>Test Accu</td></tr><tr><td>Unlexicalized features (Bowman et al.,2015)</td><td></td><td>49.4</td><td>50.4</td></tr><tr><td>+ Unigram and bigram features (Bowman et al., 2015)</td><td></td><td>99.7</td><td>78.2</td></tr><tr><td>100D LSTM encoders (Bowman et al., 2015)</td><td>0.2m</td><td>84.8</td><td>77.6</td></tr><tr><td>300D LSTM encoders (Bowman et al.,2016)</td><td>3.0m</td><td>83.9</td><td>80.6</td></tr><tr><td>1024D GRU encoders (Vendrov et al., 2016)</td><td>15.0m</td><td>98.8</td><td>81.4</td></tr><tr><td>300D Tree-based CNN encoders (Mou et al.,2016)</td><td>3.5m</td><td>83.3</td><td>82.1</td></tr><tr><td>300D SPINN-PI encoders (Bowman et al., 2016)</td><td>3.7m</td><td>89.2</td><td>83.2</td></tr><tr><td>600D Bi-LSTM encoders (Liu et al.,2016)</td><td>2.0m</td><td>86.4</td><td>83.3</td></tr><tr><td>300D NTI-SLSTM-LSTMencoders (Munkhdalai &amp; Yu,2017b)</td><td>4.0m</td><td>82.5</td><td>83.4</td></tr><tr><td>600D Bi-LSTM encoders+intra-attention (Liu et al., 2016)</td><td>2.8m</td><td>84.5</td><td>84.2</td></tr><tr><td>300D NSE encoders (Munkhdalai &amp; Yu,2017a)</td><td>3.0m</td><td>86.2</td><td>84.6</td></tr><tr><td>600D (300+300) Deep Gated Attn. (Chen et al., 2017)</td><td>11.6m</td><td>90.5</td><td>85.5</td></tr><tr><td>Bi-LSTM (Graves et al.,2013)</td><td>2.9m</td><td>90.4</td><td>85.0</td></tr><tr><td>Bi-GRU (Chung et al., 2014)</td><td>2.5m</td><td>91.9</td><td>84.9</td></tr><tr><td>Bi-SRU (Lei &amp; Zhang,2017)</td><td>2.0m</td><td>88.4</td><td>84.8</td></tr><tr><td>Multi-CNN (Kim,2014)</td><td>1.4m</td><td>89.3</td><td>83.2</td></tr><tr><td>Hrchy-CNN (Gehring et al., 2017)</td><td>3.4m</td><td>91.3</td><td>83.9</td></tr><tr><td>Multi-head (Vaswani et al.,2017)</td><td>2.0m</td><td>89.6</td><td>84.2</td></tr><tr><td>DiSAN (Shen et al.,2017)</td><td>2.3m</td><td>91.1</td><td>85.6</td></tr><tr><td>480DBi-BloSAN</td><td>2.8m</td><td>91.7</td><td>85.7</td></tr></table>
+
+Following the method of applying sentence-encoding to NLI task given in Bowman et al. (2016), two parameter-tied sentence-encoding models are applied to the premise and the hypothesis sentences respectively, to generate the premise encoding $s ^ { p }$ and the hypothesis encoding ${ \dot { s } } ^ { h }$ . A relation representation concatenating $s ^ { p }$ , $s ^ { h }$ , $s ^ { p } - s ^ { h }$ and $s ^ { p } \odot s ^ { h }$ is passed into a 300D fully connected layer, whose output is given to a 3-unit output layer with softmax to calculate the probability distribution over the three classes.
+
+Training Setup: The optimization objective is the cross-entropy loss plus L2 regularization penalty. We minimize the objective by Adadelta (Zeiler, 2012) optimizer which is empirically more stable than Adam (Kingma & Ba, 2015) on SNLI. The batch size is set to 64 for all methods. The training phase takes 50 epochs to converge. All weight matrices are initialized by Glorot Initialization (Glorot & Bengio, 2010), and the biases are initialized with 0. We use 300D GloVe 6B pre-trained vectors (Pennington et al., 2014) to initialize the word embeddings in $_ { \textbf { \em x } }$ . The Out-of-Vocabulary words in the training set are randomly initialized by uniform distribution between $\left( - 0 . 0 5 , 0 . 0 5 \right)$ . The word embeddings are fine-tuned during the training. The Dropout (Srivastava et al., 2014) keep probability and the L2 regularization weight decay factor $\gamma$ are set to 0.75 and $5 { \times } 1 0 ^ { - 5 }$ , respectively. The number of hidden units is 300. The unspecified activation functions in all models are set to Relu (Glorot et al., 2011).
+
+In Table 1, we report the number of parameters, and training/test accuracies of all baselines plus the methods from the official leaderboard. For fair comparison, we use 480D Bi-BloSAN, which leads to the similar parameter number with that of baseline encoders. Bi-BloSAN achieves the best test accuracy (similar to DiSAN) among all the sentence encoding models on SNLI. In particular, compared to the RNN models, Bi-BloSAN outperforms Bi-LSTM encoder, Bi-LSTM with attention and deep gated attention by $2 . 4 \%$ , $1 . 5 \%$ and $0 . 2 \%$ , respectively. Bi-BloSAN can even perform better than the semantic tree based models: SPINN-PI encoder $( + 2 . 5 \%$ )&Tree-based CNN encoder $( + 3 . 6 \% )$ , and the memory network based model: NSE encoder $( + 1 . 1 \% )$ . Additionally, Bi-BloSAN achieves the best performance among the baselines which are based on RNN/CNN/SAN. It outperforms Bi-LSTM $\left( + 0 . 7 \% \right)$ , Bi-GRU $( + 0 . 8 \% )$ , Bi-SRU $( + 0 . 9 \% )$ , multi-CNN $( + 2 . 5 \% )$ ), Hrchy-CNN $( + 1 . 8 \% )$ and multi-head attention $( + 1 . 5 \% )$ .
+
+Table 2: Time cost and memory consumption of the different methods on SNLI. Time(s)/epoch: average training time (second) per epoch. Memory(MB): Training GPU memory consumption (Megabyte). Inference Time(s): average inference time (second) for all dev data on SNLI with test batch size of 100.   
+
+<table><tr><td>Model</td><td>Time(s)/epoch</td><td>Memory(MB)</td><td>Inference Time(s)</td><td>Test Accuracy</td></tr><tr><td>Bi-LSTM (Graves et al., 2013)</td><td>2080</td><td>1245</td><td>9.2</td><td>85.0</td></tr><tr><td>Bi-GRU (Chung et al., 2014)</td><td>1728</td><td>1259</td><td>9.3</td><td>84.9</td></tr><tr><td>Bi-SRU (Lei &amp; Zhang,2017)</td><td>1630</td><td>731</td><td>8.2</td><td>84.8</td></tr><tr><td>Multi-CNN (Kim,2014)</td><td>284</td><td>529</td><td>2.4</td><td>83.2</td></tr><tr><td>Hrchy-CNN (Gehring et al., 2017)</td><td>343</td><td>2341</td><td>2.9</td><td>83.9</td></tr><tr><td>Multi-head (Vaswani et al., 2017)</td><td>345</td><td>1245</td><td>3.0</td><td>84.2</td></tr><tr><td>DiSAN (Shen et al., 2017)</td><td>587</td><td>2267</td><td>7.0</td><td>85.6</td></tr><tr><td>480D Bi-BloSAN</td><td>508</td><td>1243</td><td>3.4</td><td>85.7</td></tr></table>
+
+In addition, we compare time cost and memory consumption of all the baselines in Table 2. Compared to DiSAN with the same test accuracy, Bi-BloSAN is much faster and more memory efficient. In terms of training and inference time, Bi-BloSAN is $3 \sim 4 \times$ faster than the RNN models (BiLSTM, Bi-GRU, etc.). It is as fast as CNNs and multi-head attention but substantially outperforms them in test accuracy. In terms of training memory, Bi-BloSAN requires similar GPU memory to the RNN-based models and multi-head attention, which is much less than that needed by DiSAN.
+
+Finally, we conduct an ablation study of Bi-BloSAN in Table 3. In particular, we evaluate the contribution of each part of Bi-BloSAN by the change of test accuracy after removing the part. The removed part could be: 1) local context representations $^ { h }$ , 2) global context representations $E , 3 )$ ) the context fusion module (mBloSA) or 4) all fundamental modules appeared in this paper. The results show that both the local and global context representations play significant roles in BiBloSAN. They make Bi-BloSAN surpass the state-of-the-art models. Moreover, mBloSA improves the test accuracy from $8 3 . 1 \%$ to $8 5 . 7 \%$ . Source2token self-attention performs much better than vanilla attention, and improves the test accuracy by $3 . 3 \%$ .
+
+Table 3: An ablation study of Bi-BloSAN. “Local” denotes the local context representations $^ { h }$ and “Global” denotes the global context representations $E$ . “Bi-BloSAN w/o mBloSA” equals to word embeddings directly followed by a source2token attention and “Bi-BloSAN w/o mBloSA & source2token self-attn.” equals to word embeddings plus a vanilla attention without $q$ .   
+
+<table><tr><td>Model</td><td>|0</td><td>Test Accuracy</td></tr><tr><td>Bi-BloSAN</td><td>2.8m</td><td>85.7</td></tr><tr><td>Bi-BloSAN w/o Local</td><td>2.5m</td><td>85.2</td></tr><tr><td>Bi-BloSANw/o Global</td><td>1.8m</td><td>85.3</td></tr><tr><td>Bi-BloSAN w/o mBloSA</td><td>0.54m</td><td>83.1</td></tr><tr><td>Bi-BloSAN w/o mBloSA&amp; source2token self-attn.</td><td>0.45m</td><td>79.8</td></tr></table>
+
+# 4.2 READING COMPREHENSION
+
+Given a passage and a corresponding question, the goal of reading comprehension is to find the correct answer from the passage for the question. We use the Stanford Question Answering Dataset (Rajpurkar et al., 2016) $\mathbf { \bar { \Gamma } } ( \mathbf { S } \mathbf { Q u } \mathbf { \bar { A } } \mathbf { D } ) ^ { 3 }$ to evaluate all models. $\mathrm { S Q u A D }$ consists of questions posed by crowdworkers on a set of Wikipedia articles, where the answer to each question is a segment of text, or a span, from the corresponding passage.
+
+Since Bi-BloSAN and other baselines are designed for sequence encoding, such as sentence embedding, we change the task from predicting the answer span to locating the sentence containing the correct answer. We build a network structure to test the power of sequence encoding in different models to find the correct answers. The details are given in Appendix B.
+
+Training Setup: We use Adadelta optimizer to minimize the cross-entropy loss plus L2 regularization penalty, with batch size of 32. The network parameters and word embeddings initialization methods are same as those for SNLI, except that both the word embedding dimension and the number of hidden units are set to 100. We use 0.8 dropout keep probability and $1 0 ^ { - 4 }$ L2 regularization weight decay factor.
+
+We evaluate the Bi-BloSAN and the baselines except DiSAN because the memory required by DiSAN largely exceeds the GPU memory of GTX 1080Ti (11GB). The number of parameters, per epoch training time and the prediction accuracy on development set are given in Table 4.
+
+Table 4: Experimental results for different methods on modified SQuAD task.   
+
+<table><tr><td>Context Fusion Method</td><td>0</td><td>Time(s)/Epoch</td><td>Dev Accuracy</td></tr><tr><td>Bi-LSTM (Graves et al., 2013)</td><td>0.71m</td><td>857</td><td>68.01</td></tr><tr><td>Bi-GRU (Chung et al., 2014)</td><td>0.57m</td><td>782</td><td>67.98</td></tr><tr><td>Bi-SRU (Lei&amp; Zhang,2017)</td><td>0.32m</td><td>737</td><td>67.32</td></tr><tr><td>Multi-CNN (Kim, 2014)</td><td>0.60m</td><td>114</td><td>63.58</td></tr><tr><td>Multi-head (Vaswani et al., 2017)</td><td>0.45m</td><td>140</td><td>64.82</td></tr><tr><td>Bi-BloSAN</td><td>0.82m</td><td>293</td><td>68.38</td></tr></table>
+
+Compared to RNN/CNN models, Bi-BloSAN achieves state-of-the-art prediction accuracy in this modified task. Bi-BloSAN shows its competitive context fusion and sequence encoding capability compared to Bi-LSTM, Bi-GRU, Bi-SRU but is much more time-efficient. In addition, Bi-BloSAN significantly outperforms multi-CNN and multi-head attention.
+
+# 4.3 SEMANTIC RELATEDNESS
+
+The goal of semantic relatedness is to predict the similarity degree of a given pair of sentences. Unlike the classification problems introduced above, predicting the semantic relatedness of sentences is a regression problem. We use $s ^ { 1 }$ and $s ^ { 2 }$ to denote the encodings of the two sentences, and assume that the similarity degree is between $[ 1 , K ]$ . Following the method introduced by Tai et al. (2015), the concatenation of $s ^ { 1 } \odot s ^ { 2 }$ and $| s ^ { 1 } - s ^ { 2 } |$ is used as the representation of sentence relatedness. This representation is fed into a 300D fully connected layer, followed by a K-unit output layer with softmax to calculate a probability distribution $\hat { p }$ . The details of this regression problem can be found in Appendix C. We evaluate all models on Sentences Involving Compositional Knowledge $\mathrm { ( S I C K ) ^ { 4 } }$ dataset, where the similarity degree is denoted by a real number in the range of [1, 5]. SICK comprises 9,927 sentence pairs with 4,500/500/4,927 instances for training/dev/test sets.
+
+Training Setup: The optimization objective of this regression problem is the KL-divergence plus the L2 regularization penalty. We minimize the objective using Adadelta with batch size of 64. The network parameters and word embeddings are initialized as in SNLI experiment. The keep probability of dropout is set to 0.7, and the L2 regularization weight decay factor is set to $1 0 ^ { - 4 }$ .
+
+Table 5: Experimental results for different methods on SICK sentence relatedness dataset. The reported accuracies are the mean of five runs (standard deviations in parentheses).   
+
+<table><tr><td>Model</td><td>Pearson&#x27;s r</td><td>Spearman&#x27;s p</td><td>MSE</td></tr><tr><td>Meaning Factory (Bjerva et al., 2014)</td><td>0.8268</td><td>0.7721</td><td>0.3224</td></tr><tr><td>ECNU (Zhao et al., 2014)</td><td>0.8414</td><td>/</td><td>/</td></tr><tr><td>DT-RNN (Socher et al., 2014)</td><td>0.7923 (0.0070)</td><td>0.7319 (0.0071)</td><td>0.3822 (0.0137)</td></tr><tr><td>SDT-RNN (Socher et al.,2014)</td><td>0.7900 (0.0042)</td><td>0.7304 (0.0042)</td><td>0.3848 (0.0042)</td></tr><tr><td>Constituency Tree-LSTM (Tai et al., 2015)</td><td>0.8582 (0.0038)</td><td>0.7966 (0.0053)</td><td>0.2734 (0.0108)</td></tr><tr><td>Dependency Tree-LSTM(Tai et al.,2015)</td><td>0.8676 (0.0030)</td><td>0.8083 (0.0042)</td><td>0.2532 (0.0052)</td></tr><tr><td>Bi-LSTM (Graves et al.,2013)</td><td>0.8473 (0.0013)</td><td>0.7913 (0.0019)</td><td>0.3276 (0.0087)</td></tr><tr><td>Multi-CNN (Kim,2014)</td><td>0.8374 (0.0021)</td><td>0.7793 (0.0028)</td><td>0.3395 (0.0086)</td></tr><tr><td>Hrchy-CNN(Gehring et al.,2017)</td><td>0.8436 (0.0014)</td><td>0.7874 (0.0022)</td><td>0.3162 (0.0058)</td></tr><tr><td>Multi-head (Vaswani et al.,2017)</td><td>0.8521 (0.0013)</td><td>0.7942 (0.0050)</td><td>0.3258 (0.0149)</td></tr><tr><td>DiSAN (Shen et al., 2017)</td><td>0.8695 (0.0012)</td><td>0.8139 (0.0012)</td><td>0.2879 (0.0036)</td></tr><tr><td>Bi-BloSAN</td><td>0.8616 (0.0012)</td><td>0.8038 (0.0012)</td><td>0.3008 (0.0091)</td></tr></table>
+
+The performances of all models are shown in Table 5, which shows that Bi-BloSAN achieves stateof-the-art prediction quality. Although Dependency Tree-LSTM and DiSAN obtain the best performance, the Tree-LSTM needs external semantic parsing tree as the recursive input and expensive recursion computation, and DiSAN requires much larger memory for self-attention calculation. By contrast, Bi-BloSAN, as a RNN/CNN-free model, shows appealing advantage in terms of memory and time efficiency. Note that, performance of Bi-BloSAN is still better than some common models, including Bi-LSTM, CNNs and multi-head attention.
+
+# 4.4 SENTENCE CLASSIFICATIONS
+
+The goal of sentence classification is to correctly predict the class label of a given sentence in various scenarios. We evaluate the models on six sentence classification benchmarks for various NLP tasks, such as sentiment analysis and question-type classification. They are listed as follows.
+
+• $\mathbf { C } \mathbf { R } ^ { 5 }$ : Customer reviews (Hu & Liu, 2004) of various products (cameras etc.). This task is to predict whether the review is positive or negative.   
+• MPQA6: Opinion polarity detection subtask of the MPQA dataset (Wiebe et al., 2005).   
+• $\mathbf { S U B J } ^ { 7 }$ : Subjectivity dataset (Pang & Lee, 2004), which includes a set of sentences. The corresponding label indicates whether each sentence is subjective or objective.   
+• TREC8: TREC question-type classification dataset (Li & Roth, 2002) which coarsely classifies the question sentences into six types.
+
+• SST-19: Stanford Sentiment Treebank (Socher et al., 2013), which is a dataset consisting of movie reviews with five fine-grained sentiment labels, i.e., very positive, positive, neutral, negative and very negative. SST-2: Stanford Sentiment Treebank (Socher et al., 2013) with binary sentiment labels. Compared to SST-1, SST-2 removes the neutral instances, and labels the rest with either negative or positive.
+
+Note that only SST-1 and SST-2 have the standard training/dev/test split, and TREC has the training/dev split. We implement 10-fold cross validation on SUBJ, CR and MPQA because the original datasets do not provide any split. We do not use the Movie Reviews (Pang & Lee, 2005) dataset because the SST-1/2 are extensions of it.
+
+Training Setup: We use the cross-entropy loss plus L2 regularization penalty as the optimization objective. We minimize it by Adam with training batch size of 32 (except DiSAN, which uses batch size of 16 due to the limit of GPU memory). The network parameters and word embeddings are initialized as in SNLI experiment. To avoid overfitting on small datasets, we decrease the dropout keep probability and the L2 regularization weight decay factor $\gamma$ to 0.6 and $1 0 ^ { - 4 }$ , respectively.
+
+Table 6: Experimental results for different methods on various sentence classification benchmarks. The reported accuracies on CR, MPQA and SUBJ are the mean of 10-fold cross validation, the accuracies on TREC are the mean of dev accuracies of five runs, and the accuracies on SST-1 and SST-2 are the mean of test accuracies of five runs. All standard deviations are in parentheses.   
+
+<table><tr><td>Model</td><td>CR</td><td>MPQA</td><td>SUBJ</td><td>TREC</td><td>SST-1</td><td>SST-2</td></tr><tr><td>cBoW (Mikolov et al., 2013a)</td><td>79.9</td><td>86.4</td><td>91.3</td><td>87.3</td><td>/</td><td>/</td></tr><tr><td>Skip-thought (Kiros et al., 2015)</td><td>81.3</td><td>87.5</td><td>93.6</td><td>92.2</td><td>/</td><td>/</td></tr><tr><td>DCNN (Kalchbrenner et al., 2014)</td><td>/</td><td>/</td><td>/</td><td>93.0</td><td>86.8</td><td>48.5</td></tr><tr><td>AdaSent (Zhao et al., 2015)</td><td>83.6 (1.6)</td><td>90.4 (0.7)</td><td>92.2 (1.2)</td><td>91.1 (1.0)</td><td>/</td><td>/</td></tr><tr><td>SRU (Lei &amp; Zhang,2017)</td><td>84.8 (1.3)</td><td>89.7 (1.1)</td><td>93.4 (0.8)</td><td>93.9 (0.6)</td><td>89.1 (0.3)</td><td>/</td></tr><tr><td>Wide CNNs (Lei&amp; Zhang,2017)</td><td>82.2 (2.2)</td><td>88.8 (1.2)</td><td>92.9 (0.7)</td><td>93.2 (0.5)</td><td>85.3 (0.4)</td><td>/</td></tr><tr><td>Bi-LSTM(Graves et al., 2013)</td><td>84.6 (1.6)</td><td>90.2 (0.9)</td><td>94.7 (0.7)</td><td>94.4 (0.3)</td><td>87.7 (0.6)</td><td>49.9 (0.8)</td></tr><tr><td>Multi-head (Vaswani et al., 2017)</td><td>82.6 (1.9)</td><td>89.8 (1.2)</td><td>94.0 (0.8)</td><td>93.4 (0.4)</td><td>83.9 (0.4)</td><td>48.2 (0.6)</td></tr><tr><td>DiSAN (Shen et al., 2017)</td><td>84.8 (2.0)</td><td>90.1 (0.4)</td><td>94.2 (0.6)</td><td>94.2 (0.1)</td><td>87.8 (0.3)</td><td>51.0 (0.7)</td></tr><tr><td>Bi-BloSAN</td><td>84.8 (0.9)</td><td>90.4 (0.8)</td><td>94.5 (0.5)</td><td>94.8 (0.2)</td><td>87.4 (0.2)</td><td>50.6 (0.5)</td></tr></table>
+
+The prediction accuracies of different models on the six benchmark datasets are given in Table 6. Bi-BloSAN achieves the best prediction accuracies on CR, MPQA and TREC, and state-of-theart performances on SUBJ, SST-1 and SST-2 datasets (slightly worse than the best performances). Although Bi-BloSAN performs a little bit worse than the RNN models on SUBJ and SST-1, it is much more time-efficient than them. Additionally, on the SST-2 dataset, Bi-BloSAN performs slightly worse than DiSAN in terms of prediction accuracy $( - 0 . 4 \% )$ but obtains a significantly higher memory utility rate.
+
+We visualize the progress of training models on CR dataset in Figure 5. The convergence speed of Bi-BloSAN is $\sim 6 \times$ and ${ \sim } 2 \times$ faster than Bi-LSTM and DiSAN respectively. Although Bi-BloSAN is less time-efficient than CNN and multi-head attention, it has much better prediction quality.
+
+# 4.5 ANALYSES OF TIME COST AND MEMORY CONSUMPTION
+
+To compare the efficiency-memory trade-off for each model on sequences of different lengths, we generate random tensor data, and feed them into the different sequence encoding models. The models we evaluate include Bi-LSTM, Bi-GRU, Bi-SRU, CNN, multi-head attention, DiSAN and Bi-BloSAN. The shape of the random data is [batch size, sequence length, features number]. We fix the batch size to 64 and the features number to 300, then change the sequence length from 16 to 384 with a step size 16.
+
+We first discuss the time cost vs. the sequence length. As shown in Figure 6(a), the inference time of Bi-BloSAN is similar to those of multi-head attention and multi-CNN, but Bi-BloSAN outperforms both by a large margin on prediction quality in previous experiments. Moreover, Bi-BloSAN is much faster than the RNN models (Bi-LSTM, Bi-GRU, BI-SRU). In addition, although DiSAN requires less training time than the RNN models in the experiments above, it is much slower during the inference phase because the large memory allocation consumes a great amount of time. By contrast, the block structure of Bi-BloSAN significantly reduces the inference time.
+
+![](images/776a5d89d9ad5e56eb82df816d04ed2f6a6f51c9b7353e1ab65101fd7c7d2ec0.jpg)  
+Figure 5: Validation accuracy vs. training time (second) of Bi-LSTM, CNN, multi-head attention, DiSAN and Bi-BloSAN for 800 training steps on CR dataset. (The Bi-LSTM for 800 steps consumes 279s in total.)
+
+![](images/ffd239a83482134dbcc23db770733f8a769becc14bbb42785776e0aa180c7b71.jpg)  
+Figure 6: (a) Inference time cost and (b) GPU memory consumption of the sequence encoding models vs. the sequence length with the batch size of 64 and the features number of 300.
+
+The GPU memory consumption vs. the sequence length for each model is visualized in Figure 6(b). DiSAN is not scalable because its memory grows explosively with the sequence length. Bi-BloSAN is more memory-efficient and scalable than DiSAN as the growth of its memory is nearly linear. Although Bi-BloSAN consumes more memory than the RNN models, it experimentally has better time efficiency and prediction quality. Since multi-head attention uses multiplicative attention, it requires less memory than all additive attention based models, such as DiSAN and Bi-BloSAN, but multiplicative attention based models usually perform worse than additive attention based models.
+
+# 5 CONCLUSIONS
+
+This paper presents an attention network, called bi-directional block self-attention network (BiBloSAN), for fast, memory-efficient and RNN/CNN-free sequence modeling. To overcome large memory consumption of existing self-attention networks, Bi-BloSAN splits the sequence into several blocks and employs intra-block and inter-block self-attentions to capture both local and longrange context dependencies, respectively. To encode temporal order information, Bi-BloSAN applies forward and backward masks to the alignment scores between tokens for asymmetric selfattentions.
+
+Our experiments on nine benchmark datasets for various different NLP tasks show that Bi-BloSAN can achieve the best or state-of-the-art performance with better efficiency-memory trade-off than existing RNN/CNN/SAN models. Bi-BloSAN is much more time-efficient than the RNN models (e.g., Bi-LSTM, Bi-GRU, etc.), requires much less memory than DiSAN, and significantly outperforms the CNN models and multi-head attention on prediction quality.
+
+# 6 ACKNOWLEDGMENTS
+
+This research was funded by the Australian Government through the Australian Research Council (ARC) under grants 1) LP160100630 partnership with Australia Government Department of Health and 2) LP150100671 partnership with Australia Research Alliance for Children and Youth (ARACY) and Global Business College Australia (GBCA). We also acknowledge the support of NVIDIA Corporation with the donation of GPU used for this research.
+
+# REFERENCES
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR, 2015.
+
+Johannes Bjerva, Johan Bos, Rob Van der Goot, and Malvina Nissim. The meaning factory: Formal semantics for recognizing textual entailment and determining semantic similarity. In SemEval@ COLING, pp. 642– 646, 2014.
+
+Samuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. A large annotated corpus for learning natural language inference. In EMNLP, 2015.
+
+Samuel R Bowman, Jon Gauthier, Abhinav Rastogi, Raghav Gupta, Christopher D Manning, and Christopher Potts. A fast unified model for parsing and sentence understanding. In ACL, 2016.
+
+Qian Chen, Xiaodan Zhu, Zhen-Hua Ling, Si Wei, Hui Jiang, and Diana Inkpen. Recurrent neural networkbased sentence encoder with gated attention for natural language inference. In RepEval@ EMNLP, 2017.
+
+Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation of gated recurrent neural networks on sequence modeling. In NIPS, 2014.
+
+Yann N Dauphin, Angela Fan, Michael Auli, and David Grangier. Language modeling with gated convolutional networks. arXiv preprint arXiv:1612.08083, 2016.
+
+Xuanyi Dong, Junshi Huang, Yi Yang, and Shuicheng Yan. More is less: A more complicated network with less inference complexity. In Computer Vision and Pattern Recognition, 2017.
+
+Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. Convolutional sequence to sequence learning. arXiv preprint arXiv:1705.03122, 2017.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 249– 256, 2010.
+
+Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 315–323, 2011.
+
+Yichen Gong and Samuel R Bowman. Ruminating reader: Reasoning with gated multi-hop attention. arXiv preprint arXiv:1704.07415, 2017.
+
+Alex Graves, Navdeep Jaitly, and Abdel-rahman Mohamed. Hybrid speech recognition with deep bidirectional lstm. In Automatic Speech Recognition and Understanding (ASRU), 2013 IEEE Workshop on, pp. 273–278. IEEE, 2013.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8):1735–1780, 1997.
+
+Minghao Hu, Yuxing Peng, and Xipeng Qiu. Reinforced mnemonic reader for machine comprehension. arXiv preprint arXiv:1705.02798, 2017.
+
+Minqing Hu and Bing Liu. Mining and summarizing customer reviews. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 168–177. ACM, 2004.
+
+Nal Kalchbrenner, Edward Grefenstette, and Phil Blunsom. A convolutional neural network for modelling sentences. arXiv preprint arXiv:1404.2188, 2014.
+
+Nal Kalchbrenner, Lasse Espeholt, Karen Simonyan, Aaron van den Oord, Alex Graves, and Koray Kavukcuoglu. Neural machine translation in linear time. arXiv preprint arXiv:1610.10099, 2016.
+
+Yoon Kim. Convolutional neural networks for sentence classification. In EMNLP, 2014.
+
+Yoon Kim, Carl Denton, Luong Hoang, and Alexander M Rush. Structured attention networks. In ICLR, 2017.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
+
+Ryan Kiros, Yukun Zhu, Ruslan R Salakhutdinov, Richard Zemel, Raquel Urtasun, Antonio Torralba, and Sanja Fidler. Skip-thought vectors. In NIPS, 2015.
+
+Filippos Kokkinos and Alexandros Potamianos. Structural attention neural networks for improved sentiment analysis. arXiv preprint arXiv:1701.01811, 2017.
+
+Tao Lei and Yu Zhang. Training rnns as fast as cnns. arXiv preprint arXiv:1709.02755, 2017.
+
+Tao Lei, Regina Barzilay, and Tommi Jaakkola. Molding cnns for text: non-linear, non-consecutive convolutions. In EMNLP, 2015.
+
+Xin Li and Dan Roth. Learning question classifiers. In ACL, 2002.
+
+Zheng Li, Yu Zhang, Ying Wei, Yuxiang Wu, and Qiang Yang. End-to-end adversarial memory network for cross-domain sentiment classification. In IJCAI, pp. 2237–2243, 2017.
+
+Zhouhan Lin, Minwei Feng, Cicero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen Zhou, and Yoshua Bengio. A structured self-attentive sentence embedding. In ICLR, 2017.
+
+Yang Liu, Chengjie Sun, Lei Lin, and Xiaolong Wang. Learning natural language inference using bidirectional lstm model and inner-attention. arXiv preprint arXiv:1605.09090, 2016.
+
+Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013a.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In NIPS, 2013b.
+
+Lili Mou, Rui Men, Ge Li, Yan Xu, Lu Zhang, Rui Yan, and Zhi Jin. Natural language inference by tree-based convolution and heuristic matching. In ACL, 2016.
+
+Tsendsuren Munkhdalai and Hong Yu. Neural semantic encoders. In EACL, 2017a.
+
+Tsendsuren Munkhdalai and Hong Yu. Neural tree indexers for text understanding. In EACL, 2017b.
+
+Bo Pang and Lillian Lee. A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts. In ACL, 2004.
+
+Bo Pang and Lillian Lee. Seeing stars: Exploiting class relationships for sentiment categorization with respect to rating scales. In ACL, 2005.
+
+Jeffrey Pennington, Richard Socher, and Christopher D. Manning. Glove: Global vectors for word representation. In EMNLP, 2014.
+
+Qiao Qian, Minlie Huang, and Xiaoyan Zhu. Linguistically regularized lstms for sentiment classification. In ACL, 2017.
+
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. Squad: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In EMNLP, 2016.
+
+Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. In EMNLP, 2015.
+
+Lifeng Shang, Zhengdong Lu, and Hang Li. Neural responding machine for short-text conversation. In ACL, 2015.
+
+Tao Shen, Tianyi Zhou, Guodong Long, Jing Jiang, Shirui Pan, and Chengqi Zhang. Disan: Directional selfattention network for rnn/cnn-free language understanding. arXiv preprint arXiv:1709.04696, 2017.
+
+Richard Socher, Alex Perelygin, Jean Y Wu, Jason Chuang, Christopher D Manning, Andrew Y Ng, Christopher Potts, et al. Recursive deep models for semantic compositionality over a sentiment treebank. In EMNLP, 2013.
+
+Richard Socher, Andrej Karpathy, Quoc V Le, Christopher D Manning, and Andrew Y Ng. Grounded compositional semantics for finding and describing images with sentences. Transactions of the Association for Computational Linguistics, 2:207–218, 2014.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1): 1929–1958, 2014.
+
+Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In NIPS, 2015.
+
+Kai Sheng Tai, Richard Socher, and Christopher D Manning. Improved semantic representations from treestructured long short-term memory networks. In ACL, 2015.
+
+Zhiyang Teng and Yue Zhang. Bidirectional tree-structured lstm with head lexicalization. In ACL, 2017.
+
+Ashish Vaswani, Shazeer, Noam, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NIPS, 2017.
+
+Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. In ICLR, 2016.
+
+Wenhui Wang, Nan Yang, Furu Wei, Baobao Chang, and Ming Zhou. Gated self-matching networks for reading comprehension and question answering. In ACL, 2017.
+
+Zhiguo Wang, Haitao Mi, Wael Hamza, and Radu Florian. Multi-perspective context matching for machine comprehension. arXiv preprint arXiv:1612.04211, 2016.
+
+Janyce Wiebe, Theresa Wilson, and Claire Cardie. Annotating expressions of opinions and emotions in language. Language resources and evaluation, 39(2):165–210, 2005.
+
+Zichao Yang, Diyi Yang, Chris Dyer, Xiaodong He, Alexander J Smola, and Eduard H Hovy. Hierarchical attention networks for document classification. In HLT-NAACL, 2016.
+
+Matthew D Zeiler. Adadelta: an adaptive learning rate method. arXiv preprint arXiv:1212.5701, 2012.
+
+Han Zhao, Zhengdong Lu, and Pascal Poupart. Self-adaptive hierarchical sentence model. In IJCAI, 2015.
+
+Jiang Zhao, Tiantian Zhu, and Man Lan. Ecnu: One stone two birds: Ensemble of heterogenous measures for semantic relatedness and textual entailment. In SemEval@ COLING, pp. 271–277, 2014.
+
+# A THE SELECTION OF BLOCK LENGTH
+
+In mBloSA, the length $r$ of each block is a hyper-parameter that determines memory consumption of mBloSA. To minimize the memory consumption, we propose an approach that calculates the optimized block length $r$ as follows.
+
+We first introduce the method for determining $r$ for a dataset that has fixed sentence length $n$ . Given the sentence length $n$ and the block number $m = n / r$ , we have the following facts: 1) the major memory consumption in mBloSA is dominated by the masked self-attentions $g ^ { m } ( \cdot , M )$ ; 2) the memory consumption of the masked self-attention is proportional to the square of the sentence length; and 3) mBloSA contains $m$ masked self-attention with a sequence length of $r$ , and 1 masked self-attention with a sequence length of $m$ . Therefore, the memory $\xi$ required by mBloSA can be calculated by
+
+$$
+\begin{array} { r } { \xi \propto r ^ { 2 } \cdot m + m ^ { 2 } \cdot 1 } \\ { = r ^ { 2 } \cdot \frac { n } { r } + ( \frac { n } { r } ) ^ { 2 } . } \end{array}
+$$
+
+By setting the gradient of √ $\xi$ w.r.t. $r$ to zero, we know that the memory consumption $\xi$ is minimum when $r = \sqrt [ 3 ] { 2 n }$ .
+
+Second, we propose a method for selecting $r$ given a dataset with the sentence lengths that follow a normal distribution $N ( \mu , \sigma ^ { 2 } )$ . We consider the case where mini-batch SGD with a batch size of $B$ or its variant is used for training. We need to calculate the upper bound of the expectation of the maximal sentence length for each mini-batch. Let us first consider $B$ random variables $[ X _ { 1 } , X _ { 2 } , \ldots , X _ { B } ]$ in the distribution $N ( 0 , \sigma ^ { 2 } )$ . The goal is to find the upper bound of the expectation of random variable $Z + \mu$ , where $Z$ is defined as
+
+$$
+Z = \operatorname* { m a x } _ { i } X _ { i } , { \mathrm { f o r } } i = 1 , 2 , \ldots , B .
+$$
+
+By Jensen’s inequality,
+
+$$
+\begin{array} { r } { e ^ { t \mathbb { E } [ Z ] } \le \mathbb { E } [ e ^ { t Z } ] = E [ \operatorname* { m a x } _ { i } e ^ { t X _ { i } } ] } \\ { \le \displaystyle \sum _ { i = 1 } ^ { B } \mathbb { E } [ e ^ { t X _ { i } } ] = n e ^ { t ^ { 2 } \frac { \sigma ^ { 2 } } { 2 } } } \end{array}
+$$
+
+Eq.(21) leads to
+
+$$
+\mathbb { E } [ Z ] \leq \frac { \ln B } { t } + \frac { t \sigma ^ { 2 } } { 2 } .
+$$
+
+Let t = 2 ln B and we obtain the following upper bound.
+
+$$
+\mathbb { E } [ Z ] \le \sigma \sqrt { 2 \ln B }
+$$
+
+Hence, the upper bound of the expectation of the maximal sentence length among all the √ $B$ sentences in each mini-batch is $\sigma { \sqrt { 2 \ln B } } + \mu$ . Therefore, the block length $r$ is computed by
+
+$$
+r = { \sqrt [ 3 ] { 2 n } } = { \sqrt [ 3 ] { 2 ( \sigma { \sqrt { 2 \ln B } } + \mu ) } } .
+$$
+
+# B NETWORK SETUP FOR MACHINE COMPREHENSION
+
+Each sample in the Stanford Question Answering Dataset (SQuAD) (Rajpurkar et al., 2016) is composed of three parts, i.e., a passage consisting of multiple sentences, a question sentence and a span in the passage indicating the position of the answer. In order to evaluate the performance of sentence embedding models, we change the task from predicting the span of the answer to finding the sentence containing the correct answer.
+
+Given a passage consisting of $m$ sentences $\left[ s ^ { 1 } , s ^ { 2 } , \ldots , s ^ { m } \right]$ where $\pmb { s } ^ { k } = [ x _ { k 1 } , x _ { k 2 } , \pmb { . . . } , x _ { k n } ]$ , and the embedded question token sequence $\pmb { q } = [ q _ { 1 } , q _ { 2 } , \dots , q _ { l } ]$ , the goal is to predict which sentence in the $m$ sentences contains the correct answer to the question $\pmb q$ .
+
+The neural net we use to evaluate different sequence encoding models is given in Figure 7. First, we process each sentence from the passage by a context fusion layer with shared parameters, followed by a source2token self-attention with shared parameters, which outputs a vector representation of the sentence. Therefore, the $m$ sentences are represented by $m$ vectors $[ u _ { 1 } , u _ { 2 } , \ldots , u _ { m } ]$ . The question sentence $\pmb q$ is compressed into a vector representation $q$ using source2token self-attention. Second, we combine each sentence $u _ { k }$ with $q$ by concatenating $u _ { k }$ , q, $u _ { k } - q$ and $u _ { k } \odot q$ , i.e.,
+
+$$
+c _ { k } = [ u _ { k } ; q ; u _ { k } - q ; u _ { k } \odot q ] , \mathrm { f o r } \ k = 1 , 2 , \ldots , m .
+$$
+
+Then $\pmb { c } = [ c _ { 1 } , c _ { 2 } , \dots , c _ { m } ]$ is fed into another context fusion layer that explores sentence-level dependencies. Finally, the resultant output representation of each sentence is separately fed into a fully connected layer to compute a scalar score indicating the possibility of the sentence containing the answer. A softmax function is applied to the scores of all $m$ sentences, to generate a probability distribution $\hat { p } \in \mathbb { R } ^ { m }$ for cross-entropy loss function. The sentence with the largest probability is predicted as the sentence containing the answer.
+
+![](images/e071f8f33f303b03d9f2570c0c0c36eee33ed3c64dd170e7bd80ec35924fccd1.jpg)  
+Figure 7: The structure of a neural network for machine comprehension. The candidates of the context fusion layer include Bi-LSTM, Bi-GRU, Bi-SRU, multi-CNN, multi-head attention and Bi-BloSA. Unlike the original multi-CNN for sentence embedding, we use padding and remove the max-pooling along the time axis to obtain an output of the same length as input. DiSA is not considered due to memory limitation.
+
+# C LOSS OF REGRESSION PROBLEM
+
+Following the setting introduced by Tai et al. (2015) and given a predicted probability distribution $\hat { p }$ as the output of a feedforward network, the regression model predicts the similarity degree as
+
+$$
+\hat { y } = \beta ^ { T } \hat { p } ,
+$$
+
+where $\beta = [ 1 , 2 , \dots , K ]$ . The ground-truth similarity degree $y$ should be mapped to a probability distribution $p = [ p _ { i } ] _ { i = 1 } ^ { K }$ as the training target, where $p$ needs to fulfill $y = \beta ^ { \hat { T } } \overset { \cdot } { p }$ . The mapping can be defined as
+
+$$
+p _ { i } = { \left\{ \begin{array} { l l } { y - \lfloor y \rfloor , } & { i = \lfloor y \rfloor + 1 } \\ { \lfloor y \rfloor - y + 1 , } & { i = \lfloor y \rfloor } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } , i = 1 , 2 , \ldots , K .
+$$
+
+We use KL-divergence between $p$ and $\hat { p }$ as our loss function, i.e.,
+
+$$
+L = \frac { 1 } { M } \sum _ { k = 1 } ^ { M } K L ( p ^ { ( k ) } | | \hat { p } ^ { ( k ) } ) ,
+$$
+
+where the $p ^ { ( k ) }$ and $\hat { p } ^ { ( k ) }$ represent the target and predicted probability distributions of the $k$ -th sample, respectively.
+
+# D RELATED WORKS
+
+Recently, several structured attention mechanisms (Kim et al., 2017; Kokkinos & Potamianos, 2017) are proposed for capturing structural information from input sequence(s). When applied to selfattention, structured attentions share a similar idea to self-alignment attention (Hu et al., 2017) and multi-head attention (Vaswani et al., 2017) with one head, which aims to model the dependencies between the tokens. Similar to the attention from multiple perspectives in multi-head attention, multi-perspective context matching (Wang et al., 2016) explores the dependencies between passage and question from multiple perspectives for reading comprehension, while self-attentive structure (Lin et al., 2017) embeds sentences from various perspectives to produce matrix representations of the sentences. In recursive models, self-attention over children nodes (Teng & Zhang, 2017) can provide effective input for their parent node that has no standard input (Tai et al., 2015) as long as it is a non-leaf node in a semantic constituency parsing tree. Li et al. (2017) applies the multi-hop attention mechanism to transfer learning for cross-domain sentiment analysis without any RNN/CNN structure.
+
+Bi-BloSA and hierarchical attention network (Yang et al., 2016) have similar structure, i.e., both stack two-layer attention mechanisms from bottom to top. However, they are different in three respects: 1) Bi-BloSA aims to learn context-aware representation for each token, while hierarchical attention is designed for document embedding; 2) the input of Bi-BloSA is a sentence, while the intput of hierarchical attention is a document composed of multiple sentences; and 3) the hierarchical attention performs vanilla attention twice, i.e., token-level attention on each sentence and sentence-level attention. Bi-BloSA, however, applies masked self-attention twice, i.e., intra-block self-attention and inter-block self-attention. Additionally, Bi-BloSA uses a feature fusion gate for each token to combine local and global context, and positional masks to encode temporal order information.

parse/train/QZpx42n0BWr/QZpx42n0BWr.md

@@ -0,0 +1,253 @@
+# FINE Samples for Learning with Noisy Labels
+
+Taehyeon Kim ∗   
+KAIST AI   
+KAIST   
+Daejeon, South Korea   
+potter32@kaist.ac.kr   
+Jongwoo Ko ∗   
+KAIST AI   
+KAIST   
+Daejeon, South Korea   
+jongwoo.ko@kaist.ac.kr   
+Sangwook Cho   
+KAIST AI   
+KAIST   
+Daejeon, South Korea   
+sangwookcho@kaist.ac.kr   
+Jinhwan Choi   
+KAIST AI   
+KAIST   
+Daejeon, South Korea   
+jinhwanchoi@kaist.ac.kr   
+Se-Young Yun   
+KAIST AI   
+KAIST   
+Daejeon, South Korea   
+yunseyoung@kaist.ac.kr
+
+# Abstract
+
+Modern deep neural networks (DNNs) become weak when the datasets contain noisy (incorrect) class labels. Robust techniques in the presence of noisy labels can be categorized into two types: developing noise-robust functions or using noisecleansing methods by detecting the noisy data. Recently, noise-cleansing methods have been considered as the most competitive noisy-label learning algorithms. Despite their success, their noisy label detectors are often based on heuristics more than a theory, requiring a robust classifier to predict the noisy data with loss values. In this paper, we propose a novel detector for filtering label noise. Unlike most existing methods, we focus on each data point’s latent representation dynamics and measure the alignment between the latent distribution and each representation using the eigen decomposition of the data gram matrix. Our framework, coined as filtering noisy instances via their eigenvectors (FINE), provides a robust detector using derivative-free simple methods with theoretical guarantees. Under our framework, we propose three applications of the FINE: sample-selection approach, semi-supervised learning (SSL) approach, and collaboration with noiserobust loss functions. Experimental results show that the proposed methods consistently outperform corresponding baselines for all three applications on various benchmark datasets 1.
+
+# 1 Introduction
+
+Deep neural networks (DNNs) have achieved remarkable success in numerous tasks as the amount of accessible data has dramatically increased [21, 15]. On the other hand, accumulated datasets are typically labeled by a human, a labor-intensive job or through web crawling [48] so that they may be easily corrupted (label noise) in real-world situations. Recent studies have shown that deep neural networks have the capacity to memorize essentially any labeling of the data [49]. Even a small amount of such noisy data can hinder the generalization of DNNs owing to their strong memorization of noisy labels [49, 29]. Hence, it becomes crucial to train DNNs that are robust to corrupted labels. As label noise problems may appear anywhere, such robustness increases reliability in many applications such as the e-commerce market [9], medical fields [45], on-device AI [46], and autonomous driving systems [11].
+
+To improve the robustness against noisy data, the methods for learning with noisy labels (LNL) have been evolving in two main directions [18]: (1) designing noise-robust objective functions or regularizations and (2) detecting and cleansing the noisy data. In general, the former noise-robust direction uses explicit regularization techniques [6, 52, 50] or robust loss functions [38, 13, 40, 51], but their performance is far from state-of-the-art [49, 26] on datasets with severe noise rates. Recently, researchers have designed noise-cleansing algorithms focused on segregating the clean data (i.e., samples with uncorrupted labels) from the corrupted data [19, 14, 47, 18, 32, 42]. One of the popular criteria for the segregation process is the loss value between the prediction of the noisy classifier and its noisy label, where it is generally assumed that the noisy data have a large loss [19, 14, 47, 18] or the magnitude of the gradient during training [51, 40]. However, these methods may still be biased by the corrupted linear classifier towards label noise because their criterion (e.g., loss values or weight gradient) uses the posterior information of such a linear classifier [24]. Maennel et al. [31] analytically showed that the principal components of the weights of a neural network align with the randomly labeled data; this phenomenon can yield more negative effects on the classifier as the number of randomly labeled classes increases. Recently, Wu et al. [42] used an inherent geometric structure induced by nearest neighbors (NN) in latent space and filtered out isolated data in such topology, and its quality was sensitive to its hyperparameters regarding NN clustering in the presence of severe noise rates.
+
+![](images/5dc76db62bc5fdf618d00e6a7419d8677909c74d57178c02c103d2a86cb6ad74.jpg)  
+Figure 1: Illustration of (a) basic concept of this work and (b) proposed detection framework, FINE. Noise-cleansing learning generally separates clean data from the original dataset by using prediction outputs. We propose a novel derivative-free detector based on an unsupervised clustering algorithm on the high-order topological space. FINE measures the alignment of pre-logits (i.e., penultimate layer representation vectors) toward the class-representative vector that is extracted through the eigen decomposition of the gram matrix of data representations.
+
+To mitigate such issues for label noise detectors, we provide a novel yet simple detector framework, filtering noisy labels via their eigenvectors (FINE) with theoretical guarantees to provide a high-quality splitting of clean and corrupted examples (without the need to estimate noise rates). Instead of using the neural network’s linear classifier, FINE utilizes the principal components of latent representations made by eigen decomposition which is one of the most widely used unsupervised learning algorithms and separates clean data and noisy data by these components (Figure 1a). To motivate our approach, as Figure 1b shows, we find that the clean data (blue points) are mainly aligned on the principal component (black dotted line), whereas the noisy data (orange points) are not; thus, the dataset is well clustered with the alignment of representations toward the principal component by fitting them into Gaussian mixture models (GMM). We apply our framework to various LNL methods: the sample selection approach, a semi-supervised learning (SSL) approach, and collaboration with noise-robust loss functions. The key contributions of this work are summarized as follows:
+
+• We propose a novel framework, termed FINE (filtering noisy labels via their eigenvectors), for detecting clean instances from noisy datasets. FINE makes robust decision boundary for the high-order topological information of data in latent space by using eigen decomposition of their gram matrix.   
