Add files using upload-large-folder tool

parse/train/GSXEx6iYd0/GSXEx6iYd0.md

@@ -0,0 +1,355 @@
+# Numerical Composition of Differential Privacy∗
+
+Sivakanth Gopi Microsoft Research sigopi@microsoft.com
+
+Yin Tat Lee University of Washington yintat@uw.edu
+
+Lukas Wutschitz Microsoft lukas.wutschitz@microsoft.com
+
+# Abstract
+
+We give a fast algorithm to optimally compose privacy guarantees of differentially private (DP) algorithms to arbitrary accuracy. Our method is based on the notion of privacy loss random variables to quantify the privacy loss of DP algorithms. The running time and memory needed for our algorithm to approximate the privacy√ curve of a DP algorithm composed with itself $k$ times is $\bar { \tilde { O } } ( { \sqrt { k } } )$ . This improves over the best prior method by [KH21] which requires $\tilde { \Omega } ( k ^ { 1 . 5 } )$ running time. We demonstrate the utility of our algorithm by accurately computing the privacy loss of DP-SGD algorithm of Abadi et al. $[ \mathsf { A C G } ^ { + } 1 6 ]$ and showing that our algorithm speeds up the privacy computations by a few orders of magnitude compared to prior work, while maintaining similar accuracy.
+
+# 1 Introduction
+
+Differential privacy (DP) introduced by [DMNS06] provides a provable and quantifiable guarantee of privacy when the results of an algorithm run on private data are made public. Formally, we can define an $( \varepsilon , \delta )$ -differentially private algorithm as follows.
+
+Definition 1.1 ( $( \varepsilon , \delta )$ -DP [DMNS06, $\mathrm { D K M ^ { + } 0 6 ] }$ ). An algorithm $\mathcal { M }$ is $( \varepsilon , \delta )$ -DP if for any two neighboring databases $D , D ^ { \prime }$ differing in exactly one user and any subset $S$ of outputs, we have $\operatorname* { P r } [ \mathcal { M } ( D ) \in S ] \le e ^ { \varepsilon } \operatorname* { P r } [ \mathcal { M } ( D ^ { \prime } ) \in S ] + \delta$ .
+
+Intuitively, it says that looking at the outcome of $\mathcal { M }$ , we cannot tell whether it was run on $D$ or $D ^ { \prime }$ . Hence an adversary cannot infer the existence of any particular user in the input database, and therefore cannot learn any personal data of any particular user.
+
+DP algorithms have an important property called composition. Suppose $M _ { 1 }$ and $M _ { 2 }$ are DP algorithms and say $M ( D ) = ( M _ { 1 } ( D ) , M _ { 2 } ( D ) )$ , i.e., $M$ runs both the algorithms on $D$ and outputs their results. Then $M$ is also a DP algorithm.
+
+Proposition 1.2 (Simple composition $\mathrm { [ D K M ^ { + } 0 6 }$ , DL09]). If $M _ { 1 }$ is $( \varepsilon _ { 1 } , \delta _ { 1 } )$ -DP and $M _ { 2 }$ is $( \varepsilon _ { 2 } , \delta _ { 2 } )$ - $D P ,$ then $M ( D ) = ( \bar { M } _ { 1 } ( D ) , \bar { M } _ { 2 } ( D ) )$ is $( \varepsilon _ { 1 } + \varepsilon _ { 2 } , \delta _ { 1 } + \delta _ { 2 } ) – D P .$ .
+
+This also holds under adaptive composition (denoted by $M = M _ { 2 } \circ M _ { 1 } )$ , where $M _ { 2 }$ can look at both the database and the output of $M _ { 1 }$ (here $M ( D ) = ( M _ { 1 } ( D ) , M _ { 2 } ( D , M _ { 1 } ( D ) ) ) )$ . It turns out that both compositions enjoy much better DP guarantees than this simple composition rule. Let $M$ be an $( \varepsilon , \delta )$ -DP algorithm and let $M ^ { \circ k }$ denote the (adaptive) composition of $M$ with itself $k$ times. The naive composition rule shows that $M ^ { \circ k }$ is $( k \varepsilon , k \delta )$ -DP. This was significantly improved in [DRV10].