+• We provide provable evidence that FINE allows a meaningful decision boundary made by eigenvectors in latent space. We support our theoretical analysis with various experimental results regarding the characteristics of the principal components extracted by our FINE detector.   
+• We develop a simple sample-selection method by replacing the existing detector method with FINE. We empirically validate that a sample-selection learning with FINE provides consistently superior detection quality and higher test accuracy than other existing alternative methods such as the Co-teaching family [14, 47], TopoFilter [42], and CRUST [32].
+
+• We experimentally show that our detection framework can be applied in various ways to existing LNL methods and validate that ours consistently improves the generalization in the presence of noisy data: sample-selection approach [14, 47], SSL approach [25], and collaboration with noise-robust loss functions [51, 40, 29].
+
+Organization. The remainder of this paper is organized as follows. In Section 2, we discuss the recent literature on LNL solutions and meaningful detectors. In Section 3, we address our motivation for creating a noisy label detector with theoretical insights and provide our main method, filtering the noisy labels via their eigenvectors (FINE). In Section 4, we present the experimental results. Finally, Section 5 concludes the paper.
+
+# 2 Related Works
+
+Zhang et al. [49] empirically showed that any convolutional neural networks trained using stochastic gradient methods easily fit a random labeling of the training data. To tackle this issue, numerous works have examined the classification task with noisy labels. We do not consider the works that assumed the availability of small subsets of training data with clean labels [17, 36, 39, 53, 3].
+
+Noise-Cleansing-based Approaches. Noise-cleansing methods have evolved following the improvement of noisy detectors. Han et al. [14] suggested a noisy detection approach, named coteaching, that utilizes two networks, extracts subsets of instances with small losses from each network, and trains each network with subsets of instances filtered by another network. Yu et al. [47] combined a disagreement training procedure with co-teaching, which only selects instances predicted differently by two networks. Huang et al. [18] provided a simple noise-cleansing framework, training-filtering-training; the empirical efficacy was improved by first finding label errors, then training the model only on data predicted as clean. Recently, new noisy detectors with theoretical support have been developed. Wu et al. [42] proposed a method called TopoFilter that filters noisy data by utilizing the $\mathbf { k }$ -nearest neighborhood algorithm and Euclidean distance between pre-logits. Mirzasoleiman et al. [32] introduced an algorithm that selects subsets of clean instances that provide an approximately low-rank Jacobian matrix and proved that gradient descent applied to the subsets prevents overfitting to noisy labels. Pleiss et al. [34] proposed an area under margin (AUM) statistic that measures the average difference between the logit values of the assigned class and its highest non-assigned class to divide clean and noisy samples. Cheng et. al [8] progressively filtered out corrupted instances using a novel confidence regularization term. The noise-cleansing method was also developed in a semi-supervised learning (SSL) manner. Li et al. [25] modeled the per-sample loss distribution and divide it into a labeled set with clean samples and an unlabeled set with noisy samples, and they leverage the noisy samples through the well-known SSL technique MixMatch [4].
+
+Noise-Robust Models. Noise-robust models have been studied in the following directions: robustloss functions, regularizations, and strategies. First, for robust-loss functions, Ghosh et al. [13] showed that the mean absolute error (MAE) might be robust against noisy labels. Zhang & Sabuncu et al. [51] argued that MAE performed poorly with DNNs and proposed a GCE loss function, which can be seen as a generalization of MAE and cross-entropy (CE). Wang et al. [40] introduced the reverse version of the cross-entropy term (RCE) and suggested that the SCE loss function is a weighted sum of the CE and RCE. Some studies have stated that the early-stopped model can prevent the memorization phenomenon for noisy labels [2, 49] and theoretically analyzed it [26]. Based on this hypothesis, Liu et al. [29] proposed an early-learning regularization (ELR) loss function to prohibit memorizing noisy data by leveraging the semi-supervised learning techniques. Xia et al. [43] clarified which neural network parameters cause memorization and proposed a robust training strategy for these parameters. Efforts have been made to develop regularizations on the prediction level by smoothing the one-hot vector [30], using linear interpolation between data instances [50], and distilling the rescaled prediction of other models [20]. However, these works have limitations in terms of performance as the noise rate of the dataset increases.
+
+Dataset Resampling. Label-noise detection may be a category of data resampling which is a common technique in the machine learning community that extracts a “helpful” dataset from the distribution of the original dataset to remove the dataset bias. In class-imbalance tasks, numerous studies have conducted over-sampling of minority classes [7, 1] or undersampling the majority classes [5] to balance the amount of data per class. Li & Vasconcelos et al. [27] proposed a resampling procedure to reduce the representation bias of the data by learning a weight distribution that favors difficult instances for a given feature representation. Le Bras et al. [22] suggested an adversarial filteringbased approach to remove spurious artifacts in a dataset. Analogously, in anomaly detection and
+
+INPUT : Noisy training data $\mathcal { D }$ , feature extractor $g$ , number of classes $K$ , clean probability threshold $\zeta$ , set of FINE scores for class $k \mathcal { F } _ { k }$   
+OUTPUT : Collected clean data $\mathcal { C }$   
+1: Initialize $c \gets \emptyset$ , $\hat { \mathcal { D } } \gets \mathcal { D }$ , $\Sigma _ { k } \gets \mathbf { 0 }$ for all $k = 1 , \ldots , K$ $/ { * }$ Update the convariance matrices for all classes $^ { * }$   
+2: for $( \boldsymbol { x } _ { i } , \boldsymbol { y } _ { i } ) \in \mathcal { D }$ do   
+3: ${ z _ { i } \gets g ( { \pmb x } _ { i } ) }$   
+4: Update the gram matrix $\Sigma _ { y _ { i } } \gets \Sigma _ { y _ { i } } + z _ { i } z _ { i } ^ { \top }$   
+5: end for $/ { * }$ Generate the principal component with eigen decomposition $^ { * }$   
+6: for $k = 1 , \ldots , K$ do   
+7: $\mathbf { U } _ { k } , \mathbf { \Lambda } \Lambda _ { k } \gets$ EIGEN DECOMPOSITION OF $\Sigma _ { k }$   
+8: $\mathbf { u } _ { k } \gets$ THE FIRST COLUMN OF $\mathbf { U } _ { k }$   
+9: end for $/ { * }$ Compute the alignment score and get clean subset $\mathcal { C }$ \*/   
+10: for $( \boldsymbol { x } _ { i } , \boldsymbol { y } _ { i } ) \in \mathcal { D }$ do   
+11: Compute the FINE score $f _ { i } = \left. \mathbf { u } _ { y _ { i } } , z _ { i } \right. ^ { 2 }$ and $\mathcal { F } _ { y _ { i } } \gets \mathcal { F } _ { y _ { i } } \cup \{ f _ { i } \}$   
+12: end for $/ { * }$ Finding the samples whose clean probability is larger than $\zeta$ \*/   
+13: ${ \mathcal { C } } \gets { \mathcal { C } } \cup$ GMM $( \mathcal { F } _ { k } , \zeta )$ for all $k = 1 , \ldots , K$
+
+out-of-distribution detection problems [16, 28, 23], the malicious data are usually detected by examining the loss value or negative behavior in the feature representation space. While our research is motivated by such previous works, this paper focuses on the noisy image classification task.
+
+# 3 Method
+
+In this section, we present our detector framework and the theoretical motivation behind using the detector in high-dimensional classification. To segregate the clean data, we utilize the degree of alignment between the representations and the eigenvector of the representations’ gram matrices for all classes, called FINE (FIltering Noisy instnaces via their Eigenvectors). Our algorithm is as follows (Algorithm 1). FINE first creates a gram matrix of the representation in the noisy training dataset for each class and conducts the eigen decomposition for those gram matrices. Then, FINE finds clean and noisy instances using the square of inner product values between the representations and the first eigenvector having the largest eigenvalue. In this manner, we treat the data as clean if aligned onto the first eigenvector, while most of the noisy instances are not. Here, we formally define ‘alignment’ and ‘alignment clusterability’ in Definition 1 and Definition 2, respectively.
+
+# Definition 1. (Alignment) $D$
+
+Definition 2. (Alignment Clusterability) For all features labeled as class $k$ in dataset $\mathcal { D }$ , let fit a Gaussian Mixture Model (GMM) on their alignment (Definition 1) distribution to divide current samples into a clean set and a noisy set; the set having larger mean value is treated as a clean set, and another one is a noisy set. Then, we say a dataset $\mathcal { D }$ satisfies alignment clusterability if the representation $_ z$ labeled as the same true class belongs to the clean set.
+
+As an empirical evaluation, the quality of our detector for noisy data is measured with the $\boldsymbol { F }$ -score, a widely used criterion in noisy label detection, anomaly detection and out-of-distribution detection [8, 16, 28, 23]. We treat the selected clean samples as the positive class and the noisy samples as negative class. The $F _ { \mathrm { \ell } }$ -score is the harmonic mean of the precision and the recall; the precision indicates the fraction of clean samples among all samples that are predicted as clean, and the recall indicates the portion of clean samples that are identified correctly.
+
+# 3.1 Alignment Analysis for Noisy Label Detector
+
+To design a robust label noise filtering framework, we explore the linear nature of the topological space of feature vectors for data resampling techniques and deal with the classifier contamination due to random labels. Recent studies on the distribution of latent representations in DNNs provide insight regarding how correctly the outlier samples can be filtered with the hidden space’s geometrical information. For instance, in [23, 24], the authors proposed frameworks for novelty detection using the topological information of pre-logit based on the Mahalanobis distance, and, in [42], the authors filtered the noisy data based on the Euclidean distance between pre-logits. Maennel et al. [31] analytically showed that an alignment between the principal components of network parameters and those of data takes place when training with random labels. This finding points out that random labels can corrupt a classifier, and thus building a robust classifier is required.
+
+Motivated by these works, we aim to design a novel detector using the principal components of latent features to satisfy Definition 2. However, it is intractable to find the optimal classifier to maximize the separation of alignment clusterability because clean data distribution and noisy data distribution are inaccessible. To handle this issue, we attempt to approximate the clean eigenvector to maximize the alignment values of clean data rather than to maximize the separation; the algorithm utilizes the eigenvector of the data for each class (Figure 2). Below, we provide the upper bound for the perturbation toward the clean data’s eigenvector under simple problem settings with noisy labels referred to in other studies [29, 42]. We first introduce notations. Next, we establish the theoretical evidence that our FINE algorithm approximates the clean data’s eigenvectors under some assumptions for its analytic tractability (Theorem 1). We mainly present the theorem and its interpretation; details of the proofs can be found in the Appendix.
+
+![](images/2b530cf95684b0b0d61d6b3d5be3a3529386939349751e8aa4ec2aa62c5d612a.jpg)  
+Figure 2: Illustration for the problem settings and Theorem 1. The perturbation (green shade) is the angle between the first eigenvector of clean instances (blue line) and the estimated first eigenvector (green line) which is perturbed by that of noisy instances (orange line). Note that blue and orange points are clean instances and noisy instances, respectively.
+
+Notations. Consider a binary classification task. Assume that the data points and labels lie in $\mathcal { X } \times \mathcal { V }$ , where the feature space $\mathcal { X } \subset \mathbb { R } ^ { d }$ and label space $\mathcal { Y } = \{ - 1 , + 1 \}$ . A single data point $_ { \textbf { \em x } }$ and its true label $y$ follow a distribution $( \pmb { x } , y ) \sim P _ { { \pmb { \chi } } \times { \pmb { y } } }$ . Denote by $\tilde { y }$ the observed label (potentially corrupted). Without loss of generality, we focus on the set of data points whose observed label is $\tilde { y } = + 1$ .
+
+Let $\mathbf x \subset \mathcal X$ be the finite set of features with clean instances whose true label is $y = + 1$ . Similarly, let $\tilde { \textbf { X } } ( ~ \mathcal { X } ~ $ be the set of noisy instances whose true label is $y = - 1$ . To establish our theorem, we assume the following reasonable conditions referred to other works using linear discriminant analysis (LDA) assumptions [24, 12]:
+
+Assumption 1. The feature distribution is comprised of two Gaussians, each identified as a clean cluster and a noisy cluster.
+
+Assumption 2. The features of all instances with $y = + 1$ are aligned on the unit vector v with the white noise, i.e., $\mathbb { E } _ { \pmb { x } \in \mathbf { X } } \left[ \pmb { x } \right] = \pmb { v }$ . Similarly, features of all instances with $y = - 1$ are aligned on the unit vector $\pmb { w }$ , i.e., $\mathbb { E } _ { { \pmb x } \in \tilde { \bf X } } \left[ { \pmb x } \right] = { \pmb w }$ .
+
+Theorem 1. (Upper bound for the perturbation towards the clean data’s eigenvector v) Let $N _ { + }$ and $N _ { - }$ be the number of clean instances and noisy instances, respectively, and u be the FINE’s eigenvector which is the first column of U from the eigen decomposition of the whole data’s matrix $\pmb { \Sigma }$ . For any $\delta \in ( 0 , 1 )$ , its perturbation towards the v in assumption 2 (i.e., 2-norm for difference of projection matrices; left hand side of Eq. $( l )$ ) holds the following with probability $1 - \delta$ :
+
+$$
+\| \pmb { u } \pmb { u } ^ { \top } - \pmb { v } \pmb { v } ^ { \top } \| _ { 2 } \leq \frac { 3 \tau \cos \theta + \mathcal { O } ( \sigma ^ { 2 } \sqrt { \frac { d + \log ( 4 / \delta ) } { N _ { + } } } ) } { 1 - \tau ( \sin \theta + 3 \cos \theta ) - \mathcal { O } ( \sigma ^ { 2 } \sqrt { \frac { d + \log ( 4 / \delta ) } { N _ { + } } } ) }
+$$
+
+where w is the first eigenvector of noisy instances, $\tau$ is the fraction between noisy and clean instances $( \frac { N _ { - } } { N _ { + } } ) , \theta$ is $\scriptstyle \angle ( w , v )$ , and $\sigma ^ { 2 }$ is a variance of white noise.
+
+Theorem 1 states that the upper bound for the perturbation toward $\textbf {  { v } }$ are dependent on both the ratio $\tau$ and the angle $\theta$ between $\pmb { w }$ and $\textbf {  { v } }$ ; small upper bound can be guaranteed as the number of clean data increases, $\tau$ decreases, and $\theta$ approaches $\frac { \pi } { 2 }$ . We also derive the lower bound for the precision and the recall when using the eigenvector $\textbf { \em u }$ in Appendix. In this theoretical analysis, we can ensure that such lower bound values become larger as $\pmb { v }$ and $\textbf { \em w }$ become orthogonal to each other. To verify these assumptions, we provide various experimental results for the separation of alignment clusterability, the perturbation values, the scalability to the number of samples, the quality of our detector in the application of sample selection approach, and the comparison with an alternative clustering-based estimator [24].
+
+![](images/8a7dfe8ad4826fde753b055955563adfdc0f0a32d7816be67610bdec671492af.jpg)  
+Figure 3: (a), (b): Heatmaps of Eq. (2) values on unit circle in random hyperplane. We evaluate this visualization on the ResNet34 model trained with common cross-entropy loss on CIFAR-10 with asymmetric noise $40 \%$ and CIFAR-100 with symmetric noise $80 \%$ , respectively. Colors closer to yellow indicate larger the values; (c): comparison of perturbations of Eq. (1) on CIFAR-10 with symmetric noise $20 \%$ ; (d): comparison of cosine similarity values between FINE’s principal components and approximated principal components using fraction of data on CIFAR-10 with symmetric noise $80 \%$ .
+
+Validation for our Estimated Eigenvector. To validate our FINE’s principal components, we first propose a simple visualization scheme based on the following steps: (1) Pick the first eigenvector $( { \pmb u } )$ extracted by FINE algorithm, (2) Generate a random hyperplane spanned by such eigenvector $( { \pmb u } )$ and a random vector, (3) Calculate the value of the following Eq. (2) on any unit vectors $\mathbf { \Pi } ( \mathbf { a } )$ in such hyperplane and plot a heatmap with them:
+
+$$
+{ \frac { 1 } { \left| \mathbf { X } \right| } } \sum _ { \pmb { x } _ { i } \in \mathbf { X } } \left. \pmb { a } , \pmb { x } _ { i } \right. ^ { 2 } - { \frac { 1 } { \left| \tilde { \mathbf { X } } \right| } } \sum _ { \pmb { x } _ { j } \in \tilde { \mathbf { X } } } \left. \pmb { a } , \pmb { x } _ { j } \right. ^ { 2 }
+$$
+
+Eq. (2) is maximized when the unit vector $^ { a }$ not only maximizes the FINE scores of clean data for the first term in Eq. (2), but also minimizes those of noisy data for the second term in Eq. (2). This visualization shows in 2-D how FINE’s first eigenvector $( \pmb { u } )$ optimizes such values in the presence of noisy instances (Figure 3a and 3b). As the figures show, the FINE’s eigenvector $\textbf { \em u }$ (red dotted line) has almost maximum value of Eq. (2). Furthermore, we empirically evaluate the perturbation values in Theorem 1 as the noise rate changes (Figure 3c); FINE has small perturbation values even in a severe noise rate.
+
+Scalability to Number of Samples. Despite FINE’s superiority, it may require high computational costs if the whole dataset is used for eigen decomposition. To address this issue, we approximate the eigenvector with a small portion of the dataset and measure the cosine similarity values between the approximated term and the original one $( u )$ (Figure 3d). Interestingly, we verify that far accurate eigenvector is computed even using $1 \%$ data (i.e., a cosine similarity value is 0.99), and thus the eigenvector can be accurately estimated with little computation time.
+
+Validation for Dynamics of Sample-selection Approach. We evaluate the F-score dynamics of every training epoch on the symmetric and the asymmetric label noise in Figure 4. We compare FINE with the following sample-selection approaches: Co-teaching [14] and TopoFilter [42]. In Figure 4, during the training process, F-scores of FINE becomes consistently higher on both symmetric noise and asymmetric noise settings, while Co-teaching and TopoFilter achieve lower quality. Unlike TopoFilter and FINE, Co-teaching even performs the sample-selection with the access of noise rate. This evidence show that FINE is also applicable to the naive sample-selection approaches.
+
+![](images/035e0c7836c4f96bc84ff561c9be9cff0497e1c1c5a8f0a792ed9bc537024004.jpg)  
+Figure 4: Comparisons of F-scores on CIFAR10 and CIFAR100 under symmetric and asymmetric label noise. C10 and C100 denote CIFAR-10 and CIFAR-100, respectively.
+
+Comparison for Mahalanobis Distance Estimator Under similar conditions, Lee et al. [24] measured the Mahalanobis distance of pre-logits using the minimum covariance determinant (MCD) estimator and selected clean samples based on this distance. While they also utilized the LDA assumptions on pre-logits, FINE consistently outperforms MCD in both precision and recall, thus yielding better Fscore (Figure 5). The experimental results justify our proposed detector, in comparison with a similar alternative.
+
+![](images/26d68f38364785b989815bb5939e6ce9c08f14c73d88e7bfeec9f205465683ce.jpg)  
+Figure 5: Comparisons of F-scores on CIFAR-10 and CIFAR-100 under symmetric (S) and asymmetric noise (A) settings.
+
+# 4 Experiment
+
+In this section, we demonstrate the effectiveness of our FINE detector for three applications: sample selection approach, SSL, and collaboration with noise-robust loss functions.
+
+# 4.1 Experimental Settings
+
+Noisy Benchmark Dataset. Following the previous setups [25, 29], we artificially generate two types of random noisy labels: injecting uniform randomness into a fraction of labels (symmetric) and corrupting a label only to a specific class (asymmetric). For example, we generate noise by mapping TRUCK AUTOMOBILE, BIRD AIRPLANE, DEER $ \mathrm { H O R S E }$ , $\mathbf { C A T }  \mathbf { D O G }$ to make asymmetric noise for CIFAR-10. For CIFAR-100, we create 20 five-size super-classes and generate asymmetric noise by changing each class to the next class within super-classes circularly. For a real-world dataset, Clothing1M [44] containing inherent noisy labels is used. This dataset contains 1 million clothing images obtained from online shopping websites with 14 classes2. The dataset provides $5 0 k , 1 4 k$ , and $1 0 k$ verified as clean data for training, validation, and testing. Instead of using the $5 0 k$ clean training data, we use a randomly sampled pseudo-balanced subset as a training set with $1 2 0 k$ images. For evaluation, we compute the classification accuracy on the $1 0 k$ clean dataset.
+
+Networks and Hyperparameter Settings. We use the architectures and hyperparameter settings for all baseline experiments following the setup of Liu et al. [29] except with SSL approaches. For SSL approaches, we follow the setup of Li et al. [25]. We set the threshold $\zeta$ as 0.5.
+
+# 4.2 Application of FINE
+
+# 4.2.1 Sample Selection-Based Approaches
+
+We apply our FINE detector for various sample selection algorithms. In detail, after warmup training, at every epoch, FINE selects the clean data with the eigenvectors generated from the gram matrices of data predicted to be clean in the previous round, and then the neural networks are trained with them. We compare our proposed method with the following sample selection approaches: (1) Bootstrap [35], (2) Forward [33], (3) Co-teaching [14]; (4) Co-teaching $^ +$ [47]; (5) TopoFilter [42]; (6) CRUST [32]. We evaluate these algorithms three times and report error bars.
+
+Table 1: Test accuracies $( \% )$ on CIFAR-10 and CIFAR-100 under different noisy types and fractions. All comparison methods are reproduced with publicly available code, while the results for Bootstrap [35] and Forward [33] are taken from [29]. For CRUST [32], we experiment without mixup to compare the intrinsic sample selection effect of each method. The average accuracies and standard deviations over three trials are reported. Here, we substitute the sample selection method of Co-teaching [14, 47] with FINE (i.e., F-Co-teaching). The best results sharing the noisy fraction and method are highlighted in bold.   
+
+<table><tr><td>Dataset</td><td colspan="4">CIFAR-10</td><td colspan="4">CIFAR-100</td></tr><tr><td>Noisy Type</td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td>Noise Ratio</td><td>20</td><td>50</td><td>80</td><td>40</td><td>20</td><td>50</td><td>80</td><td>40</td></tr><tr><td>Standard</td><td>87.0 ± 0.1</td><td>78.2 ±0.8</td><td>53.8 ±1.0</td><td>85.0±0.0</td><td>58.7 ±0.3</td><td>42.5± 0.3</td><td>18.1 ± 0.8</td><td>42.7 ± 0.6</td></tr><tr><td>Bootstrap [35]</td><td>86.2 ±0.2</td><td>1</td><td>54.1 ± 1.3</td><td>81.2 ± 1.5</td><td>58.3 ±0.2</td><td>-</td><td>21.6 ± 1.0</td><td>45.1 ± 0.6</td></tr><tr><td>Forward [33]</td><td>88.0±0.4</td><td></td><td>54.6 ± 0.4</td><td>83.6± 0.6</td><td>39.2 ± 2.6</td><td></td><td>9.0±0.6</td><td>34.4 ± 1.9</td></tr><tr><td>Co-teaching[14]</td><td>89.3 ±0.3</td><td>83.3 ±0.6</td><td>66.3 ± 1.5</td><td>88.4±2.8</td><td>63.4 ± 0.0</td><td>49.1 ± 0.4</td><td>20.5 ± 1.3</td><td>47.7 ± 1.2</td></tr><tr><td>Co-teaching+[47]</td><td>89.1 ± 0.5</td><td>84.9 ±0.4</td><td>63.8± 2.3</td><td>86.5 ± 1.2</td><td>59.2 ± 0.4</td><td>47.1 ± 0.3</td><td>20.2 ± 0.9</td><td>44.7 ± 0.6</td></tr><tr><td>TopoFilter[42]</td><td>90.4± 0.2</td><td>86.8 ±0.3</td><td>46.8 ± 1.0</td><td>87.5± 0.4</td><td>66.9 ± 0.4</td><td>53.4 ±1.8</td><td>18.3 ± 1.7</td><td>56.6±0.5</td></tr><tr><td>CRUST[32]</td><td>89.4 ±0.2</td><td>87.0 ± 0.1</td><td>64.8 ± 1.5</td><td>82.4± 0.0</td><td>69.3 ±0.2</td><td>62.3 ± 0.2</td><td>21.7 ± 0.7</td><td>56.1 ± 0.5</td></tr><tr><td>FINE</td><td>91.0 ± 0.1</td><td>87.3 ± 0.2</td><td>69.4 ± 1.1</td><td>89.5 ± 0.1</td><td>70.3 ± 0.2</td><td>64.2 ± 0.5</td><td>25.6 ± 1.2</td><td>61.7 ± 1.0</td></tr><tr><td>F-Coteaching</td><td>92.0 ± 0.1</td><td>87.5 ± 0.1</td><td>74.2 ± 0.8</td><td>90.5 ± 0.2</td><td>71.1 ± 0.2</td><td>64.7 ± 0.3</td><td>31.6 ± 1.0</td><td>64.8 ± 0.7</td></tr></table>
+
+Table 1 summarizes the performances of different sample selection approaches on various noise distribution and datasets. We observe that our FINE method consistently outperforms the competitive methods over the various noise rates. Our FINE methods can filter the clean instances without losing essential information, leading to training the robust network.
+
+To go further, we improve the performance of Co-teaching [14] by substituting its sample selection state with our FINE algorithm. To combine FINE and the Co-teaching family, unlike the original methods that utilize the small loss instances to train with clean labels, we train one model with extracted samples by conducting FINE on another model. The results of the experiments are shown in the eighth and ninth rows of Table 1.
+
+# 4.2.2 SSL-Based Approaches
+
+SSL approaches [25, 10, 41] divide the training data into clean instances as labeled instances and noisy instances as unlabeled instances and use both the labeled and unlabeled samples to train the networks in SSL. Recently, methods belonging to this category have shown the best performance among the various LNL methods, and these methods can train robust networks for even extremely high noise rates. We compare the performances of the existing semi-supervised approaches and that in which the sample selection state of DivideMix [25] is substituted with our FINE algorithm (i.e., F-DivideMix). The results of the experiments are shown in Table 2. We achieve consistently higher performance than DivideMix by utilizing FINE instead of its loss-based filtering method and show comparable performance to the state-of-the-art SSL methods such as DST [41] and LongReMix [10]. Interestingly, as Figure 6 shows, clean and noisy data are well classified in F-DivideMix under extreme noise cases.
+
+![](images/11adb8995afaef343b648fa666cd40c5a34b839ed05e8c31a9be3bce97d46429.jpg)  
+Figure 6: Comparisons of F-scores on CIFAR-10 under symmetric $90 \%$ noise. Blue line indicates the error bar of two networks’ F-score used in Dividemix [25], and Orange line indicates those replaced by our FINE detector.
+
+# 4.2.3 Collaboration with Noise-Robust Loss Functions
+
+The goal of the noise-robust loss function is to achieve a small risk for unseen clean data even when noisy labels exist in the training data. There have been few collaboration studies of the noise-robust loss function methodology and dynamic sample selection. Most studies have selected clean and noisy data based on cross-entropy loss.
+
+Here, we state the collaboration effects of FINE with various noise-robust loss functions: generalized cross entropy (GCE) [51], symmetric cross entropy (SCE) [40], and early-learning regularization (ELR) [29]. Figure 7 shows that FINE facilitates generalization in the application of noise-robust loss functions on severe noise rate settings. The detailed results are reported in the Appendix. Unlike other methods, it is still theoretically supported because FINE extracts clean data with a robust classifier using representation.
+
+![](images/f283614e95a0f1cd3f6fa0c1544c9102377f47d401ff9597b6134bd5b8d79b1a.jpg)  
+Figure 7: Test accuracies $( \% )$ on CIFAR-10 and CIFAR-100 under different noisy types and fractions for noise-robust loss approaches. Note that the blue and orange bars are results for without and with FINE, respectively. The average accuracies and standard deviations over three trials are reported.
+
+# 4.3 Experiments on Real-World Dataset
+
+As Table 3 shows, FINE and F-DivideMix work fairly well on the Clothing 1M dataset compared to other approaches when we reproduce the experimental results under the same settings.
+
+Table 2: Comparison of test accuracies $( \% )$ for FINE collaborating with DivideMix and existing semi-supervised approaches on CIFAR-10 and CIFAR-100 under different noisy types and fractions. The results for all comparison methods are taken from their original works.   
+Table 3: Test accuracy on Clothing1M dataset   
+
+<table><tr><td>Method</td><td>Standard</td><td>GCE[51]</td><td>SCE[40]</td><td>ELR [29]</td><td>DivideMix [25]</td><td>CORES² [8]</td><td>FINE</td><td>F-DivideMix</td></tr><tr><td>Accuracy</td><td>68.94</td><td>69.75</td><td>71.02</td><td>72.87</td><td>74.30</td><td>73.24</td><td>72.91</td><td>74.37</td></tr></table>
+
+# 5 Conclusion
+
+This paper introduces FINE for detecting label noise by designing a robust noise detector. Our main idea is utilizing the principal components of latent representations made by eigen decomposition. Most existing detection methods are dependent on the loss values, while such losses may be biased by corrupted classifier [24, 31]. Our methodology alleviates this issue by extracting key information from representations without using explicit knowledge of the noise rates. We show that the FINE detector has an excellent ability to detect noisy labels in theoretical and experimental results. We propose three applications of the FINE detector: sample-selection approach, SSL approach, and collaboration with noise-robust loss functions. FINE yields strong results on standard benchmarks and a real-world dataset for various LNL approaches.
+
+We believe that our work opens the door to detecting samples having noisy labels with explainable results. It is a non-trivial task and of social significance, and thus, our work will have a substantial social impact on DL practitioners because it avoids the a labor-intensive job of checking data label quality. As future work, we hope that our work will trigger interest in the design of new labelnoise detectors and bring a fresh perspective for other data-resampling approaches (e.g., anomaly detection and novelty detection). The development of robustness against label noise even leads to an improvement in the performance of network trained with data collected through web crawling. We believe that our contribution will lower the barriers to entry for developing robust models for DL practitioners and greatly impact the internet industry. On the other hand, we are concerned that it can be exploited to train robust models using data collected illegally and indiscriminately on the dark web (e.g., web crawling), and thus it may raise privacy concerns (e.g., copyright).
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) [No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)] and [No. 2021-0-00907, Development of Adaptive and Lightweight Edge-Collaborative Analysis Technology for Enabling Proactively Immediate Response and Rapid Learning]. We thank Seongyoon Kim for discussing about the concept of perturbation.
+
+# References
+
+[1] Shin Ando and Chun Yuan Huang. Deep over-sampling framework for classifying imbalanced data. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 770–785. Springer, 2017.   
+[2] Devansh Arpit, Stanisław Jastrz˛ebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al. A closer look at memorization in deep networks. In International Conference on Machine Learning, pages 233–242. PMLR, 2017.   
+[3] Dara Bahri, Heinrich Jiang, and Maya Gupta. Deep k-nn for noisy labels. In International Conference on Machine Learning, pages 540–550. PMLR, 2020.   
+[4] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In Advances in Neural Information Processing Systems, pages 5049–5059, 2019.   
+[5] Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. Neural Networks, 106:249–259, 2018.   
+[6] Kaidi Cao, Yining Chen, Junwei Lu, Nikos Arechiga, Adrien Gaidon, and Tengyu Ma. Heteroskedastic and imbalanced deep learning with adaptive regularization. arXiv preprint arXiv:2006.15766, 2020.   
+[7] Nitesh V Chawla, Kevin W Bowyer, Lawrence O Hall, and W Philip Kegelmeyer. Smote: synthetic minority over-sampling technique. Journal of artificial intelligence research, 16:321– 357, 2002.   
+[8] Hao Cheng, Zhaowei Zhu, Xingyu Li, Yifei Gong, Xing Sun, and Yang Liu. Learning with instance-dependent label noise: A sample sieve approach. arXiv preprint arXiv:2010.02347, 2020.   
+[9] Charles Corbiere, Hedi Ben-Younes, Alexandre Ramé, and Charles Ollion. Leveraging weakly annotated data for fashion image retrieval and label prediction. In Proceedings of the IEEE International Conference on Computer Vision Workshops, pages 2268–2274, 2017.   
+[10] Filipe R Cordeiro, Ragav Sachdeva, Vasileios Belagiannis, Ian Reid, and Gustavo Carneiro. Longremix: Robust learning with high confidence samples in a noisy label environment. arXiv preprint arXiv:2103.04173, 2021.   
+[11] Di Feng, Christian Haase-Schutz, Lars Rosenbaum, Heinz Hertlein, Claudius Glaser, Fabian Timm, Werner Wiesbeck, and Klaus Dietmayer. Deep multi-modal object detection and semantic segmentation for autonomous driving: Datasets, methods, and challenges. IEEE Transactions on Intelligent Transportation Systems, 22(3):1341–1360, Mar 2021.   
+[12] Ronald A Fisher. The use of multiple measurements in taxonomic problems. Annals of eugenics, 7(2):179–188, 1936.   
+[13] Aritra Ghosh, Himanshu Kumar, and PS Sastry. Robust loss functions under label noise for deep neural networks. In AAAI, 2017.   
+[14] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In Advances in neural information processing systems, pages 8527–8537, 2018.   
+[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   
+[16] Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and out-ofdistribution examples in neural networks. arXiv preprint arXiv:1610.02136, 2016.   
+[17] Dan Hendrycks, Mantas Mazeika, Duncan Wilson, and Kevin Gimpel. Using trusted data to train deep networks on labels corrupted by severe noise. arXiv preprint arXiv:1802.05300, 2018.   
+[18] Jinchi Huang, Lie Qu, Rongfei Jia, and Binqiang Zhao. O2u-net: A simple noisy label detection approach for deep neural networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 3326–3334, 2019.   
+[19] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels, 2018.   
+[20] Taehyeon Kim, Jaehoon Oh, NakYil Kim, Sangwook Cho, and Se-Young Yun. Comparing kullback-leibler divergence and mean squared error loss in knowledge distillation, 2021.   
+[21] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems, 25:1097– 1105, 2012.   
+[22] Ronan Le Bras, Swabha Swayamdipta, Chandra Bhagavatula, Rowan Zellers, Matthew Peters, Ashish Sabharwal, and Yejin Choi. Adversarial filters of dataset biases. In International Conference on Machine Learning, pages 1078–1088. PMLR, 2020.   
+[23] Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin. A simple unified framework for detecting out-of-distribution samples and adversarial attacks. In Advances in Neural Information Processing Systems, pages 7167–7177, 2018.   
+[24] Kimin Lee, Sukmin Yun, Kibok Lee, Honglak Lee, Bo Li, and Jinwoo Shin. Robust inference via generative classifiers for handling noisy labels. In International Conference on Machine Learning, pages 3763–3772. PMLR, 2019.   
+[25] Junnan Li, Richard Socher, and Steven C.H. Hoi. Dividemix: Learning with noisy labels as semi-supervised learning. In International Conference on Learning Representations, 2020.   
+[26] Mingchen Li, Mahdi Soltanolkotabi, and Samet Oymak. Gradient descent with early stopping is provably robust to label noise for overparameterized neural networks. In International Conference on Artificial Intelligence and Statistics, pages 4313–4324. PMLR, 2020.   
+[27] Yi Li and Nuno Vasconcelos. Repair: Removing representation bias by dataset resampling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9572–9581, 2019.   
+[28] Shiyu Liang, Yixuan Li, and Rayadurgam Srikant. Enhancing the reliability of out-ofdistribution image detection in neural networks. arXiv preprint arXiv:1706.02690, 2017.   
+[29] Sheng Liu, Jonathan Niles-Weed, Narges Razavian, and Carlos Fernandez-Granda. Early-learning regularization prevents memorization of noisy labels. arXiv preprint arXiv:2007.00151, 2020.   
+[30] Michal Lukasik, Srinadh Bhojanapalli, Aditya Menon, and Sanjiv Kumar. Does label smoothing mitigate label noise? In International Conference on Machine Learning, pages 6448–6458. PMLR, 2020.   
+[31] Hartmut Maennel, Ibrahim Alabdulmohsin, Ilya Tolstikhin, Robert JN Baldock, Olivier Bousquet, Sylvain Gelly, and Daniel Keysers. What do neural networks learn when trained with random labels? arXiv preprint arXiv:2006.10455, 2020.   
+[32] Baharan Mirzasoleiman, Kaidi Cao, and Jure Leskovec. Coresets for robust training of neural networks against noisy labels. arXiv preprint arXiv:2011.07451, 2020.   
+[33] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1944–1952, 2017.   
+[34] Geoff Pleiss, Tianyi Zhang, Ethan R Elenberg, and Kilian Q Weinberger. Identifying mislabeled data using the area under the margin ranking. arXiv preprint arXiv:2001.10528, 2020.   
+[35] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014.   
+[36] Mengye Ren, Wenyuan Zeng, Bin Yang, and Raquel Urtasun. Learning to reweight examples for robust deep learning. In International Conference on Machine Learning, pages 4334–4343. PMLR, 2018.   
+[37] Hwanjun Song, Minseok Kim, Dongmin Park, and Jae-Gil Lee. Prestopping: How does early stopping help generalization against label noise?, 2020.   
+[38] Brendan Van Rooyen, Aditya Krishna Menon, and Robert C Williamson. Learning with symmetric label noise: The importance of being unhinged. arXiv preprint arXiv:1505.07634, 2015.   
+[39] Andreas Veit, Neil Alldrin, Gal Chechik, Ivan Krasin, Abhinav Gupta, and Serge Belongie. Learning from noisy large-scale datasets with minimal supervision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 839–847, 2017.   
+[40] Yisen Wang, Xingjun Ma, Zaiyi Chen, Yuan Luo, Jinfeng Yi, and James Bailey. Symmetric cross entropy for robust learning with noisy labels. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 322–330, 2019.   
+[41] Yi Wei, Xue Mei, Xin Liu, and Pengxiang Xu. Dst: Data selection and joint training for learning with noisy labels. arXiv preprint arXiv:2103.00813, 2021.   
+[42] Pengxiang Wu, Songzhu Zheng, Mayank Goswami, Dimitris Metaxas, and Chao Chen. A topological filter for learning with label noise. arXiv preprint arXiv:2012.04835, 2020.   
+[43] Xiaobo Xia, Tongliang Liu, Bo Han, Chen Gong, Nannan Wang, Zongyuan Ge, and Yi Chang. Robust early-learning: Hindering the memorization of noisy labels. In International Conference on Learning Representations, 2021.   
+[44] Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang. Learning from massive noisy labeled data for image classification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2691–2699, 2015.   
+[45] Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L_dmi: A novel information-theoretic loss function for training deep nets robust to label noise. In NeurIPS, pages 6222–6233, 2019.   
+[46] Seunghan Yang, Hyoungseob Park, Junyoung Byun, and Changick Kim. Robust federated learning with noisy labels, 2020.   
+[47] Xingrui Yu, Bo Han, Jiangchao Yao, Gang Niu, Ivor W Tsang, and Masashi Sugiyama. How does disagreement help generalization against label corruption? arXiv preprint arXiv:1901.04215, 2019.   
+[48] Xiyu Yu, Tongliang Liu, Mingming Gong, and Dacheng Tao. Learning with biased complementary labels. In Proceedings of the European conference on computer vision (ECCV), pages 68–83, 2018.   
+[49] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.   
+[50] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.   
+[51] Zhilu Zhang and Mert R Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. arXiv preprint arXiv:1805.07836, 2018.   
+[52] Zizhao Zhang, Han Zhang, Sercan O Arik, Honglak Lee, and Tomas Pfister. Distilling effective supervision from severe label noise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9294–9303, 2020.   
+[53] Zizhao Zhang, Han Zhang, Sercan O. Arik, Honglak Lee, and Tomas Pfister. Distilling effective supervision from severe label noise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.

parse/train/QZpx42n0BWr/QZpx42n0BWr_content_list.json

@@ -0,0 +1,1008 @@
+[
+    {
+        "type": "text",
+        "text": "FINE Samples for Learning with Noisy Labels ",
+        "text_level": 1,
+        "bbox": [
+            215,
+            122,
+            781,
+            148
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Taehyeon Kim ∗   \nKAIST AI   \nKAIST   \nDaejeon, South Korea   \npotter32@kaist.ac.kr   \nJongwoo Ko ∗   \nKAIST AI   \nKAIST   \nDaejeon, South Korea   \njongwoo.ko@kaist.ac.kr   \nSangwook Cho   \nKAIST AI   \nKAIST   \nDaejeon, South Korea   \nsangwookcho@kaist.ac.kr   \nJinhwan Choi   \nKAIST AI   \nKAIST   \nDaejeon, South Korea   \njinhwanchoi@kaist.ac.kr   \nSe-Young Yun   \nKAIST AI   \nKAIST   \nDaejeon, South Korea   \nyunseyoung@kaist.ac.kr ",
+        "bbox": [
+            264,
+            200,
+            441,
+            271
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            539,
+            200,
+            730,
+            271
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            184,
+            291,
+            383,
+            361
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            403,
+            291,
+            601,
+            361
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            620,
+            291,
+            812,
+            362
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            397,
+            535,
+            412
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modern deep neural networks (DNNs) become weak when the datasets contain noisy (incorrect) class labels. Robust techniques in the presence of noisy labels can be categorized into two types: developing noise-robust functions or using noisecleansing methods by detecting the noisy data. Recently, noise-cleansing methods have been considered as the most competitive noisy-label learning algorithms. Despite their success, their noisy label detectors are often based on heuristics more than a theory, requiring a robust classifier to predict the noisy data with loss values. In this paper, we propose a novel detector for filtering label noise. Unlike most existing methods, we focus on each data point’s latent representation dynamics and measure the alignment between the latent distribution and each representation using the eigen decomposition of the data gram matrix. Our framework, coined as filtering noisy instances via their eigenvectors (FINE), provides a robust detector using derivative-free simple methods with theoretical guarantees. Under our framework, we propose three applications of the FINE: sample-selection approach, semi-supervised learning (SSL) approach, and collaboration with noiserobust loss functions. Experimental results show that the proposed methods consistently outperform corresponding baselines for all three applications on various benchmark datasets 1. ",
+        "bbox": [
+            233,
+            415,
+            766,
+            662
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            669,
+            312,
+            685
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks (DNNs) have achieved remarkable success in numerous tasks as the amount of accessible data has dramatically increased [21, 15]. On the other hand, accumulated datasets are typically labeled by a human, a labor-intensive job or through web crawling [48] so that they may be easily corrupted (label noise) in real-world situations. Recent studies have shown that deep neural networks have the capacity to memorize essentially any labeling of the data [49]. Even a small amount of such noisy data can hinder the generalization of DNNs owing to their strong memorization of noisy labels [49, 29]. Hence, it becomes crucial to train DNNs that are robust to corrupted labels. As label noise problems may appear anywhere, such robustness increases reliability in many applications such as the e-commerce market [9], medical fields [45], on-device AI [46], and autonomous driving systems [11]. ",
+        "bbox": [
+            173,
+            693,
+            825,
+            832
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To improve the robustness against noisy data, the methods for learning with noisy labels (LNL) have been evolving in two main directions [18]: (1) designing noise-robust objective functions or regularizations and (2) detecting and cleansing the noisy data. In general, the former noise-robust direction uses explicit regularization techniques [6, 52, 50] or robust loss functions [38, 13, 40, 51], but their performance is far from state-of-the-art [49, 26] on datasets with severe noise rates. Recently, researchers have designed noise-cleansing algorithms focused on segregating the clean data (i.e., samples with uncorrupted labels) from the corrupted data [19, 14, 47, 18, 32, 42]. One of the popular criteria for the segregation process is the loss value between the prediction of the noisy classifier and its noisy label, where it is generally assumed that the noisy data have a large loss [19, 14, 47, 18] or the magnitude of the gradient during training [51, 40]. However, these methods may still be biased by the corrupted linear classifier towards label noise because their criterion (e.g., loss values or weight gradient) uses the posterior information of such a linear classifier [24]. Maennel et al. [31] analytically showed that the principal components of the weights of a neural network align with the randomly labeled data; this phenomenon can yield more negative effects on the classifier as the number of randomly labeled classes increases. Recently, Wu et al. [42] used an inherent geometric structure induced by nearest neighbors (NN) in latent space and filtered out isolated data in such topology, and its quality was sensitive to its hyperparameters regarding NN clustering in the presence of severe noise rates. ",
+        "bbox": [
+            176,
+            838,
+            821,
+            866
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/5dc76db62bc5fdf618d00e6a7419d8677909c74d57178c02c103d2a86cb6ad74.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of (a) basic concept of this work and (b) proposed detection framework, FINE. Noise-cleansing learning generally separates clean data from the original dataset by using prediction outputs. We propose a novel derivative-free detector based on an unsupervised clustering algorithm on the high-order topological space. FINE measures the alignment of pre-logits (i.e., penultimate layer representation vectors) toward the class-representative vector that is extracted through the eigen decomposition of the gram matrix of data representations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            89,
+            816,
+            208
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            310,
+            825,
+            530
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To mitigate such issues for label noise detectors, we provide a novel yet simple detector framework, filtering noisy labels via their eigenvectors (FINE) with theoretical guarantees to provide a high-quality splitting of clean and corrupted examples (without the need to estimate noise rates). Instead of using the neural network’s linear classifier, FINE utilizes the principal components of latent representations made by eigen decomposition which is one of the most widely used unsupervised learning algorithms and separates clean data and noisy data by these components (Figure 1a). To motivate our approach, as Figure 1b shows, we find that the clean data (blue points) are mainly aligned on the principal component (black dotted line), whereas the noisy data (orange points) are not; thus, the dataset is well clustered with the alignment of representations toward the principal component by fitting them into Gaussian mixture models (GMM). We apply our framework to various LNL methods: the sample selection approach, a semi-supervised learning (SSL) approach, and collaboration with noise-robust loss functions. The key contributions of this work are summarized as follows: ",
+        "bbox": [
+            173,
+            537,
+            825,
+            715
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a novel framework, termed FINE (filtering noisy labels via their eigenvectors), for detecting clean instances from noisy datasets. FINE makes robust decision boundary for the high-order topological information of data in latent space by using eigen decomposition of their gram matrix.   \n• We provide provable evidence that FINE allows a meaningful decision boundary made by eigenvectors in latent space. We support our theoretical analysis with various experimental results regarding the characteristics of the principal components extracted by our FINE detector.   \n• We develop a simple sample-selection method by replacing the existing detector method with FINE. We empirically validate that a sample-selection learning with FINE provides consistently superior detection quality and higher test accuracy than other existing alternative methods such as the Co-teaching family [14, 47], TopoFilter [42], and CRUST [32]. ",
+        "bbox": [
+            217,
+            731,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We experimentally show that our detection framework can be applied in various ways to existing LNL methods and validate that ours consistently improves the generalization in the presence of noisy data: sample-selection approach [14, 47], SSL approach [25], and collaboration with noise-robust loss functions [51, 40, 29]. ",
+        "bbox": [
+            218,
+            92,
+            823,
+            147
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Organization. The remainder of this paper is organized as follows. In Section 2, we discuss the recent literature on LNL solutions and meaningful detectors. In Section 3, we address our motivation for creating a noisy label detector with theoretical insights and provide our main method, filtering the noisy labels via their eigenvectors (FINE). In Section 4, we present the experimental results. Finally, Section 5 concludes the paper. ",
+        "bbox": [
+            174,
+            161,
+            825,
+            231
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            238,
+            330,
+            255
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Zhang et al. [49] empirically showed that any convolutional neural networks trained using stochastic gradient methods easily fit a random labeling of the training data. To tackle this issue, numerous works have examined the classification task with noisy labels. We do not consider the works that assumed the availability of small subsets of training data with clean labels [17, 36, 39, 53, 3]. ",
+        "bbox": [
+            176,
+            257,
+            825,
+            311
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Noise-Cleansing-based Approaches. Noise-cleansing methods have evolved following the improvement of noisy detectors. Han et al. [14] suggested a noisy detection approach, named coteaching, that utilizes two networks, extracts subsets of instances with small losses from each network, and trains each network with subsets of instances filtered by another network. Yu et al. [47] combined a disagreement training procedure with co-teaching, which only selects instances predicted differently by two networks. Huang et al. [18] provided a simple noise-cleansing framework, training-filtering-training; the empirical efficacy was improved by first finding label errors, then training the model only on data predicted as clean. Recently, new noisy detectors with theoretical support have been developed. Wu et al. [42] proposed a method called TopoFilter that filters noisy data by utilizing the $\\mathbf { k }$ -nearest neighborhood algorithm and Euclidean distance between pre-logits. Mirzasoleiman et al. [32] introduced an algorithm that selects subsets of clean instances that provide an approximately low-rank Jacobian matrix and proved that gradient descent applied to the subsets prevents overfitting to noisy labels. Pleiss et al. [34] proposed an area under margin (AUM) statistic that measures the average difference between the logit values of the assigned class and its highest non-assigned class to divide clean and noisy samples. Cheng et. al [8] progressively filtered out corrupted instances using a novel confidence regularization term. The noise-cleansing method was also developed in a semi-supervised learning (SSL) manner. Li et al. [25] modeled the per-sample loss distribution and divide it into a labeled set with clean samples and an unlabeled set with noisy samples, and they leverage the noisy samples through the well-known SSL technique MixMatch [4]. ",
+        "bbox": [
+            174,
+            314,
+            825,
+            577
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Noise-Robust Models. Noise-robust models have been studied in the following directions: robustloss functions, regularizations, and strategies. First, for robust-loss functions, Ghosh et al. [13] showed that the mean absolute error (MAE) might be robust against noisy labels. Zhang & Sabuncu et al. [51] argued that MAE performed poorly with DNNs and proposed a GCE loss function, which can be seen as a generalization of MAE and cross-entropy (CE). Wang et al. [40] introduced the reverse version of the cross-entropy term (RCE) and suggested that the SCE loss function is a weighted sum of the CE and RCE. Some studies have stated that the early-stopped model can prevent the memorization phenomenon for noisy labels [2, 49] and theoretically analyzed it [26]. Based on this hypothesis, Liu et al. [29] proposed an early-learning regularization (ELR) loss function to prohibit memorizing noisy data by leveraging the semi-supervised learning techniques. Xia et al. [43] clarified which neural network parameters cause memorization and proposed a robust training strategy for these parameters. Efforts have been made to develop regularizations on the prediction level by smoothing the one-hot vector [30], using linear interpolation between data instances [50], and distilling the rescaled prediction of other models [20]. However, these works have limitations in terms of performance as the noise rate of the dataset increases. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Dataset Resampling. Label-noise detection may be a category of data resampling which is a common technique in the machine learning community that extracts a “helpful” dataset from the distribution of the original dataset to remove the dataset bias. In class-imbalance tasks, numerous studies have conducted over-sampling of minority classes [7, 1] or undersampling the majority classes [5] to balance the amount of data per class. Li & Vasconcelos et al. [27] proposed a resampling procedure to reduce the representation bias of the data by learning a weight distribution that favors difficult instances for a given feature representation. Le Bras et al. [22] suggested an adversarial filteringbased approach to remove spurious artifacts in a dataset. Analogously, in anomaly detection and ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "INPUT : Noisy training data $\\mathcal { D }$ , feature extractor $g$ , number of classes $K$ , clean probability threshold $\\zeta$ , set of FINE scores for class $k \\mathcal { F } _ { k }$   \nOUTPUT : Collected clean data $\\mathcal { C }$   \n1: Initialize $c \\gets \\emptyset$ , $\\hat { \\mathcal { D } } \\gets \\mathcal { D }$ , $\\Sigma _ { k } \\gets \\mathbf { 0 }$ for all $k = 1 , \\ldots , K$ $/ { * }$ Update the convariance matrices for all classes $^ { * }$   \n2: for $( \\boldsymbol { x } _ { i } , \\boldsymbol { y } _ { i } ) \\in \\mathcal { D }$ do   \n3: ${ z _ { i } \\gets g ( { \\pmb x } _ { i } ) }$   \n4: Update the gram matrix $\\Sigma _ { y _ { i } } \\gets \\Sigma _ { y _ { i } } + z _ { i } z _ { i } ^ { \\top }$   \n5: end for $/ { * }$ Generate the principal component with eigen decomposition $^ { * }$   \n6: for $k = 1 , \\ldots , K$ do   \n7: $\\mathbf { U } _ { k } , \\mathbf { \\Lambda } \\Lambda _ { k } \\gets$ EIGEN DECOMPOSITION OF $\\Sigma _ { k }$   \n8: $\\mathbf { u } _ { k } \\gets$ THE FIRST COLUMN OF $\\mathbf { U } _ { k }$   \n9: end for $/ { * }$ Compute the alignment score and get clean subset $\\mathcal { C }$ \\*/   \n10: for $( \\boldsymbol { x } _ { i } , \\boldsymbol { y } _ { i } ) \\in \\mathcal { D }$ do   \n11: Compute the FINE score $f _ { i } = \\left. \\mathbf { u } _ { y _ { i } } , z _ { i } \\right. ^ { 2 }$ and $\\mathcal { F } _ { y _ { i } } \\gets \\mathcal { F } _ { y _ { i } } \\cup \\{ f _ { i } \\}$   \n12: end for $/ { * }$ Finding the samples whose clean probability is larger than $\\zeta$ \\*/   \n13: ${ \\mathcal { C } } \\gets { \\mathcal { C } } \\cup$ GMM $( \\mathcal { F } _ { k } , \\zeta )$ for all $k = 1 , \\ldots , K$ ",
+        "bbox": [
+            178,
+            111,
+            781,
+            396
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "out-of-distribution detection problems [16, 28, 23], the malicious data are usually detected by examining the loss value or negative behavior in the feature representation space. While our research is motivated by such previous works, this paper focuses on the noisy image classification task. ",
+        "bbox": [
+            174,
+            426,
+            825,
+            468
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            486,
+            269,
+            503
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present our detector framework and the theoretical motivation behind using the detector in high-dimensional classification. To segregate the clean data, we utilize the degree of alignment between the representations and the eigenvector of the representations’ gram matrices for all classes, called FINE (FIltering Noisy instnaces via their Eigenvectors). Our algorithm is as follows (Algorithm 1). FINE first creates a gram matrix of the representation in the noisy training dataset for each class and conducts the eigen decomposition for those gram matrices. Then, FINE finds clean and noisy instances using the square of inner product values between the representations and the first eigenvector having the largest eigenvalue. In this manner, we treat the data as clean if aligned onto the first eigenvector, while most of the noisy instances are not. Here, we formally define ��alignment’ and ‘alignment clusterability’ in Definition 1 and Definition 2, respectively. ",
+        "bbox": [
+            174,
+            517,
+            825,
+            656
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. (Alignment) $D$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            660,
+            364,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. (Alignment Clusterability) For all features labeled as class $k$ in dataset $\\mathcal { D }$ , let fit a Gaussian Mixture Model (GMM) on their alignment (Definition 1) distribution to divide current samples into a clean set and a noisy set; the set having larger mean value is treated as a clean set, and another one is a noisy set. Then, we say a dataset $\\mathcal { D }$ satisfies alignment clusterability if the representation $_ z$ labeled as the same true class belongs to the clean set. ",
+        "bbox": [
+            174,
+            678,
+            825,
+            747
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As an empirical evaluation, the quality of our detector for noisy data is measured with the $\\boldsymbol { F }$ -score, a widely used criterion in noisy label detection, anomaly detection and out-of-distribution detection [8, 16, 28, 23]. We treat the selected clean samples as the positive class and the noisy samples as negative class. The $F _ { \\mathrm { \\ell } }$ -score is the harmonic mean of the precision and the recall; the precision indicates the fraction of clean samples among all samples that are predicted as clean, and the recall indicates the portion of clean samples that are identified correctly. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            840
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Alignment Analysis for Noisy Label Detector ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            849,
+            524,
+            864
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To design a robust label noise filtering framework, we explore the linear nature of the topological space of feature vectors for data resampling techniques and deal with the classifier contamination due to random labels. Recent studies on the distribution of latent representations in DNNs provide insight regarding how correctly the outlier samples can be filtered with the hidden space’s geometrical information. For instance, in [23, 24], the authors proposed frameworks for novelty detection using the topological information of pre-logit based on the Mahalanobis distance, and, in [42], the authors filtered the noisy data based on the Euclidean distance between pre-logits. Maennel et al. [31] analytically showed that an alignment between the principal components of network parameters and those of data takes place when training with random labels. This finding points out that random labels can corrupt a classifier, and thus building a robust classifier is required. ",
+        "bbox": [
+            174,
+            869,
+            823,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            189
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Motivated by these works, we aim to design a novel detector using the principal components of latent features to satisfy Definition 2. However, it is intractable to find the optimal classifier to maximize the separation of alignment clusterability because clean data distribution and noisy data distribution are inaccessible. To handle this issue, we attempt to approximate the clean eigenvector to maximize the alignment values of clean data rather than to maximize the separation; the algorithm utilizes the eigenvector of the data for each class (Figure 2). Below, we provide the upper bound for the perturbation toward the clean data’s eigenvector under simple problem settings with noisy labels referred to in other studies [29, 42]. We first introduce notations. Next, we establish the theoretical evidence that our FINE algorithm approximates the clean data’s eigenvectors under some assumptions for its analytic tractability (Theorem 1). We mainly present the theorem and its interpretation; details of the proofs can be found in the Appendix. ",
+        "bbox": [
+            174,
+            195,
+            549,
+            457
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/2b530cf95684b0b0d61d6b3d5be3a3529386939349751e8aa4ec2aa62c5d612a.jpg",
+        "image_caption": [
+            "Figure 2: Illustration for the problem settings and Theorem 1. The perturbation (green shade) is the angle between the first eigenvector of clean instances (blue line) and the estimated first eigenvector (green line) which is perturbed by that of noisy instances (orange line). Note that blue and orange points are clean instances and noisy instances, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            573,
+            196,
+            810,
+            378
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Notations. Consider a binary classification task. Assume that the data points and labels lie in $\\mathcal { X } \\times \\mathcal { V }$ , where the feature space $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ and label space $\\mathcal { Y } = \\{ - 1 , + 1 \\}$ . A single data point $_ { \\textbf { \\em x } }$ and its true label $y$ follow a distribution $( \\pmb { x } , y ) \\sim P _ { { \\pmb { \\chi } } \\times { \\pmb { y } } }$ . Denote by $\\tilde { y }$ the observed label (potentially corrupted). Without loss of generality, we focus on the set of data points whose observed label is $\\tilde { y } = + 1$ . ",
+        "bbox": [
+            174,
+            470,
+            549,
+            541
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            540,
+            823,
+            568
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf x \\subset \\mathcal X$ be the finite set of features with clean instances whose true label is $y = + 1$ . Similarly, let $\\tilde { \\textbf { X } } ( ~ \\mathcal { X } ~ $ be the set of noisy instances whose true label is $y = - 1$ . To establish our theorem, we assume the following reasonable conditions referred to other works using linear discriminant analysis (LDA) assumptions [24, 12]: ",
+        "bbox": [
+            173,
+            574,
+            825,
+            632
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. The feature distribution is comprised of two Gaussians, each identified as a clean cluster and a noisy cluster. ",
+        "bbox": [
+            174,
+            635,
+            823,
+            662
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2. The features of all instances with $y = + 1$ are aligned on the unit vector v with the white noise, i.e., $\\mathbb { E } _ { \\pmb { x } \\in \\mathbf { X } } \\left[ \\pmb { x } \\right] = \\pmb { v }$ . Similarly, features of all instances with $y = - 1$ are aligned on the unit vector $\\pmb { w }$ , i.e., $\\mathbb { E } _ { { \\pmb x } \\in \\tilde { \\bf X } } \\left[ { \\pmb x } \\right] = { \\pmb w }$ . ",
+        "bbox": [
+            174,
+            666,
+            823,
+            709
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. (Upper bound for the perturbation towards the clean data’s eigenvector v) Let $N _ { + }$ and $N _ { - }$ be the number of clean instances and noisy instances, respectively, and u be the FINE’s eigenvector which is the first column of U from the eigen decomposition of the whole data’s matrix $\\pmb { \\Sigma }$ . For any $\\delta \\in ( 0 , 1 )$ , its perturbation towards the v in assumption 2 (i.e., 2-norm for difference of projection matrices; left hand side of Eq. $( l )$ ) holds the following with probability $1 - \\delta$ : ",
+        "bbox": [
+            174,
+            710,
+            825,
+            781
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/268519fa80210437c96f58a0a46a89566c2fb531a03a2a0324687638e1e68aa0.jpg",
+        "text": "$$\n\\| \\pmb { u } \\pmb { u } ^ { \\top } - \\pmb { v } \\pmb { v } ^ { \\top } \\| _ { 2 } \\leq \\frac { 3 \\tau \\cos \\theta + \\mathcal { O } ( \\sigma ^ { 2 } \\sqrt { \\frac { d + \\log ( 4 / \\delta ) } { N _ { + } } } ) } { 1 - \\tau ( \\sin \\theta + 3 \\cos \\theta ) - \\mathcal { O } ( \\sigma ^ { 2 } \\sqrt { \\frac { d + \\log ( 4 / \\delta ) } { N _ { + } } } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            785,
+            714,
+            838
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where w is the first eigenvector of noisy instances, $\\tau$ is the fraction between noisy and clean instances $( \\frac { N _ { - } } { N _ { + } } ) , \\theta$ is $\\scriptstyle \\angle ( w , v )$ , and $\\sigma ^ { 2 }$ is a variance of white noise. ",
+        "bbox": [
+            169,
+            840,
+            823,
+            872
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 states that the upper bound for the perturbation toward $\\textbf {  { v } }$ are dependent on both the ratio $\\tau$ and the angle $\\theta$ between $\\pmb { w }$ and $\\textbf {  { v } }$ ; small upper bound can be guaranteed as the number of clean data increases, $\\tau$ decreases, and $\\theta$ approaches $\\frac { \\pi } { 2 }$ . We also derive the lower bound for the precision and the recall when using the eigenvector $\\textbf { \\em u }$ in Appendix. In this theoretical analysis, we can ensure that such lower bound values become larger as $\\pmb { v }$ and $\\textbf { \\em w }$ become orthogonal to each other. To verify these assumptions, we provide various experimental results for the separation of alignment clusterability, the perturbation values, the scalability to the number of samples, the quality of our detector in the application of sample selection approach, and the comparison with an alternative clustering-based estimator [24]. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/8a7dfe8ad4826fde753b055955563adfdc0f0a32d7816be67610bdec671492af.jpg",
+        "image_caption": [
+            "Figure 3: (a), (b): Heatmaps of Eq. (2) values on unit circle in random hyperplane. We evaluate this visualization on the ResNet34 model trained with common cross-entropy loss on CIFAR-10 with asymmetric noise $40 \\%$ and CIFAR-100 with symmetric noise $80 \\%$ , respectively. Colors closer to yellow indicate larger the values; (c): comparison of perturbations of Eq. (1) on CIFAR-10 with symmetric noise $20 \\%$ ; (d): comparison of cosine similarity values between FINE’s principal components and approximated principal components using fraction of data on CIFAR-10 with symmetric noise $80 \\%$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            90,
+            825,
+            227
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            352,
+            825,
+            450
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Validation for our Estimated Eigenvector. To validate our FINE’s principal components, we first propose a simple visualization scheme based on the following steps: (1) Pick the first eigenvector $( { \\pmb u } )$ extracted by FINE algorithm, (2) Generate a random hyperplane spanned by such eigenvector $( { \\pmb u } )$ and a random vector, (3) Calculate the value of the following Eq. (2) on any unit vectors $\\mathbf { \\Pi } ( \\mathbf { a } )$ in such hyperplane and plot a heatmap with them: ",
+        "bbox": [
+            174,
+            467,
+            825,
+            536
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7849fbb7514cf0364e4f78019af906a39f6f93506cc2e02582cd78f9fef7b173.jpg",
+        "text": "$$\n{ \\frac { 1 } { \\left| \\mathbf { X } \\right| } } \\sum _ { \\pmb { x } _ { i } \\in \\mathbf { X } } \\left. \\pmb { a } , \\pmb { x } _ { i } \\right. ^ { 2 } - { \\frac { 1 } { \\left| \\tilde { \\mathbf { X } } \\right| } } \\sum _ { \\pmb { x } _ { j } \\in \\tilde { \\mathbf { X } } } \\left. \\pmb { a } , \\pmb { x } _ { j } \\right. ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            544,
+            632,
+            585
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Eq. (2) is maximized when the unit vector $^ { a }$ not only maximizes the FINE scores of clean data for the first term in Eq. (2), but also minimizes those of noisy data for the second term in Eq. (2). This visualization shows in 2-D how FINE’s first eigenvector $( \\pmb { u } )$ optimizes such values in the presence of noisy instances (Figure 3a and 3b). As the figures show, the FINE’s eigenvector $\\textbf { \\em u }$ (red dotted line) has almost maximum value of Eq. (2). Furthermore, we empirically evaluate the perturbation values in Theorem 1 as the noise rate changes (Figure 3c); FINE has small perturbation values even in a severe noise rate. ",
+        "bbox": [
+            173,
+            599,
+            825,
+            698
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Scalability to Number of Samples. Despite FINE’s superiority, it may require high computational costs if the whole dataset is used for eigen decomposition. To address this issue, we approximate the eigenvector with a small portion of the dataset and measure the cosine similarity values between the approximated term and the original one $( u )$ (Figure 3d). Interestingly, we verify that far accurate eigenvector is computed even using $1 \\%$ data (i.e., a cosine similarity value is 0.99), and thus the eigenvector can be accurately estimated with little computation time. ",
+        "bbox": [
+            174,
+            707,
+            825,
+            791
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Validation for Dynamics of Sample-selection Approach. We evaluate the F-score dynamics of every training epoch on the symmetric and the asymmetric label noise in Figure 4. We compare FINE with the following sample-selection approaches: Co-teaching [14] and TopoFilter [42]. In Figure 4, during the training process, F-scores of FINE becomes consistently higher on both symmetric noise and asymmetric noise settings, while Co-teaching and TopoFilter achieve lower quality. Unlike TopoFilter and FINE, Co-teaching even performs the sample-selection with the access of noise rate. This evidence show that FINE is also applicable to the naive sample-selection approaches. ",
+        "bbox": [
+            174,
+            808,
+            825,
+            905
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/035e0c7836c4f96bc84ff561c9be9cff0497e1c1c5a8f0a792ed9bc537024004.jpg",
+        "image_caption": [
+            "Figure 4: Comparisons of F-scores on CIFAR10 and CIFAR100 under symmetric and asymmetric label noise. C10 and C100 denote CIFAR-10 and CIFAR-100, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            88,
+            807,
+            190
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparison for Mahalanobis Distance Estimator Under similar conditions, Lee et al. [24] measured the Mahalanobis distance of pre-logits using the minimum covariance determinant (MCD) estimator and selected clean samples based on this distance. While they also utilized the LDA assumptions on pre-logits, FINE consistently outperforms MCD in both precision and recall, thus yielding better Fscore (Figure 5). The experimental results justify our proposed detector, in comparison with a similar alternative. ",
+        "bbox": [
+            174,
+            237,
+            483,
+            401
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/26d68f38364785b989815bb5939e6ce9c08f14c73d88e7bfeec9f205465683ce.jpg",
+        "image_caption": [
+            "Figure 5: Comparisons of F-scores on CIFAR-10 and CIFAR-100 under symmetric (S) and asymmetric noise (A) settings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            228,
+            812,
+            327
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Experiment ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            305,
+            430
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we demonstrate the effectiveness of our FINE detector for three applications: sample selection approach, SSL, and collaboration with noise-robust loss functions. ",
+        "bbox": [
+            174,
+            439,
+            823,
+            468
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental Settings ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            481,
+            367,
+            496
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Noisy Benchmark Dataset. Following the previous setups [25, 29], we artificially generate two types of random noisy labels: injecting uniform randomness into a fraction of labels (symmetric) and corrupting a label only to a specific class (asymmetric). For example, we generate noise by mapping TRUCK AUTOMOBILE, BIRD AIRPLANE, DEER $ \\mathrm { H O R S E }$ , $\\mathbf { C A T }  \\mathbf { D O G }$ to make asymmetric noise for CIFAR-10. For CIFAR-100, we create 20 five-size super-classes and generate asymmetric noise by changing each class to the next class within super-classes circularly. For a real-world dataset, Clothing1M [44] containing inherent noisy labels is used. This dataset contains 1 million clothing images obtained from online shopping websites with 14 classes2. The dataset provides $5 0 k , 1 4 k$ , and $1 0 k$ verified as clean data for training, validation, and testing. Instead of using the $5 0 k$ clean training data, we use a randomly sampled pseudo-balanced subset as a training set with $1 2 0 k$ images. For evaluation, we compute the classification accuracy on the $1 0 k$ clean dataset. ",
+        "bbox": [
+            173,
+            501,
+            825,
+            666
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Networks and Hyperparameter Settings. We use the architectures and hyperparameter settings for all baseline experiments following the setup of Liu et al. [29] except with SSL approaches. For SSL approaches, we follow the setup of Li et al. [25]. We set the threshold $\\zeta$ as 0.5. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            715
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Application of FINE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            734,
+            354,
+            750
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 Sample Selection-Based Approaches ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            761,
+            475,
+            776
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We apply our FINE detector for various sample selection algorithms. In detail, after warmup training, at every epoch, FINE selects the clean data with the eigenvectors generated from the gram matrices of data predicted to be clean in the previous round, and then the neural networks are trained with them. We compare our proposed method with the following sample selection approaches: (1) Bootstrap [35], (2) Forward [33], (3) Co-teaching [14]; (4) Co-teaching $^ +$ [47]; (5) TopoFilter [42]; (6) CRUST [32]. We evaluate these algorithms three times and report error bars. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            869
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b36766b855550dacb14bc4ad6afa728d46690aa006feb762d5155501612652f9.jpg",
+        "table_caption": [
+            "Table 1: Test accuracies $( \\% )$ on CIFAR-10 and CIFAR-100 under different noisy types and fractions. All comparison methods are reproduced with publicly available code, while the results for Bootstrap [35] and Forward [33] are taken from [29]. For CRUST [32], we experiment without mixup to compare the intrinsic sample selection effect of each method. The average accuracies and standard deviations over three trials are reported. Here, we substitute the sample selection method of Co-teaching [14, 47] with FINE (i.e., F-Co-teaching). The best results sharing the noisy fraction and method are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"4\">CIFAR-10</td><td colspan=\"4\">CIFAR-100</td></tr><tr><td>Noisy Type</td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td></tr><tr><td>Noise Ratio</td><td>20</td><td>50</td><td>80</td><td>40</td><td>20</td><td>50</td><td>80</td><td>40</td></tr><tr><td>Standard</td><td>87.0 ± 0.1</td><td>78.2 ±0.8</td><td>53.8 ±1.0</td><td>85.0±0.0</td><td>58.7 ±0.3</td><td>42.5± 0.3</td><td>18.1 ± 0.8</td><td>42.7 ± 0.6</td></tr><tr><td>Bootstrap [35]</td><td>86.2 ±0.2</td><td>1</td><td>54.1 ± 1.3</td><td>81.2 ± 1.5</td><td>58.3 ±0.2</td><td>-</td><td>21.6 ± 1.0</td><td>45.1 ± 0.6</td></tr><tr><td>Forward [33]</td><td>88.0±0.4</td><td></td><td>54.6 ± 0.4</td><td>83.6± 0.6</td><td>39.2 ± 2.6</td><td></td><td>9.0±0.6</td><td>34.4 ± 1.9</td></tr><tr><td>Co-teaching[14]</td><td>89.3 ±0.3</td><td>83.3 ±0.6</td><td>66.3 ± 1.5</td><td>88.4±2.8</td><td>63.4 ± 0.0</td><td>49.1 ± 0.4</td><td>20.5 ± 1.3</td><td>47.7 ± 1.2</td></tr><tr><td>Co-teaching+[47]</td><td>89.1 ± 0.5</td><td>84.9 ±0.4</td><td>63.8± 2.3</td><td>86.5 ± 1.2</td><td>59.2 ± 0.4</td><td>47.1 ± 0.3</td><td>20.2 ± 0.9</td><td>44.7 ± 0.6</td></tr><tr><td>TopoFilter[42]</td><td>90.4± 0.2</td><td>86.8 ±0.3</td><td>46.8 ± 1.0</td><td>87.5± 0.4</td><td>66.9 ± 0.4</td><td>53.4 ±1.8</td><td>18.3 ± 1.7</td><td>56.6±0.5</td></tr><tr><td>CRUST[32]</td><td>89.4 ±0.2</td><td>87.0 ± 0.1</td><td>64.8 ± 1.5</td><td>82.4± 0.0</td><td>69.3 ±0.2</td><td>62.3 ± 0.2</td><td>21.7 ± 0.7</td><td>56.1 ± 0.5</td></tr><tr><td>FINE</td><td>91.0 ± 0.1</td><td>87.3 ± 0.2</td><td>69.4 ± 1.1</td><td>89.5 ± 0.1</td><td>70.3 ± 0.2</td><td>64.2 ± 0.5</td><td>25.6 ± 1.2</td><td>61.7 ± 1.0</td></tr><tr><td>F-Coteaching</td><td>92.0 ± 0.1</td><td>87.5 ± 0.1</td><td>74.2 ± 0.8</td><td>90.5 ± 0.2</td><td>71.1 ± 0.2</td><td>64.7 ± 0.3</td><td>31.6 ± 1.0</td><td>64.8 ± 0.7</td></tr></table>",
+        "bbox": [
+            173,
+            200,
+            838,
+            359
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the performances of different sample selection approaches on various noise distribution and datasets. We observe that our FINE method consistently outperforms the competitive methods over the various noise rates. Our FINE methods can filter the clean instances without losing essential information, leading to training the robust network. ",
+        "bbox": [
+            173,
+            376,
+            825,
+            431
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To go further, we improve the performance of Co-teaching [14] by substituting its sample selection state with our FINE algorithm. To combine FINE and the Co-teaching family, unlike the original methods that utilize the small loss instances to train with clean labels, we train one model with extracted samples by conducting FINE on another model. The results of the experiments are shown in the eighth and ninth rows of Table 1. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            507
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 SSL-Based Approaches ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            522,
+            388,
+            536
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "SSL approaches [25, 10, 41] divide the training data into clean instances as labeled instances and noisy instances as unlabeled instances and use both the labeled and unlabeled samples to train the networks in SSL. Recently, methods belonging to this category have shown the best performance among the various LNL methods, and these methods can train robust networks for even extremely high noise rates. We compare the performances of the existing semi-supervised approaches and that in which the sample selection state of DivideMix [25] is substituted with our FINE algorithm (i.e., F-DivideMix). The results of the experiments are shown in Table 2. We achieve consistently higher performance than DivideMix by utilizing FINE instead of its loss-based filtering method and show comparable performance to the state-of-the-art SSL methods such as DST [41] and LongReMix [10]. Interestingly, as Figure 6 shows, clean and noisy data are well classified in F-DivideMix under extreme noise cases. ",
+        "bbox": [
+            176,
+            545,
+            581,
+            780
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/11adb8995afaef343b648fa666cd40c5a34b839ed05e8c31a9be3bce97d46429.jpg",
+        "image_caption": [
+            "Figure 6: Comparisons of F-scores on CIFAR-10 under symmetric $90 \\%$ noise. Blue line indicates the error bar of two networks’ F-score used in Dividemix [25], and Orange line indicates those replaced by our FINE detector. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            594,
+            541,
+            823,
+            672
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.3 Collaboration with Noise-Robust Loss Functions ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            790,
+            563,
+            804
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The goal of the noise-robust loss function is to achieve a small risk for unseen clean data even when noisy labels exist in the training data. There have been few collaboration studies of the noise-robust loss function methodology and dynamic sample selection. Most studies have selected clean and noisy data based on cross-entropy loss. ",
+        "bbox": [
+            174,
+            808,
+            825,
+            863
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here, we state the collaboration effects of FINE with various noise-robust loss functions: generalized cross entropy (GCE) [51], symmetric cross entropy (SCE) [40], and early-learning regularization (ELR) [29]. Figure 7 shows that FINE facilitates generalization in the application of noise-robust loss functions on severe noise rate settings. The detailed results are reported in the Appendix. Unlike other methods, it is still theoretically supported because FINE extracts clean data with a robust classifier using representation. ",
+        "bbox": [
+            176,
+            869,
+            823,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f283614e95a0f1cd3f6fa0c1544c9102377f47d401ff9597b6134bd5b8d79b1a.jpg",
+        "image_caption": [
+            "Figure 7: Test accuracies $( \\% )$ on CIFAR-10 and CIFAR-100 under different noisy types and fractions for noise-robust loss approaches. Note that the blue and orange bars are results for without and with FINE, respectively. The average accuracies and standard deviations over three trials are reported. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            145,
+            812,
+            431
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            496,
+            825,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 Experiments on Real-World Dataset ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            554,
+            464,
+            570
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As Table 3 shows, FINE and F-DivideMix work fairly well on the Clothing 1M dataset compared to other approaches when we reproduce the experimental results under the same settings. ",
+        "bbox": [
+            171,
+            580,
+            823,
+            608
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/1513438badf07225201a025f5f60420d4525cd1a5b634121c99721f89d346990.jpg",
+        "table_caption": [
+            "Table 2: Comparison of test accuracies $( \\% )$ for FINE collaborating with DivideMix and existing semi-supervised approaches on CIFAR-10 and CIFAR-100 under different noisy types and fractions. The results for all comparison methods are taken from their original works. ",
+            "Table 3: Test accuracy on Clothing1M dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Standard</td><td>GCE[51]</td><td>SCE[40]</td><td>ELR [29]</td><td>DivideMix [25]</td><td>CORES² [8]</td><td>FINE</td><td>F-DivideMix</td></tr><tr><td>Accuracy</td><td>68.94</td><td>69.75</td><td>71.02</td><td>72.87</td><td>74.30</td><td>73.24</td><td>72.91</td><td>74.37</td></tr></table>",
+        "bbox": [
+            192,
+            632,
+            803,
+            670
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            693,
+            299,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces FINE for detecting label noise by designing a robust noise detector. Our main idea is utilizing the principal components of latent representations made by eigen decomposition. Most existing detection methods are dependent on the loss values, while such losses may be biased by corrupted classifier [24, 31]. Our methodology alleviates this issue by extracting key information from representations without using explicit knowledge of the noise rates. We show that the FINE detector has an excellent ability to detect noisy labels in theoretical and experimental results. We propose three applications of the FINE detector: sample-selection approach, SSL approach, and collaboration with noise-robust loss functions. FINE yields strong results on standard benchmarks and a real-world dataset for various LNL approaches. ",
+        "bbox": [
+            173,
+            723,
+            825,
+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We believe that our work opens the door to detecting samples having noisy labels with explainable results. It is a non-trivial task and of social significance, and thus, our work will have a substantial social impact on DL practitioners because it avoids the a labor-intensive job of checking data label quality. As future work, we hope that our work will trigger interest in the design of new labelnoise detectors and bring a fresh perspective for other data-resampling approaches (e.g., anomaly detection and novelty detection). The development of robustness against label noise even leads to an improvement in the performance of network trained with data collected through web crawling. We believe that our contribution will lower the barriers to entry for developing robust models for DL practitioners and greatly impact the internet industry. On the other hand, we are concerned that it can be exploited to train robust models using data collected illegally and indiscriminately on the dark web (e.g., web crawling), and thus it may raise privacy concerns (e.g., copyright). ",
+        "bbox": [
+            174,
+            856,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            189
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            207,
+            553,
+            224
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) [No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)] and [No. 2021-0-00907, Development of Adaptive and Lightweight Edge-Collaborative Analysis Technology for Enabling Proactively Immediate Response and Rapid Learning]. We thank Seongyoon Kim for discussing about the concept of perturbation. ",
+        "bbox": [
+            174,
+            238,
+            825,
+            320
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            340,
+            266,
+            357
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] Shin Ando and Chun Yuan Huang. Deep over-sampling framework for classifying imbalanced data. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 770–785. Springer, 2017.   \n[2] Devansh Arpit, Stanisław Jastrz˛ebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al. A closer look at memorization in deep networks. In International Conference on Machine Learning, pages 233–242. PMLR, 2017.   \n[3] Dara Bahri, Heinrich Jiang, and Maya Gupta. Deep k-nn for noisy labels. In International Conference on Machine Learning, pages 540–550. PMLR, 2020.   \n[4] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In Advances in Neural Information Processing Systems, pages 5049–5059, 2019.   \n[5] Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. Neural Networks, 106:249–259, 2018.   \n[6] Kaidi Cao, Yining Chen, Junwei Lu, Nikos Arechiga, Adrien Gaidon, and Tengyu Ma. Heteroskedastic and imbalanced deep learning with adaptive regularization. arXiv preprint arXiv:2006.15766, 2020.   \n[7] Nitesh V Chawla, Kevin W Bowyer, Lawrence O Hall, and W Philip Kegelmeyer. Smote: synthetic minority over-sampling technique. Journal of artificial intelligence research, 16:321– 357, 2002.   \n[8] Hao Cheng, Zhaowei Zhu, Xingyu Li, Yifei Gong, Xing Sun, and Yang Liu. Learning with instance-dependent label noise: A sample sieve approach. arXiv preprint arXiv:2010.02347, 2020.   \n[9] Charles Corbiere, Hedi Ben-Younes, Alexandre Ramé, and Charles Ollion. Leveraging weakly annotated data for fashion image retrieval and label prediction. In Proceedings of the IEEE International Conference on Computer Vision Workshops, pages 2268–2274, 2017.   \n[10] Filipe R Cordeiro, Ragav Sachdeva, Vasileios Belagiannis, Ian Reid, and Gustavo Carneiro. Longremix: Robust learning with high confidence samples in a noisy label environment. arXiv preprint arXiv:2103.04173, 2021.   \n[11] Di Feng, Christian Haase-Schutz, Lars Rosenbaum, Heinz Hertlein, Claudius Glaser, Fabian Timm, Werner Wiesbeck, and Klaus Dietmayer. Deep multi-modal object detection and semantic segmentation for autonomous driving: Datasets, methods, and challenges. IEEE Transactions on Intelligent Transportation Systems, 22(3):1341–1360, Mar 2021.   \n[12] Ronald A Fisher. The use of multiple measurements in taxonomic problems. Annals of eugenics, 7(2):179–188, 1936.   \n[13] Aritra Ghosh, Himanshu Kumar, and PS Sastry. Robust loss functions under label noise for deep neural networks. In AAAI, 2017.   \n[14] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In Advances in neural information processing systems, pages 8527–8537, 2018.   \n[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[16] Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and out-ofdistribution examples in neural networks. arXiv preprint arXiv:1610.02136, 2016.   \n[17] Dan Hendrycks, Mantas Mazeika, Duncan Wilson, and Kevin Gimpel. Using trusted data to train deep networks on labels corrupted by severe noise. arXiv preprint arXiv:1802.05300, 2018.   \n[18] Jinchi Huang, Lie Qu, Rongfei Jia, and Binqiang Zhao. O2u-net: A simple noisy label detection approach for deep neural networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 3326–3334, 2019.   \n[19] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels, 2018.   \n[20] Taehyeon Kim, Jaehoon Oh, NakYil Kim, Sangwook Cho, and Se-Young Yun. Comparing kullback-leibler divergence and mean squared error loss in knowledge distillation, 2021.   \n[21] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems, 25:1097– 1105, 2012.   \n[22] Ronan Le Bras, Swabha Swayamdipta, Chandra Bhagavatula, Rowan Zellers, Matthew Peters, Ashish Sabharwal, and Yejin Choi. Adversarial filters of dataset biases. In International Conference on Machine Learning, pages 1078–1088. PMLR, 2020.   \n[23] Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin. A simple unified framework for detecting out-of-distribution samples and adversarial attacks. In Advances in Neural Information Processing Systems, pages 7167–7177, 2018.   \n[24] Kimin Lee, Sukmin Yun, Kibok Lee, Honglak Lee, Bo Li, and Jinwoo Shin. Robust inference via generative classifiers for handling noisy labels. In International Conference on Machine Learning, pages 3763–3772. PMLR, 2019.   \n[25] Junnan Li, Richard Socher, and Steven C.H. Hoi. Dividemix: Learning with noisy labels as semi-supervised learning. In International Conference on Learning Representations, 2020.   \n[26] Mingchen Li, Mahdi Soltanolkotabi, and Samet Oymak. Gradient descent with early stopping is provably robust to label noise for overparameterized neural networks. In International Conference on Artificial Intelligence and Statistics, pages 4313–4324. PMLR, 2020.   \n[27] Yi Li and Nuno Vasconcelos. Repair: Removing representation bias by dataset resampling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9572–9581, 2019.   \n[28] Shiyu Liang, Yixuan Li, and Rayadurgam Srikant. Enhancing the reliability of out-ofdistribution image detection in neural networks. arXiv preprint arXiv:1706.02690, 2017.   \n[29] Sheng Liu, Jonathan Niles-Weed, Narges Razavian, and Carlos Fernandez-Granda. Early-learning regularization prevents memorization of noisy labels. arXiv preprint arXiv:2007.00151, 2020.   \n[30] Michal Lukasik, Srinadh Bhojanapalli, Aditya Menon, and Sanjiv Kumar. Does label smoothing mitigate label noise? In International Conference on Machine Learning, pages 6448–6458. PMLR, 2020.   \n[31] Hartmut Maennel, Ibrahim Alabdulmohsin, Ilya Tolstikhin, Robert JN Baldock, Olivier Bousquet, Sylvain Gelly, and Daniel Keysers. What do neural networks learn when trained with random labels? arXiv preprint arXiv:2006.10455, 2020.   \n[32] Baharan Mirzasoleiman, Kaidi Cao, and Jure Leskovec. Coresets for robust training of neural networks against noisy labels. arXiv preprint arXiv:2011.07451, 2020.   \n[33] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1944–1952, 2017.   \n[34] Geoff Pleiss, Tianyi Zhang, Ethan R Elenberg, and Kilian Q Weinberger. Identifying mislabeled data using the area under the margin ranking. arXiv preprint arXiv:2001.10528, 2020.   \n[35] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014.   \n[36] Mengye Ren, Wenyuan Zeng, Bin Yang, and Raquel Urtasun. Learning to reweight examples for robust deep learning. In International Conference on Machine Learning, pages 4334–4343. PMLR, 2018.   \n[37] Hwanjun Song, Minseok Kim, Dongmin Park, and Jae-Gil Lee. Prestopping: How does early stopping help generalization against label noise?, 2020.   \n[38] Brendan Van Rooyen, Aditya Krishna Menon, and Robert C Williamson. Learning with symmetric label noise: The importance of being unhinged. arXiv preprint arXiv:1505.07634, 2015.   \n[39] Andreas Veit, Neil Alldrin, Gal Chechik, Ivan Krasin, Abhinav Gupta, and Serge Belongie. Learning from noisy large-scale datasets with minimal supervision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 839–847, 2017.   \n[40] Yisen Wang, Xingjun Ma, Zaiyi Chen, Yuan Luo, Jinfeng Yi, and James Bailey. Symmetric cross entropy for robust learning with noisy labels. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 322–330, 2019.   \n[41] Yi Wei, Xue Mei, Xin Liu, and Pengxiang Xu. Dst: Data selection and joint training for learning with noisy labels. arXiv preprint arXiv:2103.00813, 2021.   \n[42] Pengxiang Wu, Songzhu Zheng, Mayank Goswami, Dimitris Metaxas, and Chao Chen. A topological filter for learning with label noise. arXiv preprint arXiv:2012.04835, 2020.   \n[43] Xiaobo Xia, Tongliang Liu, Bo Han, Chen Gong, Nannan Wang, Zongyuan Ge, and Yi Chang. Robust early-learning: Hindering the memorization of noisy labels. In International Conference on Learning Representations, 2021.   \n[44] Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang. Learning from massive noisy labeled data for image classification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2691–2699, 2015.   \n[45] Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L_dmi: A novel information-theoretic loss function for training deep nets robust to label noise. In NeurIPS, pages 6222–6233, 2019.   \n[46] Seunghan Yang, Hyoungseob Park, Junyoung Byun, and Changick Kim. Robust federated learning with noisy labels, 2020.   \n[47] Xingrui Yu, Bo Han, Jiangchao Yao, Gang Niu, Ivor W Tsang, and Masashi Sugiyama. How does disagreement help generalization against label corruption? arXiv preprint arXiv:1901.04215, 2019.   \n[48] Xiyu Yu, Tongliang Liu, Mingming Gong, and Dacheng Tao. Learning with biased complementary labels. In Proceedings of the European conference on computer vision (ECCV), pages 68–83, 2018.   \n[49] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.   \n[50] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.   \n[51] Zhilu Zhang and Mert R Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. arXiv preprint arXiv:1805.07836, 2018.   \n[52] Zizhao Zhang, Han Zhang, Sercan O Arik, Honglak Lee, and Tomas Pfister. Distilling effective supervision from severe label noise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9294–9303, 2020.   \n[53] Zizhao Zhang, Han Zhang, Sercan O. Arik, Honglak Lee, and Tomas Pfister. Distilling effective supervision from severe label noise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. ",
+        "bbox": [
+            179,
+            357,
+            826,
+            914
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            54,
+            826,
+            915
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            40,
+            826,
+            919
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            89,
+            828,
+            349
+        ],
+        "page_idx": 12
+    }
+]

parse/train/e8WWUBeafM/e8WWUBeafM.md

@@ -0,0 +1,377 @@
+# Bellman-consistent Pessimism for Offline Reinforcement Learning
+
+Tengyang Xie UIUC tx10@illinois.edu
+
+Ching-An Cheng Microsoft Research chinganc@microsoft.com
+
+Nan Jiang UIUC nanjiang@illinois.edu
+
+Paul Mineiro Microsoft Research pmineiro@microsoft.com
+
+Alekh Agarwal Google Research alekhagarwal@google.com
+
+# Abstract
+
+The use of pessimism, when reasoning about datasets lacking exhaustive exploration, has recently gained prominence in offline reinforcement learning. Despite the robustness it adds to the algorithm, overly pessimistic reasoning can be equally damaging in precluding the discovery of good policies, which is an issue for the popular bonus-based pessimism. In this paper, we introduce the notion of Bellmanconsistent pessimism for general function approximation: instead of calculating a point-wise lower bound for the value function, we implement pessimism at the initial state over the set of functions consistent with the Bellman equations. Our theoretical guarantees only require Bellman closedness as standard in the exploratory setting, in which case bonus-based pessimism fails to provide guarantees. Even in the special case of linear function approximation where stronger expressivity assumptions hold, our result improves upon a recent bonus-based approach by $\mathcal O ( d )$ in its sample complexity when the action space is finite and small. Remarkably, our algorithms automatically adapt to the best bias-variance tradeoff in the hindsight, whereas most prior approaches require tuning extra hyperparameters a priori.
+
+# 1 Introduction
+
+Using past experiences to learn improved behavior for future interactions is a critical capability for a Reinforcement Learning (RL) agent. However, robustly extrapolating knowledge from a historical dataset for sequential decision making is highly challenging, particularly in settings where function approximation is employed to generalize across related observations. In this paper, we provide a systematic treatment of such scenarios with general function approximation, and devise algorithms that can provably leverage an arbitrary historical dataset to discover the policy that obtains the largest guaranteed rewards, amongst all possible scenarios consistent with the dataset.
+
+The problem of learning a good policy from historical datasets, typically called batch or offline RL, has a long history [see e.g., Precup et al., 2000; Antos et al., 2008; Levine et al., 2020, and references therein]. Many prior works [e.g., Precup et al., 2000; Antos et al., 2008; Chen and Jiang, 2019] make the so-called coverage assumptions on the dataset, requiring the dataset to contain any possible state, action pair or trajectory with a lower bounded probability. These assumptions are evidently prohibitive in practice, particularly for problems with large state and/or action spaces. Furthermore, the methods developed under these assumptions routinely display unstable behaviors such as lack of convergence or error amplification, when coverage assumptions are violated [Wang et al., 2020, 2021].
+
+Driven by these instabilities, a growing body of recent literature has pursued a so-called best effort style of guarantee instead. The key idea is to replace the stringent assumptions on the dataset with a dataset-dependent performance bound, which gracefully degrades from guaranteeing a near-optimal policy under standard coverage assumptions to offering no improvement over the data collection policy in the most degenerate case. Algorithmically, these works all leverage the principle of pessimistic extrapolation from offline data and aim to maximize the rewards the trained agent would obtain in the worst possible MDP that is consistent with the observed dataset. These methods have been shown to be typically more robust to the violation of coverage assumptions in practice, and their theoretical guarantees often provide non-trivial conclusions in settings where the previous results did not apply.
+
+Even though many such best-effort methods have now been developed, very few works provide a comprehensive theory for using generic function approximation, unlike the setting where the dataset satisfies the coverage assumptions [Antos et al., 2008; Munos, 2003; Szepesvári and Munos, 2005; Munos and Szepesvári, 2008; Farahmand et al., 2010; Chen and Jiang, 2019; Xie and Jiang, 2020]. For example, [Kidambi et al., 2020] provides a partial theory under the assumption of an uncertainty quantification oracle, which however is highly nontrivial to obtain for general function approximation. [Fujimoto et al., 2019; Kumar et al., 2020] develop sound theoretical arguments in the tabular setting, which were only heuristically extended to the function approximation setting. The works that explicitly consider function approximation in their design either use an ad-hoc truncation of Bellman backups [Liu et al., 2020] or strongly rely on particular parameterizations such as linear function approximation [Jin et al., 2021]. In particular, [Liu et al., 2020] additionally requires the ability to approximate stationary distribution of the behavior policy, which is a challenging density estimation problem for complex state spaces and cannot be provably performed in the standard linear MDP setting (see Section 3.1).
+
+Our paper takes an important step in this direction. We provide a systematic way to encode pessimism compatible with an arbitrary function approximation class and MDP and give strong theoretical guarantees without requiring any coverage assumptions on the dataset. Our first contribution is an information theoretic algorithm that returns a policy with a small regret to any comparator policy, for which coverage assumptions (approximately) hold with respect to the data collection policy. This regret bound is identical to what can be typically obtained when the coverage assumptions hold for all policies [Antos et al., 2008; Chen and Jiang, 2019]. But our algorithm requires neither the coverage assumptions, nor additional assumptions such as reliable density estimation for the data generating distribution used by existing best-effort approaches [Liu et al., 2020]. We furthermore instantiate these results in the special case of linear parameterization; under the linear MDP assumption, our sample complexity bound leads to a factor of $\mathcal O ( d )$ improvement for a $d$ -dimensional linear MDP, compared with the best known result translated to our discounted setting [Jin et al., 2021], when the action set is small in size. In addition to the information theoretic algorithm, we also develop a computationally practical version of our algorithm using a Lagrangian relaxation combined with recent advances in soft policy iteration [Even-Dar et al., 2009; Geist et al., 2019; Agarwal et al., 2019]. We show that this algorithm can be executed efficiently by querying a (regularized) loss minimization oracle over the value function class, although it has slightly worse theoretical guarantees than the information theoretic version. Both our algorithms display an adaptive property in selecting the best possible form of a bias-variance decomposition, where most prior approaches had to commit to a particular point through their choice of hyperparameters (see the discussion following Theorem 3.1).
+
+# 2 Preliminaries
+
+Markov Decision Processes We consider dynamical systems modeled as Markov Decision Processes (MDPs). An MDP is specified by $( S , \mathcal { A } , P , R , \gamma , s _ { 0 } )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $P : \mathcal { S } \times \mathcal { A }  \Delta ( \mathcal { S } )$ is the transition function with $\Delta ( \cdot )$ being the probability simplex, $R : \mathcal { S } \times \mathcal { A }  [ 0 , R _ { \mathrm { m a x } } ]$ is the reward function, $\gamma \in [ 0 , 1 )$ is the discount factor, and $s _ { 0 }$ is a deterministic initial state, which is without loss of generality. We assume the state and the action spaces are finite but can be arbitrarily large. A (stochastic) policy $\pi : { \mathcal { S } }  \Delta ( { \mathcal { A } } )$ specifies a decision-making strategy, and induces a random trajectory $s _ { 0 } , a _ { 0 } , r _ { 0 } , s _ { 1 } , a _ { 1 } , r _ { 1 } , . . . _$ , where $a _ { t } \sim \pi ( \cdot | s _ { t } ) , r _ { t } = R ( \bar { s _ { t } } , a _ { t } ) , \bar { s _ { t + 1 } } \sim P ( \cdot | s _ { t } , a _ { t } ) ,$ $\forall t \geq 0$ . We denote the expected discounted return of a policy $\pi$ as $\begin{array} { r } { J ( \pi ) : = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } | \pi ] } \end{array}$ , and the learning goal is to find the maximizer of this value: $\pi ^ { \star } : = \operatorname { a r g m a x } _ { \pi } J ( \pi )$ . A related concept is the policy-specific $Q$ -function, $Q ^ { \pi } : S \times \mathcal { A }  \mathbb { R }$ $Q ^ { \pi } ( s , a )$ is the discounted return when the trajectory starts with $( s , a )$ and all remaining actions are taken according to $\pi$ . $Q ^ { \pi }$ is the unique fixed point of the (policy-specific) Bellman operator $\mathcal { T } ^ { \pi } : \mathbb { R } ^ { S \times A }  \mathbb { R } ^ { S \times A }$ , defined as:
+
+$$
+\forall f , \quad ( T ^ { \pi } f ) ( s , a ) = R ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot \vert s , a ) } [ f ( s ^ { \prime } , \pi ) ] ,
+$$
+
+where $f ( s ^ { \prime } , \pi )$ is a shorthand for $\mathbb { E } _ { a ^ { \prime } \sim \pi ( \cdot | s ^ { \prime } ) } [ f ( s ^ { \prime } , a ^ { \prime } ) ]$ .