+
+Proposition 1.3 (Advanced composition [DRV10, $\mathrm { D R } ^ { + } 1 4 ]$ ). If $M$ is $( \varepsilon , \delta )$ -DP, then $M ^ { \circ k }$ is $( \varepsilon ^ { \prime } , k \delta +$ $\delta ^ { \prime }$ )-DP where
+
+$$
+\varepsilon ^ { \prime } = \varepsilon \sqrt { 2 k \log \left( \frac { 1 } { \delta ^ { \prime } } \right) } + k \varepsilon ( e ^ { \varepsilon } - 1 ) .
+$$
+
+Note that if $\begin{array} { r } { \varepsilon = O \left( \frac { 1 } { \sqrt { k } } \right) } \end{array}$ and $\begin{array} { r } { \delta = o \left( \frac { 1 } { k } \right) } \end{array}$ , then $M ^ { \circ k }$ satisfies $( O _ { \delta ^ { \prime } } ( 1 ) , \delta ^ { \prime } )$ -DP. Using simple composition (Proposition 1.2), we can only claim that $M ^ { \circ k }$ is $( O ( { \sqrt { k } } ) , o ( 1 ) )$ -DP. Thus advanced composition often results in $\sqrt { k }$ -factor savings in privacy which is significant in practice. The optimal DP guarantees for $k$ -fold composition of an $( \varepsilon , \delta )$ -DP algorithm were finally obtained by [KOV15]. For composing different algorithms, the situation is more complicated. If $M _ { 1 } , M _ { 2 } , \ldots , M _ { k }$ are DP algorithms such that $M _ { i }$ is $( \varepsilon _ { i } , \delta _ { i } )$ -DP, then it is shown by [MV16] that computing the exact DP guarantees for $M = M _ { 1 } \circ M _ { 2 } \circ \cdot \cdot \cdot \circ M _ { k }$ is #P-complete. They also give an algorithm to approximate the DP guarantees of $M$ to desired accuracy $\eta$ which runs in
+
+$$
+\tilde { O } \left( \frac { k ^ { 3 } \bar { \varepsilon } ( 1 + \bar { \varepsilon } ) } { \eta } \right)
+$$
+
+time where $\textstyle { \bar { \varepsilon } } = ( \sum _ { i = 1 } ^ { k } \varepsilon _ { i } ) / k$ .2 If each $\begin{array} { r } { \varepsilon _ { i } \approx \frac { 1 } { \sqrt { k } } } \end{array}$ (so that $M$ will satisfy reasonable privacy guarantees by advanced composition), then the running time is $\tilde { O } ( k ^ { 2 . 5 } / \eta )$ .
+
+In most situations, DP algorithms come with a collection of $( \varepsilon , \delta )$ -DP guarantees, i.e., for each value of $\varepsilon$ , there exists $\delta$ such that the algorithm is $( \varepsilon , \delta )$ -DP.
+
+Definition 1.4 (Privacy curve). A DP algorithm $M$ is said to have privacy curve $\delta : \mathbb { R }  [ 0 , 1 ] ,$ if for every $\varepsilon \in \mathbb { R }$ , $M$ is $( \varepsilon , \delta ( \varepsilon ) ) \lrcorner D P .$
+
+For example the privacy curve of a Gaussian mechanism (with sensitivity 1 and noise scale $\sigma$ ) is given by $\delta ( \varepsilon ) = \Phi \left( - \varepsilon \sigma + 1 / 2 \sigma \right) - e ^ { \varepsilon } \Phi \left( - \varepsilon \sigma - 1 / 2 \sigma \right)$ where $\Phi ( \cdot )$ is the Gaussian CDF [BW18]. Suppose we want to compose several Gaussian mechanisms, which $( \varepsilon , \delta )$ -DP guarantee should we choose for each mechanism? Any choice will lead to suboptimal DP guarantees for the final composition. Instead, we need a way to compose the privacy curves directly. This was suggested through the use of privacy region in [KOV15] and explicitly studied in the $f$ -DP framework of [DRS19]. $f$ -DP is a dual way (and equivalent) to look at the privacy curve $\delta ( \varepsilon )$ .