+
+Another important concept is the notion of discounted state-action occupancy, $d _ { \pi } \in \Delta ( S \times \mathcal { A } )$ , defined as $\begin{array} { r } { \hat { d } _ { \pi } ( s , a ) : = ( \hat { 1 ^ { ' } } - \gamma ) \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { 1 } [ s _ { t } = s , a _ { t } = a ] | \pi ] } \end{array}$ , which characterizes the states and actions visited by a policy $\pi$ .
+
+Offline RL In the offline setting, the learner only has access to a pre-collected dataset and cannot directly interact with the environment. We assume the standard i.i.d. data generation protocol in our theoretical derivations, that the offline dataset $\mathcal { D }$ consists of $n$ i.i.d. $( s , a , r , s ^ { \prime } )$ tuples generated as $( s , a ) \sim \mu , r = R ( s , a ) , s ^ { \prime } \sim P ( \cdot | s , a )$ for some data distribution $\mu$ . We will also use $\mathbb { E } _ { \mu } [ \cdot ]$ for taking expectation with respect to $\mu$ . We will frequently use the data-weighted 2-norm (squared) $\| f \| _ { 2 , \mu } : =$ $\mathbb { E } _ { \mu } [ f ^ { 2 } ]$ , and the definition extends when we replace $\mu$ with any other state-action distribution $\nu$ . The empirical approximation of $\| f \| _ { 2 , \mu } ^ { 2 }$ is $\begin{array} { r } { \| f \| _ { 2 , \mathcal { D } } ^ { 2 } : = \frac { 1 } { n } \sum _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } } f ( s , a ) ^ { 2 } . } \end{array}$ .
+
+Function Approximation Function approximation is crucial to generalizing over large and complex state and action spaces. In this work, we search for a good policy in a policy class $\Pi \subset ( \bar { S } $ $\Delta ( \mathcal { A } ) )$ with the help of a value-function class ${ \mathcal { F } } \subset ( S \times { \mathcal { A } }  [ 0 , V _ { \operatorname* { m a x } } ] )$ to model $Q ^ { \pi }$ , where $V _ { \mathrm { m a x } } = R _ { \mathrm { m a x } } / ( 1 - \gamma )$ . Such a combination is commonly found in approximate policy iteration and actor-critic algorithms [e.g., Bertsekas and Tsitsiklis, 1996; Konda and Tsitsiklis, 2000]. For most part of the paper we do not make any structural assumptions on $\Pi$ and $\mathcal { F }$ , making our approach and guarantees applicable to generic function approximators. For simplicity we will assume that these function classes are finite but exponentially large, and use log-cardinality to measure their statistical complexities in the generic results (Section 3 and Section 4). These guarantees easily extend to continuous function classes where log-cardinalities are replaced by the appropriate notions of covering numbers, which we demonstrate when we instantiate our results in the linear function approximation setting and work with continuous linear classes (Section 3.1).
+
+We now recall two standard expressivity assumptions on $\mathcal { F }$ [e.g., Antos et al., 2008]. To our knowledge, no existing works on offline RL with insufficient data coverage have provided guarantees under these standard assumptions for general function approximation, and they often require stronger or tweaked assumptions (see Section 1).
+
+Assumption 1 (Realizability). For any $\pi \in \Pi$ , we have
+
+$$
+\operatorname* { i n f } _ { f \in \mathcal { F } } \operatorname* { s u p } _ { a d m i s s i b l e \nu } \| f - \mathcal { T } ^ { \pi } f \| _ { 2 , \nu } ^ { 2 } \leq \varepsilon _ { \mathcal { F } } ,
+$$
+
+where an admissible distribution $\nu$ means that $\nu \in \{ d _ { \pi ^ { \prime } } : \pi ^ { \prime } \in \Pi \}$ .
+
+Assumption 1 requires that for every $\pi \in \Pi$ , there exists $f \in { \mathcal { F } }$ that well-approximates $Q ^ { \pi }$ . This assumption is often called realizability.1 Technically this is asserted by requiring $f$ to have small Bellman error w.r.t. $\mathcal { T } ^ { \pi }$ under all possible admissible distributions. As a sufficient condition, we have $\varepsilon _ { \mathcal { F } } = 0$ if $Q ^ { \pi } \in { \mathcal { F } } , \forall \pi \in \Pi$ .
+
+Assumption 2 (Completeness). For any $\pi \in \Pi$ , we have
+
+$$
+\underset { f \in \mathcal { F } } { \operatorname* { s u p } } \operatorname* { i n f } _ { f ^ { \prime } \in \mathcal { F } } \left. f ^ { \prime } - \mathcal { T } ^ { \pi } f \right. _ { 2 , \mu } ^ { 2 } \leq \varepsilon _ { \mathcal { F } , \mathcal { F } } .
+$$
+
+Assumption 2 asserts that $\mathcal { F }$ is approximately closed under $\tau ^ { \pi }$ .2 Such an assumption is widely used in RL theory and can be only avoided in some rare cases [Xie and Jiang, 2021], and the hardness of learning with realizability alone has been established in various settings (e.g., [Weisz et al., 2021; Zanette, 2021]). We also emphasize that we only measure the violation of completeness under $\mu$ and do not need to reason about all admissible distributions.
+
+Distribution shift A unique challenge in RL is that the learned policy may induce a state (and action) distribution that is different from the data distribution $\mu$ , and the issue is particularly salient when we do not impose any coverage assumption on $\mu$ . Therefore, it is important to carefully characterize the distribution shift, which we measure using the following definition, which generalizes prior definitions specific to linear function approximation [Agarwal et al., 2019; Duan et al., 2020]:
+
+Definition 1. We define $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi )$ as follows to measure the distribution shift from an arbitrary distribution $\nu$ to the data distribution $\mu$ , w.r.t. $\mathcal { F }$ and $\pi$ ,
+
+$$
+\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi ) : = \operatorname* { m a x } _ { f \in \mathcal { F } } \frac { \| f - \mathcal { T } ^ { \pi } f \| _ { 2 , \nu } ^ { 2 } } { \| f - \mathcal { T } ^ { \pi } f \| _ { 2 , \mu } ^ { 2 } } .
+$$
+
+Intuitively, $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi )$ measures how well Bellman errors under $\pi$ transfer between the distributions $\nu$ and $\mu$ . For instance, a small value of $\mathcal { C } ( d _ { \pi } ; \mu , \mathcal { F } , \pi )$ enables accurate policy evaluation for $\pi$ using data collected under $\mu$ . More generally, we observe that
+
+$$
+\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi ) \leq \| \nu / \mu \| _ { \infty } : = \operatorname* { s u p } _ { s , a } \frac { \nu ( s , a ) } { \mu ( s , a ) } , \quad \mathrm { f o r ~ a n y } \pi , \mathcal { F } .
+$$
+
+and the RHS is a classical notion of bounded distribution ratio for error transfer (e.g., [Munos and Szepesvári, 2008; Chen and Jiang, 2019; Xie and Jiang, 2020]). Moreover, our measure can be tighter than $\| \nu / \mu \| _ { \infty }$ : Even two distributions $\nu$ and $\mu$ that are sufficiently disparate might admit a reasonable transfer, so long as this difference is not detected by $\pi$ and $\mathcal { F }$ . To this end, our definition better captures the crucial role of function approximation in generalizing across different states. As an example, in the special case of linear MDPs, full coverage under our definition (i.e., boundedness of $\mathcal { C }$ for all admissible $\nu$ ) can be implied from the standard coverage assumption for linear MDPs that considers the spectrum of the feature covariance matrix under $\mu$ ; see Section 3.1 for more details.
+
+# 3 Information-Theoretic Results with Bellman-consistent Pessimism
+
+In this section, we provide our first theoretical result which is information-theoretic, in that it uses a computationally inefficient algorithm. The approach uses the offline dataset to first compute a lower bound on the value of each policy $\pi \in \Pi$ , and then returns the policy with the highest pessimistic value estimate. While this high-level template is at the heart of many recent approaches [e.g., Fujimoto et al., 2019; Kumar et al., 2019; Liu et al., 2020; Kidambi et al., 2020; Yu et al., 2020; Kumar et al., 2020], our main novelty is in the design and analysis of Bellman-consistent pessimism for general function approximation.
+
+For a policy $\pi$ , we first form a version space of all the functions $f \in { \mathcal { F } }$ which have a small Bellman error under the evaluation operator $\tau ^ { \pi }$ . We then return the predicted value of $\pi$ in the initial state $s _ { 0 }$ by the functions in this version space. The use of pessimism at the initial state, while maintaining Bellman consistency (by virtue of having a small Bellman error) limits over pessimism, which is harder to preclude in the pointwise pessimistic penalties used in some other works [Jin et al., 2021]. More formally, given a dataset $\mathcal { D }$ , let us define
+
+$$
+\mathcal { L } ( f ^ { \prime } , f , \pi ; \mathcal { D } ) : = \frac { 1 } { n } \sum _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } } \left( f ^ { \prime } ( s , a ) - r - \gamma f ( s ^ { \prime } , \pi ) \right) ^ { 2 } ,
+$$
+
+and an empirical estimate of the Bellman error $\mathcal { E } ( f , \pi ; \mathcal { D } )$ is
+
+$$
+\mathcal { E } ( f , \pi ; \mathcal { D } ) : = \mathcal { L } ( f , f , \pi ; \mathcal { D } ) - \operatorname* { m i n } _ { f ^ { \prime } \in \mathcal { F } } \mathcal { L } ( f ^ { \prime } , f , \pi ; \mathcal { D } ) .
+$$
+
+Our algorithm. With this notation, our information-theoretic approach finds a policy by optimizing:
+
+$$
+\widehat { \pi } = \underset { \pi \in \Pi } { \operatorname { a r g m a x } } \ \underset { f \in \mathcal { F } _ { \pi , \varepsilon } } { \operatorname* { m i n } } f ( s _ { 0 } , \pi ) , \quad \mathrm { w h e r e } \ \mathcal { F } _ { \pi , \varepsilon } = \big \{ f \in \mathcal { F } : \mathcal { E } ( f , \pi ; \mathcal { D } ) \leq \varepsilon \big \} ,
+$$
+
+In the formulation above, $\mathcal { F } _ { \pi , \varepsilon }$ is the version space of policy $\pi$ . To better understand the intuition behind the estimator in Equation 3.2, let us define
+
+$$
+{ \bf \Pi } _ { \pi , \mathrm { m i n } } ^ { \mathrm { r } } : = \operatorname * { a r g m i n } _ { f \in \mathcal { F } _ { \pi , \epsilon } } f ( s _ { 0 } , \pi ) , f _ { \pi , \mathrm { m a x } } : = \operatorname * { a r g m a x } _ { f \in \mathcal { F } _ { \pi , \epsilon } } f ( s _ { 0 } , \pi ) , \mathrm { a n d } \Delta f _ { \pi } ( s , a ) : = f _ { \pi , \mathrm { m a x } } ( s , a ) - f _ { \pi , \mathrm { m i n } } ( s , a ) .
+$$
+
+Intuitively, if the parameter $\varepsilon$ is defined to ensure that $Q ^ { \pi }$ (or its best approximation in $\mathcal { F }$ ) is in $\mathcal { F } _ { \pi , \varepsilon }$ , we easily see that $\Delta f _ { \pi } ( s _ { 0 } , \pi )$ is an upper bound on the error in our estimate of $J ( \pi )$ for any $\pi \in \Pi$ . In fact, an easy argument in our analysis shows that if $Q ^ { \pi } \in { \mathcal { F } } _ { \pi , \varepsilon }$ for all $\pi \in \Pi$ , then $\Delta f ( s _ { 0 } , \pi )$ is an upper bound on the regret $J ( \pi ) - J ( \widehat { \pi } )$ of our estimator relative to any $\pi$ we wish to compete with.
+
+Theoretical analysis. To leverage this observation, we first define the a critical threshold $\varepsilon _ { r }$ which ensures that (the best approximation of) $Q ^ { \pi }$ is indeed contained in our version spaces for all $\pi$ :
+
+$$
+\varepsilon _ { r } : = \frac { 1 3 9 V _ { \mathrm { m a x } } ^ { 2 } \log \frac { | \mathcal { F } | | \Pi | } { \delta } } { n } + 3 9 \varepsilon _ { \mathcal { F } } .
+$$
+
+With this definition, we now give a more refined bound on the regret of our algorithm (3.2) by further splitting the error estimate $\Delta f _ { \pi } ( s _ { 0 } , \pi )$ which is random owing to its dependence on the version space, and analyze it through a novel decomposition into on-support and off-support components. While we bound the on-support error using standard techniques, the off-support error is akin to a bias term which captures the interplay between the data collection distribution and function approximation in the quality of the final solution. Also note that our choice of $\varepsilon _ { r }$ requires the knowledge of $\varepsilon { \mathcal { F } }$ , which is a common characteristic of version-space-based algorithms [e.g., Jiang et al., 2017]. The challenge of unknown $\varepsilon _ { F }$ can be possibly addressed using model-selection techniques in practice and we leave further investigation to future work.
+
+Theorem 3.1. Let $\varepsilon = \varepsilon _ { r }$ where is $\varepsilon _ { r }$ defined in Eq.(3.3) and $\widehat { \pi }$ be obtained by Eq.(3.2). Then, for any policy $\pi \in \Pi$ and any constant $C _ { 2 } \geq 1$ b, with probability at least $1 - \delta$ ,
+
+$$
+\begin{array} { r l } & { I ( \pi ) - J ( \widehat \pi ) \leq \underbrace { \mathcal { O } \left( \displaystyle \frac { V _ { \operatorname* { m a x } } \sqrt { C _ { 2 } } } { 1 - \gamma } \sqrt { \frac { \log \frac { | \mathcal { F } | | \Pi | } { \delta } } { n } } + \frac { \sqrt { C _ { 2 } ( \varepsilon _ { \mathcal { F } , \mathcal { F } } + \varepsilon _ { \mathcal { F } } ) } } { 1 - \gamma } \right) } _ { \mathrm { e r r } _ { \operatorname* { m i n } } ( \pi ) : o n \cdot u p p o r t e r o r } } \\ & { \quad \quad \quad + \frac { 1 } { 1 - \gamma } \cdot \underset { \nu \cdot \mathcal { O } ( \nu ; \mu , \mathcal { F } , \pi ) \leq C _ { 2 } } { \operatorname* { m i n } } \underset { ( s , a ) \in S \times A } { \sum } ( d _ { \pi } \setminus \nu ) ( s , a ) \left[ \Delta f _ { \pi } ( s , a ) - \gamma ( \mathcal { P } ^ { \pi } \Delta f _ { \pi } ) ( s , a ) \right] , } \end{array}
+$$
+
+where $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi )$ is defined in Definition $I$ , $( d _ { \pi } \setminus \nu ) ( s , a ) : = \operatorname* { m a x } ( d _ { \pi } ( s , a ) - \nu ( s , a ) , 0 )$ and $( \mathcal P ^ { \pi } f ) ( s , a ) = \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot | s , a ) } [ f ( s ^ { \prime } , \pi ) ]$ for any $f$ .
+
+Bias-variance decomposition. Note that decomposition of our error bound into on-support and off-support parts effectively achieves a bias-variance tradeoff. A small value of the concentrability threshold $C _ { 2 }$ requires the choice of the distribution $\nu$ closer to $\mu$ , which results in better estimation error guarantee (which is $\mathcal { O } ( \sqrt { C _ { 2 } / n } ) )$ when we transfer from $\mu$ to $\nu$ , but potentially pays a high bias due to the mismatch between $d _ { \pi }$ and $\nu$ . A larger threshold permits more flexibility in choosing $\nu$ similar to $d _ { \pi }$ for a smaller bias, but results in a larger variance and estimation error. Rather than commit to a particular tradeoff, our estimator automatically adapts to the best possible splitting (Figure 1 illustrates this concept) by allowing us to choose the best threshold $C _ { 2 }$ . The on-support part matches the $n$ rate (fast rate error bound) of API or AVI analysis (e.g., [Pires and Szepesvári, 2012; Lazaric et al., 2012; Chen and Jiang, 2019]). The dependency on horizon is only linear and matches the best previous result with concentrability assumption [Xie and Jiang, 2020]. For the off-support part, it depends on the off-support mass $d _ { \pi } \setminus \nu$ and the “quality” of the off-support estimation: if all value functions in the version space are close to each other in the off-support region for policy $\pi$ , the gap between $J ( \pi )$ and $J ( \widehat { \pi } )$ can still be small even with a large boff-support mass. The following corollary formally states this property.
+
+![](images/9efc325511f863aac091fd74b7ddb94076412c3673defbcfbbcbe122b0ea0604.jpg)  
+Figure 1: An example illustrating different on-support and off-support splittings (denoted by two different vertical lines). Different splitting has different $C _ { 2 }$ values, and further yields different bias-variance trade-offs.
+
+Corollary 1 (“Double Robustness”). Under conditions of Theorem 3.1, for any $\pi$ and $C _ { 2 } \geq 0$ $\mathrm { e r r } _ { \mathrm { o f f } } ( \pi ) = 0$ when either (1) $\mathcal { C } ( d _ { \pi } ; \mu , \mathcal { F } , \pi ) \le C _ { 2 }$ , or, $( 2 ) \Delta f _ { \pi } - \gamma \mathcal { P } ^ { \pi } \Delta f _ { \pi } \equiv 0$ .
+
+Adaptive guarantees by algorithm design. As mentioned above, Theorem 3.1 implicitly selects the best bias-variance decomposition through the best choice of $C _ { 2 }$ in hindsight, with this decomposition being purely a proof technique, not an knob in the algorithm. In contrast, many prior approaches [Liu et al., 2020; Fujimoto et al., 2019; Kumar et al., 2019] employ explicit thresholds to control density ratios in their algorithms, which makes the tradeoff a hyperparameter in their algorithms. Since choosing hyperparameters is particularly challenging in offline RL, where even policy evaluation can be unreliable, this novel axis of adaptivity is an extremely desirable property of our approach.
+
+Comparison to guarantees in the exploratory setting. When a dataset with full coverage is given, classical analyses provide near-optimality guarantees that compete with the optimal policy $\pi ^ { \star }$ with a polynomial sample complexity, when $\pi ^ { \star } \in \Pi$ and both realizability and completeness hold for $\mathcal { F }$ ; see [Antos et al., 2008] for a representative analysis. As mentioned earlier, such analysis often requires boundedness of $\| \nu / \mu \| _ { \infty }$ for all admissible distributions $\nu \in \{ d _ { \pi } : \pi \in \Pi \}$ . On the other hand, it is easily seen that we can compete with $\pi ^ { \star }$ under much weaker conditions.
+
+Corollary 2 (Competing with optimal policy). Under conditions of Theorem 3.1, if $\mathcal { C } ( d _ { \pi ^ { \star } } ; \mu , \mathcal { F } , \pi ^ { \star } ) \le C _ { 2 }$ , we have
+
+$$
+J ( \pi ^ { \star } ) - J ( \widehat \pi ) \leq \mathcal O \left( \frac { V _ { \operatorname* { m a x } } \sqrt { C _ { 2 } } } { 1 - \gamma } \sqrt { \frac { \log \frac { | \mathcal F | | \Pi | } { \delta } } { n } + \frac { \sqrt { C _ { 2 } ( \varepsilon _ { \mathcal F , \mathcal F } + \varepsilon _ { \mathcal F } ) } } { 1 - \gamma } } \right) .
+$$
+
+Notably, these milder coverage assumptions in Corollaries 1 and 2 provide offline RL counterparts to the benefits of policy-gradient-style methods with online access to the environment [Kakade and Langford, 2002; Scherrer, 2014; Agarwal et al., 2019].
+
+Comparison with Liu et al. [2020]. The closest prior result to our work is that of [Liu et al., 2020], who develop a pessimistic estimator that truncates Bellman backups from state-action pairs infrequently visited by $\mu$ , and analyzes the resulting pessimistic policy and value iteration algorithms under general function approximation. For truncating Bellman backups, however, their work requires estimating the state-action distribution of data, which can be challenging in high-dimensions and they incur additional errors from density estimation which we avoid. Further, their algorithms only compete with policies $\pi$ where $\| d _ { \pi } / \mu \| _ { \infty }$ is bounded instead of the more general result that we provide, and makes their results vacuous in a linear MDP setting under typical feature coverage assumptions.
+
+Safe Policy Improvement. Some prior works [e.g. Laroche et al., 2019; Liu et al., 2020] discuss the scenario where the dataset $\mathcal { D }$ is collected with a behavior policy $\pi _ { b }$ with $\mu = d _ { \pi _ { b } }$ , and demonstrate that their algorithms always return a policy competitive with $\pi _ { b }$ . In our setup, this is straightforward as $d _ { \pi _ { b } }$ is always covered, as shown next.
+
+Corollary 3 (Bounded degradation from behavior policy). Under conditions of Theorem 3.1, if $\mu = d _ { \pi _ { b } }$ for some policy $\pi _ { b } \in \Pi$ , we have
+
+$$
+J ( \pi _ { b } ) - J ( \widehat { \pi } ) \leq \mathcal { O } \left( \frac { V _ { \operatorname* { m a x } } } { 1 - \gamma } \sqrt { \frac { \log \frac { | \mathcal { F } | | \Pi | } { \delta } } { n } } + \frac { \sqrt { \varepsilon _ { \mathcal { F } , \mathcal { F } } + \varepsilon _ { \mathcal { F } } } } { 1 - \gamma } \right) .
+$$
+
+Proof sketch of Theorem 3.1. We now briefly describe the core ideas in the proof. More detailed arguments are deferred to the full proof in Appendix B.1.
+
+The key to prove Theorem 3.1 is to translate the $J ( \pi ) - J ( \widehat { \pi } )$ to the Bellman error of value functions in the version space $\mathcal { F } _ { \pi , \varepsilon }$ b. Our main proving strategies are as follows:
+
+1. As the selection of $\varepsilon = \varepsilon _ { r }$ ensures the accurate estimation $Q ^ { \pi }$ is contained in the version space $\mathcal { F } _ { \widehat { \pi } , \varepsilon }$ for any $\pi$ , we can obtain $\begin{array} { r } { J ( \pi ) - J ( \widehat \pi ) \leq \operatorname* { m a x } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) - \operatorname* { m i n } _ { f \in \mathcal { F } _ { \widehat \pi , \varepsilon } } f ( s _ { 0 } , \widehat \pi ) + } \end{array}$ bapproximation error. 2. By the optimality of $\widehat { \pi }$ , we have $\begin{array} { r } { \operatorname* { m i n } _ { f \in \mathcal { F } _ { \widehat { \pi } , \varepsilon } } f ( s _ { 0 } , \widehat { \pi } ) \geq \operatorname* { m i n } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) } \end{array}$ . This indicates that $\begin{array} { r } { J ( \pi ) - J ( \widehat { \pi } ) \leq \operatorname* { m a x } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) - \operatorname* { m i n } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) + \mathrm { a p p r o x i m a t i } } \end{array}$ ion error.
+
+3. By using a standard telescoping argument (e.g., [Xie and Jiang, 2020, Lemma 1]), $\begin{array} { r } { \operatorname* { m a x } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) \ - \ \operatorname* { m i n } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } \bar { f } ( s _ { 0 } , \pi ) } \end{array}$ can be upper bounded by the Bellman error of $\mathrm { a r g m a x } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f \mathopen { } \mathclose \bgroup \left( s _ { 0 } , \pi \aftergroup \egroup \right)$ and $\begin{array} { r } { \operatorname * { a r g m i n } _ { f \in \mathcal { F } _ { \pi , \varepsilon } } f ( s _ { 0 } , \pi ) } \end{array}$ over distribution $d _ { \pi }$ .
+
+After combining all the three steps above together and considering the distribution shift effect, we complete the proof.
+
+# 3.1 Results for Linear Function Approximation
+
+Here we perform a case study in linear function approximation. We will show that our results—when instantiated under linear function approximation (with realizability and completeness assumptions)— automatically provides state-of-the-art guarantees, improving over existing results specialized to this setting by a factor of $\mathcal O ( d )$ [Jin et al., 2021] when the action space is finite and small.
+
+We recall the linear function approximation setup (we set $R _ { \operatorname* { m a x } } = 1$ and $\begin{array} { r } { V _ { \mathrm { m a x } } = \frac { 1 } { 1 - \gamma } } \end{array}$ for consistency with literature).
+
+Definition 2 (Linear Function Approximation). Let $\phi : S \times \mathcal { A }  \mathbb { R } ^ { d }$ be a feature mapping. Without loss of generality, we assume $\| \bar { \phi ( s , a ) } \| _ { 2 } \le 1 , \forall ( s , a ) \in \mathcal { S } \times \mathcal { A }$ . We define the value-function class $\mathcal { F } _ { \Phi }$ as $\mathcal { F } _ { \Phi } : = \{ \phi ( \cdot , \cdot ) ^ { \top } \theta : \theta \in \mathbb { R } ^ { d } , \phi ( \cdot , \cdot ) ^ { \top } \theta \in [ 0 , V _ { \operatorname* { m a x } } ] \}$ , and the policy class $\Pi _ { \Phi }$ consists of the greedy policies of each value function in $\mathcal { F } _ { \Phi }$ .
+
+Assumption 3 (Realizability and Completeness). $\varepsilon _ { \mathcal { F } , \mathcal { F } } = \varepsilon _ { \mathcal { F } } = 0$ .
+
+Note that, when the feature mapping $\phi ( \cdot , \cdot )$ is the one induced by the linear MDP [Jin et al., 2020], it automatically ensure that $\pi ^ { \star } \in \Pi _ { \Phi }$ and $\mathcal { F } _ { \Phi }$ satisfies Assumptions 3. In contrast, we highlight that the standard linear function approximation or linear MDP setup does not entail all the assumptions needed by Liu et al. [2020] as mentioned earlier.
+
+Below is our main result in the linear function approximation setting.
+
+Theorem 3.2. Suppose the value-function class $\mathcal { F }$ is a linear function class that satisfies realizability and completeness (Definition 2 and Assumption $^ 3$ ) and $\widehat { \pi }$ is the output of Eq.(3.2) using valuefunction class $\mathcal { F } _ { \Phi }$ and policy class $\Pi _ { \Phi }$ . If we choose $\begin{array} { r } { \varepsilon = c V _ { \mathrm { m a x } } ^ { 2 } d \log { \frac { V _ { \mathrm { m a x } } | A | d } { \delta } } / n } \end{array}$ , then, for any policy $\pi : S  \Delta ( { \mathcal { A } } )$ , we have
+
+$$
+J ( \pi ) - J ( \widehat { \pi } ) \leq \mathcal { O } \left( \frac { V _ { \operatorname* { m a x } } } { 1 - \gamma } \sqrt { \frac { d \log \frac { V _ { \operatorname* { m a x } } | A | d } { \delta } } { n } } \mathbb { E } _ { d _ { \pi } } \left[ \sqrt { \phi ( s , a ) ^ { \mathsf { T } } \Sigma _ { \mathcal { D } } ^ { - 1 } \phi ( s , a ) } \right] \right) ,
+$$
+
+where c is an absolute constant, and $\Sigma _ { \mathcal { D } } : = \mathbb { E } _ { \mathcal { D } } \left[ \phi ( s , a ) \phi ( s , a ) ^ { \mathsf { T } } \right]$ .
+
+The detailed proof of Theorem 3.2 is provided in Appendix B.2. Our guarantee is structurally very similar to that of Jin et al. [2021, Theorem 4.4], except that we only need linear function approximation with realizability and completeness assumptions, and they consider the finite-horizon linear MDP setting. If we translate their result to the discounted setting by setting $H = \mathcal { O } ( 1 / ( 1 - \gamma ) )$ , we enjoy a net improvement of order $\mathcal O ( d )$ in sample complexity when the action space is finite and small. To make it concrete, that is $\mathcal { O } ( \sqrt { d ^ { 2 } \log ( d n / \delta ) / n } )$ vs. $\mathcal { O } ( \sqrt { d \log ( d \vert A \vert / \delta ) / n } )$ error bounds. The bound also shows that having a full-rank $\Sigma _ { \mathcal { D } }$ (which is ensured by a full-rank covariance under $\mu$ ) is sufficient for consistent offline RL in linear function approximation. Crucially, the full-rank covariance is an easily checkable condition on data, as opposed to unverifiable concentrability assumptions. As a caveat, our results do not imply a computationally efficient algorithm, as a naïve implementation involves evaluating each policy pessimistically to pick the best. We discuss a computationally efficient adaptation of our approach in the next section.
+
+# 4 Practical Algorithm — Regularized Offline Policy Optimization
+
+A major challenge using the proposed algorithm in Section 3 in practice is that searching the policy with the best pessimistic evaluation over the policy space $\Pi$ is not computationally tractable. In this section, we present a practical algorithm that is computationally efficient assuming access to a (regularized) loss minimization oracle over the value function class $\mathcal { F }$ , and also comes with rigorous theoretical guarantees.
+
+Our practical algorithm is summarized in Algorithm 1. It has three key differences from the information-theoretic version in Eq.(3.2):
+
+Algorithm 1 PSPI: Pessimistic Soft Policy Iteration
+
+Input: Batch data $\mathcal { D }$ , regularization coefficient $\lambda$ .
+
+1: Initialize policy $\pi _ { 1 }$ as the uniform policy.   
+2: for $t = 1 , 2 , \dots , T$ do   
+3: Obtain the pessimistic estimation for $\pi _ { t }$ as $f _ { t }$ ,
+
+$$
+f _ { t } \gets \underset { f \in \mathcal { F } } { \mathrm { a r g m i n } } \left( f ( s _ { 0 } , \pi _ { t } ) + \lambda \mathcal { E } ( f , \pi _ { t } ; \mathcal { D } ) \right) ,
+$$
+
+where $\mathcal { E } ( f , \pi _ { t } ; \mathcal { D } )$ is defined in Eq.(3.1).
+
+4: Calculate $\pi _ { t + 1 }$ by,
+
+$$
+\pi _ { t + 1 } ( a | s ) \propto \pi _ { t } ( a | s ) \exp \left( \eta f _ { t } ( s , a ) \right) , \forall s , a \in \mathcal { S } \times \mathcal { A } .
+$$
+
+5: end for
+
+6: Output $\bar { \pi } : = \mathsf { U n i f } ( \pi _ { [ 1 : T ] } )$
+
+▷ uniformly mix $\pi _ { 1 } , \ldots , \pi _ { T }$ at the trajectory level
+
+1. The pessimistic policy evaluation is now performed via regularization (Line 3) instead of constrained optimization.
+
+2. Instead of searching over an explicit policy space Π, we search over a policy class implicitly induced from $\mathcal { F }$ (defined in Eq.(4.2)) and therefore no longer have a policy class independent of $\mathcal { F }$ separately, which is a common practice in API-style algorithms [Munos, 2003; Antos et al., 2008; Lazaric et al., 2012].
+
+3. We optimize the policy using mirror descent updates, which yields computationally tractable optimization over the implicit policy class. This property has been leveraged in many prior works, although typically in online RL settings [Even-Dar et al., 2009; Agarwal et al., 2019; Geist et al., 2019; Cai et al., 2020; Shani et al., 2020].
+
+Note that the use of a specific policy class above can be relaxed if a stronger structural assumption is made on the MDP (e.g., linear MDPs [Jin et al., 2020, 2021]).
+
+# 4.1 Analysis of Algorithm 1
+
+We now provide the analysis of Algorithm 1 in this section. For ease of presentation, we formally define the implicit policy class for this section:
+
+$$
+\Pi _ { \mathrm { S P I } } : = \{ \pi ^ { \prime } ( \cdot | s ) \propto \exp ( \eta \sum _ { i = 1 } ^ { t } f ^ { ( t ) } ( s , \cdot ) ) : 1 \le t \le T , f ^ { ( 1 ) } , \ldots , f ^ { ( i ) } \in \mathcal { F } \} ,
+$$
+
+which is the natural policy class for soft policy-iteration approaches. The following theorem describes the performance guarantee of $\bar { \pi }$ .
+
+Theorem 4.1. Let $\begin{array} { r } { \lambda = \sqrt [ 3 ] { V _ { \mathrm { m a x } } \big / ( 1 - \gamma ) { } ^ { 2 } \varepsilon _ { r } ^ { 2 } } } \end{array}$ with $\varepsilon _ { r }$ in Eq.(3.3), $\begin{array} { r } { \eta = \sqrt { \frac { \log | \mathcal { A } | } { 2 V _ { \operatorname* { m a x } } ^ { 2 } T } } } \end{array}$ , and $\bar { \pi }$ be obtained from Algorithm 1. For any policy $\pi : S  \Delta ( { \mathcal { A } } )$ we wish to compete with, suppose Assumptions 1 and 2 hold with respect to the policy class $\Pi _ { S P I } \cup \{ \pi \}$ . Then, for any constant $C _ { 2 } \geq 1$ , we have with probability at least $1 - \delta$ ,
+
+$$
+\begin{array} { r l } & { J ( \pi ) - J ( \overline { { \pi } } ) } \\ & { \leq \underbrace { \mathcal { O } \left( \sqrt { C _ { 2 } } \left( \frac { \sqrt { \varepsilon _ { \mathcal { F } , \mathcal { F } } + \varepsilon _ { \mathcal { F } } } } { 1 - \gamma } + \frac { V _ { \operatorname* { m a x } } } { 1 - \gamma } \sqrt [ 3 ] { \frac { T \log { \frac { | \mathcal { F } | } { \delta } } } { n } } + \sqrt [ 3 ] { \frac { V _ { \operatorname* { m a x } } \varepsilon _ { \mathcal { F } } } { ( 1 - \gamma ) ^ { 2 } } } \right) \right) } _ { \mathrm { e r r } _ { \operatorname* { m i n } } ( \pi ) : o n \cdot s u p p o r e r m r } + \underbrace { \mathcal { O } \left( \frac { V _ { \operatorname* { m a x } } } { 1 - \gamma } \sqrt { \frac { \log { | \mathcal { A } | } } { T } } \right) } _ { \displaystyle 1 - \gamma } } \\ & { + \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \underset { \nu : \mathcal { C } ( \nu ; \mu , F , \pi _ { t } ) \leq C _ { 2 } } { \operatorname* { m i n } } \left| \underset { ( s , a ) \in S \times A } { \sum _ { m } } \frac { \left( d _ { \pi } \setminus \nu \right) ( s , a ) [ f _ { t } ( s , a ) - ( \mathcal { T } ^ { \pi _ { t } } f _ { t } ) ( s , a ) ] } { 1 - \gamma } \right| \right) , } \end{array}
+$$
+
+$\operatorname { e r r } _ { \mathrm { o f f } } ( \pi ) .$ {z: off-support error
+
+where $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi _ { t } )$ is defined in Definition $^ { l }$ , $( d _ { \pi } \setminus \nu ) ( s , a ) : = \operatorname* { m a x } ( d _ { \pi } ( s , a ) - \nu ( s , a ) , 0 ) .$ .
+
+We provide a proof sketch of Theorem 4.1 at the end of this section, and defer the full proof to Appendix C. We now make a few remarks about the results in Theorem 4.1.
+
+Measurement of distribution shift effect. Compared with the information-theoretical result (provided in Theorem 3.1), the measurement of distribution shift in Theorem 4.1 depends on the optimization trajectory. That is, it measures the distance between two distribution $\nu$ and $\mu$ by $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi _ { t } )$ $( \pi _ { [ 1 : T ] }$ is the sequence of policies produced by the algorithm) whereas Theorem 3.1 uses $\mathcal { C } ( \nu ; \mu , \mathcal { F } , \pi )$ ( $\pi$ is the baseline policy we compete with). We remark that both of these two measurements are weaker then traditional density-ratio definitions (e.g., [Munos and Szepesvári, 2008; Chen and Jiang, 2019; Xie and Jiang, 2020]) as we demonstrated before, as the dependence of $\mathcal { C }$ on $\pi$ is relatively secondary.
+
+Dependence on $T$ . The number of optimization rounds $T$ affects the bound in two opposite ways: as $T$ increases, the optimization error term decreases, whereas the second term of the on-support error increases. The latter increase is due to the complexity of the implicit policy class $\Pi$ growing exponentially with $T$ , which affects our concentration bounds. To optimize the bound, the optimal choice is $T = \mathcal { O } ( n ^ { 2 / 5 } )$ , leading to an overall $\mathcal { O } ( n ^ { - 1 / 5 } )$ rate. While such a rate is relatively slow, we remark that the complexity bound of $\Pi$ is conservative, and in certain cases it is possible to obtain much sharper bounds: for example, in linear function approximation (Section 3.1), $\Pi _ { \mathrm { S P I } }$ are a priori captured by the space of softmax policies, whose complexity has no dependence on $T$ (up to mild logarithmic dependence due to norms). That is, the $\operatorname { e r r } _ { \mathrm { o n } } ( \pi )$ term in Theorem 4.1 reduces to $\widetilde { \mathcal { O } } ( \frac { V _ { \mathrm { m a x } } } { 1 - \gamma } \sqrt [ 3 ] { d / n } )$ $( \varepsilon _ { \mathcal { F } , \mathcal { F } } = \varepsilon _ { \mathcal { F } } = 0$ in linear function approximation), and yields an overall $\mathcal { O } ( n ^ { - 1 / 3 } )$ rate.
+
+Bias-variance decomposition. Similar to Theorem 3.1, Theorem 4.1 also allows arbitrary decomposition of the error bound into on-support and off-support components by setting the concentrability threshold $C _ { 2 }$ , which serves as a bias-variance tradeoff as before. In fact, the splitting can be done separately for each $\pi _ { t }$ in $1 \leq t \leq T$ and we omit such flexibility for readability. The optimization error does not depend on the splitting. Our performance guarantee naturally adapts to the best possible decomposition as before. As in Theorem 3.1, if the estimation on the off-support region is “highquality”, we can further simplify the performance guarantees, but the requirement of “high-quality” is different from that of Corollary 1. We make it formal in the following corollary.
+
+Corollary 4 (“Double Robustness”). For any $\pi$ and $C _ { 2 } \geq 0$ , $\mathrm { e r r } _ { \mathrm { o f f } } ( \pi ) = 0$ when either (1) $\mathcal { C } ( d _ { \pi } ; \mu , \mathcal { F } , \pi _ { t } ) \le C _ { 2 }$ for all $t \in [ T ]$ , or, (2) $f _ { t } - \mathcal { T } ^ { \pi _ { t } } \Delta f _ { t } \equiv 0$ for all $t \in [ T ]$ .
+
+We note that the conditions above depend on the optimization trajectory through their dependence on $\pi _ { t }$ , but can be made algorithm-independent by instead asserting the stronger requirement that $\mathcal { C } ( d _ { \pi } , \mu , \mathcal { F } , \pi ^ { \prime } ) \le C _ { 2 }$ for all $\pi ^ { \prime } \in \Pi _ { \mathrm { S P I } }$ in the first condition.
+
+Competing with the optimal policy. As before, we can provide a guarantee for competing with the optimal policy, under coverage assumptions weaker than the typical batch RL literature, albeit slightly stronger than those of Corollary 2. We state the formal result below.
+
+Corollary 5 (Competing with optimal policy). Under conditions of Theorem 4.1, if $\mathcal { C } ( d _ { \pi ^ { \star } } ; \mu , \mathcal { F } , \pi ) \le C _ { 2 }$ for all $\pi \in \Pi _ { S P I } ,$ , we have
+
+$$
+J ( \pi ^ { \star } ) - J ( \widehat \pi ) \leq \mathcal O \left( \frac { V _ { \operatorname* { m a x } } \sqrt { C _ { 2 } } } { 1 - \gamma } \left( \frac { \log \frac { | \mathcal F | } { \delta } \log | A | } { n } \right) ^ { 1 / 5 } + \frac { \sqrt { C _ { 2 } ( \varepsilon _ { \mathcal F } , \mathcal F + \varepsilon _ { \mathcal F } ) } } { 1 - \gamma } \right) .
+$$
+
+Note that the conditions of Corollary 5 are satisfied as before whenever $\| d _ { \pi ^ { \star } } / \mu \| _ { \infty } \leq C _ { 2 }$
+
+Computationally-efficient implementation with linear function approximation We remark that our algorithm is computationally efficient when the value-function class $\mathcal { F }$ is linear, that is, $\mathcal { F } : = \{ \phi ( \cdot , \cdot ) ^ { \top } \theta : \theta \in \mathbb { R } ^ { d } \}$ . In this case, the objective of Eq.(4.1) has a closed-form expression which is quadratic in $\theta$ . In addition, under additional matrix invertibility conditions, Eq.(4.1) has a closed-form solution which generalizes LSTDQ [Lagoudakis and Parr, 2003; Sutton et al., 2009; Dann et al., 2014]. A similar connection has been made by Antos et al. [2008], but our derivation is more general. See Appendix Appendix D for further details.
+
+We conclude the section with a proof sketch showing the key insights used in establishing the proof.
+
+Proof sketch of Theorem 4.1. Our proof constructs a corresponding MDP $\mathcal { M } _ { t }$ for every $f _ { t } , \pi _ { t }$ pair. Each $\mathcal { M } _ { t }$ has the same dynamics as the ground-truth MDP, but chooses a different reward function, such that $f _ { t }$ is the $Q$ -function of $\pi _ { t }$ in $\mathcal { M } _ { t }$ , $Q _ { \mathcal { M } _ { t } } ^ { \pi _ { t } }$ (we use the subscript of $\mathcal { M } _ { t }$ to denote the corresponding value or operator in MDP $\mathcal { M } _ { t }$ ). Our proof relies on some key properties of $\mathcal { M } _ { t }$ , such as $Q ^ { \pi ^ { - } } - T _ { \mathcal { M } _ { t } } ^ { \pi ^ { } } \bar { Q } ^ { \pi } = f _ { t } - \bar { T } ^ { \pi _ { t } } f _ { t }$ . We decompose $J ( \pi ) - J ( \bar { \pi } )$ as follows.
+
+$$
+\begin{array} { r l } & { J ( \pi ) - J ( \bar { \pi } ) \le \underbrace { \cfrac { 1 } { T } \displaystyle \sum _ { t = 1 } ^ { T } \left( J _ { \mathcal M _ { t } } ( \pi ) - J _ { \mathcal M _ { t } } ( \pi _ { t } ) \right) } _ { \mathrm { o p t i m i z a t i o n ~ e r r o r ~ } } + \underbrace { \cfrac { 1 } { T } \displaystyle \sum _ { t = 1 } ^ { T } ( J ( \pi ) - J _ { \mathcal M _ { t } } ( \pi ) ) } _ { \mathrm { c o n t r o l e d b y ~ } \| Q ^ { \pi } - T _ { \mathcal M _ { t } } ^ { \pi } Q ^ { \pi } \| _ { 2 , d \pi } = \| f _ { t } - \mathcal T ^ { \pi } t f _ { t } \| _ { 2 , d \pi } } } \\ & { \quad \quad + \underset { \mathrm { a p p r o x i m a t i o n / s t a t i s t i c a l ~ e r r o r s . } } { \mathrm { c o n t r o l e x . } } } \end{array}
+$$
+
+The proof is completed by bounding $\| f _ { t } - T ^ { \pi _ { t } } f _ { t } \| _ { 2 , d _ { \pi } }$ on both on-support and off-support regions.
+
+# 5 Conclusions
+
+This paper investigates sample-efficient offline reinforcement learning without data coverage assumptions (e.g., concentrability). To achieve that goal, our paper contributes several crucial improvements to the literature. We introduce the concept of Bellman-consistent pessimism. It enables the sample-efficient guarantees with only the Bellman-completeness assumption which is standard in the exploratory setting, whereas the point-wise/bonus-based pessimism popularly adopted in the literature usually requires stronger and/or extra assumptions. Algorithmically, we demonstrate how to implicitly infer a policy value lower bound through a version space and provide a tractable implementation. A particularly important aspect of our results is the ability to adapt to the best bias-variance tradeoff in the hindsight, which no prior algorithms achieve to the best of our knowledge. When applying our results in linear function approximation, we attain an $\mathcal O ( d )$ improvement in sample complexity, compared with the best-known recent work of offline RL in linear MDPs, whenever the action space is finite and small.
+
+As of limitations and future work, the sample complexity of our practical algorithm is worse than that of the information-theoretic approach, and it will be interesting to close this gap. Another future direction is to empirically evaluate PSPI on benchmarks and compare it to existing approaches.
+
+# Acknowledgment
+
+Part of this work was carried out while TX and AA worked at Microsoft Research. NJ acknowledges funding support from the ARL Cooperative Agreement W911NF-17-2-0196, NSF IIS-2112471, and Adobe Data Science Research Award.
+
+# References
+
+Doina Precup, Richard S Sutton, and Satinder P Singh. Eligibility traces for off-policy policy evaluation. In Proceedings of the Seventeenth International Conference on Machine Learning, pages 759–766, 2000.   
+András Antos, Csaba Szepesvári, and Rémi Munos. Learning near-optimal policies with bellmanresidual minimization based fitted policy iteration and a single sample path. Machine Learning, 71 (1):89–129, 2008.   
+Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020.   
+Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. In International Conference on Machine Learning, pages 1042–1051, 2019.   
+Ruosong Wang, Dean Foster, and Sham M Kakade. What are the statistical limits of offline rl with linear function approximation? In International Conference on Learning Representations, 2020.   
+Ruosong Wang, Yifan Wu, Ruslan Salakhutdinov, and Sham M Kakade. Instabilities of offline rl with pre-trained neural representation. arXiv preprint arXiv:2103.04947, 2021.
+
+Rémi Munos. Error bounds for approximate policy iteration. In Proceedings of the Twentieth International Conference on International Conference on Machine Learning, pages 560–567, 2003.
+
+Csaba Szepesvári and Rémi Munos. Finite time bounds for sampling based fitted value iteration. In Proceedings of the 22nd international conference on Machine learning, pages 880–887, 2005.
+
+Rémi Munos and Csaba Szepesvári. Finite-time bounds for fitted value iteration. Journal of Machine Learning Research, 9(5), 2008.
+
+Amir Massoud Farahmand, Rémi Munos, and Csaba Szepesvári. Error propagation for approximate policy and value iteration. In Advances in Neural Information Processing Systems, 2010.
+
+Tengyang Xie and Nan Jiang. $\mathbf { Q } ^ { * }$ approximation schemes for batch reinforcement learning: A theoretical comparison. In Conference on Uncertainty in Artificial Intelligence, pages 550–559. PMLR, 2020.
+
+Rahul Kidambi, Aravind Rajeswaran, Praneeth Netrapalli, and Thorsten Joachims. Morel: Modelbased offline reinforcement learning. In NeurIPS, 2020.
+
+Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. In International Conference on Machine Learning, pages 2052–2062, 2019.
+
+Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative q-learning for offline reinforcement learning. arXiv preprint arXiv:2006.04779, 2020.
+
+Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforce ment learning without great exploration. arXiv preprint arXiv:2007.08202, 2020.
+
+Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? In International Conference on Machine Learning, pages 5084–5096. PMLR, 2021.
+
+Eyal Even-Dar, Sham M Kakade, and Yishay Mansour. Online markov decision processes. Mathematics of Operations Research, 34(3):726–736, 2009.
+
+Matthieu Geist, Bruno Scherrer, and Olivier Pietquin. A theory of regularized markov decision processes. In International Conference on Machine Learning, pages 2160–2169. PMLR, 2019.
+
+Alekh Agarwal, Sham M Kakade, Jason D Lee, and Gaurav Mahajan. On the theory of policy gradient methods: Optimality, approximation, and distribution shift. arXiv preprint arXiv:1908.00261, 2019.
+
+Dimitri P. Bertsekas and John N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1st edition, 1996. ISBN 1886529108.
+
+Vijay R Konda and John N Tsitsiklis. Actor-critic algorithms. In Advances in neural information processing systems, pages 1008–1014. Citeseer, 2000.
+
+Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. In International Conference on Machine Learning, pages 11404–11413. PMLR, 2021.
+
+Gellért Weisz, Philip Amortila, and Csaba Szepesvári. Exponential lower bounds for planning in mdps with linearly-realizable optimal action-value functions. In Algorithmic Learning Theory, pages 1237–1264. PMLR, 2021.
+
+Andrea Zanette. Exponential lower bounds for batch reinforcement learning: Batch rl can be exponentially harder than online rl. In International Conference on Machine Learning, pages 12287–12297. PMLR, 2021.
+
+Yaqi Duan, Zeyu Jia, and Mengdi Wang. Minimax-optimal off-policy evaluation with linear function approximation. In International Conference on Machine Learning, pages 2701–2709. PMLR, 2020.
+
+Aviral Kumar, Justin Fu, Matthew Soh, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. Advances in Neural Information Processing Systems, 32:11784–11794, 2019.
+
+Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon, James Y Zou, Sergey Levine, Chelsea Finn, and Tengyu Ma. Mopo: Model-based offline policy optimization. Advances in Neural Information Processing Systems, 33:14129–14142, 2020.
+
+Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. Contextual decision processes with low bellman rank are pac-learnable. In International Conference on Machine Learning, pages 1704–1713. PMLR, 2017.
+
+Bernardo Ávila Pires and Csaba Szepesvári. Statistical linear estimation with penalized estimators: an application to reinforcement learning. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 1755–1762, 2012.
+
+Alessandro Lazaric, Mohammad Ghavamzadeh, and Rémi Munos. Finite-sample analysis of leastsquares policy iteration. Journal of Machine Learning Research, 13:3041–3074, 2012.
+
+Sham Kakade and John Langford. Approximately optimal approximate reinforcement learning. In ICML, volume 2, pages 267–274, 2002.
+
+Bruno Scherrer. Approximate policy iteration schemes: a comparison. In International Conference on Machine Learning, pages 1314–1322. PMLR, 2014.
+
+Romain Laroche, Paul Trichelair, and Remi Tachet Des Combes. Safe policy improvement with baseline bootstrapping. In International Conference on Machine Learning, pages 3652–3661, 2019.
+
+Chi Jin, Zhuoran Yang, Zhaoran Wang, and Michael I Jordan. Provably efficient reinforcement learning with linear function approximation. In Conference on Learning Theory, pages 2137–2143. PMLR, 2020.
+
+Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. In International Conference on Machine Learning, pages 1283–1294. PMLR, 2020.
+
+Lior Shani, Yonathan Efroni, Aviv Rosenberg, and Shie Mannor. Optimistic policy optimization with bandit feedback. In International Conference on Machine Learning, pages 8604–8613. PMLR, 2020.
+
+Michail G Lagoudakis and Ronald Parr. Least-squares policy iteration. The Journal of Machine Learning Research, 4:1107–1149, 2003.
+
+Richard S Sutton, Hamid Reza Maei, Doina Precup, Shalabh Bhatnagar, David Silver, Csaba Szepesvári, and Eric Wiewiora. Fast gradient-descent methods for temporal-difference learning with linear function approximation. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 993–1000, 2009.
+
+Christoph Dann, Gerhard Neumann, Jan Peters, et al. Policy evaluation with temporal differences: A survey and comparison. Journal of Machine Learning Research, 15:809–883, 2014.
+
+Amit Daniely, Sivan Sabato, Shai Ben-David, and Shai Shalev-Shwartz. Multiclass learnability and the erm principle. In Proceedings of the 24th Annual Conference on Learning Theory, pages 207–232. JMLR Workshop and Conference Proceedings, 2011.
+
+Shai Ben-David, Nicolo Cesa-Bianchi, David Haussler, and Philip M Long. Characterizations of learnability for classes of $\{ 0 , \ldots , n \}$ -valued functions. Journal of Computer and System Sciences, 50(1):74–86, 1995.
+
+David Haussler and Philip M Long. A generalization of sauer’s lemma. Journal of Combinatorial Theory, Series A, 71(2):219–240, 1995.
+
+David Haussler. Sphere packing numbers for subsets of the boolean n-cube with bounded vapnikchervonenkis dimension. Journal of Combinatorial Theory, Series A, 69(2):217–232, 1995.
+
+Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of machine learning. MIT press, 2018.

parse/train/e8WWUBeafM/e8WWUBeafM_content_list.json

@@ -0,0 +1,1894 @@
+[
+    {
+        "type": "text",
+        "text": "Bellman-consistent Pessimism for Offline Reinforcement Learning ",
+        "text_level": 1,
+        "bbox": [
+            295,
+            122,
+            700,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie UIUC tx10@illinois.edu ",
+        "bbox": [
+            202,
+            226,
+            349,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ching-An Cheng Microsoft Research chinganc@microsoft.com ",
+        "bbox": [
+            385,
+            226,
+            578,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nan Jiang UIUC nanjiang@illinois.edu ",
+        "bbox": [
+            611,
+            226,
+            794,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Paul Mineiro Microsoft Research pmineiro@microsoft.com ",
+        "bbox": [
+            258,
+            289,
+            449,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alekh Agarwal Google Research alekhagarwal@google.com ",
+        "bbox": [
+            539,
+            289,
+            740,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            367,
+            535,
+            382
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The use of pessimism, when reasoning about datasets lacking exhaustive exploration, has recently gained prominence in offline reinforcement learning. Despite the robustness it adds to the algorithm, overly pessimistic reasoning can be equally damaging in precluding the discovery of good policies, which is an issue for the popular bonus-based pessimism. In this paper, we introduce the notion of Bellmanconsistent pessimism for general function approximation: instead of calculating a point-wise lower bound for the value function, we implement pessimism at the initial state over the set of functions consistent with the Bellman equations. Our theoretical guarantees only require Bellman closedness as standard in the exploratory setting, in which case bonus-based pessimism fails to provide guarantees. Even in the special case of linear function approximation where stronger expressivity assumptions hold, our result improves upon a recent bonus-based approach by $\\mathcal O ( d )$ in its sample complexity when the action space is finite and small. Remarkably, our algorithms automatically adapt to the best bias-variance tradeoff in the hindsight, whereas most prior approaches require tuning extra hyperparameters a priori. ",
+        "bbox": [
+            233,
+            401,
+            766,
+            608
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            637,
+            310,
+            655
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Using past experiences to learn improved behavior for future interactions is a critical capability for a Reinforcement Learning (RL) agent. However, robustly extrapolating knowledge from a historical dataset for sequential decision making is highly challenging, particularly in settings where function approximation is employed to generalize across related observations. In this paper, we provide a systematic treatment of such scenarios with general function approximation, and devise algorithms that can provably leverage an arbitrary historical dataset to discover the policy that obtains the largest guaranteed rewards, amongst all possible scenarios consistent with the dataset. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            767
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The problem of learning a good policy from historical datasets, typically called batch or offline RL, has a long history [see e.g., Precup et al., 2000; Antos et al., 2008; Levine et al., 2020, and references therein]. Many prior works [e.g., Precup et al., 2000; Antos et al., 2008; Chen and Jiang, 2019] make the so-called coverage assumptions on the dataset, requiring the dataset to contain any possible state, action pair or trajectory with a lower bounded probability. These assumptions are evidently prohibitive in practice, particularly for problems with large state and/or action spaces. Furthermore, the methods developed under these assumptions routinely display unstable behaviors such as lack of convergence or error amplification, when coverage assumptions are violated [Wang et al., 2020, 2021]. ",
+        "bbox": [
+            174,
+            775,
+            825,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Driven by these instabilities, a growing body of recent literature has pursued a so-called best effort style of guarantee instead. The key idea is to replace the stringent assumptions on the dataset with a dataset-dependent performance bound, which gracefully degrades from guaranteeing a near-optimal policy under standard coverage assumptions to offering no improvement over the data collection policy in the most degenerate case. Algorithmically, these works all leverage the principle of pessimistic extrapolation from offline data and aim to maximize the rewards the trained agent would obtain in the worst possible MDP that is consistent with the observed dataset. These methods have been shown to be typically more robust to the violation of coverage assumptions in practice, and their theoretical guarantees often provide non-trivial conclusions in settings where the previous results did not apply. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Even though many such best-effort methods have now been developed, very few works provide a comprehensive theory for using generic function approximation, unlike the setting where the dataset satisfies the coverage assumptions [Antos et al., 2008; Munos, 2003; Szepesvári and Munos, 2005; Munos and Szepesvári, 2008; Farahmand et al., 2010; Chen and Jiang, 2019; Xie and Jiang, 2020]. For example, [Kidambi et al., 2020] provides a partial theory under the assumption of an uncertainty quantification oracle, which however is highly nontrivial to obtain for general function approximation. [Fujimoto et al., 2019; Kumar et al., 2020] develop sound theoretical arguments in the tabular setting, which were only heuristically extended to the function approximation setting. The works that explicitly consider function approximation in their design either use an ad-hoc truncation of Bellman backups [Liu et al., 2020] or strongly rely on particular parameterizations such as linear function approximation [Jin et al., 2021]. In particular, [Liu et al., 2020] additionally requires the ability to approximate stationary distribution of the behavior policy, which is a challenging density estimation problem for complex state spaces and cannot be provably performed in the standard linear MDP setting (see Section 3.1). ",
+        "bbox": [
+            174,
+            236,
+            825,
+            428
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our paper takes an important step in this direction. We provide a systematic way to encode pessimism compatible with an arbitrary function approximation class and MDP and give strong theoretical guarantees without requiring any coverage assumptions on the dataset. Our first contribution is an information theoretic algorithm that returns a policy with a small regret to any comparator policy, for which coverage assumptions (approximately) hold with respect to the data collection policy. This regret bound is identical to what can be typically obtained when the coverage assumptions hold for all policies [Antos et al., 2008; Chen and Jiang, 2019]. But our algorithm requires neither the coverage assumptions, nor additional assumptions such as reliable density estimation for the data generating distribution used by existing best-effort approaches [Liu et al., 2020]. We furthermore instantiate these results in the special case of linear parameterization; under the linear MDP assumption, our sample complexity bound leads to a factor of $\\mathcal O ( d )$ improvement for a $d$ -dimensional linear MDP, compared with the best known result translated to our discounted setting [Jin et al., 2021], when the action set is small in size. In addition to the information theoretic algorithm, we also develop a computationally practical version of our algorithm using a Lagrangian relaxation combined with recent advances in soft policy iteration [Even-Dar et al., 2009; Geist et al., 2019; Agarwal et al., 2019]. We show that this algorithm can be executed efficiently by querying a (regularized) loss minimization oracle over the value function class, although it has slightly worse theoretical guarantees than the information theoretic version. Both our algorithms display an adaptive property in selecting the best possible form of a bias-variance decomposition, where most prior approaches had to commit to a particular point through their choice of hyperparameters (see the discussion following Theorem 3.1). ",
+        "bbox": [
+            174,
+            434,
+            825,
+            710
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            728,
+            318,
+            746
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Markov Decision Processes We consider dynamical systems modeled as Markov Decision Processes (MDPs). An MDP is specified by $( S , \\mathcal { A } , P , R , \\gamma , s _ { 0 } )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $P : \\mathcal { S } \\times \\mathcal { A }  \\Delta ( \\mathcal { S } )$ is the transition function with $\\Delta ( \\cdot )$ being the probability simplex, $R : \\mathcal { S } \\times \\mathcal { A }  [ 0 , R _ { \\mathrm { m a x } } ]$ is the reward function, $\\gamma \\in [ 0 , 1 )$ is the discount factor, and $s _ { 0 }$ is a deterministic initial state, which is without loss of generality. We assume the state and the action spaces are finite but can be arbitrarily large. A (stochastic) policy $\\pi : { \\mathcal { S } }  \\Delta ( { \\mathcal { A } } )$ specifies a decision-making strategy, and induces a random trajectory $s _ { 0 } , a _ { 0 } , r _ { 0 } , s _ { 1 } , a _ { 1 } , r _ { 1 } , . . . _$ , where $a _ { t } \\sim \\pi ( \\cdot | s _ { t } ) , r _ { t } = R ( \\bar { s _ { t } } , a _ { t } ) , \\bar { s _ { t + 1 } } \\sim P ( \\cdot | s _ { t } , a _ { t } ) ,$ $\\forall t \\geq 0$ . We denote the expected discounted return of a policy $\\pi$ as $\\begin{array} { r } { J ( \\pi ) : = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } | \\pi ] } \\end{array}$ , and the learning goal is to find the maximizer of this value: $\\pi ^ { \\star } : = \\operatorname { a r g m a x } _ { \\pi } J ( \\pi )$ . A related concept is the policy-specific $Q$ -function, $Q ^ { \\pi } : S \\times \\mathcal { A }  \\mathbb { R }$ $Q ^ { \\pi } ( s , a )$ is the discounted return when the trajectory starts with $( s , a )$ and all remaining actions are taken according to $\\pi$ . $Q ^ { \\pi }$ is the unique fixed point of the (policy-specific) Bellman operator $\\mathcal { T } ^ { \\pi } : \\mathbb { R } ^ { S \\times A }  \\mathbb { R } ^ { S \\times A }$ , defined as: ",
+        "bbox": [
+            173,
+            758,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fcc9865a3bed89c88fddcb9794f92d9d452eaf38719fcf62f659faf6d223dada.jpg",
+        "text": "$$\n\\forall f , \\quad ( T ^ { \\pi } f ) ( s , a ) = R ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( \\cdot \\vert s , a ) } [ f ( s ^ { \\prime } , \\pi ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            123,
+            681,
+            142
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f ( s ^ { \\prime } , \\pi )$ is a shorthand for $\\mathbb { E } _ { a ^ { \\prime } \\sim \\pi ( \\cdot | s ^ { \\prime } ) } [ f ( s ^ { \\prime } , a ^ { \\prime } ) ]$ . ",
+        "bbox": [
+            174,
+            146,
+            531,
+            162
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another important concept is the notion of discounted state-action occupancy, $d _ { \\pi } \\in \\Delta ( S \\times \\mathcal { A } )$ , defined as $\\begin{array} { r } { \\hat { d } _ { \\pi } ( s , a ) : = ( \\hat { 1 ^ { ' } } - \\gamma ) \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } [ s _ { t } = s , a _ { t } = a ] | \\pi ] } \\end{array}$ , which characterizes the states and actions visited by a policy $\\pi$ . ",
+        "bbox": [
+            174,
+            167,
+            826,
+            210
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Offline RL In the offline setting, the learner only has access to a pre-collected dataset and cannot directly interact with the environment. We assume the standard i.i.d. data generation protocol in our theoretical derivations, that the offline dataset $\\mathcal { D }$ consists of $n$ i.i.d. $( s , a , r , s ^ { \\prime } )$ tuples generated as $( s , a ) \\sim \\mu , r = R ( s , a ) , s ^ { \\prime } \\sim P ( \\cdot | s , a )$ for some data distribution $\\mu$ . We will also use $\\mathbb { E } _ { \\mu } [ \\cdot ]$ for taking expectation with respect to $\\mu$ . We will frequently use the data-weighted 2-norm (squared) $\\| f \\| _ { 2 , \\mu } : =$ $\\mathbb { E } _ { \\mu } [ f ^ { 2 } ]$ , and the definition extends when we replace $\\mu$ with any other state-action distribution $\\nu$ . The empirical approximation of $\\| f \\| _ { 2 , \\mu } ^ { 2 }$ is $\\begin{array} { r } { \\| f \\| _ { 2 , \\mathcal { D } } ^ { 2 } : = \\frac { 1 } { n } \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } f ( s , a ) ^ { 2 } . } \\end{array}$ . ",
+        "bbox": [
+            173,
+            223,
+            825,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Function Approximation Function approximation is crucial to generalizing over large and complex state and action spaces. In this work, we search for a good policy in a policy class $\\Pi \\subset ( \\bar { S } $ $\\Delta ( \\mathcal { A } ) )$ with the help of a value-function class ${ \\mathcal { F } } \\subset ( S \\times { \\mathcal { A } }  [ 0 , V _ { \\operatorname* { m a x } } ] )$ to model $Q ^ { \\pi }$ , where $V _ { \\mathrm { m a x } } = R _ { \\mathrm { m a x } } / ( 1 - \\gamma )$ . Such a combination is commonly found in approximate policy iteration and actor-critic algorithms [e.g., Bertsekas and Tsitsiklis, 1996; Konda and Tsitsiklis, 2000]. For most part of the paper we do not make any structural assumptions on $\\Pi$ and $\\mathcal { F }$ , making our approach and guarantees applicable to generic function approximators. For simplicity we will assume that these function classes are finite but exponentially large, and use log-cardinality to measure their statistical complexities in the generic results (Section 3 and Section 4). These guarantees easily extend to continuous function classes where log-cardinalities are replaced by the appropriate notions of covering numbers, which we demonstrate when we instantiate our results in the linear function approximation setting and work with continuous linear classes (Section 3.1). ",
+        "bbox": [
+            173,
+            340,
+            825,
+            507
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now recall two standard expressivity assumptions on $\\mathcal { F }$ [e.g., Antos et al., 2008]. To our knowledge, no existing works on offline RL with insufficient data coverage have provided guarantees under these standard assumptions for general function approximation, and they often require stronger or tweaked assumptions (see Section 1). ",
+        "bbox": [
+            174,
+            513,
+            825,
+            569
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 (Realizability). For any $\\pi \\in \\Pi$ , we have ",
+        "bbox": [
+            173,
+            571,
+            540,
+            587
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/013be798cf4fba7333759d1639d6efd86650365460445a225d14e4e599e152ce.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { f \\in \\mathcal { F } } \\operatorname* { s u p } _ { a d m i s s i b l e \\nu } \\| f - \\mathcal { T } ^ { \\pi } f \\| _ { 2 , \\nu } ^ { 2 } \\leq \\varepsilon _ { \\mathcal { F } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            592,
+            617,
+            619
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where an admissible distribution $\\nu$ means that $\\nu \\in \\{ d _ { \\pi ^ { \\prime } } : \\pi ^ { \\prime } \\in \\Pi \\}$ . ",
+        "bbox": [
+            173,
+            623,
+            612,
+            638
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 requires that for every $\\pi \\in \\Pi$ , there exists $f \\in { \\mathcal { F } }$ that well-approximates $Q ^ { \\pi }$ . This assumption is often called realizability.1 Technically this is asserted by requiring $f$ to have small Bellman error w.r.t. $\\mathcal { T } ^ { \\pi }$ under all possible admissible distributions. As a sufficient condition, we have $\\varepsilon _ { \\mathcal { F } } = 0$ if $Q ^ { \\pi } \\in { \\mathcal { F } } , \\forall \\pi \\in \\Pi$ . ",
+        "bbox": [
+            173,
+            647,
+            826,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 (Completeness). For any $\\pi \\in \\Pi$ , we have ",
+        "bbox": [
+            173,
+            707,
+            549,
+            722
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84f2d4e95a0eaeffd9ab00c988ac11d077172f830998080f9e045f684bb3fb0e.jpg",
+        "text": "$$\n\\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\operatorname* { i n f } _ { f ^ { \\prime } \\in \\mathcal { F } } \\left. f ^ { \\prime } - \\mathcal { T } ^ { \\pi } f \\right. _ { 2 , \\mu } ^ { 2 } \\leq \\varepsilon _ { \\mathcal { F } , \\mathcal { F } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            726,
+            612,
+            756
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 asserts that $\\mathcal { F }$ is approximately closed under $\\tau ^ { \\pi }$ .2 Such an assumption is widely used in RL theory and can be only avoided in some rare cases [Xie and Jiang, 2021], and the hardness of learning with realizability alone has been established in various settings (e.g., [Weisz et al., 2021; Zanette, 2021]). We also emphasize that we only measure the violation of completeness under $\\mu$ and do not need to reason about all admissible distributions. ",
+        "bbox": [
+            173,
+            767,
+            826,
+            838
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Distribution shift A unique challenge in RL is that the learned policy may induce a state (and action) distribution that is different from the data distribution $\\mu$ , and the issue is particularly salient when we do not impose any coverage assumption on $\\mu$ . Therefore, it is important to carefully characterize the distribution shift, which we measure using the following definition, which generalizes prior definitions specific to linear function approximation [Agarwal et al., 2019; Duan et al., 2020]: ",
+        "bbox": [
+            173,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. We define $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi )$ as follows to measure the distribution shift from an arbitrary distribution $\\nu$ to the data distribution $\\mu$ , w.r.t. $\\mathcal { F }$ and $\\pi$ , ",
+        "bbox": [
+            173,
+            164,
+            821,
+            193
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5335117891e07332ceaa538f7a4e3210bd3f07e23c1d26ac80b67239d234d0a0.jpg",
+        "text": "$$\n\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi ) : = \\operatorname* { m a x } _ { f \\in \\mathcal { F } } \\frac { \\| f - \\mathcal { T } ^ { \\pi } f \\| _ { 2 , \\nu } ^ { 2 } } { \\| f - \\mathcal { T } ^ { \\pi } f \\| _ { 2 , \\mu } ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            196,
+            627,
+            236
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi )$ measures how well Bellman errors under $\\pi$ transfer between the distributions $\\nu$ and $\\mu$ . For instance, a small value of $\\mathcal { C } ( d _ { \\pi } ; \\mu , \\mathcal { F } , \\pi )$ enables accurate policy evaluation for $\\pi$ using data collected under $\\mu$ . More generally, we observe that ",
+        "bbox": [
+            174,
+            246,
+            825,
+            289
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f24d7efd4c3f0087d88ead99c10bda21ffac1f8e6c45b47cc6e20f35f8bdb3b0.jpg",
+        "text": "$$\n\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi ) \\leq \\| \\nu / \\mu \\| _ { \\infty } : = \\operatorname* { s u p } _ { s , a } \\frac { \\nu ( s , a ) } { \\mu ( s , a ) } , \\quad \\mathrm { f o r ~ a n y } \\pi , \\mathcal { F } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            292,
+            692,
+            328
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and the RHS is a classical notion of bounded distribution ratio for error transfer (e.g., [Munos and Szepesvári, 2008; Chen and Jiang, 2019; Xie and Jiang, 2020]). Moreover, our measure can be tighter than $\\| \\nu / \\mu \\| _ { \\infty }$ : Even two distributions $\\nu$ and $\\mu$ that are sufficiently disparate might admit a reasonable transfer, so long as this difference is not detected by $\\pi$ and $\\mathcal { F }$ . To this end, our definition better captures the crucial role of function approximation in generalizing across different states. As an example, in the special case of linear MDPs, full coverage under our definition (i.e., boundedness of $\\mathcal { C }$ for all admissible $\\nu$ ) can be implied from the standard coverage assumption for linear MDPs that considers the spectrum of the feature covariance matrix under $\\mu$ ; see Section 3.1 for more details. ",
+        "bbox": [
+            173,
+            332,
+            826,
+            444
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Information-Theoretic Results with Bellman-consistent Pessimism ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            460,
+            754,
+            478
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide our first theoretical result which is information-theoretic, in that it uses a computationally inefficient algorithm. The approach uses the offline dataset to first compute a lower bound on the value of each policy $\\pi \\in \\Pi$ , and then returns the policy with the highest pessimistic value estimate. While this high-level template is at the heart of many recent approaches [e.g., Fujimoto et al., 2019; Kumar et al., 2019; Liu et al., 2020; Kidambi et al., 2020; Yu et al., 2020; Kumar et al., 2020], our main novelty is in the design and analysis of Bellman-consistent pessimism for general function approximation. ",
+        "bbox": [
+            173,
+            491,
+            826,
+            589
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a policy $\\pi$ , we first form a version space of all the functions $f \\in { \\mathcal { F } }$ which have a small Bellman error under the evaluation operator $\\tau ^ { \\pi }$ . We then return the predicted value of $\\pi$ in the initial state $s _ { 0 }$ by the functions in this version space. The use of pessimism at the initial state, while maintaining Bellman consistency (by virtue of having a small Bellman error) limits over pessimism, which is harder to preclude in the pointwise pessimistic penalties used in some other works [Jin et al., 2021]. More formally, given a dataset $\\mathcal { D }$ , let us define ",
+        "bbox": [
+            173,
+            593,
+            826,
+            678
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56f3a0fe4f01e1928f6554fe943e3964abfe811a6e897804864087efa20d3856.jpg",
+        "text": "$$\n\\mathcal { L } ( f ^ { \\prime } , f , \\pi ; \\mathcal { D } ) : = \\frac { 1 } { n } \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } \\left( f ^ { \\prime } ( s , a ) - r - \\gamma f ( s ^ { \\prime } , \\pi ) \\right) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            681,
+            696,
+            722
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and an empirical estimate of the Bellman error $\\mathcal { E } ( f , \\pi ; \\mathcal { D } )$ is ",
+        "bbox": [
+            173,
+            727,
+            568,
+            742
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a8e236070fe14ca20544abe40cfb629ba86e51f735171fdcb788d6a4a08f76ae.jpg",
+        "text": "$$\n\\mathcal { E } ( f , \\pi ; \\mathcal { D } ) : = \\mathcal { L } ( f , f , \\pi ; \\mathcal { D } ) - \\operatorname* { m i n } _ { f ^ { \\prime } \\in \\mathcal { F } } \\mathcal { L } ( f ^ { \\prime } , f , \\pi ; \\mathcal { D } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            746,
+            660,
+            771
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our algorithm. With this notation, our information-theoretic approach finds a policy by optimizing: ",
+        "bbox": [
+            171,
+            796,
+            825,
+            811
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e0437807e374aa2f19a2cd0e50bbef3493fe187d0c257259bc09f806bf0eae3.jpg",
+        "text": "$$\n\\widehat { \\pi } = \\underset { \\pi \\in \\Pi } { \\operatorname { a r g m a x } } \\ \\underset { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } { \\operatorname* { m i n } } f ( s _ { 0 } , \\pi ) , \\quad \\mathrm { w h e r e } \\ \\mathcal { F } _ { \\pi , \\varepsilon } = \\big \\{ f \\in \\mathcal { F } : \\mathcal { E } ( f , \\pi ; \\mathcal { D } ) \\leq \\varepsilon \\big \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            815,
+            740,
+            842
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the formulation above, $\\mathcal { F } _ { \\pi , \\varepsilon }$ is the version space of policy $\\pi$ . To better understand the intuition behind the estimator in Equation 3.2, let us define ",
+        "bbox": [
+            171,
+            847,
+            828,
+            875
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ac57a045de5d94d94e2efbd585f8850f714213975a26adb395b4670037ab4cb.jpg",
+        "text": "$$\n{ \\bf \\Pi } _ { \\pi , \\mathrm { m i n } } ^ { \\mathrm { r } } : = \\operatorname * { a r g m i n } _ { f \\in \\mathcal { F } _ { \\pi , \\epsilon } } f ( s _ { 0 } , \\pi ) , f _ { \\pi , \\mathrm { m a x } } : = \\operatorname * { a r g m a x } _ { f \\in \\mathcal { F } _ { \\pi , \\epsilon } } f ( s _ { 0 } , \\pi ) , \\mathrm { a n d } \\Delta f _ { \\pi } ( s , a ) : = f _ { \\pi , \\mathrm { m a x } } ( s , a ) - f _ { \\pi , \\mathrm { m i n } } ( s , a ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            878,
+            839,
+            907
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, if the parameter $\\varepsilon$ is defined to ensure that $Q ^ { \\pi }$ (or its best approximation in $\\mathcal { F }$ ) is in $\\mathcal { F } _ { \\pi , \\varepsilon }$ , we easily see that $\\Delta f _ { \\pi } ( s _ { 0 } , \\pi )$ is an upper bound on the error in our estimate of $J ( \\pi )$ for any $\\pi \\in \\Pi$ . In fact, an easy argument in our analysis shows that if $Q ^ { \\pi } \\in { \\mathcal { F } } _ { \\pi , \\varepsilon }$ for all $\\pi \\in \\Pi$ , then $\\Delta f ( s _ { 0 } , \\pi )$ is an upper bound on the regret $J ( \\pi ) - J ( \\widehat { \\pi } )$ of our estimator relative to any $\\pi$ we wish to compete with. ",
+        "bbox": [
+            173,
+            90,
+            826,
+            148
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theoretical analysis. To leverage this observation, we first define the a critical threshold $\\varepsilon _ { r }$ which ensures that (the best approximation of) $Q ^ { \\pi }$ is indeed contained in our version spaces for all $\\pi$ : ",
+        "bbox": [
+            173,
+            165,
+            823,
+            194
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a2c404df3073ad0a3c4b34098c0245d92d528d2fc2a4da05d21508cb86d68fc.jpg",
+        "text": "$$\n\\varepsilon _ { r } : = \\frac { 1 3 9 V _ { \\mathrm { m a x } } ^ { 2 } \\log \\frac { | \\mathcal { F } | | \\Pi | } { \\delta } } { n } + 3 9 \\varepsilon _ { \\mathcal { F } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            198,
+            614,
+            233
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With this definition, we now give a more refined bound on the regret of our algorithm (3.2) by further splitting the error estimate $\\Delta f _ { \\pi } ( s _ { 0 } , \\pi )$ which is random owing to its dependence on the version space, and analyze it through a novel decomposition into on-support and off-support components. While we bound the on-support error using standard techniques, the off-support error is akin to a bias term which captures the interplay between the data collection distribution and function approximation in the quality of the final solution. Also note that our choice of $\\varepsilon _ { r }$ requires the knowledge of $\\varepsilon { \\mathcal { F } }$ , which is a common characteristic of version-space-based algorithms [e.g., Jiang et al., 2017]. The challenge of unknown $\\varepsilon _ { F }$ can be possibly addressed using model-selection techniques in practice and we leave further investigation to future work. ",
+        "bbox": [
+            173,
+            236,
+            826,
+            361
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1. Let $\\varepsilon = \\varepsilon _ { r }$ where is $\\varepsilon _ { r }$ defined in Eq.(3.3) and $\\widehat { \\pi }$ be obtained by Eq.(3.2). Then, for any policy $\\pi \\in \\Pi$ and any constant $C _ { 2 } \\geq 1$ b, with probability at least $1 - \\delta$ , ",
+        "bbox": [
+            174,
+            363,
+            823,
+            392
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d06ca6840399e2b31e604db54a0c71b210b699f933eef7806b92aa2b1705c86.jpg",
+        "text": "$$\n\\begin{array} { r l } & { I ( \\pi ) - J ( \\widehat \\pi ) \\leq \\underbrace { \\mathcal { O } \\left( \\displaystyle \\frac { V _ { \\operatorname* { m a x } } \\sqrt { C _ { 2 } } } { 1 - \\gamma } \\sqrt { \\frac { \\log \\frac { | \\mathcal { F } | | \\Pi | } { \\delta } } { n } } + \\frac { \\sqrt { C _ { 2 } ( \\varepsilon _ { \\mathcal { F } , \\mathcal { F } } + \\varepsilon _ { \\mathcal { F } } ) } } { 1 - \\gamma } \\right) } _ { \\mathrm { e r r } _ { \\operatorname* { m i n } } ( \\pi ) : o n \\cdot u p p o r t e r o r } } \\\\ & { \\quad \\quad \\quad + \\frac { 1 } { 1 - \\gamma } \\cdot \\underset { \\nu \\cdot \\mathcal { O } ( \\nu ; \\mu , \\mathcal { F } , \\pi ) \\leq C _ { 2 } } { \\operatorname* { m i n } } \\underset { ( s , a ) \\in S \\times A } { \\sum } ( d _ { \\pi } \\setminus \\nu ) ( s , a ) \\left[ \\Delta f _ { \\pi } ( s , a ) - \\gamma ( \\mathcal { P } ^ { \\pi } \\Delta f _ { \\pi } ) ( s , a ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            397,
+            833,
+            510
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi )$ is defined in Definition $I$ , $( d _ { \\pi } \\setminus \\nu ) ( s , a ) : = \\operatorname* { m a x } ( d _ { \\pi } ( s , a ) - \\nu ( s , a ) , 0 )$ and $( \\mathcal P ^ { \\pi } f ) ( s , a ) = \\mathbb { E } _ { s ^ { \\prime } \\sim P ( \\cdot | s , a ) } [ f ( s ^ { \\prime } , \\pi ) ]$ for any $f$ . ",
+        "bbox": [
+            173,
+            546,
+            825,
+            577
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Bias-variance decomposition. Note that decomposition of our error bound into on-support and off-support parts effectively achieves a bias-variance tradeoff. A small value of the concentrability threshold $C _ { 2 }$ requires the choice of the distribution $\\nu$ closer to $\\mu$ , which results in better estimation error guarantee (which is $\\mathcal { O } ( \\sqrt { C _ { 2 } / n } ) )$ when we transfer from $\\mu$ to $\\nu$ , but potentially pays a high bias due to the mismatch between $d _ { \\pi }$ and $\\nu$ . A larger threshold permits more flexibility in choosing $\\nu$ similar to $d _ { \\pi }$ for a smaller bias, but results in a larger variance and estimation error. Rather than commit to a particular tradeoff, our estimator automatically adapts to the best possible splitting (Figure 1 illustrates this concept) by allowing us to choose the best threshold $C _ { 2 }$ . The on-support part matches the $n$ rate (fast rate error bound) of API or AVI analysis (e.g., [Pires and Szepesvári, 2012; Lazaric et al., 2012; Chen and Jiang, 2019]). The dependency on horizon is only linear and matches the best previous result with concentrability assumption [Xie and Jiang, 2020]. For the off-support part, it depends on the off-support mass $d _ { \\pi } \\setminus \\nu$ and the “quality” of the off-support estimation: if all value functions in the version space are close to each other in the off-support region for policy $\\pi$ , the gap between $J ( \\pi )$ and $J ( \\widehat { \\pi } )$ can still be small even with a large boff-support mass. The following corollary formally states this property. ",
+        "bbox": [
+            174,
+            588,
+            549,
+            796
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9efc325511f863aac091fd74b7ddb94076412c3673defbcfbbcbe122b0ea0604.jpg",
+        "image_caption": [
+            "Figure 1: An example illustrating different on-support and off-support splittings (denoted by two different vertical lines). Different splitting has different $C _ { 2 }$ values, and further yields different bias-variance trade-offs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            562,
+            589,
+            821,
+            686
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            796,
+            825,
+            881
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1 (“Double Robustness”). Under conditions of Theorem 3.1, for any $\\pi$ and $C _ { 2 } \\geq 0$ $\\mathrm { e r r } _ { \\mathrm { o f f } } ( \\pi ) = 0$ when either (1) $\\mathcal { C } ( d _ { \\pi } ; \\mu , \\mathcal { F } , \\pi ) \\le C _ { 2 }$ , or, $( 2 ) \\Delta f _ { \\pi } - \\gamma \\mathcal { P } ^ { \\pi } \\Delta f _ { \\pi } \\equiv 0$ . ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Adaptive guarantees by algorithm design. As mentioned above, Theorem 3.1 implicitly selects the best bias-variance decomposition through the best choice of $C _ { 2 }$ in hindsight, with this decomposition being purely a proof technique, not an knob in the algorithm. In contrast, many prior approaches [Liu et al., 2020; Fujimoto et al., 2019; Kumar et al., 2019] employ explicit thresholds to control density ratios in their algorithms, which makes the tradeoff a hyperparameter in their algorithms. Since choosing hyperparameters is particularly challenging in offline RL, where even policy evaluation can be unreliable, this novel axis of adaptivity is an extremely desirable property of our approach. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            189
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Comparison to guarantees in the exploratory setting. When a dataset with full coverage is given, classical analyses provide near-optimality guarantees that compete with the optimal policy $\\pi ^ { \\star }$ with a polynomial sample complexity, when $\\pi ^ { \\star } \\in \\Pi$ and both realizability and completeness hold for $\\mathcal { F }$ ; see [Antos et al., 2008] for a representative analysis. As mentioned earlier, such analysis often requires boundedness of $\\| \\nu / \\mu \\| _ { \\infty }$ for all admissible distributions $\\nu \\in \\{ d _ { \\pi } : \\pi \\in \\Pi \\}$ . On the other hand, it is easily seen that we can compete with $\\pi ^ { \\star }$ under much weaker conditions. ",
+        "bbox": [
+            173,
+            212,
+            826,
+            296
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2 (Competing with optimal policy). Under conditions of Theorem 3.1, if $\\mathcal { C } ( d _ { \\pi ^ { \\star } } ; \\mu , \\mathcal { F } , \\pi ^ { \\star } ) \\le C _ { 2 }$ , we have ",
+        "bbox": [
+            173,
+            300,
+            825,
+            329
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/895d8f82beb1059a25edcc9d0982e2859efcae71c06fdf263a9aa1f5e044a6ba.jpg",
+        "text": "$$\nJ ( \\pi ^ { \\star } ) - J ( \\widehat \\pi ) \\leq \\mathcal O \\left( \\frac { V _ { \\operatorname* { m a x } } \\sqrt { C _ { 2 } } } { 1 - \\gamma } \\sqrt { \\frac { \\log \\frac { | \\mathcal F | | \\Pi | } { \\delta } } { n } + \\frac { \\sqrt { C _ { 2 } ( \\varepsilon _ { \\mathcal F , \\mathcal F } + \\varepsilon _ { \\mathcal F } ) } } { 1 - \\gamma } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            337,
+            736,
+            387
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Notably, these milder coverage assumptions in Corollaries 1 and 2 provide offline RL counterparts to the benefits of policy-gradient-style methods with online access to the environment [Kakade and Langford, 2002; Scherrer, 2014; Agarwal et al., 2019]. ",
+        "bbox": [
+            173,
+            401,
+            826,
+            444
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Comparison with Liu et al. [2020]. The closest prior result to our work is that of [Liu et al., 2020], who develop a pessimistic estimator that truncates Bellman backups from state-action pairs infrequently visited by $\\mu$ , and analyzes the resulting pessimistic policy and value iteration algorithms under general function approximation. For truncating Bellman backups, however, their work requires estimating the state-action distribution of data, which can be challenging in high-dimensions and they incur additional errors from density estimation which we avoid. Further, their algorithms only compete with policies $\\pi$ where $\\| d _ { \\pi } / \\mu \\| _ { \\infty }$ is bounded instead of the more general result that we provide, and makes their results vacuous in a linear MDP setting under typical feature coverage assumptions. ",
+        "bbox": [
+            173,
+            459,
+            825,
+            585
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Safe Policy Improvement. Some prior works [e.g. Laroche et al., 2019; Liu et al., 2020] discuss the scenario where the dataset $\\mathcal { D }$ is collected with a behavior policy $\\pi _ { b }$ with $\\mu = d _ { \\pi _ { b } }$ , and demonstrate that their algorithms always return a policy competitive with $\\pi _ { b }$ . In our setup, this is straightforward as $d _ { \\pi _ { b } }$ is always covered, as shown next. ",
+        "bbox": [
+            173,
+            601,
+            826,
+            657
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3 (Bounded degradation from behavior policy). Under conditions of Theorem 3.1, if $\\mu = d _ { \\pi _ { b } }$ for some policy $\\pi _ { b } \\in \\Pi$ , we have ",
+        "bbox": [
+            173,
+            661,
+            823,
+            690
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/728c0b45b8023c12b05b05b18c55d2210ab6a646ba5d38617f6c79f0d8b24b89.jpg",
+        "text": "$$\nJ ( \\pi _ { b } ) - J ( \\widehat { \\pi } ) \\leq \\mathcal { O } \\left( \\frac { V _ { \\operatorname* { m a x } } } { 1 - \\gamma } \\sqrt { \\frac { \\log \\frac { | \\mathcal { F } | | \\Pi | } { \\delta } } { n } } + \\frac { \\sqrt { \\varepsilon _ { \\mathcal { F } , \\mathcal { F } } + \\varepsilon _ { \\mathcal { F } } } } { 1 - \\gamma } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            699,
+            700,
+            751
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 3.1. We now briefly describe the core ideas in the proof. More detailed arguments are deferred to the full proof in Appendix B.1. ",
+        "bbox": [
+            173,
+            765,
+            825,
+            795
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The key to prove Theorem 3.1 is to translate the $J ( \\pi ) - J ( \\widehat { \\pi } )$ to the Bellman error of value functions in the version space $\\mathcal { F } _ { \\pi , \\varepsilon }$ b. Our main proving strategies are as follows: ",
+        "bbox": [
+            171,
+            800,
+            823,
+            830
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. As the selection of $\\varepsilon = \\varepsilon _ { r }$ ensures the accurate estimation $Q ^ { \\pi }$ is contained in the version space $\\mathcal { F } _ { \\widehat { \\pi } , \\varepsilon }$ for any $\\pi$ , we can obtain $\\begin{array} { r } { J ( \\pi ) - J ( \\widehat \\pi ) \\leq \\operatorname* { m a x } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) - \\operatorname* { m i n } _ { f \\in \\mathcal { F } _ { \\widehat \\pi , \\varepsilon } } f ( s _ { 0 } , \\widehat \\pi ) + } \\end{array}$ bapproximation error. 2. By the optimality of $\\widehat { \\pi }$ , we have $\\begin{array} { r } { \\operatorname* { m i n } _ { f \\in \\mathcal { F } _ { \\widehat { \\pi } , \\varepsilon } } f ( s _ { 0 } , \\widehat { \\pi } ) \\geq \\operatorname* { m i n } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) } \\end{array}$ . This indicates that $\\begin{array} { r } { J ( \\pi ) - J ( \\widehat { \\pi } ) \\leq \\operatorname* { m a x } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) - \\operatorname* { m i n } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) + \\mathrm { a p p r o x i m a t i } } \\end{array}$ ion error. ",
+        "bbox": [
+            173,
+            837,
+            826,
+            914
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3. By using a standard telescoping argument (e.g., [Xie and Jiang, 2020, Lemma 1]), $\\begin{array} { r } { \\operatorname* { m a x } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) \\ - \\ \\operatorname* { m i n } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } \\bar { f } ( s _ { 0 } , \\pi ) } \\end{array}$ can be upper bounded by the Bellman error of $\\mathrm { a r g m a x } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f \\mathopen { } \\mathclose \\bgroup \\left( s _ { 0 } , \\pi \\aftergroup \\egroup \\right)$ and $\\begin{array} { r } { \\operatorname * { a r g m i n } _ { f \\in \\mathcal { F } _ { \\pi , \\varepsilon } } f ( s _ { 0 } , \\pi ) } \\end{array}$ over distribution $d _ { \\pi }$ . ",
+        "bbox": [
+            174,
+            90,
+            825,
+            137
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "After combining all the three steps above together and considering the distribution shift effect, we complete the proof. ",
+        "bbox": [
+            173,
+            141,
+            825,
+            170
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.1 Results for Linear Function Approximation ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            183,
+            516,
+            199
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here we perform a case study in linear function approximation. We will show that our results—when instantiated under linear function approximation (with realizability and completeness assumptions)— automatically provides state-of-the-art guarantees, improving over existing results specialized to this setting by a factor of $\\mathcal O ( d )$ [Jin et al., 2021] when the action space is finite and small. ",
+        "bbox": [
+            174,
+            207,
+            826,
+            262
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We recall the linear function approximation setup (we set $R _ { \\operatorname* { m a x } } = 1$ and $\\begin{array} { r } { V _ { \\mathrm { m a x } } = \\frac { 1 } { 1 - \\gamma } } \\end{array}$ for consistency with literature). ",
+        "bbox": [
+            174,
+            266,
+            823,
+            297
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Linear Function Approximation). Let $\\phi : S \\times \\mathcal { A }  \\mathbb { R } ^ { d }$ be a feature mapping. Without loss of generality, we assume $\\| \\bar { \\phi ( s , a ) } \\| _ { 2 } \\le 1 , \\forall ( s , a ) \\in \\mathcal { S } \\times \\mathcal { A }$ . We define the value-function class $\\mathcal { F } _ { \\Phi }$ as $\\mathcal { F } _ { \\Phi } : = \\{ \\phi ( \\cdot , \\cdot ) ^ { \\top } \\theta : \\theta \\in \\mathbb { R } ^ { d } , \\phi ( \\cdot , \\cdot ) ^ { \\top } \\theta \\in [ 0 , V _ { \\operatorname* { m a x } } ] \\}$ , and the policy class $\\Pi _ { \\Phi }$ consists of the greedy policies of each value function in $\\mathcal { F } _ { \\Phi }$ . ",
+        "bbox": [
+            173,
+            299,
+            825,
+            357
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3 (Realizability and Completeness). $\\varepsilon _ { \\mathcal { F } , \\mathcal { F } } = \\varepsilon _ { \\mathcal { F } } = 0$ . ",
+        "bbox": [
+            174,
+            359,
+            612,
+            375
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that, when the feature mapping $\\phi ( \\cdot , \\cdot )$ is the one induced by the linear MDP [Jin et al., 2020], it automatically ensure that $\\pi ^ { \\star } \\in \\Pi _ { \\Phi }$ and $\\mathcal { F } _ { \\Phi }$ satisfies Assumptions 3. In contrast, we highlight that the standard linear function approximation or linear MDP setup does not entail all the assumptions needed by Liu et al. [2020] as mentioned earlier. ",
+        "bbox": [
+            174,
+            381,
+            825,
+            438
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Below is our main result in the linear function approximation setting. ",
+        "bbox": [
+            174,
+            440,
+            627,
+            457
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. Suppose the value-function class $\\mathcal { F }$ is a linear function class that satisfies realizability and completeness (Definition 2 and Assumption $^ 3$ ) and $\\widehat { \\pi }$ is the output of Eq.(3.2) using valuefunction class $\\mathcal { F } _ { \\Phi }$ and policy class $\\Pi _ { \\Phi }$ . If we choose $\\begin{array} { r } { \\varepsilon = c V _ { \\mathrm { m a x } } ^ { 2 } d \\log { \\frac { V _ { \\mathrm { m a x } } | A | d } { \\delta } } / n } \\end{array}$ , then, for any policy $\\pi : S  \\Delta ( { \\mathcal { A } } )$ , we have ",
+        "bbox": [
+            174,
+            458,
+            826,
+            517
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3dfbaf012e4a050d94aeef36ee86c3dfde2c102ced66c8c68d375b5641cf18b3.jpg",
+        "text": "$$\nJ ( \\pi ) - J ( \\widehat { \\pi } ) \\leq \\mathcal { O } \\left( \\frac { V _ { \\operatorname* { m a x } } } { 1 - \\gamma } \\sqrt { \\frac { d \\log \\frac { V _ { \\operatorname* { m a x } } | A | d } { \\delta } } { n } } \\mathbb { E } _ { d _ { \\pi } } \\left[ \\sqrt { \\phi ( s , a ) ^ { \\mathsf { T } } \\Sigma _ { \\mathcal { D } } ^ { - 1 } \\phi ( s , a ) } \\right] \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            518,
+            753,
+            570
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where c is an absolute constant, and $\\Sigma _ { \\mathcal { D } } : = \\mathbb { E } _ { \\mathcal { D } } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\mathsf { T } } \\right]$ . ",
+        "bbox": [
+            174,
+            571,
+            607,
+            589
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The detailed proof of Theorem 3.2 is provided in Appendix B.2. Our guarantee is structurally very similar to that of Jin et al. [2021, Theorem 4.4], except that we only need linear function approximation with realizability and completeness assumptions, and they consider the finite-horizon linear MDP setting. If we translate their result to the discounted setting by setting $H = \\mathcal { O } ( 1 / ( 1 - \\gamma ) )$ , we enjoy a net improvement of order $\\mathcal O ( d )$ in sample complexity when the action space is finite and small. To make it concrete, that is $\\mathcal { O } ( \\sqrt { d ^ { 2 } \\log ( d n / \\delta ) / n } )$ vs. $\\mathcal { O } ( \\sqrt { d \\log ( d \\vert A \\vert / \\delta ) / n } )$ error bounds. The bound also shows that having a full-rank $\\Sigma _ { \\mathcal { D } }$ (which is ensured by a full-rank covariance under $\\mu$ ) is sufficient for consistent offline RL in linear function approximation. Crucially, the full-rank covariance is an easily checkable condition on data, as opposed to unverifiable concentrability assumptions. As a caveat, our results do not imply a computationally efficient algorithm, as a naïve implementation involves evaluating each policy pessimistically to pick the best. We discuss a computationally efficient adaptation of our approach in the next section. ",
+        "bbox": [
+            173,
+            594,
+            826,
+            765
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Practical Algorithm — Regularized Offline Policy Optimization ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            780,
+            736,
+            799
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A major challenge using the proposed algorithm in Section 3 in practice is that searching the policy with the best pessimistic evaluation over the policy space $\\Pi$ is not computationally tractable. In this section, we present a practical algorithm that is computationally efficient assuming access to a (regularized) loss minimization oracle over the value function class $\\mathcal { F }$ , and also comes with rigorous theoretical guarantees. ",
+        "bbox": [
+            174,
+            809,
+            825,
+            878
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our practical algorithm is summarized in Algorithm 1. It has three key differences from the information-theoretic version in Eq.(3.2): ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 PSPI: Pessimistic Soft Policy Iteration ",
+        "bbox": [
+            178,
+            90,
+            514,
+            106
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Input: Batch data $\\mathcal { D }$ , regularization coefficient $\\lambda$ . ",
+        "bbox": [
+            176,
+            108,
+            501,
+            122
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize policy $\\pi _ { 1 }$ as the uniform policy.   \n2: for $t = 1 , 2 , \\dots , T$ do   \n3: Obtain the pessimistic estimation for $\\pi _ { t }$ as $f _ { t }$ , ",
+        "bbox": [
+            181,
+            125,
+            526,
+            166
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/731638b2c31b3eb153ad7b3ae9935e21e3732f3af8bc0ec39e3f06b58e7e8caa.jpg",
+        "text": "$$\nf _ { t } \\gets \\underset { f \\in \\mathcal { F } } { \\mathrm { a r g m i n } } \\left( f ( s _ { 0 } , \\pi _ { t } ) + \\lambda \\mathcal { E } ( f , \\pi _ { t } ; \\mathcal { D } ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            172,
+            650,
+            200
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { E } ( f , \\pi _ { t } ; \\mathcal { D } )$ is defined in Eq.(3.1). ",
+        "bbox": [
+            200,
+            207,
+            465,