+
+Independently, an algorithm called Privacy Buckets for approximately composing privacy curves using the notion of was initiated in [MM18]. This algorithm depends on the notion of Privacy Loss Random Variable (PRV) [DR16], whose distribution is called Privacy Loss Distribution (PLD). For any DP-algorithm, one can associate a PRV and the privacy curve of that algorithm can be easily obtained from the PRV. The really useful property of PRVs is that under adaptive composition, they just add up; the PRV $Y$ of the composition $M = M _ { 1 } \circ M _ { 2 } \circ \cdots \circ M _ { k }$ is given by $\begin{array} { r } { \bar { Y } = \sum _ { i = 1 } ^ { k } \bar { Y _ { i } } } \end{array}$ where $Y _ { i }$ is the PRV of $M _ { i }$ .3 Therefore, one can find the distribution of $Y$ by the convolution of the distributions of $Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }$ . In an important paper, $[ \mathrm { K J H ^ { + } } 2 0 ]$ proposed that one can speed up the convolutions using Fast Fourier Transform (FFT). Explicit error bounds were obtained for the approximation obtained by their algorithm in $[ \mathrm { K J H ^ { + } } 2 0 $ , KJPH21, KH21]. The running time of this algorithm was analyzed in [KH21] where it was shown that the privacy curve $\delta _ { M } ( \varepsilon )$ of $M = M _ { 1 } \circ M _ { 2 } \circ \cdot \cdot \cdot \circ M _ { k }$ can be computed up to an additive error of $\delta _ { \mathrm { e r r o r } }$ in time
+
+$$
+\tilde { O } \left( \frac { k ^ { 3 } \bar { \varepsilon } } { \delta _ { \mathrm { e r r o r } } } \right) ,
+$$
+
+if each algorithm $M _ { i }$ is satisfies $( \varepsilon _ { i } , 0 )$ -DP and $\begin{array} { r } { \bar { \varepsilon } = \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \varepsilon _ { i } } \end{array}$ . Assuming that each $\begin{array} { r } { \varepsilon _ { i } \approx \frac { 1 } { \sqrt { k } } } \end{array}$ , we get $\tilde { O } ( k ^ { 2 . 5 } / \delta _ { \mathrm { e r r o r } } )$ running time. Note that this is slightly worse than (1), where the denominator $\eta$ is the multiplicative error in $\delta _ { M }$ . When composing the same algorithm with itself for $k$ times, the running time can be improved to $\begin{array} { r } { \tilde { O } \left( \frac { k ^ { 2 } \bar { \varepsilon } } { \delta _ { \mathrm { e r r o r } } } \right) } \end{array}$ , which is $\begin{array} { r } { \tilde { O } \left( \frac { k ^ { 1 . 5 } } { \delta _ { \mathrm { e r r o r } } } \right) } \end{array}$ when ε¯ = √1k .
+
+Moments Accountant and Renyi DP In an influential paper where they introduce Differentially Private Deep Learning, $[ \mathsf { A C G } ^ { + } 1 6 ]$ proposed a method called the Moments Accountant (MA) for giving an upper bound the privacy curve of a composition of DP algorithms. They applied their method to bound the privacy loss of differentially private Stochastic Gradient Descent (DP-SGD) algorithm which they introduced. Analyzing the privacy loss of DP-SGD involves composing the privacy curve of each iteration of training with itself $k$ times, where $k$ is the total of number of training iterations. Typical values of $k$ range from 1000 to 300000 (such as when training large models like GPT3). The Moments Accountant was subsumed into the framework of Renyi Differential Privacy (RDP) introduced by [Mir17]. The running time of these accountants are independent of $k$ , but they only give an upper bound and cannot approximate the privacy curve to arbitrary accuracy.
+
+DP-SGD is one of the most important DP algorithms in practice, because one can use it to train neural networks to achieve good privacy-vs-utility tradeoffs. Therefore obtaining accurate and tight privacy guarantees for DP-SGD is important. For example reducing $\varepsilon$ from 2 to 1, can mean that one can train the network for 4 times more epochs while staying within the same privacy budget. Therefore DP-SGD is one of the main motivations for this work.
+
+There are also situations when the PRVs do not have bounded moments and so Moments Accountant or Renyi DP cannot be applied for analyzing privacy. An example of such an algorithm is the DP-SGD-JL algorithm of $[ \mathrm { B G K } ^ { + } 2 1 ]$ which uses numerical composition of PRVs to analyze privacy.
+
+GDP Accountant [DRS19, BDLS19] introduced the notion of Gaussian Differential Privacy (GDP) and used it to develop an accountant for DP-SGD. The accountant is based on central limit theorem and only gives an approximation to the true privacy curve, where the approximation gets better with $k$ . But as we show in Figure 1, GDP accountant can significantly underreport the true epsilon value.