+            222
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4: Calculate $\\pi _ { t + 1 }$ by, ",
+        "bbox": [
+            181,
+            223,
+            348,
+            236
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/29dc398c4c2fa6b8dd250edc823c0a1dec93b88aef4fb471f2e2ed59135692c6.jpg",
+        "text": "$$\n\\pi _ { t + 1 } ( a | s ) \\propto \\pi _ { t } ( a | s ) \\exp \\left( \\eta f _ { t } ( s , a ) \\right) , \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            242,
+            689,
+            260
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5: end for ",
+        "bbox": [
+            181,
+            267,
+            253,
+            280
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6: Output $\\bar { \\pi } : = \\mathsf { U n i f } ( \\pi _ { [ 1 : T ] } )$ ",
+        "bbox": [
+            181,
+            280,
+            370,
+            296
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "▷ uniformly mix $\\pi _ { 1 } , \\ldots , \\pi _ { T }$ at the trajectory level ",
+        "bbox": [
+            495,
+            280,
+            825,
+            295
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1. The pessimistic policy evaluation is now performed via regularization (Line 3) instead of constrained optimization. ",
+        "bbox": [
+            171,
+            327,
+            825,
+            356
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "2. Instead of searching over an explicit policy space Π, we search over a policy class implicitly induced from $\\mathcal { F }$ (defined in Eq.(4.2)) and therefore no longer have a policy class independent of $\\mathcal { F }$ separately, which is a common practice in API-style algorithms [Munos, 2003; Antos et al., 2008; Lazaric et al., 2012]. ",
+        "bbox": [
+            174,
+            357,
+            826,
+            412
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3. We optimize the policy using mirror descent updates, which yields computationally tractable optimization over the implicit policy class. This property has been leveraged in many prior works, although typically in online RL settings [Even-Dar et al., 2009; Agarwal et al., 2019; Geist et al., 2019; Cai et al., 2020; Shani et al., 2020]. ",
+        "bbox": [
+            174,
+            415,
+            826,
+            470
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that the use of a specific policy class above can be relaxed if a stronger structural assumption is made on the MDP (e.g., linear MDPs [Jin et al., 2020, 2021]). ",
+        "bbox": [
+            171,
+            477,
+            823,
+            506
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 Analysis of Algorithm 1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            520,
+            377,
+            536
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now provide the analysis of Algorithm 1 in this section. For ease of presentation, we formally define the implicit policy class for this section: ",
+        "bbox": [
+            174,
+            545,
+            825,
+            574
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4104aca62971d53971b8c77516a783da0f4db9215e987e3b930b3a00ca58bf19.jpg",
+        "text": "$$\n\\Pi _ { \\mathrm { S P I } } : = \\{ \\pi ^ { \\prime } ( \\cdot | s ) \\propto \\exp ( \\eta \\sum _ { i = 1 } ^ { t } f ^ { ( t ) } ( s , \\cdot ) ) : 1 \\le t \\le T , f ^ { ( 1 ) } , \\ldots , f ^ { ( i ) } \\in \\mathcal { F } \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            579,
+            743,
+            622
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "which is the natural policy class for soft policy-iteration approaches. The following theorem describes the performance guarantee of $\\bar { \\pi }$ . ",
+        "bbox": [
+            169,
+            626,
+            825,
+            655
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1. Let $\\begin{array} { r } { \\lambda = \\sqrt [ 3 ] { V _ { \\mathrm { m a x } } \\big / ( 1 - \\gamma ) { } ^ { 2 } \\varepsilon _ { r } ^ { 2 } } } \\end{array}$ with $\\varepsilon _ { r }$ in Eq.(3.3), $\\begin{array} { r } { \\eta = \\sqrt { \\frac { \\log | \\mathcal { A } | } { 2 V _ { \\operatorname* { m a x } } ^ { 2 } T } } } \\end{array}$ , and $\\bar { \\pi }$ be obtained from Algorithm 1. For any policy $\\pi : S  \\Delta ( { \\mathcal { A } } )$ we wish to compete with, suppose Assumptions 1 and 2 hold with respect to the policy class $\\Pi _ { S P I } \\cup \\{ \\pi \\}$ . Then, for any constant $C _ { 2 } \\geq 1$ , we have with probability at least $1 - \\delta$ , ",
+        "bbox": [
+            173,
+            660,
+            825,
+            724
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccfae0d10bcf9b79f78f138898f63e11dff5c48d5502a76181e2a6901f481494.jpg",
+        "text": "$$\n\\begin{array} { r l } & { J ( \\pi ) - J ( \\overline { { \\pi } } ) } \\\\ & { \\leq \\underbrace { \\mathcal { O } \\left( \\sqrt { C _ { 2 } } \\left( \\frac { \\sqrt { \\varepsilon _ { \\mathcal { F } , \\mathcal { F } } + \\varepsilon _ { \\mathcal { F } } } } { 1 - \\gamma } + \\frac { V _ { \\operatorname* { m a x } } } { 1 - \\gamma } \\sqrt [ 3 ] { \\frac { T \\log { \\frac { | \\mathcal { F } | } { \\delta } } } { n } } + \\sqrt [ 3 ] { \\frac { V _ { \\operatorname* { m a x } } \\varepsilon _ { \\mathcal { F } } } { ( 1 - \\gamma ) ^ { 2 } } } \\right) \\right) } _ { \\mathrm { e r r } _ { \\operatorname* { m i n } } ( \\pi ) : o n \\cdot s u p p o r e r m r } + \\underbrace { \\mathcal { O } \\left( \\frac { V _ { \\operatorname* { m a x } } } { 1 - \\gamma } \\sqrt { \\frac { \\log { | \\mathcal { A } | } } { T } } \\right) } _ { \\displaystyle 1 - \\gamma } } \\\\ & { + \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left( \\underset { \\nu : \\mathcal { C } ( \\nu ; \\mu , F , \\pi _ { t } ) \\leq C _ { 2 } } { \\operatorname* { m i n } } \\left| \\underset { ( s , a ) \\in S \\times A } { \\sum _ { m } } \\frac { \\left( d _ { \\pi } \\setminus \\nu \\right) ( s , a ) [ f _ { t } ( s , a ) - ( \\mathcal { T } ^ { \\pi _ { t } } f _ { t } ) ( s , a ) ] } { 1 - \\gamma } \\right| \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            731,
+            816,
+            872
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "$\\operatorname { e r r } _ { \\mathrm { o f f } } ( \\pi ) .$ {z: off-support error ",
+        "bbox": [
+            416,
+            877,
+            553,
+            890
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi _ { t } )$ is defined in Definition $^ { l }$ , $( d _ { \\pi } \\setminus \\nu ) ( s , a ) : = \\operatorname* { m a x } ( d _ { \\pi } ( s , a ) - \\nu ( s , a ) , 0 ) .$ . ",
+        "bbox": [
+            166,
+            896,
+            782,
+            914
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We provide a proof sketch of Theorem 4.1 at the end of this section, and defer the full proof to Appendix C. We now make a few remarks about the results in Theorem 4.1. ",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Measurement of distribution shift effect. Compared with the information-theoretical result (provided in Theorem 3.1), the measurement of distribution shift in Theorem 4.1 depends on the optimization trajectory. That is, it measures the distance between two distribution $\\nu$ and $\\mu$ by $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi _ { t } )$ $( \\pi _ { [ 1 : T ] }$ is the sequence of policies produced by the algorithm) whereas Theorem 3.1 uses $\\mathcal { C } ( \\nu ; \\mu , \\mathcal { F } , \\pi )$ ( $\\pi$ is the baseline policy we compete with). We remark that both of these two measurements are weaker then traditional density-ratio definitions (e.g., [Munos and Szepesvári, 2008; Chen and Jiang, 2019; Xie and Jiang, 2020]) as we demonstrated before, as the dependence of $\\mathcal { C }$ on $\\pi$ is relatively secondary. ",
+        "bbox": [
+            173,
+            132,
+            825,
+            244
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dependence on $T$ . The number of optimization rounds $T$ affects the bound in two opposite ways: as $T$ increases, the optimization error term decreases, whereas the second term of the on-support error increases. The latter increase is due to the complexity of the implicit policy class $\\Pi$ growing exponentially with $T$ , which affects our concentration bounds. To optimize the bound, the optimal choice is $T = \\mathcal { O } ( n ^ { 2 / 5 } )$ , leading to an overall $\\mathcal { O } ( n ^ { - 1 / 5 } )$ rate. While such a rate is relatively slow, we remark that the complexity bound of $\\Pi$ is conservative, and in certain cases it is possible to obtain much sharper bounds: for example, in linear function approximation (Section 3.1), $\\Pi _ { \\mathrm { S P I } }$ are a priori captured by the space of softmax policies, whose complexity has no dependence on $T$ (up to mild logarithmic dependence due to norms). That is, the $\\operatorname { e r r } _ { \\mathrm { o n } } ( \\pi )$ term in Theorem 4.1 reduces to $\\widetilde { \\mathcal { O } } ( \\frac { V _ { \\mathrm { m a x } } } { 1 - \\gamma } \\sqrt [ 3 ] { d / n } )$ $( \\varepsilon _ { \\mathcal { F } , \\mathcal { F } } = \\varepsilon _ { \\mathcal { F } } = 0$ in linear function approximation), and yields an overall $\\mathcal { O } ( n ^ { - 1 / 3 } )$ rate. ",
+        "bbox": [
+            173,
+            257,
+            826,
+            412
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Bias-variance decomposition. Similar to Theorem 3.1, Theorem 4.1 also allows arbitrary decomposition of the error bound into on-support and off-support components by setting the concentrability threshold $C _ { 2 }$ , which serves as a bias-variance tradeoff as before. In fact, the splitting can be done separately for each $\\pi _ { t }$ in $1 \\leq t \\leq T$ and we omit such flexibility for readability. The optimization error does not depend on the splitting. Our performance guarantee naturally adapts to the best possible decomposition as before. As in Theorem 3.1, if the estimation on the off-support region is “highquality”, we can further simplify the performance guarantees, but the requirement of “high-quality” is different from that of Corollary 1. We make it formal in the following corollary. ",
+        "bbox": [
+            173,
+            425,
+            826,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4 (“Double Robustness”). For any $\\pi$ and $C _ { 2 } \\geq 0$ , $\\mathrm { e r r } _ { \\mathrm { o f f } } ( \\pi ) = 0$ when either (1) $\\mathcal { C } ( d _ { \\pi } ; \\mu , \\mathcal { F } , \\pi _ { t } ) \\le C _ { 2 }$ for all $t \\in [ T ]$ , or, (2) $f _ { t } - \\mathcal { T } ^ { \\pi _ { t } } \\Delta f _ { t } \\equiv 0$ for all $t \\in [ T ]$ . ",
+        "bbox": [
+            173,
+            540,
+            823,
+            570
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We note that the conditions above depend on the optimization trajectory through their dependence on $\\pi _ { t }$ , but can be made algorithm-independent by instead asserting the stronger requirement that $\\mathcal { C } ( d _ { \\pi } , \\mu , \\mathcal { F } , \\pi ^ { \\prime } ) \\le C _ { 2 }$ for all $\\pi ^ { \\prime } \\in \\Pi _ { \\mathrm { S P I } }$ in the first condition. ",
+        "bbox": [
+            176,
+            577,
+            821,
+            619
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Competing with the optimal policy. As before, we can provide a guarantee for competing with the optimal policy, under coverage assumptions weaker than the typical batch RL literature, albeit slightly stronger than those of Corollary 2. We state the formal result below. ",
+        "bbox": [
+            174,
+            631,
+            826,
+            674
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Corollary 5 (Competing with optimal policy). Under conditions of Theorem 4.1, if $\\mathcal { C } ( d _ { \\pi ^ { \\star } } ; \\mu , \\mathcal { F } , \\pi ) \\le C _ { 2 }$ for all $\\pi \\in \\Pi _ { S P I } ,$ , we have ",
+        "bbox": [
+            173,
+            676,
+            825,
+            705
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b572cdd740a9416cc80c97225b48e6941fd4cad98fb94ff37430e8fd60642c03.jpg",
+        "text": "$$\nJ ( \\pi ^ { \\star } ) - J ( \\widehat \\pi ) \\leq \\mathcal O \\left( \\frac { V _ { \\operatorname* { m a x } } \\sqrt { C _ { 2 } } } { 1 - \\gamma } \\left( \\frac { \\log \\frac { | \\mathcal F | } { \\delta } \\log | A | } { n } \\right) ^ { 1 / 5 } + \\frac { \\sqrt { C _ { 2 } ( \\varepsilon _ { \\mathcal F } , \\mathcal F + \\varepsilon _ { \\mathcal F } ) } } { 1 - \\gamma } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            708,
+            767,
+            758
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that the conditions of Corollary 5 are satisfied as before whenever $\\| d _ { \\pi ^ { \\star } } / \\mu \\| _ { \\infty } \\leq C _ { 2 }$ ",
+        "bbox": [
+            171,
+            767,
+            761,
+            784
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Computationally-efficient implementation with linear function approximation We remark that our algorithm is computationally efficient when the value-function class $\\mathcal { F }$ is linear, that is, $\\mathcal { F } : = \\{ \\phi ( \\cdot , \\cdot ) ^ { \\top } \\theta : \\theta \\in \\mathbb { R } ^ { d } \\}$ . In this case, the objective of Eq.(4.1) has a closed-form expression which is quadratic in $\\theta$ . In addition, under additional matrix invertibility conditions, Eq.(4.1) has a closed-form solution which generalizes LSTDQ [Lagoudakis and Parr, 2003; Sutton et al., 2009; Dann et al., 2014]. A similar connection has been made by Antos et al. [2008], but our derivation is more general. See Appendix Appendix D for further details. ",
+        "bbox": [
+            173,
+            795,
+            825,
+            892
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We conclude the section with a proof sketch showing the key insights used in establishing the proof. ",
+        "bbox": [
+            174,
+            896,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 4.1. Our proof constructs a corresponding MDP $\\mathcal { M } _ { t }$ for every $f _ { t } , \\pi _ { t }$ pair. Each $\\mathcal { M } _ { t }$ has the same dynamics as the ground-truth MDP, but chooses a different reward function, such that $f _ { t }$ is the $Q$ -function of $\\pi _ { t }$ in $\\mathcal { M } _ { t }$ , $Q _ { \\mathcal { M } _ { t } } ^ { \\pi _ { t } }$ (we use the subscript of $\\mathcal { M } _ { t }$ to denote the corresponding value or operator in MDP $\\mathcal { M } _ { t }$ ). Our proof relies on some key properties of $\\mathcal { M } _ { t }$ , such as $Q ^ { \\pi ^ { - } } - T _ { \\mathcal { M } _ { t } } ^ { \\pi ^ { } } \\bar { Q } ^ { \\pi } = f _ { t } - \\bar { T } ^ { \\pi _ { t } } f _ { t }$ . We decompose $J ( \\pi ) - J ( \\bar { \\pi } )$ as follows. ",
+        "bbox": [
+            174,
+            90,
+            826,
+            162
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56799b224cf53700e8ae8ed1d2c1aad28570949e281a092a7d2248938b4d8ac3.jpg",
+        "text": "$$\n\\begin{array} { r l } & { J ( \\pi ) - J ( \\bar { \\pi } ) \\le \\underbrace { \\cfrac { 1 } { T } \\displaystyle \\sum _ { t = 1 } ^ { T } \\left( J _ { \\mathcal M _ { t } } ( \\pi ) - J _ { \\mathcal M _ { t } } ( \\pi _ { t } ) \\right) } _ { \\mathrm { o p t i m i z a t i o n ~ e r r o r ~ } } + \\underbrace { \\cfrac { 1 } { T } \\displaystyle \\sum _ { t = 1 } ^ { T } ( J ( \\pi ) - J _ { \\mathcal M _ { t } } ( \\pi ) ) } _ { \\mathrm { c o n t r o l e d b y ~ } \\| Q ^ { \\pi } - T _ { \\mathcal M _ { t } } ^ { \\pi } Q ^ { \\pi } \\| _ { 2 , d \\pi } = \\| f _ { t } - \\mathcal T ^ { \\pi } t f _ { t } \\| _ { 2 , d \\pi } } } \\\\ & { \\quad \\quad + \\underset { \\mathrm { a p p r o x i m a t i o n / s t a t i s t i c a l ~ e r r o r s . } } { \\mathrm { c o n t r o l e x . } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            199,
+            169,
+            803,
+            253
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The proof is completed by bounding $\\| f _ { t } - T ^ { \\pi _ { t } } f _ { t } \\| _ { 2 , d _ { \\pi } }$ on both on-support and off-support regions. ",
+        "bbox": [
+            169,
+            257,
+            794,
+            273
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            291,
+            307,
+            308
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper investigates sample-efficient offline reinforcement learning without data coverage assumptions (e.g., concentrability). To achieve that goal, our paper contributes several crucial improvements to the literature. We introduce the concept of Bellman-consistent pessimism. It enables the sample-efficient guarantees with only the Bellman-completeness assumption which is standard in the exploratory setting, whereas the point-wise/bonus-based pessimism popularly adopted in the literature usually requires stronger and/or extra assumptions. Algorithmically, we demonstrate how to implicitly infer a policy value lower bound through a version space and provide a tractable implementation. A particularly important aspect of our results is the ability to adapt to the best bias-variance tradeoff in the hindsight, which no prior algorithms achieve to the best of our knowledge. When applying our results in linear function approximation, we attain an $\\mathcal O ( d )$ improvement in sample complexity, compared with the best-known recent work of offline RL in linear MDPs, whenever the action space is finite and small. ",
+        "bbox": [
+            174,
+            321,
+            825,
+            487
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As of limitations and future work, the sample complexity of our practical algorithm is worse than that of the information-theoretic approach, and it will be interesting to close this gap. Another future direction is to empirically evaluate PSPI on benchmarks and compare it to existing approaches. ",
+        "bbox": [
+            174,
+            493,
+            825,
+            535
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            554,
+            321,
+            570
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Part of this work was carried out while TX and AA worked at Microsoft Research. NJ acknowledges funding support from the ARL Cooperative Agreement W911NF-17-2-0196, NSF IIS-2112471, and Adobe Data Science Research Award. ",
+        "bbox": [
+            174,
+            584,
+            825,
+            627
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            645,
+            266,
+            661
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Doina Precup, Richard S Sutton, and Satinder P Singh. Eligibility traces for off-policy policy evaluation. In Proceedings of the Seventeenth International Conference on Machine Learning, pages 759–766, 2000.   \nAndrás Antos, Csaba Szepesvári, and Rémi Munos. Learning near-optimal policies with bellmanresidual minimization based fitted policy iteration and a single sample path. Machine Learning, 71 (1):89–129, 2008.   \nSergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020.   \nJinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. In International Conference on Machine Learning, pages 1042–1051, 2019.   \nRuosong Wang, Dean Foster, and Sham M Kakade. What are the statistical limits of offline rl with linear function approximation? In International Conference on Learning Representations, 2020.   \nRuosong Wang, Yifan Wu, Ruslan Salakhutdinov, and Sham M Kakade. Instabilities of offline rl with pre-trained neural representation. arXiv preprint arXiv:2103.04947, 2021. ",
+        "bbox": [
+            171,
+            667,
+            828,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rémi Munos. Error bounds for approximate policy iteration. In Proceedings of the Twentieth International Conference on International Conference on Machine Learning, pages 560–567, 2003. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Csaba Szepesvári and Rémi Munos. Finite time bounds for sampling based fitted value iteration. In Proceedings of the 22nd international conference on Machine learning, pages 880–887, 2005. ",
+        "bbox": [
+            173,
+            143,
+            823,
+            172
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rémi Munos and Csaba Szepesvári. Finite-time bounds for fitted value iteration. Journal of Machine Learning Research, 9(5), 2008. ",
+        "bbox": [
+            174,
+            183,
+            821,
+            213
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Amir Massoud Farahmand, Rémi Munos, and Csaba Szepesvári. Error propagation for approximate policy and value iteration. In Advances in Neural Information Processing Systems, 2010. ",
+        "bbox": [
+            173,
+            222,
+            825,
+            252
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie and Nan Jiang. $\\mathbf { Q } ^ { * }$ approximation schemes for batch reinforcement learning: A theoretical comparison. In Conference on Uncertainty in Artificial Intelligence, pages 550–559. PMLR, 2020. ",
+        "bbox": [
+            173,
+            262,
+            825,
+            304
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rahul Kidambi, Aravind Rajeswaran, Praneeth Netrapalli, and Thorsten Joachims. Morel: Modelbased offline reinforcement learning. In NeurIPS, 2020. ",
+        "bbox": [
+            171,
+            315,
+            825,
+            344
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. In International Conference on Machine Learning, pages 2052–2062, 2019. ",
+        "bbox": [
+            171,
+            354,
+            823,
+            385
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative q-learning for offline reinforcement learning. arXiv preprint arXiv:2006.04779, 2020. ",
+        "bbox": [
+            173,
+            393,
+            823,
+            422
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforce ment learning without great exploration. arXiv preprint arXiv:2007.08202, 2020. ",
+        "bbox": [
+            174,
+            433,
+            821,
+            463
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? In International Conference on Machine Learning, pages 5084–5096. PMLR, 2021. ",
+        "bbox": [
+            174,
+            472,
+            823,
+            502
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Eyal Even-Dar, Sham M Kakade, and Yishay Mansour. Online markov decision processes. Mathematics of Operations Research, 34(3):726–736, 2009. ",
+        "bbox": [
+            173,
+            512,
+            825,
+            541
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Geist, Bruno Scherrer, and Olivier Pietquin. A theory of regularized markov decision processes. In International Conference on Machine Learning, pages 2160–2169. PMLR, 2019. ",
+        "bbox": [
+            171,
+            551,
+            823,
+            582
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alekh Agarwal, Sham M Kakade, Jason D Lee, and Gaurav Mahajan. On the theory of policy gradient methods: Optimality, approximation, and distribution shift. arXiv preprint arXiv:1908.00261, 2019. ",
+        "bbox": [
+            176,
+            590,
+            825,
+            633
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dimitri P. Bertsekas and John N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1st edition, 1996. ISBN 1886529108. ",
+        "bbox": [
+            173,
+            643,
+            823,
+            674
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Vijay R Konda and John N Tsitsiklis. Actor-critic algorithms. In Advances in neural information processing systems, pages 1008–1014. Citeseer, 2000. ",
+        "bbox": [
+            174,
+            683,
+            821,
+            713
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. In International Conference on Machine Learning, pages 11404–11413. PMLR, 2021. ",
+        "bbox": [
+            173,
+            723,
+            825,
+            752
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Gellért Weisz, Philip Amortila, and Csaba Szepesvári. Exponential lower bounds for planning in mdps with linearly-realizable optimal action-value functions. In Algorithmic Learning Theory, pages 1237–1264. PMLR, 2021. ",
+        "bbox": [
+            173,
+            762,
+            826,
+            805
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Andrea Zanette. Exponential lower bounds for batch reinforcement learning: Batch rl can be exponentially harder than online rl. In International Conference on Machine Learning, pages 12287–12297. PMLR, 2021. ",
+        "bbox": [
+            173,
+            815,
+            823,
+            858
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yaqi Duan, Zeyu Jia, and Mengdi Wang. Minimax-optimal off-policy evaluation with linear function approximation. In International Conference on Machine Learning, pages 2701–2709. PMLR, 2020. ",
+        "bbox": [
+            176,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aviral Kumar, Justin Fu, Matthew Soh, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. Advances in Neural Information Processing Systems, 32:11784–11794, 2019. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon, James Y Zou, Sergey Levine, Chelsea Finn, and Tengyu Ma. Mopo: Model-based offline policy optimization. Advances in Neural Information Processing Systems, 33:14129–14142, 2020. ",
+        "bbox": [
+            174,
+            140,
+            823,
+            183
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. Contextual decision processes with low bellman rank are pac-learnable. In International Conference on Machine Learning, pages 1704–1713. PMLR, 2017. ",
+        "bbox": [
+            173,
+            189,
+            825,
+            233
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Bernardo Ávila Pires and Csaba Szepesvári. Statistical linear estimation with penalized estimators: an application to reinforcement learning. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 1755–1762, 2012. ",
+        "bbox": [
+            174,
+            239,
+            825,
+            284
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Alessandro Lazaric, Mohammad Ghavamzadeh, and Rémi Munos. Finite-sample analysis of leastsquares policy iteration. Journal of Machine Learning Research, 13:3041–3074, 2012. ",
+        "bbox": [
+            171,
+            290,
+            823,
+            319
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sham Kakade and John Langford. Approximately optimal approximate reinforcement learning. In ICML, volume 2, pages 267–274, 2002. ",
+        "bbox": [
+            174,
+            325,
+            823,
+            354
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Bruno Scherrer. Approximate policy iteration schemes: a comparison. In International Conference on Machine Learning, pages 1314–1322. PMLR, 2014. ",
+        "bbox": [
+            173,
+            361,
+            823,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Romain Laroche, Paul Trichelair, and Remi Tachet Des Combes. Safe policy improvement with baseline bootstrapping. In International Conference on Machine Learning, pages 3652–3661, 2019. ",
+        "bbox": [
+            174,
+            395,
+            825,
+            439
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Chi Jin, Zhuoran Yang, Zhaoran Wang, and Michael I Jordan. Provably efficient reinforcement learning with linear function approximation. In Conference on Learning Theory, pages 2137–2143. PMLR, 2020. ",
+        "bbox": [
+            173,
+            444,
+            825,
+            488
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. In International Conference on Machine Learning, pages 1283–1294. PMLR, 2020. ",
+        "bbox": [
+            173,
+            493,
+            823,
+            523
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lior Shani, Yonathan Efroni, Aviv Rosenberg, and Shie Mannor. Optimistic policy optimization with bandit feedback. In International Conference on Machine Learning, pages 8604–8613. PMLR, 2020. ",
+        "bbox": [
+            173,
+            530,
+            823,
+            571
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Michail G Lagoudakis and Ronald Parr. Least-squares policy iteration. The Journal of Machine Learning Research, 4:1107–1149, 2003. ",
+        "bbox": [
+            171,
+            579,
+            825,
+            608
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Richard S Sutton, Hamid Reza Maei, Doina Precup, Shalabh Bhatnagar, David Silver, Csaba Szepesvári, and Eric Wiewiora. Fast gradient-descent methods for temporal-difference learning with linear function approximation. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 993–1000, 2009. ",
+        "bbox": [
+            173,
+            614,
+            826,
+            671
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Christoph Dann, Gerhard Neumann, Jan Peters, et al. Policy evaluation with temporal differences: A survey and comparison. Journal of Machine Learning Research, 15:809–883, 2014. ",
+        "bbox": [
+            171,
+            678,
+            823,
+            707
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Amit Daniely, Sivan Sabato, Shai Ben-David, and Shai Shalev-Shwartz. Multiclass learnability and the erm principle. In Proceedings of the 24th Annual Conference on Learning Theory, pages 207–232. JMLR Workshop and Conference Proceedings, 2011. ",
+        "bbox": [
+            174,
+            713,
+            823,
+            756
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Shai Ben-David, Nicolo Cesa-Bianchi, David Haussler, and Philip M Long. Characterizations of learnability for classes of $\\{ 0 , \\ldots , n \\}$ -valued functions. Journal of Computer and System Sciences, 50(1):74–86, 1995. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            805
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "David Haussler and Philip M Long. A generalization of sauer’s lemma. Journal of Combinatorial Theory, Series A, 71(2):219–240, 1995. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            840
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "David Haussler. Sphere packing numbers for subsets of the boolean n-cube with bounded vapnikchervonenkis dimension. Journal of Combinatorial Theory, Series A, 69(2):217–232, 1995. ",
+        "bbox": [
+            174,
+            847,
+            823,
+            876
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of machine learning. MIT press, 2018. ",
+        "bbox": [
+            173,
+            882,
+            821,
+            911
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rJliMh09F7/rJliMh09F7.md

@@ -0,0 +1,414 @@
+# DIVERSITY-SENSITIVE CONDITIONAL GENERATIVE ADVERSARIAL NETWORKS
+
+Dingdong $\mathbf { Y a n g } ^ { * \dagger }$ , Seunghoon Hong∗ †, Yunseok Jang†, Tianchen Zhao†, Honglak Lee†,‡   
+† University of Michigan, Ann Arbor, MI, USA   
+‡ Google Brain, Mountain View, CA, USA
+
+# ABSTRACT
+
+We propose a simple yet highly effective method that addresses the mode-collapse problem in the Conditional Generative Adversarial Network (cGAN). Although conditional distributions are multi-modal (i.e., having many modes) in practice, most cGAN approaches tend to learn an overly simplified distribution where an input is always mapped to a single output regardless of variations in latent code. To address such issue, we propose to explicitly regularize the generator to produce diverse outputs depending on latent codes. The proposed regularization is simple, general, and can be easily integrated into most conditional GAN objectives. Additionally, explicit regularization on generator allows our method to control a balance between visual quality and diversity. We demonstrate the effectiveness of our method on three conditional generation tasks: image-to-image translation, image inpainting, and future video prediction. We show that simple addition of our regularization to existing models leads to surprisingly diverse generations, substantially outperforming the previous approaches for multi-modal conditional generation specifically designed in each individual task.
+
+# 1 INTRODUCTION
+
+The objective of conditional generative models is learning a mapping function from input to output distributions. Since many conditional distributions are inherently ambiguous (e.g. predicting the future of a video from past observations), the ideal generative model should be able to learn a multi-modal mapping from inputs to outputs. Recently, Conditional Generative Adversarial Networks (cGAN) have been successfully applied to a wide-range of conditional generation tasks, such as image-to-image translation (Isola et al., 2017; Wang et al., 2018; Zhu et al., 2017a), image inpainting (Pathak et al., 2016; Iizuka et al., 2017), text-to-image synthesis (Huang et al., 2017; Hong et al., 2018), video generation (Villegas et al., 2017), etc.. In conditional GAN, the generator learns a deterministic mapping from input to output distributions, where the multi-modal nature of the mapping is handled by sampling random latent codes from a prior distribution.
+
+However, it has been widely observed that conditional GANs are often suffered from the mode collapse problem (Salimans et al., 2016; Arjovsky & Bottou, 2017), where only small subsets of output distribution are represented by the generator. The problem is especially prevalent for highdimensional input and output, such as images and videos, since the model is likely to observe only one example of input and output pair during training. To resolve such issue, there has been recent attempts to learn multi-modal mapping in conditional generative models (Zhu et al., 2017b; Huang et al., 2018). However, they are focused on specific conditional generation tasks (e.g. image-toimage translation) and require specific network architectures and objective functions that sometimes are not easy to incorporate into the existing conditional GANs.
+
+In this work, we introduce a simple method to regularize the generator in conditional GAN to resolve the mode-collapse problem. Our method is motivated from an observation that the mode-collapse happens when the generator maps a large portion of latent codes to similar outputs. To avoid this, we propose to encourage the generator to produce different outputs depending on the latent code, so as to learn a one-to-one mapping from the latent codes to outputs instead of many-to-one. Despite the simplicity, we show that the proposed method is widely applicable to various cGAN architectures and tasks, and outperforms more complicated methods proposed to achieve multi-modal conditional generation for specific tasks. Additionally, we show that we can control a balance between visual quality and diversity of generator outputs with the proposed formulation. We demonstrate the effectiveness of the proposed method in three representative conditional generation tasks, where most existing cGAN approaches produces deterministic outputs: Image-to-image translation, image inpainting and video prediction. We show that simple addition of the proposed regularization to the existing cGAN models effectively induces stochasticity from the generator outputs.
+
+# 2 RELATED WORK
+
+Resolving the mode-collapse problem in GAN is an important research problem, and has been extensively studied in the standard GAN settings (Metz et al., 2017; Arjovsky et al., 2017; Gulrajani et al., 2017; Salimans et al., 2016; Miyato et al., 2018). These approaches include unrolling the generator gradient update steps (Metz et al., 2017), incorporating the minibatch statistics into the discriminator (Salimans et al., 2016), employing the improved divergence measure to smooth the loss landscape of the discriminator (Gulrajani et al., 2017; Arjovsky et al., 2017; Miyato et al., 2018), etc.. Although these approaches have been successful in modeling unconditional data distribution to some extent, recent studies have reported that it is still not sufficient to resolve a mode-collapse problem in many conditional generative tasks, especially for high-dimensional input and output.
+
+Recently, some approaches have been proposed to address the mode-collapse issue in conditional GAN. Zhu et al. (2017b) proposed a hybrid model of conditional GAN and Variational Autoencoder (VAE) for multi-modal image-to-image translation task. The main idea is designing the generator to be invertible by employing an additional encoder network that predicts the latent code from the generated image. The similar idea has been applied to unsupervised image-to-image translation (Huang et al., 2018) and stochastic video generation (Lee et al., 2018) but with non-trivial task-specific modifications. However, these approaches are designed to achieve multi-modal generation in each specific task, and there has been no unified solution that addresses the mode-collapse problem for general conditional GANs. Recently, (Odena et al., 2018) proposed a method that regularizes the generator by clamping the generator Jacobian within a certain range. Our method shares the similar motivation with (Odena et al., 2018) but employs a different objective function that simply maximizes the norm of the generator gradient with an optional upper-bound, which we found that works much more stable over a wide range of tasks with less number of hyper-parameters.
+
+# 3 METHOD
+
+Consider a problem of learning a conditional mapping function $G : \mathcal { X }  \mathcal { Y }$ , which generates an output $\textbf { y } \in \mathcal { V }$ conditioned on the input $\textbf { \em x } \in { \mathcal { X } }$ . Our goal is to learn a multi-modal mapping $G : \mathcal { X } \times \mathcal { Z }  \mathcal { Y }$ , such that an input $_ { \textbf { \em x } }$ can be mapped to multiple and diverse outputs in $\mathcal { V }$ depending on the latent factors encoded in $z \in { \mathcal { Z } }$ . To learn such multi-modal mapping $G$ , we consider a conditional Generative Adversarial Network (cGAN), which learns both conditional generator $G$ and discriminator $D$ by optimizing the following adversarial objective:
+
+$$
+\displaystyle \operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathcal { L } _ { c G A N } ( G , D ) = \mathbb { E } _ { x , y } [ \log D ( { \pmb x } , { \pmb y } ) ] + \mathbb { E } _ { { \pmb x } , { \pmb z } } [ \log ( 1 - D ( { \pmb x } , G ( { \pmb x } , { \pmb z } ) ) ) ] .
+$$
+
+Although conditional GAN has been proved to work well for many conditional generation tasks, it has been also reported that optimization of Eq. (1) often suffers from the mode-collapse problem, which in extreme cases leads the generator to learn a deterministic mapping from $_ { \textbf { \em x } }$ to $\textbf {  { y } }$ and ignore any stochasticity induced by $_ z$ . To address such issue, previous approaches encouraged the generator to learn an invertible mapping from latent code to output by $E \bar { ( } \bar { G } ( { \pmb x } , z ) ) = z$ (Zhu et al., 2017b; Huang et al., 2018). However, incorporating an extra encoding network $E$ into the existing conditional GANs requires non-trivial modification of network architecture and introduce the new training challenges, which limits its applicability to various models and tasks.
+
+We introduce a simple yet effective regularization on the generator that directly penalizes its modecollapsing behavior. Specifically, we add the following maximization objective to the generator:
+
+$$
+\operatorname* { m a x } _ { G } \mathcal { L } _ { z } ( G ) = \mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \operatorname* { m i n } \left( \frac { \| G ( \pmb { x } , z _ { 1 } ) - G ( \pmb { x } , z _ { 2 } ) \| } { \| z _ { 1 } - z _ { 2 } \| } , \tau \right) \right] ,
+$$
+
+where $\lVert \cdot \rVert$ indicates a norm and $\tau$ is a bound for ensuring numerical stability. The intuition behind the proposed regularization is very simple: when the generator collapses into a single mode and produces deterministic outputs based only on the conditioning variable $_ { \textbf { \em x } }$ , Eq. (2) approaches its minimum since $G ( \pmb { x } , z _ { 1 } ) \overset { \cdot } { \approx } G ( \pmb { x } , z _ { 2 } )$ for all $z _ { 1 } , z _ { 2 } \sim N ( \mathbf { 0 } , \mathbf { 1 } )$ . By regularizing generator to maximize Eq. (2), we force the generator to produce diverse outputs depending on latent code $_ z$ .
+
+Our full objective function can be written as:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathcal { L } _ { c G A N } ( G , D ) - \lambda \mathcal { L } _ { z } ( G ) ,
+$$
+
+where $\lambda$ controls an importance of the regularization, thus, the degree of stochasticity in $G$ . If $G$ has bounded outputs through a non-linear output function (e.g. sigmoid), we remove the margin from Eq. (2) in practice and control its importance only with $\lambda$ . In this case, adding our regularization introduces only one additional hyper-parameter.
+
+The proposed regularization is simple, general, and can be easily integrated into most existing conditional GAN objectives. In the experiment, we show that our method can be applied to various models under different objective functions, network architectures, and tasks. In addition, our regularization allows an explicit control over a degree of diversity via hyper-parameter $\lambda$ . We show that different types of diversity emerge with different $\lambda$ . Finally, the proposed regularization can be extended to incorporate different distance metrics to measure the diversity of samples. We show this extension using distance in feature space and for sequence data.
+
+# 4 ANALYSIS OF THE PROPOSED REGULARIZATION
+
+Connection to Generator Gradient. We show in Appendix A that the proposed regularization in Eq. (2) corresponds to a lower-bound of averaged gradient norm of $G$ over $[ z _ { 1 } , z _ { 2 } ]$ as:
+
+$$
+\mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \frac { \| G ( \pmb { x } , z _ { 2 } ) - G ( \pmb { x } , z _ { 1 } ) \| } { \| z _ { 2 } - z _ { 1 } \| } \right] \leq \mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \int _ { 0 } ^ { 1 } \| \nabla _ { z } G ( \pmb { x } , \gamma ( t ) ) \| d t \right]
+$$
+
+where $\gamma ( t ) = t z _ { 2 } + ( 1 - t ) z _ { 1 }$ is a straight line connecting $z _ { 1 }$ and $z _ { 2 }$ . It implies that optimizing our regularization (LHS of Eq. (4)) will increase the gradient norm of the generator $\Vert \nabla _ { z } \bar { G } \Vert$ .
+
+It has been known that the GAN suffers from a gradient vanishing issue (Arjovsky & Bottou, 2017) since the gradient of optimal discriminator vanishes almost everywhere $\nabla D \approx 0$ except near the true data points. To avoid this issue, many previous works had been dedicated to smoothing out the loss landscape of $D$ so as to relax the vanishing gradient problem (Arjovsky et al., 2017; Miyato et al., 2018; Gulrajani et al., 2017; Kurach et al., 2018). Instead of smoothing $\nabla D$ by regularizing discriminator, we increase $\| \nabla _ { z } G \|$ to encourage $G ( { \pmb x } , z )$ to be more spread over the output space from the fixed $z _ { j } \sim p ( z )$ , so as to capture more meaningful gradient from $D$ .
+
+Optimization Perspective. We provide another perspective to understand how the proposed method addresses the mode-collapse problem. For notational simplicity, here we omit the conditioning variable from the generator and focus on a mapping of latent code to output $G _ { \theta } : \mathcal { Z } \to \mathcal { V }$ .
+
+Let a mode $\mathcal { M }$ denotes a set of data points in an output space $\mathcal { V }$ , where all elements of the mode have very small differences that are perceptually indistinguishable. We consider that the mode-collapse happens if the generator maps a large portion of latent codes to the mode $\mathcal { M }$ .
+
+Under this definition, we are interested in a situation where the generator output $G _ { \theta } ( z _ { 1 } )$ for a certain latent code $z _ { 1 }$ moves closer to a mode $\mathcal { M }$ by a distance of $\epsilon$ via a single gradient update. Then we show in Appendix B that such gradient update at $z _ { 1 }$ will also move the generator outputs of neighbors in a neighborhood $ { \mathcal { N } } _ { r } ( z _ { 1 } )$ to the same mode $\mathcal { M }$ . In addition, the size of neighborhood $ { \mathcal { N } } _ { r } ( z _ { 1 } )$ can be arbitrarily large but is bounded by an open ball of a radius $\begin{array} { r } { r = \epsilon \cdot \left( 4 \operatorname* { i n f } _ { z } \left\{ \operatorname* { m a x } \left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z ) \| } { \| z _ { 1 } - z \| } , \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z ) \| } { \| z _ { 1 } - z \| } \right) \right\} \right) ^ { - 1 } } \end{array}$ , where $\theta _ { t }$ and $\theta _ { t + 1 }$ denote the generator parameters before and after the gradient update, respectively.
+
+Without any constraints on kG(z1)−G(z2)kkz −z k , a single gradient update can cause the generator outputs for a large amount of latent codes to be collapsed into a mode $\mathcal { M }$ . We propose to shrink the size of such neighborhood by constraining kG(z1)−G(z2)kk − k above some threshold τ > 0, therefore prevent the generator placing a large probability mass around a mode $\mathcal { M }$ .
+
+Connection with BicycleGAN (Zhu et al., 2017b). We establish an interesting connection of our regularization with Zhu et al. (2017b). Recall that the objective of BicycleGAN is encouraging an invertibility of a generator by minimizing $\| z - E ( G ( z ) ) \| _ { 1 }$ . By taking derivative with respect to $_ z$ , it implies that optimal $E$ will satisfy $I = \nabla _ { G } E ( G ( z ) ) \nabla _ { z } G ( z )$ . Because an ideal encoder $E$ should be robust against spurious perturbations from inputs, we can naturally assume that the gradient norm of
+
+$E$ should not be very large. Therefore, to maintain invertibility, we expect the gradient of $G$ should not be zero, i.e. $\| \nabla _ { z } G ( z ) \| > \tau$ for some $\tau > 0$ , which prevents a gradient of the generator being vanishing. It is related to our idea that penalizes a vanishing gradient of the generator. Contrary to BicycleGAN, however, our method explicitly optimizes a generator gradient to have a reasonably high norm. It also allows us to control a degree of diversity with a hyper-parameter $\lambda$ .
+
+# 5 EXPERIMENTS
+
+In this section, we demonstrate the effectiveness of the proposed regularization in three representative conditional generation tasks that most existing methods suffer from mode-collapse: imageto-image translation, image inpainting and future frame prediction. In each task, we choose an appropriate cGAN baseline from the previous literature, which produces realistic but deterministic outputs, and apply our method by simply adding our regularization to their objective function. We denote our method as DSGAN (Diversity-Sensitive GAN). Note that both cGAN and DSGAN use the exactly the same networks. Throughout the experiments, we use the following objective:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathcal { L } _ { c G A N } ( G , D ) + \beta \mathcal { L } _ { r e c } ( G ) - \lambda \mathcal { L } _ { z } ( G ) ,
+$$
+
+where $\mathcal { L } _ { r e c } ( G )$ is a regression (or reconstruction) loss to ensure similarity between a prediction $\hat { y }$ and ground-truth $\textbf {  { y } }$ , which is chosen differently by each baseline method. Unless otherwise stated, we use $l _ { 1 }$ distance for $\mathcal { L } _ { r e c } ( G ) = \| G ( \pmb { x } , \pmb { z } ) - \pmb { y } \|$ and $l _ { 1 }$ norm for ${ \mathcal { L } } _ { z } ( G ) ^ { 1 }$ . We provide additional video results at anonymous website: https://sites.google.com/view/iclr19-dsgan/.
+
+# 5.1 IMAGE-TO-IMAGE TRANSLATION
+
+In this section, we consider a task of image-to-image translation. Given a set of training data $( { \pmb x } , { \pmb y } ) \in ( { \pmb x } , { \pmb y } )$ , the objective of the task is learning a mapping $G$ that transforms an image in domain $\mathcal { X }$ to another image in domain $\mathcal { V }$ (e.g. sketch to photo image).
+
+As a baseline cGAN model, we employ the generator and discriminator architectures from BicycleGAN (Zhu et al., 2017b) for a fair comparison. We evaluate the results on three datasets: label image (Radim Tylecek, 2013), ˇ edge photo (Zhu et al., 2016; Yu & Grauman, 2014), map image (Isola et al., 2017). For evaluation, we measure both the quality and the diversity of generation using two metrics from the previous literature. We employed Learned Perceptual Image Path Similarity (LPIPS) (Zhang et al., 2018) to measure the diversity of samples, which computes the distance between generated samples using features extracted from the pretrained CNN. Higher LPIPS score indicates more perceptual differences in generated images. In addition, we use Frechet ´ Inception Distance (FID) (Heusel et al., 2017) to measure the distance between training and generated distributions using the features extracted by the inception network (Szegedy et al., 2015). The lower FID indicates that the two distributions are more similar. To measure realism of the generated images, we also present human evaluation results using Amazon Mechanical Turk (AMT). Detailed evaluation protocols are described in Appendix D.1.
+
+![](images/b2531f85b977b01f565310f2011d1c47a61239ad55b0a0378dc15fc900262dec.jpg)  
+Figure 1: Impact of our regularization on multi-modal conditional generation.
+
+Impact of the Proposed Regularization. To analyze the impact of our regularization on learning a multi-modal mapping, we first conduct an ablation study by varying the weights $( \lambda )$ for our regularization. We choose labe $ $ image dataset for this experiment, and summarize the results in Figure 1. From the figure, it is clearly observed that the baseline cGAN $\lambda = 0$ ) experiences a severe mode-collapse and produces deterministic outputs. By adding our regularization $( \lambda > 0 )$ , we observe that the diversity emerges from the generator outputs. Increasing the $\lambda$ increases LPIPS scores and lower the FID, which means that the generator learns a more diverse mapping from input to output, and the generated distribution is getting closer to the actual distribution. If we impose too strong constraints on diversity with high $\lambda$ , the diversity keeps increasing, but generator outputs become less realistic and deviate from the actual distribution as shown in high FID (i.e. we got $\mathrm { { F I D } = 1 9 1 }$ and $\mathrm { L P I P S = 0 . 2 0 }$ for $\lambda = 2 0$ ). It shows that there is a natural trade-off between realism and diversity, and our method can control a balance between them by controlling $\lambda$ .
+
+Comparison with BicycleGAN (Zhu et al., 2017b). Next, we conduct comparison experiments with BicycleGAN (Zhu et al., 2017b), which is proposed to achieve multi-modal conditional generation in image-to-image translation. In this experiment, we fix $\lambda = 8$ for our method across all datasets and compare it against BicycleGAN with its optimal settings. Table 1 summarizes the results. Compared to the cGAN baseline, both our method and BicycleGAN are effective to learn multi-modal output distributions as shwon in higher LPIPS scores. Compared to BicycleGAN, our method still generates much diverse outputs and distributions that are generally more closer to actual ones as shown in lower FID score. In human evaluation on perceptual realism, we found that there is no clear winning method over others. It indicates that outputs from all three methods are in similar visual quality. Note that applying BicycleGAN to baseline cGAN requires non-trivial modifications in network architecture and obejctive function, while the proposed regularization can be simply integrated into the objective function without any modifications. Figure 2 illustrates generation results by our method. See Appendix D.1.3 for qualitative comparisons to BicycleGAN and cGAN.
+
+We also conducted an experiment by varying a length of latent code $_ z$ . Table 2 summarizes the results. As discussed in Zhu et al. (2017b), generation quality of BicycleGAN degrades with highdimensional $_ z$ due to the difficulties in matching the encoder distribution $E ( { \pmb x } )$ with prior distribution $p ( z )$ . Compared to BicycleGAN, our method is less suffered from such issue by sampling $_ { z }$ from the prior distribution, thus exhibits consistent performance over various latent code sizes.
+
+<table><tr><td>Method</td><td>label→image FID</td><td>LPIPS</td><td>map→image FID</td><td>LPIPS</td><td>edge→photo FID</td><td>LPIPS</td></tr><tr><td>cGAN</td><td>85.07</td><td>0.01</td><td>90.08</td><td>0.02</td><td>31.80</td><td>0.02</td></tr><tr><td>BicycleGAN</td><td>62.95</td><td>0.15</td><td>55.53</td><td>0.11</td><td>20.27</td><td>0.11</td></tr><tr><td>DSGAN</td><td>57.20</td><td>0.18</td><td>49.92</td><td>0.13</td><td>23.06</td><td>0.12</td></tr></table>
+
+Table 1: Comparisons of cGAN baseline, BicycleGAN and DSGAN (ours).
+
+![](images/eba1c680628684dcfef37a75156fe8ee89111673cffbdf77e0ca64966b820dfb.jpg)  
+Figure 2: Diverse outputs generated by DSGAN. The first and second column shows ground-truth and input images, while the rest columns are generated images with different latent codes.
+
+Table 2: Comparisons of BicycleGAN and DSGAN (ours) using various lengths of latent code.   