+
+Several different notions of privacy were introduced for obtaining good upper bounds on the privacy curve of composition of DP algorithms such as Concentrated DP (CDP) [DR16, BS16], Truncated CDP [BDRS18] etc. None of these methods can approximate the privacy curve of compositions to arbitrary accuracy. The notion of $f$ -DP introduced by [DRS19], allows for a lossless composition theorem, but computing the privacy curve of composition seems computationally hard and they do not give any algorithms for doing it.
+
+# 1.1 Our Contributions
+
+The main contribution of this work is a new algorithm with an improved analysis for computing the privacy curve of the composition of a large number of DP algorithms.
+
+Theorem 1.5 (Informal version of Theorem 5.5). Suppose $M _ { 1 } , M _ { 2 } , \ldots , M _ { k }$ are $D P$ algorithms. Then the privacy curve $\delta _ { M } ( \varepsilon )$ of adaptive composition $M = M _ { 1 } \circ M _ { 2 } \circ \cdot \cdot \cdot \circ M _ { k }$ can be approximated in time
+
+$$
+O \left( \frac { \varepsilon _ { \mathrm { u p p e r } } k ^ { 1 . 5 } \log k \sqrt { \log \frac { 1 } { \delta _ { \mathrm { e r r o r } } } } } { \varepsilon _ { \mathrm { e r r o r } } } \right) ,
+$$
+
+where $\varepsilon _ { \mathrm { e r r o r } }$ is the additive error in $\varepsilon$ , $\delta _ { \mathrm { e r r o r } }$ is the additive error in $\delta$ and $\varepsilon _ { \mathrm { u p p e r } }$ is an upper bound on max $\begin{array} { r } { \left\{ \varepsilon _ { M } ( \delta _ { \mathrm { e r r o r } } ) , \mathrm { m a x } _ { i } \varepsilon _ { M _ { i } } \left( \frac { \delta _ { \mathrm { e r r o r } } } { k } \right) \right\} } \end{array}$ . 4
+
+If each $M _ { i }$ satisfies $\scriptstyle \left( { \frac { 1 } { \sqrt { k } } } , { \frac { o ( 1 ) } { k } } \right)$ -DP, then by advanced composition (Proposition 1.3), we can set $\varepsilon _ { \mathrm { u p p e r } } = O ( 1 )$ . Therefore the running time of our algorithm in this case is $\tilde { O } \left( \frac { k ^ { 1 . 5 } \sqrt { \log \frac { 1 } { \delta _ { \mathrm { e r r o r } } } } } { \varepsilon _ { \mathrm { e r r o r } } } \right)$ . We can save a factor of $k$ , when we compose the same algorithm with itself $k$ times.
+
+Theorem 1.6. Suppose $M$ is a DP algorithm. Then the privacy curve $\delta _ { M ^ { \circ } k } ( \varepsilon )$ of $M$ (adaptively) composed with itself $k$ times can be approximated in time
+
+$$
+O \left( \frac { \varepsilon _ { \mathrm { u p p e r } } k ^ { \frac { 1 } { 2 } } \log k \sqrt { \log \frac { 1 } { \delta _ { \mathrm { e r r o r } } } } } { \varepsilon _ { \mathrm { e r r o r } } } \right) ,
+$$
+
+where $\varepsilon _ { \mathrm { e r r o r } }$ is the additive error in $\varepsilon$ , $\delta _ { \mathrm { e r r o r } }$ is the additive error in $\delta$ and $\varepsilon _ { \mathrm { u p p e r } }$ is an upper bound on max $\begin{array} { r } { \left\{ \varepsilon _ { M ^ { \circ k } } ( \delta _ { \mathrm { e r r o r } } ) , \varepsilon _ { M } \left( \frac { \delta _ { \mathrm { e r r o r } } } { k } \right) \right\} } \end{array}$ .
+
+Thus we improve the state-of-the-art by at least a factor of $k$ in running time. We also note that our algorithm improves the memory required by a factor of $k$ . See Figure 1 for a comparison of our algorithm with that of [KJPH21]. Also note that RDP Accountant (equivalent to the Moments Accountant) significantly overestimates the true $\varepsilon$ , while the GDP Accountant significantly underestimates the true $\varepsilon$ . In contrast, the upper and lower bounds provided by our algorithm lie very close to each other.