+
+<table><tr><td></td><td colspan="2">2 =8</td><td colspan="2">|2 =32</td><td colspan="2">2 =64</td><td colspan="2">2 =256</td></tr><tr><td></td><td>FID</td><td>LPIPS</td><td>FID</td><td>LPIPS</td><td>FID</td><td>LPIPS</td><td>FID</td><td>LPIPS</td></tr><tr><td>BicycleGAN</td><td>62.95</td><td>0.15</td><td>79.31</td><td>0.16</td><td>94.47</td><td>0.17</td><td>111.45</td><td>0.17</td></tr><tr><td>DSGAN</td><td>57.20</td><td>0.18</td><td>58.34</td><td>0.18</td><td>59.81</td><td>0.18</td><td>60.79</td><td>0.18</td></tr></table>
+
+Table 3: Comparisons of high-resolution image synthesis results in Cityscape dataset.   
+
+<table><tr><td></td><td>FID</td><td>LPIPS</td><td>Segmentation acc (%)</td></tr><tr><td>cGAN (pix2pixHD)</td><td>48.85</td><td>0.00</td><td>0.93</td></tr><tr><td>BicycleGAN</td><td>89.42</td><td>0.16</td><td>0.72</td></tr><tr><td>DSGAN (pix2pixHD)</td><td>28.80</td><td>0.12</td><td>0.92</td></tr></table>
+
+![](images/1c761002908b2d77ec4d8178d8ac0bdc976b26d266cd85040e08a357f8a7ea61.jpg)  
+Figure 3: Qualitative comparison for high-resolution image synthesis $( 1 0 2 4 \times 5 1 2 \ : \mathrm { p x } . )$ .
+
+Extension to High-Resolution Image Synthesis. The proposed regularization is agnostic to the choice of network architecture and loss, therefore can be easily applicable to various methods. To demonstrate this idea, we apply our regularization to the network of pix2pixHD (Wang et al., 2018), which synthesizes a photo-realistic image of $1 0 2 4 \times 5 1 2$ resolution from a segmentation label. In addition to the network architectures, Wang et al. (2018) incorporates a feature matching loss based on the discriminator as a reconstruction loss in Eq. (5). Therefore, this experiment also demonstrates that our regularization is compatible with other choices of $\mathcal { L } _ { r e c }$ .
+
+Table 3 shows the comparison results on Cityscape dataset (Cordts et al., 2016). In addition to FID and LPIPS scores, we compute the segmentation accuracy to measure the visual quality of the generated images. We compare the pixel-wise accuracy between input segmentation label and the predicted one from the generated image using DeepLab V3. (Chen et al., 2018). Since applying BicycleGAN to this baseline requires non-trivial modifications, we compared against the original BicycleGAN. As shown in the table, applying our method to the baseline effectively increases the output diversity with a cost of slight degradation in quality. Compared to BicycleGAN, our method generates much more visually plausible images. Figure 3 illustrates the qualitative comparison.
+
+# 5.2 IMAGE INPAINTING
+
+In this section, we demonstrate an application of our regularization to image inpainting task. The objective of this task is learning a generator $G : \mathcal { X }  \mathcal { Y }$ that takes an image with missing regions $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ and generates a complete image $\mathbf { \boldsymbol { y } } \in \mathcal { V }$ by inferring the missing regions.
+
+For this task, we employ generator and discriminator networks from Iizuka et al. (2017) as a baseline cGAN model with minor modification (See Appendix for more details). To create a data for inpainting, we take $2 5 6 \times 2 5 6$ images of centered faces from the celebA dataset (Liu et al., 2015) and remove center pixels of size $1 2 8 \times 1 2 8$ which contains most parts of the face. Similar to the image-to-image task, we employ FID and LPIPS to measure the generation performance. Please refer Appendix D.2 for more details about the network architecture and implementation details.
+
+In this experiment, we also test with an extension of our regularization using a different sample distance metric. Instead of computing sample distances directly from the generator output as in Eq. (2), we use the encoder features that capture more semantically meaningful distance between samples. Similar to feature matching loss (Wang et al., 2018), we use the features from a discriminator to compute our regularization as follow:
+
+$$
+\mathcal { L } _ { z } ( G ) = \mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \frac { \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \left\| D ^ { l } ( \pmb { x } , z _ { 1 } ) - D ^ { l } ( \pmb { x } , z _ { 2 } ) \right\| } { \left\| z _ { 1 } - z _ { 2 } \right\| } \right] ,
+$$
+
+where $D ^ { l }$ indicates a feature extracted from $l _ { \mathrm { t h } }$ layer of the discriminator $D$ . We denote our methods based on Eq. (2) and Eq. (6) as $\mathrm { D S G A N _ { R G B } }$ and $\mathrm { D S G A N _ { F M } }$ , respectively. Since there is no prior work on stochastic image inpainting to our best knowledge, we present comparisons of cGAN baseline along with our variants.
+
+Analysis on Regularization. We conduct both quantitative and qualitative comparisons of our methods and summarize the results in Table 4 and Figure 4, respectively. As we observed in the previous section, adding our regularization induces multi-modal outputs from the baseline cGAN. See Figure F for qualitative impact of $\lambda$ . Interestingly, we can see that sample variations in DSGANRGB tend to be in a low-level (e.g. global skin-color). We believe that sample difference in color may not be appropriate for faces, since human reacts more sensitively to the changes in semantic features (e.g. facial landmarks) than just color. Employing perceptual distance metric in our regularizationOurs-FM leads to semantically more meaningful variations, such as expressions, identity, etc..
+
+<table><tr><td></td><td>FID</td><td>LPIPS</td></tr><tr><td>cGAN</td><td>13.99</td><td>0.00</td></tr><tr><td>DSGANRGB</td><td>13.95</td><td>0.01</td></tr><tr><td>DSGANFM</td><td>13.94</td><td>0.05</td></tr></table>
+
+Table 4: Quantitative comparisons of our variants.
+
+![](images/0e8d433247dd14ff6afa30f8bef53be4de8b1458932ae7a89dec03cbe3d1c50e.jpg)  
+Figure 4: Qualitative comparisons of our variants. We present one example for baseline as it produces deterministic outputs.
+
+Analysis on Latent Space. To further understand if our regularization encourages $_ { z }$ to encode meaningful features, we conduct qualitative analysis on $_ z$ . We employ $\mathrm { D S G A N _ { F M } }$ for this experiments. We generate multiple samples across various input conditions while fixing the latent codes $_ { z }$ . Figure 5 illustrates the results. We observe that our method generates outputs which are realistic and diverse depending on $_ z$ . More interestingly, the generator outputs given the same $_ z$ exhibit similar attributes (e.g. gaze direction, smile) but also context-specific characteristics that match the input condition (e.g. skin color, hairs). It shows that our method guides the generator to learn meaningful latent factors in $_ z$ , which are disentangled from the input context to some extent.
+
+![](images/c85ff1a8b8d5996a9b6e2fcc0b98bed8c40a9aea502435f7dc0d9c6430fd5827.jpg)  
+Figure 5: Stochastic image inpainting results. Given an input image with missing region (first row), we generate multiple faces by sampling different $_ { z }$ (second–fifth rows). Each row is generated from the same $_ { z }$ , and exhibits similar face attributes.
+
+# 5.3 VIDEO PREDICTION
+
+In this section, we apply our method to a conditional sequence generation task. Specifically, we consider a task of anticipating $T$ future frames $\left\{ { \pmb x } _ { K + 1 } , { \pmb x } _ { K + 2 } , . . . , { \pmb x } _ { K + T } \right\} \in \mathcal { V }$ conditioned on $K$ previous frames $\left\{ { \pmb x } _ { 1 } , { \pmb x } _ { 2 } , . . . , { \pmb x } _ { K } \right\} \in { \mathcal { X } }$ . Since both the input and output of the generator are sequences in this task, we simply modify our regularization by
+
+$$
+\mathcal { L } _ { z } ( G ) = \mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \frac { \frac { 1 } { T } \sum _ { t = K } ^ { K + T } \left\| G ( \pmb { x } _ { 1 : t } , z _ { 1 } ) - G ( \pmb { x } _ { 1 : t } , z _ { 2 } ) \right\| _ { 1 } } { \left\| z _ { 1 } - z _ { 2 } \right\| _ { 1 } } \right] ,
+$$
+
+![](images/2f60cff28bb565c10cb2c65d082deb963ec52496bc25d8876c4ec75fb26bbdd5.jpg)  
+Figure 6: Stochastic video prediction results. Given two input frames, we present three random samples generated by each method. Compared to the baseline that produces deterministic outputs and SAVP that has limited diversity in KTH, our method generates diverse futures in both datasets.   
+Table 5: Comparisons of cGAN, SAVP, and DSGAN (ours). Diversity: pixel-wise distance among the predicted videos. $S i m _ { m a x }$ : largest cosine similarity between the predicted video and the ground truth. $D i s t _ { m i n }$ : closest pixel-wise distance between the predicted video and the ground truth.
+
+(Base: 1.0 × 10−3)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">BAIR</td><td colspan="3">KTH</td></tr><tr><td>Diversity</td><td>Simmax</td><td>Distmin</td><td>Diversity</td><td>Simmax</td><td>Distmin</td></tr><tr><td>cGAN</td><td>2.48</td><td>861.92</td><td>22.46</td><td>0.04</td><td>802.79</td><td>5.00</td></tr><tr><td>SAVP</td><td>18.93</td><td>869.58</td><td>20.44</td><td>0.51</td><td>777.70</td><td>5.48</td></tr><tr><td>DSGAN</td><td>26.75</td><td>874.12</td><td>18.46</td><td>3.96</td><td>855.10</td><td>3.84</td></tr></table>
+
+where $\pmb { x } _ { 1 : t }$ represents a set of frames from time step 1 to $t$
+
+We compare our method against SAVP (Lee et al., 2018), which also addresses the multi-modal video prediction task. Similar to Zhu et al. (2017b), it employs a hybrid model of conditional GAN and VAE, but using the recurrent generator designed specifically for future frame prediction. We take only GAN component (generator and discriminator networks) from SAVP as a baseline cGAN model and apply our regularization with $\lambda = 5 0$ to induce stochasticity. We use $| z | = 8$ for all compared methods. See Appendix D.3.2 for more details about the network architecture.
+
+We conduct experiments on two datasets from the previous literature: the BAIR action-free robot pushing dataset (Ebert et al., 2017) and the KTH human actions dataset (Schuldt et al., 2004). To measure both the diversity and the quality, we generate 100 random samples of 28 future frames for each test video and compute the (a) Diversity (pixel-wise distance among the predicted videos) and the (b) $D i s t _ { m i n }$ (minimum pixel-wise distance between the predicted videos and the ground truth). Also, for a better understanding of quality, we additionally measured the (c) $S i m _ { m a x }$ (largest cosine similarity metric between the predicted video and the ground truth on VGGNet (Simonyan & Zisserman, 2015) feature space). An ideal stochastic video prediction model may have higher Diversity, while having lower $D i s t _ { m i n }$ with higher $S i m _ { m a x }$ so that a model still can predict similar to the ground truth as a candidate. More details about evaluation metric are described in Appendix D.3.3.
+
+We present both quantitative and qualitative comparison results in Table 5 and Figure 6, respectively. As illustrated in the results, both our method and SAVP can predict diverse futures compared to the baseline cGAN that produces deterministic outputs. As shown in Table 5, our method generates more diverse and realistic outputs than SAVP with much less number of parameters and simpler training procedures. Interestingly, as shown in KTH results, SAVP still suffers from a mode-collapse problem when the training videos have limited diversity, whereas our method generally works well in both cases. It shows that our method generalizes much better to various videos despite its simplicity.
+
+# 6 CONCLUSION
+
+In this paper, we investigate a way to resolve a mode-collapsing in conditional GAN by regularizing generator. The proposed regularization is simple, general, and can be easily integrated into existing conditional GANs with broad classes of loss function, network architecture, and data modality. We apply our regularization for three conditional generation tasks and show that simple addition of our regularization to existing cGAN objective effectively induces the diversity. We believe that achieving an appropriate balance between realism and diversity by learning $\lambda$ and $\tau$ such that the learned distribution matches an actual data distribution would be an interesting future work.
+
+Acknowledgement This work was supported in part by ONR N00014-13-1-0762, NSF CAREER IIS-1453651, DARPA Explainable AI (XAI) program #313498, and Sloan Research Fellowship.
+
+# REFERENCES
+
+Mart´ın Arjovsky and Leon Bottou. Towards Principled Methods for Training Generative Adversarial ´ Networks. In ICLR, 2017.
+
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein Generative Adversarial Net- ´ works. In ICML, 2017.
+
+Liang-Chieh Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam. EncoderDecoder with Atrous Separable Convolution for Semantic Image Segmentation. In ECCV, 2018.
+
+Marius Cordts, Mohamed Omran, Sebastian Ramos, Timo Rehfeld, Markus Enzweiler, Rodrigo Benenson, Uwe Franke, Stefan Roth, and Bernt Schiele. The Cityscapes Dataset for Semantic Urban Scene Understanding. In CVPR, 2016.
+
+Emily Denton and Rob Fergus. Stochastic Video Generation with a Learned Prior. In ICML, 2018.
+
+Frederik Ebert, Chelsea Finn, Alex X Lee, and Sergey Levine. Self-Supervised Visual Planning with Temporal Skip Connections. In CoRL, 2017.
+
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved Training of Wasserstein GANs. In NIPS, 2017.
+
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, Gunter Klambauer, and ¨ Sepp Hochreiter. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In NIPS, 2017.
+
+Seunghoon Hong, Dingdong Yang, Jongwook Choi, and Honglak Lee. Inferring Semantic Layout for Hierarchical Text-to-Image Synthesis. In CVPR, 2018.
+
+Xun Huang, Yixuan Li, Omid Poursaeed, John Hopcroft, and Serge Belongie. Stacked generative adversarial networks. In CVPR, 2017.
+
+Xun Huang, Ming-Yu Liu, Serge Belongie, and Jan Kautz. Multimodal Unsupervised Image-toImage Translation. In ECCV, 2018.
+
+Satoshi Iizuka, Edgar Simo-Serra, and Hiroshi Ishikawa. Globally and Locally Consistent Image Completion. In SIGGRAPH, 2017.
+
+Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-Image Translation with Conditional Adversarial Networks. In CVPR, 2017.
+
+Yunseok Jang, Gunhee Kim, and Yale Song. Video Prediction with Appearance and Motion Conditions. In ICML, 2018.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. ImageNet Classification with Deep Convolutional Neural Networks. In NIPS, 2012.
+
+Karol Kurach, Mario Lucic, Xiaohua Zhai, Marcin Michalski, and Sylvain Gelly. The GAN Landscape: Losses, Architectures, Regularization, and Normalization. arXiv:1807.04720, 2018.
+
+Alex X. Lee, Richard Zhang, Frederik Ebert, Pieter Abbeel, Chelsea Finn, and Sergey Levine. Stochastic Adversarial Video Prediction. arXiv:1804.01523, 2018.
+
+Chuan Li and Michael Wand. Precomputed Real-Time Texture Synthesis with Markovian Generative Adversarial Networks. In ECCV, 2016.   
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep Learning Face Attributes in the Wild. In ICCV, 2015.   
+Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled Generative Adversarial Networks. In ICLR, 2017.   
+Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. In ICLR, 2018.   
+Augustus Odena, Jacob Buckman, Catherine Olsson, Tom B Brown, Christopher Olah, Colin Raffel, and Ian Goodfellow. Is generator conditioning causally related to gan performance? arXiv preprint arXiv:1802.08768, 2018.   
+Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context Encoders: Feature Learning by Inpainting. In CVPR, 2016.   
+Radim Sˇ ara Radim Tyle ´ cek. Spatial Pattern Templates for Recognition of Objects with Regular ˇ Structure. In GCPR, 2013.   
+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional Networks for Biomedical Image Segmentation. In MICCAI, 2015.   
+Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved Techniques for Training GANs. In NIPS, 2016.   
+Christian Schuldt, Ivan Laptev, and Barbara Caputo. Recognizing Human Actions: A Local SVM Approach. In ICPR, 2004.   
+Karen Simonyan and Andrew Zisserman. Very Deep Convolutional Networks for Large-Scale Image Recognition. In ICLR, 2015.   
+Akash Srivastava, Lazar Valkoz, Chris Russell, Michael U Gutmann, and Charles Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In NIPS, 2017.   
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going Deeper with Convolutions. In CVPR, 2015.   
+Ruben Villegas, Jimei Yang, Seunghoon Hong, Xunyu Lin, and Honglak Lee. Decomposing Motion and Content for Natural Video Sequence Prediction. In ICLR, 2017.   
+Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Andrew Tao, Jan Kautz, and Bryan Catanzaro. HighResolution Image Synthesis and Semantic Manipulation with Conditional GANs. In CVPR, 2018.   
+Tom White. Sampling Generative Networks. arXiv:1609.04468, 2016.   
+SHI Xingjian, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting. In NIPS, 2015.   
+Aron Yu and Kristen Grauman. Fine-Grained Visual Comparisons with Local Learning. In CVPR, 2014.   
+Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The Unreasonable Effectiveness of Deep Features as a Perceptual Metric. In CVPR, 2018.   
+Jun-Yan Zhu, Philipp Krahenb ¨ uhl, Eli Shechtman, and Alexei A Efros. Generative Visual Manipu- ¨ lation on the Natural Image Manifold. In ECCV, 2016.   
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks. In ICCV, 2017a.   
+Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward Multimodal Image-to-Image Translation. In NIPS, 2017b.
+
+# APPENDIX
+
+# A DERIVATION OF LOWER-BOUND OF GRADIENT NORM
+
+In this section, we provide a derivation of our regularization term from a true gradient norm of the generator. Given arbitrary latent samples $z _ { 1 } , z _ { 2 }$ , from gradient theorem we have
+
+$$
+\begin{array} { r l } { \| G ( x , z _ { 2 } ) - G ( x , z _ { 1 } ) \| } & { = \frac { \| \int _ { \gamma [ z _ { 1 } , z _ { 2 } ] } \nabla _ { z } G ( x , z ) \cdot d z \| } { \| z _ { 2 } - z _ { 1 } \| } } \\ & { = \frac { \| \int _ { 0 } ^ { 1 } \nabla _ { z } G ( x , \gamma ( l ) ) \cdot \gamma ^ { \prime } ( l ) d l \| } { \| z _ { 2 } - z _ { 1 } \| } } \\ & { = \frac { \| \int _ { 0 } ^ { 1 } \nabla _ { z } G ( x , \gamma ( l ) ) \cdot ( z _ { 2 } - z _ { 1 } ) d l \| } { \| z _ { 2 } - z _ { 1 } \| } } \\ & { \leq \frac { \int _ { 0 } ^ { 1 } \| \nabla _ { z } G ( x , \gamma ( l ) ) \| \| z _ { 2 } - z _ { 1 } \| d l } { \| z _ { 2 } - z _ { 1 } \| } } \\ & { = \int _ { 0 } ^ { 1 } \| \nabla _ { z } G ( x , \gamma ( l ) ) \| d l , } \end{array}
+$$
+
+where $\gamma$ is a straight line connecting $z _ { 1 }$ and $z _ { 2 }$ , where $\gamma ( 0 ) = z _ { 1 }$ and $\gamma ( 1 ) = z _ { 2 }$ .
+
+Apply expectation on both sides of (8) with respect to $z _ { 1 } , z _ { 2 }$ from standard Gaussian distribution gives Eqn. 4:
+
+$$
+\mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \frac { \| G ( x , z _ { 2 } ) - G ( { \pmb x } , z _ { 1 } ) \| } { \| z _ { 2 } - z _ { 1 } \| } \right] \leq \mathbb { E } _ { z _ { 1 } , z _ { 2 } } \left[ \int _ { 0 } ^ { 1 } \| \nabla _ { z } G ( { \pmb x } , \gamma ( t ) ) \| d t \right] \ .
+$$
+
+# B COLLAPSING TO A MODE AS A GROUP
+
+For notational simplicity, we omit the conditioning variable from the generator and focus on a mapping of latent code to output $G _ { \theta } : \mathcal { Z }  \mathcal { V }$ where $\mathcal { V }$ is the image space.
+
+Definition B.1. A mode $\mathcal { M }$ is a subset of $\mathcal { V }$ satisfying $\operatorname* { m a x } _ { { \pmb y } \in \mathcal { M } } \| { \pmb y } - { \pmb y } ^ { * } \| < \alpha$ for some image $\boldsymbol { y } ^ { * }$ and $\alpha > 0$ . Let $z _ { 1 }$ be a sample in latent space, we say $z _ { 1 }$ is attracted to a mode $\mathcal { M }$ by $\epsilon$ from a gradient step if $\| \pmb { y } ^ { * } - G _ { \theta _ { t + 1 } } ( \bar { z } _ { 1 } ) \| + \epsilon < \| \bar { \pmb { y } } ^ { * } - G _ { \theta _ { t } } ( \bar { z _ { 1 } } ) \|$ , where ${ \boldsymbol { \mathbf { \mathit { y } } } } ^ { * } \in { \mathcal { M } }$ is an image in a mode, $\theta _ { t }$ and $\theta _ { t + 1 }$ are the generator parameters before and after the gradient updates respectively.
+
+In other words, we define modes as sets consisting of images that are close to some real images, and we consider a situation where the generator output $G _ { \theta _ { t } } ( \bar { z } _ { 1 } )$ at certain $z _ { 1 }$ is attracted to a mode $\mathcal { M }$ by a single gradient update.
+
+With Definition B.1, we are now ready to state and prove the following proposition.
+
+Proposition B.1. Suppose $z _ { 1 }$ is attracted to the mode $\mathcal { M }$ by $\epsilon$ , then there exists a neighborhood $\bar { \mathcal { N } } _ { r } ( z _ { 1 } )$ of $z _ { 1 }$ such that $z _ { 2 }$ is attracted to $\mathcal { M }$ by $\epsilon / 2$ , for all $z _ { 2 } \doteq \mathcal { N } _ { r } ( z _ { 1 } )$ . The size of $\bar { \mathcal { N } } _ { r } ( z _ { 1 } )$ can be arbitrarily large but is bounded by an open ball of radius $r$ where
+
+$$
+r = \epsilon \cdot \left( 4 \operatorname* { i n f } _ { z } \left. \operatorname* { m a x } \left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z ) \| } { \| z _ { 1 } - z \| } , \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z ) \| } { \| z _ { 1 } - z \| } \right) \right. \right) ^ { - 1 } .
+$$
+
+Proof. Consider the following expansion.
+
+$$
+\begin{array} { r l } & { \| y ^ { * } - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| \leq \| y ^ { * } - \bar { G } _ { \theta _ { t + 1 } } ( z _ { 1 } ) \| + \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| } \\ & { \qquad < \| y ^ { * } - G _ { \theta _ { t } } ( z _ { 1 } ) \| + \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| - \epsilon } \\ & { \qquad \leq \| y ^ { * } - G _ { \theta _ { t } } ( z _ { 2 } ) \| + \| G _ { \theta _ { t } } ( z _ { 2 } ) - G _ { \theta _ { t } } ( z _ { 1 } ) \| + \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| - \epsilon } \\ & { \qquad = \| y ^ { * } - G _ { \theta _ { t } } ( z _ { 2 } ) \| } \\ & { \qquad + \left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z _ { 2 } ) \| } { \| z _ { 1 } - z _ { 2 } \| } + \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| } { \| z _ { 1 } - z _ { 2 } \| } \right) \| z _ { 1 } - z _ { 2 } \| - \epsilon . } \end{array}
+$$
+
+Eq. (11) implies that
+
+$$
+\lVert \pmb { y } ^ { * } - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \rVert + \frac { \epsilon } { 2 } < \lVert \pmb { y } ^ { * } - G _ { \theta _ { t } } ( z _ { 2 } ) \rVert ,
+$$
+
+for all $z _ { 2 }$ that satisfies
+
+$$
+\left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z _ { 2 } ) \| } { \| z _ { 1 } - z _ { 2 } \| } + \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z _ { 2 } ) \| } { \| z _ { 1 } - z _ { 2 } \| } \right) \| z _ { 1 } - z _ { 2 } \| \leq \frac { \epsilon } { 2 } .
+$$
+
+Define $\begin{array} { r } { \mathcal { N } _ { \tau } ( z _ { 1 } ) = \Big \{ z : \operatorname* { m a x } \Big ( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z ) \| } { \| z _ { 1 } - z \| } , \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z ) \| } { \| z _ { 1 } - z \| } \Big ) \leq \tau \Big \} } \end{array}$ , then Eq. (13) holds for any $z _ { 2 } \in \bigcup _ { \tau > 0 } \left\{ B _ { \epsilon / 4 \tau } ( z _ { 1 } ) \cap \mathcal { N } _ { \tau } ( z _ { 1 } ) \right\} = \mathcal { N } _ { r } ( z _ { 1 } )$ . We have $\mathcal { N } _ { r } ( z _ { 1 } ) \neq \emptyset$ since $z _ { 1 } \in \mathcal { N } _ { r } ( z _ { 1 } )$ .
+
+The size of $ { \mathcal { N } } _ { r } ( z _ { 1 } )$ can be arbitrarily large if $B _ { \epsilon / 4 \tau } ( z _ { 1 } ) \subseteq { \mathcal { N } } _ { \tau } ( z _ { 1 } )$ for arbitrarily small $\tau$ . And for any $\tau > 0$ such that $\begin{array} { r } { \tau \leq \operatorname* { m a x } \left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z ) \| } { \| z _ { 1 } - z \| } , \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z ) \| } { \| z _ { 1 } - z \| } \right) } \end{array}$ for all $_ z$ , we have $\mathcal { N } _ { r } ( z _ { 1 } ) \subseteq B _ { \epsilon / 4 \tau } ( z _ { 1 } )$ . In particular, pick the largest $\tau$ possible yields the bound $\mathcal { N } _ { r } ( z _ { 1 } ) \subseteq B _ { r } ( z _ { 1 } )$ ,
+
+$$
+r = \epsilon \cdot \left( 4 \operatorname* { i n f } _ { z } \left. \operatorname* { m a x } \left( \frac { \| G _ { \theta _ { t } } ( z _ { 1 } ) - G _ { \theta _ { t } } ( z ) \| } { \| z _ { 1 } - z \| } , \frac { \| G _ { \theta _ { t + 1 } } ( z _ { 1 } ) - G _ { \theta _ { t + 1 } } ( z ) \| } { \| z _ { 1 } - z \| } \right) \right. \right) ^ { - 1 } .
+$$
+
+# C ABLATION STUDY
+
+# C.1 APPLICATION TO UNCONDITIONAL GAN
+
+Since the proposed regularization is not only limited to conditional GAN, we further analyze its impact on unconditional GAN. To this end, we adopt synthetic datasets from (Srivastava et al., 2017; Metz et al., 2017), a mixture of eight 2D Gaussian distributions arranged in a ring. For unconditional generator and discriminator, we adopt the vanilla GAN implementation from (Srivastava et al., 2017), and train the model with and without our regularization $\lambda = 0 . 1$ ). We follow the same evaluation protocols used in (Srivastava et al., 2017). It generates 2,500 samples by the generator, and counts a sample as high-quality if it is within three standard deviations of the nearest mode. Then the performance is reported by 1) counting the number of modes containing at least one high-quality sample and 2) computing the portion of high-quality samples from all generated ones. We summarize the qualitative and quantitative results (10-run average) in Figure A and Table A, respectively.
+
+As illustrated in Figure A, we observe that vanilla GAN experiences a severe mode collapse, which puts a significant probability mass around a single output mode. Contrary to results reported in (Srivastava et al., 2017), we observed that the mode captured by the generator is still not close enough to the actual mode, resulted in 0 high-quality samples as shown in Table A. On the other hand, applying our regularization effectively alleviates the mode-collapse problem by encouraging the generator to efficiently explore the data space, enabling the generator to capture much more modes compared to vanilla GAN setting.
+
+![](images/78004cb91c3e538c538fe66b3644b8f0597f4ac38a27ee58fa930973a01799ce.jpg)  
+Figure A: Density plots of ground-truth, vanilla GAN and DSGAN.   
+Step 0k Step 8k SteTable A: Quantitative evaluation results ( $^ *$ 16k Step 24k Step 32k Step 40k denotes the results reported in Srivastava et al. (2017)).
+
+<table><tr><td>Method</td><td colspan="2">2D Ring Modes High Quality</td></tr><tr><td>Vanilla GAN</td><td>(Max 8) 0(1*）</td><td>Samples (%) 0.0 (99.3*)</td></tr><tr><td>Unrolled GAN</td><td>7.6</td><td>35.6</td></tr><tr><td>VEEGAN</td><td>8</td><td>52.9</td></tr><tr><td>DSGAN</td><td>8</td><td>81.4</td></tr></table>
+
+# C.2 IMPACT ON UNBALANCED CGAN
+
+In principle, our regularization can help the generator to cope with vanishing gradient from the discriminator to some extent, as it spreads out the generator landscape thus increases the chance to capture useful gradient signals around true data points. To verify its impact, we simulate the vanishing gradient problem by training the baseline cGAN until it converges and retraining the model with our regularization while initializing the discriminator with the pre-trained weights. Empirically we observed that the pre-trained discriminator can distinguish the real data and generated samples from the randomly initialized generator almost perfectly, and the generator experiences a severe vanishing gradient problem at the beginning of the training. We use the image-to-image translation model on label image dataset for this experiment.
+
+In the experiment, we found that the generator converges to the FID and LPIPS scores of 52.32 and 0.16, respectively, which are close to the ones we achieved with the balanced discriminator (FID: 57.20, LPIPS: 0.18). We observed that our regularization loss goes down very quickly in the early stage of training, which helps the generator to efficiently explorer its output space when the discriminator gradients are vanishing. Together with the reconstruction loss, we found that it helps the generator to capture useful learning signals from the discriminator and learn both realistic and diverse modes in the conditional distribution.
+
+# D ADDITIONAL EXPERIMENT RESULTS
+
+This section provides additional experiment details and results that could not be accommodated in the main paper due to space restriction. We are going to release the code and datasets upon the acceptance of the paper.
+
+# D.1 IMAGE-TO-IMAGE TRANSLATION
+
+# D.1.1 BASELINE MODELS
+
+BicycleGAN’s Generator and Discriminator. We use BicycleGAN’s generator and discriminator structures as a baseline cGAN. The baseline model has exactly the same hyperparameters as BicycleGAN including the weight of pixel-wise L1 loss and GAN loss. The baseline model setting then basically becomes a pix2pix image-to-image translation setting (Isola et al., 2017). The generator architecture is a U-Net style network (Ronneberger et al., 2015) and the discriminator is a two-scale patchGAN-style (Li & Wand, 2016) network.
+
+Pix2pixHD Baseline. In this setting, we adopt the generator and discriminator networks from pix2pixHD (Wang et al., 2018) as a baseline cGAN. Compared to the original pix2pixHD network that employs two nested generators, we use only one generator for simplicity. Since the original generator does not contain stochastic component, we modified the generator network by injecting the latent code after the downsampling layers by spatial tiling and depth-wise concatenation. The discriminator is a two-scale patchGAN-style (Li & Wand, 2016) network. Following the original setting, we employ feature matching loss based on discriminator (Wang et al., 2018) and perceptual loss based on the pre-trained VGGNet (Simonyan & Zisserman, 2015) as a reconstruction loss in Eq. (5).
+
+# D.1.2 EVALUATION METRICS
+
+Here we provide a detailed descriptions for evaluation metrics and evaluation protocols.
+
+Learned Perceptual Image Patch Similarity, LPIPS (Zhang et al., 2018). LPIPS score measures the diversity of the generated samples using the L1 distance of features extracted from pretrained AlexNet (Krizhevsky et al., 2012). We generate 20 samples for each validation image, and compute the average of pairwise distances between all samples generated from the same input. Then we report the average of LPIPS scores over all validation images.
+
+Frechet Inception Distance, FID (Heusel et al., 2017). ´ For each method, we compute FID score on the validation dataset. For each input from the validation dataset, we sample 20 randomly generated output. We take the generated images as a generated dataset and compute the FID score between the generated dataset and training dataset. If the size of an image is different between the training dataset and generated dataset, we resize training images to the size of generated images. We use the features from the final average pooling layer of the InceptionV3 (Szegedy et al., 2015) network to compute Frechet Distance. ´
+
+Human Evaluation via Amazon Mechanical Turk (AMT). To compare the perceptual quality of generations among different methods, we conduct human evaluation via AMT. We conduct sideby-side comparisons between our method and a competitor (i.e. baseline cGAN and BicycleGAN). Specifically, we present two sets of images generated by each compared method given the same input condition, and ask turkers to choose the set that is visually more plausible and matches the input condition. Each set has 6 randomly sampled images. We collect answers over 100 examples for each dataset, where each question is answered by 5 unique turkers.
+
+# D.1.3 QUALITATIVE RESULTS
+
+Qualitative Comparison. We present the qualitative comparisons of various methods presented in Table 1 and Table 3 in the main paper. Figure B illustrates the qualitative comparison results of DSGAN (ours), baseline cGAN and BicycleGAN in Table 1. In the example of edges photo dataset, the input edge images miss some of the features in the ground-truth images (e.g. shoelace). While both DSGAN and baseline cGAN are able to capture such missing parts in an input, we observe that some of BicycleGAN’s outputs are missing it. Also in the example of maps images dataset, both DSGAN and baseline cGAN can generate natural and variable vegetation in the corresponding area while BicycleGAN tends to generate plain texture with global color variations in such area. These two examples show how our regularization can help cGAN to learn more visually reasonable and diverse results. We additionally present the qualitative comparison results of Table 3. We observe that the generation results from BicycleGAN suffers from low-visual quality, while our method is able to generate fine details of the objects and scene by exploiting the network for high-resolution image synthesis.
+
+![](images/6f85e831bc3179da703baf0e33640eabbff0739081bf1bf456f04fbf0eb9f9d8.jpg)  
+Figure B: Qualitative comparisons of Table 1.
+
+![](images/386b0b5836c17e10260e6fa03b30bcd218631629eb300626202a6c1d74d8143c.jpg)  
+Figure C: Qualitative comparisons of Table 3.
+
+Analysis on Latent Space. To better understand the latent space learned with the proposed regularization, we generate images in Cityscape dataset by interpolating two randomly sampled latent vectors by spherical linear interpolation (White, 2016). Figure D illustrates the interpolation results. As shown in the figure, the intermediate generation results are all reasonable and exhibit smooth transitions, which implies that the learned latent space has a smooth manifold.
+
+![](images/bbe33f85a9e4d2a0089f85be65748624f98ffdd9f572d9c57d39f1b34db17af3.jpg)  
+Figure D: Interpolation in latent space of DSGAN on Cityscapes dataset.
+
+We also present the comparison of interpolation results between DSGAN and BicycleGAN on maps images dataset. As shown in Figure E, DSGAN generates meaningful and diverse predictions on ambiguous regions (e.g. forest on a map) and has a smooth transition from one latent code to another. On contrary, the BicyceGAN does not show meaningful changes within the interpolations and sometimes has a sudden changes on its output (e.g. last generated image). We also observe similar patterns across many examples in this dataset. It shows an example that DSGAN learns better latent space than BicycleGAN.
+
+![](images/048dcb30854f78689edff7a15874db32807030bd3a67df036c4288b5317fc73d.jpg)  
+Figure E: Comparison of latent space interpolation results between DSGAN and BicycleGAN.
+
+# D.2 IMAGE INPAINTING
+
+In this section, we provide details of image inpainting experiment.
+
+Network Architecture. We employ the generator and discriminator networks from Iizuka et al. (2017) as baseline conditional GAN. Our generator takes $2 5 6 \times 2 5 6$ image with the masked region $_ { \textbf { \em x } }$ as an input and produces $2 5 6 \times 2 5 6$ prediction of the missing region $\hat { \pmb { y } }$ as an output. Then we combine the predicted image with the input by $\pmb { y } = ( 1 - M ) \odot \pmb { x } + M \odot \hat { \pmb { y } }$ as an output of the network, where $M$ is a binary mask indicating the missing region. Then the combined output $\textbf {  { y } }$ is passed as an input to the discriminator. We apply two modifications to the baseline model to achieve better generation quality. First, compared to the original model that employs the Mean Squared Error (MSE) as a reconstruction loss $\mathcal { L } _ { r e c } ( G )$ in Eq. (5), we apply the feature matching loss based on the discriminator (Wang et al., 2018). Second, compared to the original model that employs two discriminators applied independently to the inpainted region and entire image, we employ only one discriminator on the inpainted region but using patchGAN-style discriminator (Li & Wand, 2016). Please note that these modifications are to achieve better image quality but irrelevant to our regularization.
+
+Analysis on Regularization. First, we conduct qualitative analysis on how the proposed regularization controls a diversity of the generator outputs. To this end, we train the model $( \mathrm { D S G A N _ { F M } } )$ ) by varying the weights for our regularization, and present the results in Figure F. As already observed in Section 5.1, imposing stronger constraints on the generator by our regularization indeed increases the diversity in the generator outputs. With small weights (e.g. $\lambda = 2$ ), we observe limited visual differences among samples, such as subtle changes in facial expressions or makeup. By increasing $\lambda$ (e.g. $\lambda = 5$ ), we can see that more meaningful diversity emerges such as hair-style, age, and even identity while maintaining the visual quality and alignment to input condition. It shows more intuitively how our regularization can effectively help the model to discover more meaningful modes in the output space.
+
+![](images/42903023a7455349dffb384506ff773c8d0f51a8869158a87b54c9ac268e0f77.jpg)  
+Figure F: Image inpainting results with different $\lambda$ . We observe more diversity emerges from the genrator outputs as we increase the weights for our regularization.
+
+Analysis on Latent Space. We further conduct a qualitative analysis on the learned latent space. To verify that the model learns a continuous conditional distribution with our regularization, we conduct the interpolation experiment similar to the previous section. Specifically, we sample two random latent codes from the prior distribution and generate images by linearly interpolating the latent code between two samples. Figure G illustrates the results. As it shows, the generator outputs exhibit a smooth transition between two samples, while most intermediate samples also look realistic.
+
+![](images/f3f9f4aa234e3fe1f4cffd6f065fc336dd2c0130dbf58860aad055a762e324b2.jpg)  
+Figure G: Interpolation results on image inpainting task. For each row, we sample the two latent codes (leftmost and rightmost images), and generate the images from the interpolated latent codes from one latent code to another.
+
+# D.3 VIDEO PREDICTION
+
+In this section, we provide more details on network architecture, datasets and evaluation metrics on the video prediction task.
+
+# D.3.1 DATASET
+
+We measure the effectiveness of our method based on two real-world datasets: the BAIR action-free robot pushing dataset (Ebert et al., 2017) and the KTH human actions dataset (Schuldt et al., 2004). For both of the dataset, we provide two frames as the condition and train the model to predict 10 future frames ( $k = 2$ , $T = 1 0$ in Eq. (7)). In testing time, we run each model to predict 28 frames $k = 2$ , $T = 2 8$ ). Following Lee et al. (2018), we used $6 4 \times 6 4$ frames for both datasets. The details of data pre-processing are described in below.
+
+BAIR Action-Free (Ebert et al., 2017). This dataset contains randomly moving robot arms on a table with a static background. This dataset contains the diverse movement of a robot arm with a diverse set of objects. We downloaded the pre-processed data provided by the authors (Lee et al., 2018) and used it directly for our experiment.
+
+KTH (Schuldt et al., 2004). Each video in this dataset contains a single person in a static background performing one of six activities: walking, jogging, running, boxing, hand waving, and hand clapping. We download the pre-processed videos from Villegas et al. (2017); Denton & Fergus (2018), which contains the frames with reasonable motions. Following Jang et al. (2018), we added a diversity to videos by randomly skipping frames in a range of [1,3].
+
+# D.3.2 NETWORK ARCHITECTURE
+
+We compare our method against SAVP (Lee et al., 2018) which is proposed to achieve stochastic video prediction. SAVP addresses a mode-collapse problem using the hybrid model of conditional GAN and VAE. For a fair comparison, we construct our baseline cGAN by taking GAN component from SAVP (the generator and discriminator networks). In below, we provide more details of the generator and discriminator architectures used in our baseline cGAN model.
+
+The generator is based on the encoder-decoder network with convolutional LSTM (Xingjian et al., 2015). At each step, it takes a frame together with a latent code as inputs and produces the next frame as an output. Contrary to the original SAVP that takes a latent code at each step to encode framewise stochasticity, we modified the generator to take one latent code per sequence that encodes the global dynamics of a video. Then the discriminator takes the entire video as an input and produces a prediction on real or fake through 3D convolution operations.
+
+# D.3.3 EVALUATION METRICS
+
+We provide more details about the evaluation metrics used in our experiment. For each test video, we generate 100 random samples with a length of 28 frames and evaluate the performance based on the following metrics:
+
+• Diversity: To measure the degree of diversity of the generated samples, we computed the frame-wise distance between each pair of the generated videos based on Mean Squared Error (MSE). Then we reported the average distance over all pairs as a result. $D i s t _ { m i n }$ : Following Lee et al. (2018), we evaluate the quality of generations by measuring the distance of the closest sample among the all generated ones to the ground-truth. Specifically, for each test video, we computed the minimum distance between the generated samples and the ground-truth based on MSE and reported the average of the distances over the entire test videos.   
+• $S i m _ { m a x }$ : As another measure for the generation quality, we compute the similarity of the closest sample to the ground-truth similar to $D i s t _ { m i n }$ but using the cosine similarity of features extracted from VGGNet (Simonyan & Zisserman, 2015). We report the average of the computed similarity for entire test videos.
+
+# D.3.4 MORE EXAMPLES
+
+We present more video prediction results on both BAIR and KTH datasets in Figure H, which corresponds to Figure 6 in the main paper. As discussed in the main paper, the baseline cGAN produces realistic but deterministic outputs, whereas both SAVP and our method generate far more diverse future predictions. SAVP fails to generate the diverse outputs in KTH datasets, mainly because the dataset contains many examples with small motions. On the contrary, our method generates diverse outputs in both datasets, since our regularization directly penalizes the mode-collapsing behavior and force the model to discover various modes. Interestingly, we found that our model sometimes generates actions different from the input video when the motion in input frames are ambiguous (e.g. hand-clapping to hand-waving in the highlighted example). It shows that our method can generate diverse and meaningful futures.
+
+Figure I presents more detailed qualitative comparison in BAIR robot arm dataset. Both baseline cGAN and SAVP often suffer from the noise predictions in the background, since they fail to predict the correct motion of foreground objects. On the other hand, our method can generate more clear outputs and sometimes even an interaction between foreground and background objects by predicting more meaningful dynamics of videos from latent code $_ z$ . See captions of Figure I for more detailed discussions.
+
+![](images/c0e0efdaf1656a28fe410a3d79c715f0b813d66f48b8d2a43403b375a1650b22.jpg)  
+Figure H: Stochastic video prediction results. In both datasets, our method presents diverse prediction, whereas SAVP generate less diverse result especially in the KTH dataset. Interestingly, as you can see from the dotted orange box, our model can explore not only the original condition (hand clapping) but also other cases (hand waving) if the context is not too strong. Please check our web page to see the videos: https://sites.google.com/view/iclr19-dsgan/
+
+![](images/f0b3c8d2c4b1e86afd67587bef06de9f02d476043fd5abc967bf0814f3cd3f97.jpg)  
+Figure I: Qualitative comparison of various video prediction methods. Both baseline cGAN and SAVP exhibit some noises in the predicted videos due to the failures in separating the moving foreground object from the background clutters (red arrow). Compared to this, our method tends to generate more clear predictions on both foreground and background. Interestingly, SAVP sometimes fail to predict interaction between objects (magenta arrows). For instance, the objects on a table stay in the same position even after pushed by the robot arm. On the other hand, our method is able to capture such interactions more precisely (blue arrows).

parse/train/ud-WYSo9JSL/ud-WYSo9JSL.md

@@ -0,0 +1,282 @@
+# Can contrastive learning avoid shortcut solutions?
+
+Joshua RobinsonMIT CSAIL & LIDSjoshrob@mit.edu
+
+Li Sun University of Pittsburgh lis118@pitt.edu
+
+Ke Yu University of Pittsburgh yu.ke@pitt.edu
+
+Kayhan Batmanghelich University of Pittsburgh kayhan@pitt.edu
+
+Stefanie Jegelka MIT CSAIL stefje@csail.mit.edu
+
+Suvrit Sra  
+MIT LIDS  
+suvrit@mit.edu
+
+# Abstract
+
+The generalization of representations learned via contrastive learning depends crucially on what features of the data are extracted. However, we observe that the contrastive loss does not always sufficiently guide which features are extracted, a behavior that can negatively impact the performance on downstream tasks via “shortcuts”, i.e., by inadvertently suppressing important predictive features. We find that feature extraction is influenced by the difficulty of the so-called instance discrimination task (i.e., the task of discriminating pairs of similar points from pairs of dissimilar ones). Although harder pairs improve the representation of some features, the improvement comes at the cost of suppressing previously well represented features. In response, we propose implicit feature modification (IFM), a method for altering positive and negative samples in order to guide contrastive models towards capturing a wider variety of predictive features. Empirically, we observe that IFM reduces feature suppression, and as a result improves performance on vision and medical imaging tasks. The code is available at: https://github. com/joshr17/IFM.
+
+# 1 Introduction
+
+Representations trained with contrastive learning are adept at solving various vision tasks including classification, object detection, instance segmentation, and more [5, 15, 44]. In contrastive learning, encoders are trained to discriminate pairs of positive (similar) inputs from a selection of negative (dissimilar) pairs. This task is called instance discrimination: It is often framed using the InfoNCE loss [14, 33], whose minimization forces encoders to extract input features that are sufficient to discriminate similar and dissimilar pairs.
+
+However, learning features that are discriminative during training does not guarantee a model will generalize. Many studies find inductive biases in supervised learning toward simple “shortcut” features and decision rules [16, 21, 32] which result in unpredictable model behavior under perturbations [22, 43] and failure outside the training distribution [2, 37]. Simplicity bias has various potential sources [11] including training methods [8, 29, 41] and architecture design [10, 17]. Bias towards shortcut decision rules also hampers transferability in contrastive learning [4], where it is in addition influenced by the instance discrimination task. These difficulties lead us to ask: can the contrastive instance discrimination task itself be modified to avoid learning shortcut solutions?
+
+We approach this question by studying the relation between contrastive instance discrimination and feature learning. First, we theoretically explain why optimizing the InfoNCE loss alone does not guarantee avoidance of shortcut solutions that suppress (i.e., discard) certain input features [4, 11]. Second, despite this negative result, we show that it is still possible to trade off representation of one feature for another using simple methods for adjusting the difficulty of instance discrimination. However, these methods have an important drawback: improved learning of one feature often comes at the cost of harming another. That is, feature suppression is still prevalent. In response, we propose implicit feature modification, a technique that encourages encoders to discriminate instances using multiple input features. Our method introduces no computational overhead, reduces feature suppression (without trade-offs), and improves generalization on various downstream tasks.
+
+![](images/d6fbf1d29fd523c967273ded3a7ee69208a51fd99337cc1bb6d7349925dbcce9.jpg)  
+Figure 1: An ideal encoder would discriminate between instances using multiple distinguishing features instead of finding simple shortcuts that suppress features. We show that InfoNCE-trained encoders can suppress features (Sec. 2.2). However, making instance discrimination harder during training can trade off representation of different features (Sec. 2.3). To avoid the need for trade-offs we propose implicit feature modification (Sec. 3), which reduces suppression in general, and improves generalization (Sec. 4).
+
+Contributions. In summary, this paper makes the following main contributions:
+
+1. It analyzes feature suppression in contrastive learning, and explains why feature suppression can occur when optimizing the InfoNCE loss. 2. It studies the relation between instance discrimination tasks and feature learning; concretely, adjustments to instance discrimination difficulty leads to different features being learned. 3. It proposes implicit feature modification, a simple and efficient method that reduces the tendency to use feature suppressing shortcut solutions and improves generalization.
+
+# 1.1 Related work
+
+Unsupervised representation learning is enjoying a renaissance driven by steady advances in effective frameworks [3, 5, 15, 18, 33, 44, 45, 51]. As well as many effective contrastive methods, Siamese approaches that avoid representation collapse without explicitly use of negatives have also been proposed [6, 13, 51]. Pretext task design has been at the core of progress in self-supervised learning. Previously popular tasks include image colorization [54] and inpainting [35], and theoretical work shows pre-trained encoders can provably generalize if a pretext task necessitates the learning of features that solve downstream tasks [27, 39]. In contrastive learning, augmentation strategies are a key design component [5, 48, 50], as are negative mining techniques [9, 15, 25, 40]. While feature learning in contrastive learning has received less attention, recent work finds that low- and mid-level features are more important for transfer learning [55], and feature suppression can occur [4] just as with supervised learning [10, 16]. Combining contrastive learning with an auto-encoder has also been considered [28], but was found to harm representation of some features in order to avoid suppression of others. Our work is distinguished from prior work through our focus on how the design of the instance discrimination task itself affects which features are learned.
+
+# 2 Feature suppression in contrastive learning
+
+Feature suppression refers to the phenomenon where, in the presence of multiple predictive input features, a model uses only a subset of them and ignores the others. The selected subset often corresponds to intuitively “simpler” features, e.g., color as opposed to shape. Such features lead to “shortcut” decision rules that might perform well on training data, but can harm generalization and lead to poor robustness to data shifts. Feature suppression has been identified as a common problem in deep learning [11], and both supervised and contrastive learning suffer from biases induced by the choice of optimizer and architecture. However, contrastive learning bears an additional potential source of bias: the choice of instance discrimination task. Which positive and negative pairs are presented critically affects which features are discriminative, and hence which features are learned. In this work we study the relation between feature suppression and instance discrimination.
+
+First, we explain why optimizing the InfoNCE loss is insufficient in general to avoid feature suppression, and show how it can lead to counter-intuitive generalization (Sec. 2.2). Given this negative result, we then ask if it is at least possible to control which features a contrastive encoder learns? We find that this is indeed the case, and that adjustments to the instance discrimination task lead to different features being learned (Sec. 2.3). However, the primary drawback of these adjustments is that improving one feature often comes at the cost of harming representation of another. That is, feature suppression is still prevalent. Addressing this drawback is the focus of Sec. 3.
+
+# 2.1 Setup and definition of feature suppression
+
+Formally, we assume that the data has underlying feature spaces ${ \mathcal { Z } } ^ { 1 } , \ldots , { \mathcal { Z } } ^ { n }$ with a distribution $p _ { j }$ on each $\mathcal { Z } ^ { j }$ . Each $j \in [ n ]$ , corresponding to a latent space $\mathcal { Z } ^ { j }$ , models a distinct feature. We write the product as $\begin{array} { r } { \mathcal { Z } ^ { S } = \prod _ { j \in S } \mathcal { Z } ^ { j } } \end{array}$ , and simply write $\mathcal { Z }$ instead of $\mathcal { Z } ^ { [ n ] }$ where $[ n ] = \{ 1 , \dots , n \}$ . A set of features $z = ( z ^ { j } ) _ { j \in [ n ] } \in \mathcal { Z }$ is generated by sampling each coordinate $z ^ { j } \in \mathcal { Z } ^ { j }$ independently, and we denote the measure on $\mathcal { Z }$ induced by $z$ by $\lambda$ . Further, let $\lambda ( \cdot | z ^ { S } )$ denote the conditional measure on $\mathcal { Z }$ for fixed $z ^ { S }$ . For $S \subseteq [ n ]$ we use $\dot { z } ^ { S }$ to denote the projection of $z$ onto $\mathcal { Z } ^ { S }$ . Finally, an injective map $g : { \mathcal { Z } }  { \mathcal { X } }$ produces observations $x = g ( z )$ .
+
+Our aim is to train an encoder $f : \mathcal { X } \to \mathbb { S } ^ { d - 1 }$ to map input data $x$ to the surface of the unit sphere $\mathbb { S } ^ { d - 1 } = \{ u \in \mathbb { R } ^ { d } : \| u \| _ { 2 } = 1 \}$ in such a way that $f$ extracts useful information. To formally define feature suppression, we need the pushforward $h \# \nu ( V ) = \nu ( h ^ { - 1 } ( V ) )$ of a measure $\nu$ on a space $\mathcal { U }$ for a measurable map $h : \mathcal { U } \to \mathcal { V }$ and measurable $V \subseteq \nu$ , where $\dot { h } ^ { - 1 } ( V )$ denotes the preimage.
+
+Consider an encoder $f : \mathcal { X } \to \mathbb { S } ^ { d - 1 }$ and features $S \subseteq [ n ]$ . For each $z ^ { S } \in \mathcal { Z } ^ { S }$ , let $\mu ( \cdot | z ^ { S } ) =$ $( f \circ g ) \# \lambda ( \cdot | z ^ { S } )$ be the pushforward measure on $\mathbb { S } ^ { d - 1 }$ by $f \circ g$ of the conditional $\lambda ( \cdot | z ^ { S } )$ .
+
+1. $f$ suppresses $S$ if for any pair $z ^ { S } , { \bar { z } } ^ { S } \in { \mathcal { Z } } ^ { S }$ , we have $\mu ( \cdot | z ^ { S } ) = \mu ( \cdot | \bar { z } ^ { S } )$ .
+
+2. $f$ distinguishes $S$ if for any pair of distinct $z ^ { S } , { \bar { z } } ^ { S } \in { \mathcal { Z } } ^ { S }$ , measures $\mu ( \cdot | z ^ { S } ) , \mu ( \cdot | \bar { z } ^ { S } )$ have disjoint support.
+
+Feature suppression is thus captured in a distributional manner, stating that $S$ is suppressed if the encoder distributes inputs in a way that is invariant to the value $z ^ { S }$ . Distinguishing features, meanwhile, asks that the encoder $f$ separates points with different features $z ^ { S }$ into disjoint regions. We consider training an encoder $f : \mathcal { X } \to \mathbb { S } ^ { d - 1 }$ to optimize the InfoNCE loss [33, 14],
+
+$$
+\begin{array} { r } { \mathcal { L } _ { m } ( f ) = \mathbb { E } _ { x , x ^ { + } , \{ x _ { i } ^ { - } \} _ { i = 1 } ^ { m } } \bigg [ - \log \frac { e ^ { f ( x ) ^ { \top } f ( x ^ { + } ) / \tau } } { e ^ { f ( x ) ^ { \top } f ( x ^ { + } ) / \tau } + \sum _ { i = 1 } ^ { m } e ^ { f ( x ) ^ { \top } f ( x _ { i } ^ { - } ) / \tau } } \bigg ] , } \end{array}
+$$
+
+where $\tau$ is known as the temperature. Positive pairs $x , x ^ { + }$ are generated by first sampling $z \sim \lambda$ , then independently sampling two random augmentations $a , a ^ { + } \sim A$ , $a : \mathcal { X } \to \mathcal { X }$ from a distribution $\mathcal { A }$ , and setting $x = a ( g ( z ) )$ and $x ^ { + } = a ^ { + } ( g ( z ) )$ . We assume $\mathcal { A }$ samples the identity function $a ( x ) = x$ with non-zero probability (“ $x$ is similar to itself”), and that there are no collisions: $a ( x ) \neq a ^ { \prime } ( x ^ { \prime } )$ for all $a , a ^ { \prime }$ , and all $x \neq x ^ { \prime }$ . Each negative example $\boldsymbol { x } _ { i } ^ { - }$ is generated as $x _ { i } ^ { - } = a _ { i } ( g ( z _ { i } ) )$ , by independently sampling features $z _ { i } \sim \lambda$ and an augmentation $a _ { i } \sim { \mathcal { A } }$ .
+
+# 2.2 Why optimizing the InfoNCE loss can still lead to feature suppression
+
+Do optimal solutions to the InfoNCE loss automatically avoid shortcut solutions? Unfortunately, as we show in this section, this is not the case in general; there exist both optimal solutions of the InfoNCE loss that do and solutions that do not suppress a given feature. Following previous work [40, 49, 56], we analyze the loss as the number of negatives goes to infinity,
+
+$$
+\begin{array} { r l } { \stackrel { \cdot } { = } \underset { m  \infty } { \operatorname* { l i m } } \{ \mathcal { L } _ { m } ( f ) - \log m - \frac { 2 } { \tau } \} = \frac { 1 } { 2 \tau } \mathbb { E } _ { x , x ^ { + } } \Vert f ( x ) - f ( x ^ { + } ) \Vert ^ { 2 } + \mathbb { E } _ { x ^ { + } } \log [ \mathbb { E } _ { x ^ { - } } e ^ { f ( x ^ { + } ) ^ { \top } f ( x ^ { - } ) / \tau } ] . } \end{array}
+$$
+
+We subtract $\log m$ to ensure the limit is finite, and use $x ^ { - }$ to denote a random sample with the same distribution as $\boldsymbol { x } _ { i } ^ { - }$ . Prop. 2.2 (proved in App. A) shows that, assuming the marginals $p _ { j }$ are uniform, the InfoNCE loss is optimized both by encoders that suppress feature $j$ , and by encoders that distinguish $j$ .
+
+Suppose that $p _ { j }$ is uniform on $\mathcal { Z } ^ { j } = \mathbb { S } ^ { d - 1 }$ for all $j \in [ n ]$ . Then for any feature $j \in [ n ]$ there exists an encoder $f _ { \mathrm { s u p p } }$ that suppresses feature $j$ and encoder $f _ { \mathrm { d i s c } }$ that discriminates $j$ but both attain $\operatorname* { m i n } { } _ { f }$ : measurable $\mathcal { L } ( f )$ .
+
+The condition that $p _ { j }$ is uniformly distributed on $\mathcal { Z } ^ { j } = \mathbb { S } ^ { d - 1 }$ is similar to conditions used in previous work [56]. Prop. 2.2 shows that empirical observations of feature suppression [4] (see also Fig. 3)
+
+are not simply due to a failure to sufficiently optimize the loss, but that the possibility of feature suppression is built into the loss. What does Prop. 2.2 imply for the generalization behavior of encoders? Besides explaining why feature suppression can occur, Prop. 2.2 also suggests another counter-intuitive possibility: lower InfoNCE loss may actually lead to worse performance on some tasks.
+
+To empirically study whether this possibility manifests in practice, we use two datasets with known semantic features: (1) In the Trifeature data, [16] each image is $1 2 8 \times 1 2 8$ and has three features: color, shape, and texture, each taking possible 10 values. See Fig. 10, App. C for sample images. (2) In the STL-digits data, samples combine MNIST digits and STL10 objects by placing copies of a randomly selected MNIST digit on top of an STL10 image. See Fig. 11 App. C for sample images.
+
+We train encoders with ResNet-18 backbone using SimCLR [5]. To study correlations between the loss value and error on downstream tasks, we train 33 encoders on Trifea
+
+![](images/58d9620e1c56307182ac968752d1752ee45a3c23d4152d77dc12bb6e84348c5e.jpg)  
+Figure 2: Linear readout error on different downstream tasks can be negatively correlated. Further, lower InfoNCE loss does not always yield not lower error: error rates on texture, shape and STL10 prediction are negatively correlated with InfoNCE loss.
+
+ture and 7 encoders on STL-digits with different hyperparameter settings (see App. C.2 for full details on training and hyperparameters). For Trifeature, we compute the Pearson correlation between InfoNCE loss and linear readout error when predicting {color, shape, texture}. Likewise, for STL-digits we compute correlations between the InfoNCE loss and MNIST and STL10 prediction error.
+
+Fig. 2 shows that performance on different downstream tasks is not always positively correlated. For Trifeature, color error is negatively correlated with shape and texture, while for STL-digits there is a strong negative correlation between MNIST digit error and STL10 error. Importantly, lower InfoNCE loss is correlated with lower prediction error for color and MNIST-digit, but with larger error for shape, texture and STL10. Hence, lower InfoNCE loss can improve representation of some features (color, MNIST digit), but may actually hurt others. This conflict is likely due to the simpler color and MNIST digit features being used as shortcuts. Our observation is an important addition to the statement of Wang and Isola [49] that lower InfoNCE loss improves generalization: the situation is more subtle – whether lower InfoNCE helps generalization on a task depends on the use of shortcuts.
+
+# 2.3 Controlling feature learning via the difficulty of instance discrimination
+
+The previous section showed that the InfoNCE objective has solutions that suppress features. Next, we ask what factors determine which features are suppressed? Is there a way to target specific features and ensure they are encoded? One idea is to use harder positive and negative examples. Hard examples are precisely those that are not easily distinguishable using the currently extracted features. So, a focus on hard examples may change the scope of the captured features. To test this hypothesis, we consider two methods for adjusting the difficulty of positive and negative samples:
+
+1. Temperature $\tau$ in the InfoNCE loss (Eqn. 1). Smaller $\tau$ places higher importance on positive an negative pairs with high similarity [47]. 2. Hard negative sampling method of Robinson et al. [40], which uses importance sampling to sample harder negatives. The method introduces a hardness concentration parameter $\beta$ , with larger $\beta$ corresponding to harder negatives (see [40] for full details).
+
+Results reported in Fig. 3 (also Fig. 13 in App. C.2) show that varying instance discrimination difficulty—i.e., varying temperature $\tau$ or hardness concentration $\beta$ —enables trade-offs between which features are represented. On Trifeature, easier instance discrimination (large $\tau$ , small $\beta$ ) yields good performance on ‘color’—an “easy” feature for which a randomly initialized encoder already has high linear readout accuracy—while generalization on the harder texture and shape features is poor. The situation reverses for harder instance discrimination (small $\tau$ , large $\beta$ ). We hypothesize that the use of “easy” features with easy instance discrimination is analogous to simplicity biases in supervised deep networks [17, 21]. As with supervised learning [10, 17], we observe a bias for texture over shape in convolutional networks, with texture prediction always outperforming shape.
+
+![](images/ec74850ae8dbb37af5c9996ed86223161b43646369ceb4f7b0680fc8bb16fcb2.jpg)  
+Figure 3: Trifeature dataset [16]. The difficulty of instance discrimination affects which features are learned (Sec. 2.3). When instance discrimination is easy (big $\tau$ , small $\beta$ ), encoders represent color well and other features badly. When instance discrimination is hard (small $\tau$ , big $\beta$ ), encoders represent more challenging shape and texture features well, at the expense of color.
+
+That there are simple levers for controlling which features are learned already distinguishes contrastive learning from supervised learning, where attaining such control is less easy (though efforts have been made [23]). However, these results show that representation of one feature must be sacrificed in exchange for learning another one better. To understand how to develop methods for improving feature representation without suppressing others, the next result (proof in App. A) examines more closely why there is a relationship between (hard) instance discrimination tasks and feature learning.
+
+[Informal] Suppose that $p _ { j }$ is uniform on $\mathcal { Z } ^ { j } = \mathbb { S } ^ { d - 1 }$ for all $j \in [ n ]$ . Further, for $S \subseteq [ n ]$ suppose that $x , x ^ { + } , \{ x _ { i } ^ { - } \} _ { i }$ are conditioned on the event that they have the same features $S$ . Then any $f$ that minimizes the (limiting) InfoNCE loss suppresses features $S$ .
+
+The positive and negative instances in Prop. 2.3 must be distinguished with features in $S ^ { c }$ . Relating this point to the above observations, assume that an encoder exclusively uses features $S$ . Any positives and negatives that do not (much) differ in features $S$ are difficult for the encoder. By Prop. 2.3, focusing the training on these difficult examples pushes the encoder to instead use features in $S ^ { c }$ , i.e., to learn new features. But at the same time, the proposition also says that a strong focus on such hard negative pairs leads to suppressing the originally used features $S$ , explaining the results in Fig. 3. While the two techniques for adjusting instance difficulty we studied were unable to avoid feature suppression, this insight forms the motivation for implicit feature modification, which we introduce next.
+
+# 3 Implicit feature modification for reducing feature suppression
+
+The previous section found that simple adjustments to instance discrimination difficulty could significantly alter which features a model learns. Prop. 2.3 suggests that this ability to modify which features are learned stems from holding features constant across positive and negative samples. However, these methods were unable to avoid trade-offs in feature representation (Fig. 3) since features that are held constant are themselves suppressed (Prop. 2.3).
+
+To avoid this effect, we develop a technique that adaptively modifies samples to remove whichever features are used to discriminate a particular positive pair from negatives, then trains an encoder to discriminate instances using both the original features, and the features left over after modification. While a natural method for modifying features is to directly transform raw input data, it is very challenging to modify the semantics of an input in this way. So instead we propose modifying features by applying transformations to encoded samples $v = f ( x )$ . Since we modify the encoded samples, instead of raw inputs $x$ , we describe our method as implicit.
+
+We set up our notation. Given batch $x , x ^ { + } , \{ x _ { i } ^ { - } \} _ { i = 1 } ^ { m }$ we write $ { \boldsymbol { v } } \ = \ f (  { \boldsymbol { { x } } } )$ , $v ^ { + } ~ = ~ f ( x ^ { + } )$ , and $v _ { i } ^ { - } = f ( x _ { i } ^ { - } )$ to denote the corresponding embeddings. As in Eqn. 1, the point-wise InfoNCE loss is,
+
+$$
+\ell ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = - \log \frac { e ^ { v ^ { \top } v ^ { + } / \tau } } { e ^ { v ^ { \top } v ^ { + } / \tau } + \sum _ { i = 1 } ^ { m } e ^ { v ^ { \top } v _ { i } ^ { - } / \tau } } .
+$$
+
+[Implicit feature modification] Given budget $\varepsilon \in \mathbb { R } _ { + } ^ { m }$ , and encoder $f : \mathcal { X } \to \mathbb { S } ^ { d }$ , an adversary removes features from $f$ that discriminates batch $x , \stackrel { \cdot } { x } ^ { + } , \{ x _ { i } ^ { - } \} _ { i = 1 } ^ { m }$ by maximizing the point-wise InfoNCE loss, \`"(v, v+, {vi }mi=1) = max+2B + ,{i 2B" }mi=1 \` $\begin{array} { r } { \ell _ { \varepsilon } ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = \operatorname* { m a x } _ { \delta ^ { + } \in \mathcal { B } _ { \varepsilon ^ { + } } , \{ \delta _ { i } ^ { - } \in \mathcal { B } _ { \varepsilon _ { i } } \} _ { i = 1 } ^ { m } } \ell ( v , v ^ { + } + \delta ^ { + } , \{ v _ { i } ^ { - } + \delta _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) } \end{array}$ .
+
+Here $B _ { \varepsilon }$ denotes the $\ell _ { 2 }$ -ball of radius $\varepsilon$ . Implicit feature modification (IFM) removes components of the current representations that are used to discriminate positive and negative pairs. In other words, the embeddings of positive and negative samples are modified to remove well represented features. So, if the encoder is currently using a simple shortcut solution, IFM removes the features used, thereby encouraging the encoder to also discriminate instances using other features. By applying perturbations in the embedding space IFM can modify high level semantic features (see Fig. 4), which is extremely challenging when applying perturbations in input space. In order to learn new features using the perturbed loss while still learning potentially complementary information using the original InfoNCE objective, we propose optimizing the the multi-task objective $\mathrm { m i n } _ { f } \{ \mathcal { L } ( f ) \dot { + } \alpha \mathcal { L } _ { \varepsilon } ( \dot { f } ) \} / 2$ where $\mathcal { L } _ { \varepsilon } = \mathbb { E } \ell _ { \varepsilon }$ is the adversarial perturbed loss, and $\mathcal { L }$ the standard InfoNCE loss. For simplicity, all experiments set the balancing parameter $\alpha = 1$ unless explicitly noted, and all take $\varepsilon ^ { + } , \varepsilon _ { i } ^ { - }$ to be equal, and denote this single value by $\varepsilon$ . Crucially, $\ell _ { \varepsilon }$ can be computed analytically and efficiently. For any $v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } \in \bar { \mathbb { R } } ^ { d }$ we have,
+
+$$
+\nabla _ { v _ { j } ^ { - } } \ell = \frac { e ^ { v ^ { \top } v _ { j } ^ { - } / \tau } } { e ^ { v ^ { \top } v + / \tau } + \sum _ { i = 1 } ^ { m } e ^ { v ^ { \top } v _ { i } ^ { - } / \tau } } \cdot \frac { v } { \tau } \quad \mathrm { a n d } \quad \nabla _ { v ^ { + } } \ell = \left( \frac { e ^ { v ^ { \top } v ^ { + } / \tau } } { e ^ { v ^ { \top } v ^ { + } / \tau } + \sum _ { i = 1 } ^ { m } e ^ { v ^ { \top } v _ { i } ^ { - } / \tau } } - 1 \right) \cdot \frac { v } { \tau } .
+$$
+
+In particular, $\nabla _ { v _ { \bot \ldots } ^ { - } } \ell \propto v$ and $\nabla _ { v ^ { + } } \ell \propto - v$ . This expression shows that the adversary perturbs $\cdot ^ { v _ { j } ^ { - } }$ (resp. $v ^ { + }$ j) in the direction of the anchor $v$ (resp $- v ,$ ). Since the derivative directions are independent of $\{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m }$ and $v ^ { + }$ , we can analytically compute optimal perturbations in $B _ { \varepsilon }$ . Indeed, following the constant ascent direction shows the optimal updates are simply $v _ { i } ^ { - }  v _ { i } ^ { - } + \varepsilon _ { i } v$ and $v ^ { + }  v ^ { + } - \varepsilon ^ { + } v$ . The positive (resp. negative) perturbations increase (resp. decrease) cosine similarity to the anchor $\sin ( v , v _ { i } ^ { - } + \varepsilon _ { i } v )  1$ as $\varepsilon _ { i } \to \infty$ (resp. $\sin ( v , v ^ { + } - \bar { \varepsilon } ^ { + } v )  - 1$ as $\varepsilon ^ { + } \to \infty$ ). In Fig. 4 we visualize the newly synthesized $v _ { i } ^ { - } , v ^ { + }$ and find meaningful interpolation of semantics. Plugging the update rules for $v ^ { + }$ and ${ \boldsymbol v } _ { i } ^ { - }$ into the point-wise InfoNCE loss yields,
+
+$$
+\begin{array} { r } { \ell _ { \varepsilon } ( v , v ^ { + } , \{ v _ { i } ^ { - } \} _ { i = 1 } ^ { m } ) = - \log \frac { e ^ { ( v ^ { \top } v ^ { + } - \varepsilon ^ { + } ) / \tau } } { e ^ { ( v ^ { \top } v ^ { + } - \varepsilon ^ { + } ) / \tau } + \sum _ { i = 1 } ^ { m } e ^ { ( v ^ { \top } v _ { i } ^ { - } + \varepsilon _ { i } ) / \tau } } . } \end{array}
+$$
+
+In other words, IFM amounts to simply perturbing the logits – reduce the positive logit by $\varepsilon ^ { + } / \tau$ and increase negative logits by $\varepsilon _ { i } / \tau$ . From this we see that $\ell _ { \varepsilon }$ is automatically symmetrized in the positive samples: perturbing $v$ instead of $v ^ { + }$ results in the exact same objective. Eqn. 2 shows that IFM re-weights each negative sample by a factor $e ^ { \varepsilon _ { i } / \tau }$ and positive samples by $e ^ { - \varepsilon ^ { + } / \tau }$ .
+
+# 3.1 Visualizing implicit feature modification
+
+With implicit feature modification, newly synthesized data points do not directly correspond to any “true” input data point. However it is still possible to visualize the effects of implicit feature modification. To do this, assume access to a memory bank of input data $\mathcal { M } = \{ x _ { i } \} _ { i }$ . A newly synthesized sample $s$ can be approximately visualized by retrieving the 1-nearest neighbour using cosine similarity arg $\operatorname* { m i n } _ { x \in { \mathcal { M } } }$ $\sin ( s , f ( x ) )$ and viewing the image $x$ as an approximation to $s$ .
+
+Fig. 4 shows results using a ResNet-50 encoder trained using MoCo-v2 on ImageNet1K using the training set as the memory bank. For positive pair $v , v ^ { + }$ increasing $\varepsilon$ causes the semantics of $v$ and $v ^ { + }$ to diverge. For $\varepsilon = 0 . 1$ a different car with similar pose and color is generated, for $\varepsilon = 0 . 2$ the pose and color then changes, and finally for $\varepsilon = 1$ the pose, color and type of vehicle changes. For negative pair $v , v ^ { - }$ the reverse occurs. For $\varepsilon = 0 . 1$ , $v ^ { - }$ is a vehicle with similar characteristics (number of windows, color etc.), and with $\varepsilon = 0 . 2$ , the pose of the vehicle $v ^ { + }$ aligns with $v$ . Finally for $\varepsilon = 1$ the pose and color of the perturbed negative sample become aligned to the anchor $v$ . In summary, implicit feature modification successfully modifies the feature content in positive and negative samples, thereby altering which features can be used to discriminate instances.
+
+Related Work. Several works consider adversarial contrastive learning [19, 24, 26] using PGD (e.g. FGSM) attacks to alter samples in input space. Unlike our approach, PGD-based attacks require costly inner-loop optimization. Other work takes an adversarial viewpoint in input space for other self-supervised tasks e.g., rotations and jigsaws but uses an image-to-image network to simulate FGSM/PGD attacks [31], introducing comparable computation overheads. They note that low-level (i.e., pixel-level) shortcuts can be avoided using their method. All of these works differ from ours by applying attacks in input space, thereby focusing on lower-level features, whereas ours aims to modify high-level features. Fig. 5 compares IFM to this family of input-space adversarial methods by comparing to a top performing method ACL(DS) [24]. We find that ACL improves robust accuracy under $\ell _ { \infty }$ -attack on input space (see [24] for protocol details), whereas IFM improves standard accuracy (full details and discussion in Appdx. C.3). Synthesizing harder negatives in latent space using Mixup [53] has also been considered [25] but does not take an adversarial perspective. Other work, AdCo [20], also takes an adversarial viewpoint in latent space. There are several differences to our approach. AdCo perturbs all negatives using the same weighted combination of all the queries, whereas IFM perturbations are query specific. In other words, IFM makes instance discrimination harder point-wise, whereas AdCo perturbation makes the InfoNCE loss larger on average (see Fig. 4 for visualizations of instance dependent perturbation using IFM). AdCo also treats the negatives as learnable parameters, introducing $\sim 1 M$ more parameters and $\sim 7 \%$ computational overhead, while IFM has no computational overhead and is implemented with only two lines of code (see Tab. 1 for empirical comparison). Finally, no previous work makes the connection between suppression of semantic features and adversarial methods in contrastive learning (see Fig. 6).
+
+![](images/5afd711f6d1c8ba5155a029a38cb357006a49104919af8921fef36588c01ef36.jpg)  
+Figure 4: Visualizing implicit feature modification. Top row: progressively moving positive sample away from anchor. Bottom row: progressively moving negative sample away from anchor. In both cases, semantics such as color, orientation, and vehicle type are modified, showing the suitability of implicit feature modification for altering instance discrimination tasks.
+
+![](images/12e76d56fbd4c437b958a68c9ba5a8527e9b6f78defb2f0d01b3e203aa5892c9.jpg)  
+Figure 5: Comparison between IFM and ACL(DS). Under standard linear evaluation IFM performs best. ACL is suited to adversarial evaluation.
+
+![](images/13aac9c5a8c8da9fb88ce77335c13c805d3c88c58d0b67da7c450ce2eb3ce0da.jpg)  
+Figure 6: Trifeature dataset. Implicit feature modification reduces feature suppression, enhancing the representation of texture, shape and color features simultaneously. All results are average linear readout accuracy over three seeds and use a fixed value $\varepsilon = 0 . 1$ to illustrate robustness to $\varepsilon$ .
+
+# 4 Experimental results
+
+Implicit feature modification (IFM) can be used with any InfoNCE-based contrastive framework, and we write IFM-SimCLR, IFM-MoCo-v2 etc. to denote IFM applied within a specific framework. Code for IFM will be released publicly, and is also available in the supplementary material.
+
+# 4.1 Does implicit feature modification help avoid feature suppression?
+
+We study the effect IFM has on feature suppression by training ResNet-18 encoders for 200 epochs with $\tau \in \{ 0 . 0 5 , 0 . 2 , 0 . 5 \}$ on the Trifeature dataset [16]. Results are averaged over three seeds, with IFM using $\varepsilon = 0 . 1$ for simplicity. Fig. 6 shows that IFM improves the linear readout accuracy across all three features for all temperature settings. The capability of IFM to enhance the representation of all features – i.e. reduce reliance on shortcut solutions – is an important contrast with tuning temperature $\tau$ or using hard negatives, which Fig. 3 shows only trades-off which features are learned.
+
+![](images/26ba8877d15efa834c609010064f2443c1db3c4db9cb586a0504757a21b985bb.jpg)  
+Figure 7: IFM improves linear readout performance on all datasets for all $\varepsilon \in \{ 0 . 0 5 , 0 . 1 , 0 . 2 \}$ compared to baselines. Protocol uses 400 epochs of training with ResNet-50 backbone.
+
+# 4.2 Performance on downstream tasks
+
+Sec. 3.1 and Sec. 4.1 demonstrate that implicit feature modification is adept at altering high-level features of an input, and combats feature suppression. This section shows that these desirable traits translate into improved performance on object classification and medical imaging tasks.
+
+Experimental setup for classification tasks. Having observed the positive effect IFM has on feature suppression, we next test if this feeds through to improved performance on real tasks of interest. We benchmark using both SimCLR and MoCo-v2 [5, 7] with standard data augmentation [5]. All encoders have ResNet-50 backbones and are trained for 400 epochs (with the exception of on ImageNet100, which is trained for 200 epochs). All encoders are evaluated using the test accuracy of a linear classifier trained on the full training dataset (see Appdx. C.4 for full setup details).
+
+Classification tasks. Results given in Fig. 7 and Tab. 1 find that every value of $0 ~ < ~ \varepsilon ~ \le ~ 0 . 2$ improves performance across all datasets using both MoCo- $\nu 2$ and SimCLR frameworks. We find
+
+<table><tr><td></td><td>MoCo-v2</td><td>AdCo [20]</td><td colspan="3">IFM-MoCo-v2</td></tr><tr><td>m</td><td>N/A</td><td>N/A</td><td>0.05</td><td>0.1</td><td>0.2</td></tr><tr><td>top-1</td><td>80.4±0.11</td><td>78.9±0.21</td><td>81.1±0.02</td><td>80.9±0.25</td><td>80.7±0.13</td></tr></table>
+
+Table 1: Linear readout $( \% )$ ) on ImageNet100, averaged over five seeds.   
+IFM improves over MoCo-v2 for all settings of $\varepsilon$ .
+
+that optimizing $\mathcal { L } _ { \varepsilon }$ $7 6 . 0 \%$ average score across all eight runs in Fig. 7) performs similarly to the standard contrastive loss $( 7 5 . 9 \%$ average score), and does worse than the IFM loss $( \mathcal { L } + \mathcal { L } _ { \varepsilon } ) \dot { / } 2$ . This suggests that $\mathcal { L }$ and $\mathcal { L } _ { \varepsilon }$ learn complementary features. Tab. 1 benchmarks IFM on ImageNet100 [44] using MoCo-v2, observing improvements of $0 . 9 \%$ . We also compare results on ImageNet100 to AdCo [20], another adversarial method for contrastive learning. We adopt the official code and use the exact same training and finetuning hyperparameters as for MoCo-v2 and IFM. For the AdCo-specific hyperparamters – negatives learning rate $l r _ { \mathrm { n e g } }$ and negatives temperature $\tau _ { \mathrm { n e g } } - \mathrm { w e }$ use a grid search over all combinations $l r _ { \mathrm { n e g } } \in \{ 1 , \bar { 2 } , 3 , 4 \}$ and $\tau _ { \mathrm { n e g } } \in \mathsf { \bar { \{ 0 . 0 2 , 0 . 1 \} } }$ , which includes the AdCo default ImageNet1K recommendations $l r _ { \mathrm { n e g } } = 3$ and $\tau _ { \mathrm { n e g } } = 0 . 0 2$ [20]. The resulting AdCo performance of $7 8 . 9 \%$ is slightly below MoCo-v2. However using their respective ImageNet1K default parameters AdCo and MoCo-v2 achieve $7 2 . 4 \%$ and $7 1 . 8 \%$ respectively, suggesting that the discrepancy between AdCo and MoCo-v2 may in part be due to the use of improved hyperparameters tuned on MoCo-v2. Note importantly, IFM is robust to the choice of $\varepsilon$ : all values $\varepsilon \in \{ 0 . 0 5 , 0 . 1 , 0 . 2 \}$ were found to boost performance across all datasets and all frameworks. We emphasize that the MoCo-v2 baseline performance of $8 0 . 5 \%$ on ImageNet100 is strong. Our hyperparameters, which we detail in Appdx. C.4.1, may be of interest to other works benchmarking MoCo-v2 on ImageNet100.
+
+Medical images. To evaluate our method on a modality differing significantly from object-based images we consider the task of learning representations of medical images. We benchmark using the approach proposed by [42] which is a variant of MoCo-v2 that incorporates the anatomical context in the medical images. We evaluate our method on the COPDGene dataset [38], which is a multi-center observational study focused on the genetic epidemiology of Chronic obstructive pulmonary disease (COPD). See Appdx. C.5 for full background details on the COPDGene dataset, the five COPD related outcomes we use for evaluation, and our implementation. We perform regression analysis for continuous outcomes in terms of coefficient of determination (R-square), and logistic regression to predict ordinal outcomes and report the classification accuracy and the 1-off accuracy, i.e., the probability of the predicted category is within one class of true value.
+
+Tab. 2 reports results. For fair comparison we use same experimental configuration for the baseline approach [42] and our method. We find that IFM yields improvements on all outcome predictions.
+
+<table><tr><td>Method</td><td>logFEV1pp</td><td>logFEV1FVC</td><td>CLE</td><td>CLE1-off</td><td></td><td>Para-septalPara-septal l-off</td><td>mMRC</td><td>mMRC1-off</td></tr><tr><td>Loss</td><td colspan="2">R-Square</td><td colspan="6">Accuracy (%)</td></tr><tr><td>L(baseline)</td><td>0.566±.005</td><td>0.661±.005</td><td>49.6±0.4</td><td>81.8±0.5</td><td>55.7±0.3</td><td>84.4±0.2</td><td>50.4±0.5</td><td>72.5±0.3</td></tr><tr><td>Lg,ε=0.1</td><td>0.591±.008</td><td>0.681±.008</td><td>49.4±0.4</td><td>81.9±0.3</td><td>55.6±0.3</td><td>85.1±0.2</td><td>50.3±0.8</td><td>72.7±0.4</td></tr><tr><td>IFM,ε=0.1</td><td>0.615±.005</td><td>0.691±.006</td><td>48.2±0.8</td><td>80.6±0.4</td><td>55.3±0.4</td><td>84.7±0.3</td><td>50.4±0.5</td><td>72.8±0.2</td></tr><tr><td>IFM,ε = 0.2</td><td>0.595±.006</td><td>0.683±.006</td><td>48.5±0.6</td><td>80.5±0.6</td><td>55.3±0.3</td><td>85.1±0.1</td><td>49.8±0.8</td><td>72.0±0.3</td></tr><tr><td>IFM,ε= 0.5</td><td>0.607±.006</td><td>0.683±.005</td><td>49.6±0.4</td><td>82.0±0.3</td><td>54.9±0.2</td><td>84.7±0.2</td><td>50.6±0.4</td><td>73.1±0.2</td></tr><tr><td>IFM,ε = 1.0</td><td>0.583±.005</td><td>0.675±.006</td><td>50.0±0.5</td><td>82.9±0.4</td><td>56.3±0.6</td><td>85.7±0.2</td><td>50.3±0.6</td><td>71.9±0.3</td></tr></table>
+
+Table 2: Linear readout performance on COPDGene dataset. The values are the average of 5-fold cross validation with standard deviations. The bold face indicates the best average performance. IFM yields improvements on all phenotype predictions.
+
+![](images/b3a0c0f5b45705c48026a3b939aab0886ce1f3dbd5c0de0274447ad9ce01439a.jpg)  
+Figure 8: Label $\left\{ \mathcal { D } , \mathcal { D } _ { \mathrm { R } } , \mathcal { D } _ { \mathrm { N R } } \right\}$ indicates which dataset was used to train the linear readout function. Improved performance of IFM on standard data $\mathcal { D }$ can be attributed to improved representation of robust features $\mathcal { D } _ { \mathrm { R } }$ . See Sec. 4.3 for construction of robust $( \mathcal { D } _ { \mathtt { R } } )$ and non-robust $( \mathcal { D } _ { \mathrm { N R } } )$ datasets.
+
+The gain is largest on spirometry outcome prediction, particularly logFEV1pp with improvement of $8 . { \bar { 7 } } \%$ with $\varepsilon = 0 . 1$ . We found that at least $\varepsilon = 0 . 5$ and 1.0 improve performance on all tasks. However, we note that not all features yield a statistically significant improvement with IFM.
+
+# 4.3 Further study on the impact of IFM on feature learning
+
+This section further studies the effect implicit feature modification has on what type of features are extracted. Specifically, we consider the impact on learning of robust (higher-level) vs. non-robust features (pixel-level features). Our methodology, which is similar to that of Ilyas et al. [22] for deep supervised learning, involves carefully perturbing inputs to obtain non-robust features.
+
+Constructing non-robust features. Given encoder $f$ we finetune a linear probe (classifier) $h$ ontop of $f$ using training data (we do not use data augmentation). Once $h$ is trained, we consider each labeled example $( x , y )$ from training data $\mathcal { D } _ { \mathrm { t r a i n } } \in \{ \mathrm { t i n y I m a g e N e t , S T L 1 0 , C I F A R 1 0 , C I F A R 1 0 0 } \}$ . A hallucinated target label $t$ is sampled uniformly at random, and we perturb $x = x _ { 0 }$ until $h \circ f$ predicts $t$ using repeated FGSM attacks [12] $x _ { k } \gets x _ { k - 1 } - \varepsilon \mathrm { s i g n } ( \nabla _ { x } \ell ( h \circ f ( x _ { k - 1 } ) , t ) )$ . At each step we check if arg maxi $h \circ f ( x _ { k } ) _ { i } = t$ (we use the maximum of logits for inference) and stop iterating and set $x _ { \mathrm { a d v } } = x _ { k }$ for the first $k$ for which the prediction is $t$ . This usually takes no more than a few FGSM steps with $\varepsilon = 0 . 0 1$ . We form a dataset of “robust” features by adding $( x _ { \mathrm { a d v } } , y )$ to $\mathcal { D } _ { R }$ , and a dataset of “non-robust” features by adding $( x _ { \mathrm { a d v } } , t )$ to $\mathcal { D } _ { N R }$ . To a human the pair $( x _ { \mathrm { a d v } } , t )$ will look mislabeled, but for the encoder $x _ { \mathrm { a d v } }$ contains features predictive of $t$ . Finally, we re-finetune (i.e. re-train) linear classifier $g$ using $\mathcal { D } _ { R }$ (resp. $\mathcal { D } _ { N R }$ ).
+
+Fig. 8 compares accuracy of the re-finetuned models on a test set of standard $\mathcal { D } _ { \mathrm { t e s t } }$ examples (no perturbations are applied to the test set). Note that $\mathcal { D } _ { R }$ , $\mathcal { D } _ { N R }$ depend on the original encoder $f$ . When re-finetuning $f$ we always use datasets $\mathcal { D } _ { R }$ , $\mathcal { D } _ { N R }$ formed via FGSM attacks on $f$ itself. So there is one set $\mathcal { D } _ { R } , \mathcal { D } _ { N R }$ for SimCLR, and another set for IFM. Fig. 8 shows that IFM achieves superior generalization $( \mathcal { D } )$ compared to SimCLR by better representing robust features $( \mathcal { D } _ { R } )$ . Representation of non-robust features $( \mathcal { D } _ { N R } )$ is similar for IFM $( 5 \bar { 5 } . 5 \%$ average across all datasets) and SimCLR $( 5 6 . 7 \%$ average). IFM is juxtaposed to the supervised adversarial training of Madry et al., which sacrifices standard supervised performance in exchange for not using non-robust features [30, 46].
+
+# 5 Discussion
+
+This work studies the relation between contrastive instance discrimination and feature learning. While we focus specifically on contrastive learning, it would be of interest to also study feature learning for other empirically successful self-supervised methods [1, 6, 13, 51]. Understanding differences in feature learning biases between different methods may inform which methods are best suited for a given task, as well as point the way to further improved self-supervised techniques.
+
+Acknowledgments SJ was supported by NSF BIGDATA award IIS-1741341, NSF Convergence Accelerator Track D 2040636. SS acknowledges support from NSF-TRIPODS $^ +$ X:RES (1839258). JR was partially supported by a Two Sigma fellowship. KB acknowledges support from NIH (1R01HL141813-01), NSF (1839332 Tripod $+ \mathrm { X }$ ), and a research grant from SAP SE Commonwealth Universal Research Enhancement (CURE) program awards research grants from the Pennsylvania Department of Health. Finally, we warmly thank Katherine Hermann and Andrew Lampinen for making the Trifeature dataset available for our use.
+
+# References
+
+[1] Adrien Bardes, Jean Ponce, and Yann LeCun. VICReg: Variance-invariance-covariance regularization for self-supervised learning. preprint arXiv:2105.04906, 2021.   
+[2] Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision (ECCV), pages 456–473, 2018.   
+[3] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. In Advances in Neural Information Processing Systems (NeurIPS), pages 9912–9924, 2020.   
+[4] Ting Chen and Lala Li. Intriguing properties of contrastive losses. preprint arXiv:2011.02803, 2020.   
+[5] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In Int. Conference on Machine Learning (ICML), pages 10709–10719, 2020.   
+[6] Xinlei Chen and Kaiming He. Exploring simple siamese representation learning. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[7] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. preprint arXiv:2003.04297, 2020.   
+[8] Lenaic Chizat and Francis Bach. Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss. In Conference on Learning Theory (COLT), pages 1305–1338, 2020.   
+[9] Ching-Yao Chuang, Joshua Robinson, Lin Yen-Chen, Antonio Torralba, and Stefanie Jegelka. Debiased contrastive learning. In Advances in Neural Information Processing Systems (NeurIPS), pages 8765–8775, 2020.   
+[10] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and Wieland Brendel. ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness. In Int. Conf. on Learning Representations (ICLR), 2019.   
+[11] Robert Geirhos, Jorn-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel, ¨ Matthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):665–673, 2020.   
+[12] Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In Int. Conf. on Learning Representations (ICLR), 2015.   
+[13] Jean-Bastien Grill, Florian Strub, Florent Altche, Corentin Tallec, Pierre H Richemond, Elena ´ Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. In Advances in Neural Information Processing Systems (NeurIPS), pages 21271–21284, 2020.   
+[14] Michael Gutmann and Aapo Hyvarinen. Noise-contrastive estimation: A new estimation ¨ principle for unnormalized statistical models. In Proc. Int. Conference on Artificial Intelligence and Statistics (AISTATS), pages 297–304, 2010.   
+[15] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 9729–9738, 2020.   
+[16] Katherine L Hermann and Andrew K Lampinen. What shapes feature representations? Exploring datasets, architectures, and training. In Advances in Neural Information Processing Systems (NeurIPS), pages 9995–10006, 2020.   
+[17] Katherine L Hermann, Ting Chen, and Simon Kornblith. The origins and prevalence of texture bias in convolutional neural networks. In Advances in Neural Information Processing Systems (NeurIPS), pages 19000–19015, 2019.   
+[18] R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In Int. Conf. on Learning Representations (ICLR), 2019.   
+[19] Chih-Hui Ho and Nuno Nvasconcelos. Contrastive learning with adversarial examples. In Advances in Neural Information Processing Systems (NeurIPS), pages 17081–17093, 2020.   
+[20] Qianjiang Hu, Xiao Wang, Wei Hu, and Guo-Jun Qi. Adco: Adversarial contrast for efficient learning of unsupervised representations from self-trained negative adversaries. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   
+[21] Minyoung Huh, Hossein Mobahi, Richard Zhang, Brian Cheung, Pulkit Agrawal, and Phillip Isola. The low-rank simplicity bias in deep networks. preprint arXiv:2103.10427, 2021.   
+[22] Andrew Ilyas, Shibani Santurkar, Dimitris Tsipras, Logan Engstrom, Brandon Tran, and Aleksander Madry. Adversarial examples are not bugs, they are features. In Advances in Neural Information Processing Systems (NeurIPS), pages 125–136, 2019.   
+[23] Jorn-Henrik Jacobsen, Jens Behrmann, Richard Zemel, and Matthias Bethge. Excessive ¨ invariance causes adversarial vulnerability. In Int. Conf. on Learning Representations (ICLR), 2018.   
+[24] Ziyu Jiang, Tianlong Chen, Ting Chen, and Zhangyang Wang. Robust pre-training by adversarial contrastive learning. In Advances in Neural Information Processing Systems (NeurIPS), pages 16199–16210, 2020.   
+[25] Yannis Kalantidis, Mert Bulent Sariyildiz, Noe Pion, Philippe Weinzaepfel, and Diane Larlus. Hard negative mixing for contrastive learning. In Advances in Neural Information Processing Systems (NeurIPS), pages 21798–21809, 2020.   
+[26] Minseon Kim, Jihoon Tack, and Sung Ju Hwang. Adversarial self-supervised contrastive learning. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   
+[27] Jason D Lee, Qi Lei, Nikunj Saunshi, and Jiacheng Zhuo. Predicting what you already know helps: Provable self-supervised learning. preprint arXiv:2008.01064, 2020.   
+[28] Tianhong Li, Lijie Fan, Yuan Yuan, Hao He, Yonglong Tian, and Dina Katabi. Informationpreserving contrastive learning for self-supervised representations. preprint arXiv:2012.09962, 2020.   
+[29] Kaifeng Lyu and Jian Li. Gradient descent maximizes the margin of homogeneous neural networks. In Int. Conf. on Learning Representations (ICLR), 2020.   
+[30] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In Int. Conf. on Learning Representations (ICLR), 2018.   
+[31] Matthias Minderer, Olivier Bachem, Neil Houlsby, and Michael Tschannen. Automatic shortcut removal for self-supervised representation learning. In International Conference on Machine Learning, pages 6927–6937, 2020.   
+[32] Thao Nguyen, Maithra Raghu, and Simon Kornblith. Do wide and deep networks learn the same things? uncovering how neural network representations vary with width and depth. In Int. Conf. on Learning Representations (ICLR), 2021.   
+[33] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. preprint arXiv:1807.03748, 2018.
+
+[34] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems (NeurIPS), 2019.
+
+[35] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2536–2544, 2016.
+
+[36] Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, ¨ Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikitlearn: Machine learning in python. Journal of machine learning research, pages 2825–2830, 2011.
+
+[37] Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal Shankar. Do ImageNet classifiers generalize to ImageNet? In Int. Conference on Machine Learning (ICML), pages 5389–5400, 2019.
+
+[38] Elizabeth A Regan, John E Hokanson, James R Murphy, Barry Make, David A Lynch, Terri H Beaty, Douglas Curran-Everett, Edwin K Silverman, and James D Crapo. Genetic epidemiology of COPD (COPDGene) study design. COPD: Journal of Chronic Obstructive Pulmonary Disease, 7(1):32–43, 2011.
+
+[39] Joshua Robinson, Stefanie Jegelka, and Suvrit Sra. Strength from weakness: Fast learning using weak supervision. In Int. Conference on Machine Learning (ICML), pages 8127–8136, 2020.
+
+[40] Joshua Robinson, Ching-Yao Chuang, Suvrit Sra, and Stefanie Jegelka. Contrastive learning with hard negative samples. In Int. Conf. on Learning Representations (ICLR), 2021.
+
+[41] Daniel Soudry, Elad Hoffer, Mor Shpigel Nacson, Suriya Gunasekar, and Nathan Srebro. The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1):2822–2878, 2018.
+
+[42] Li Sun, Ke Yu, and Kayhan Batmanghelich. Context matters: Graph-based self-supervised representation learning for medical images. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 4874–4882, 2021.
+
+[43] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In Int. Conf. on Learning Representations (ICLR), 2014.
+
+[44] Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. In Europ. Conference on Computer Vision (ECCV), 2020.
+
+[45] Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. In Advances in Neural Information Processing Systems (NeurIPS), pages 6827–6839, 2020.
+
+[46] Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and Aleksander Madry. Robustness may be at odds with accuracy. In Int. Conf. on Learning Representations (ICLR), 2018.
+
+[47] Feng Wang and Huaping Liu. Understanding the behaviour of contrastive loss. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2021.
+
+[48] Feng Wang, Huaping Liu, Di Guo, and Fuchun Sun. Unsupervised representation learning by invariance propagation. In Advances in Neural Information Processing Systems (NeurIPS), pages 3510–3520, 2020.
+
+[49] Tongzhou Wang and Phillip Isola. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. In Int. Conference on Machine Learning (ICML), pages 9574–9584, 2020.
+
+[50] Xiao Wang and Guo-Jun Qi. Contrastive learning with stronger augmentations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.   
+[51] Jure Zbontar, Li Jing, Ishan Misra, Yann LeCun, and Stephane Deny. Barlow twins: Self- ´ supervised learning via redundancy reduction. In Int. Conference on Machine Learning (ICML), 2021.   
+[52] Hongyang Zhang, Yaodong Yu, Jiantao Jiao, Eric Xing, Laurent El Ghaoui, and Michael Jordan. Theoretically principled trade-off between robustness and accuracy. In Int. Conference on Machine Learning (ICML), pages 7472–7482, 2019.   
+[53] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In Int. Conf. on Learning Representations (ICLR), 2018.   
+[54] Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In Europ. Conference on Computer Vision (ECCV), pages 649–666, 2016.   
+[55] Nanxuan Zhao, Zhirong Wu, Rynson WH Lau, and Stephen Lin. What makes instance discrimination good for transfer learning? In Int. Conf. on Learning Representations (ICLR), 2021.   
+[56] Roland S Zimmermann, Yash Sharma, Steffen Schneider, Matthias Bethge, and Wieland Brendel. Contrastive learning inverts the data generating process. In Int. Conference on Machine Learning (ICML), 2021.