+
+![](images/8e9f81b58d5b20927335c64d5bde49592def1d97d29b397a5ab89e225d8706ef.jpg)
+
+![](images/9d2af7302e29615a33af3e6ff3f26516bd66935578ca15397ae42aa3cec7bb70.jpg)  
+(b) Our algorithm can improve significantly over the RDP Accountant. We also see that GDP Accountant can significantly underreport the true $\varepsilon$ . We have set $\varepsilon _ { \mathrm { e r r o r } } = 0 . 1$ , $\delta _ { \mathrm { e r r o r } } = { \bar { \delta } } / 1 0 0 0$ here.
+
+(a) Our algorithm gives much closer upper and lower bounds on the true privacy curve compared to [KJPH21], under the same mesh size of $4 ~ \times$ $1 0 ^ { - 5 }$ . Our upper and lower bounds are nearly coinciding.
+
+Figure 1: Case study on DP-SGD. Sampling probability $p = 1 0 ^ { - 3 }$ , noise scale $\sigma = 0 . 8$ , $\delta = 1 0 ^ { - 7 }$
+
+Our Techniques Our algorithm (also the prior work of $\mathrm { [ K J H ^ { + } 2 0 ] }$ ) proceeds by approximating the privacy loss random variables (PRVs) by truncating and discretizing them. We then use Fast Fourier Transform (FFT) to convolve the distributions efficiently. The main difference is in the approximation procedure and the error analysis. In the approximation procedure, we correct the approximation so that the expected value of the discretization matches with the expected value of the PRV.
+
+To analyze the approximation error, we introduce the concept of coupling approximation (Definition 5.1), which is a variant of Wasserstein (optimal transport) distance specifically tailored to this application. We first show that the approximation output by our algorithm to each privacy random variable is a good coupling approximation. We then show that when independent coupling approximations are added, cancellation happens between the errors due to Hoeffding bound, producing a much better coupling approximation than one naively expects from the triangle inequality. This allows us to choose the mesh size in our discretization to be $\approx { \frac { 1 } { \sqrt { k } } }$ , whereas [KH21] choose a mesh size of $\approx \frac { 1 } { k }$ . The other improvement is the truncation procedure. We give a tight tail bound of the PRVs (Lemma 5.4). This allows us to choose the domain size for in truncation to be√ $\approx \tilde { O } ( 1 )$ , whereas [KH21] choose $\approx \tilde { O } ( \sqrt { k } )$ . Both ideas together saves a factor of $k$ in the run time and memory.
+
+For the analysis, the previous paper analyzes the discretization error by studying the stability of convolution. This leads to complicated calculations with the runtime linear in $1 / \bar { \delta } _ { \mathrm { e r r o r } } \bar { }$ (see (2)). Since $\delta _ { \mathrm { e r r o r } } \ll \delta \ll 1 / N$ is required to give meaningful privacy guarantee $N$ is the number of users), this term $1 / \delta _ { \mathrm { e r r o r } }$ is huge. In this paper, we show various facts about how coupling approximation accumulates and use them to give a runtime depending only on $\sqrt { \log ( 1 / \delta _ { \mathrm { e r r o r } } ) }$ .
+
+# 2 DP Preliminaries
+
+Given a DP algorithm $\mathcal { M }$ , for each value of $\varepsilon \geq 0$ , there exists some $\delta \in [ 0 , 1 ]$ such that $\mathcal { M }$ is $( \varepsilon , \delta )$ -DP. We can represent all these privacy guarantees by a function $\delta _ { \mathcal { M } } ( \varepsilon ) : \mathbb { R } ^ { \geq 0 }  [ 0 , 1 ]$ and say that $\delta _ { \mathcal { M } } ( \cdot )$ is the privacy curve of $\mathcal { M }$ . This inspires the following definition of a privacy curve between two random variables.
+
+Definition 2.1 (Privacy curve). Given two random variables $X , Y$ supported on some set $\Omega$ , define $\delta ( X \vert \vert Y ) : \mathbb { R } \to [ 0 , 1 ]$ as:
+
+$$
+\delta ( X | | Y ) ( \varepsilon ) = \operatorname* { s u p } _ { S \subset \Omega } \operatorname* { P r } [ Y \in S ] - e ^ { \varepsilon } \operatorname* { P r } [ X \in S ] .
+$$
+
+Therefore an algorithm $\mathcal { M }$ is $( \varepsilon , \delta )$ -DP iff $\delta \left( \mathcal { M } ( D ) \vert \vert \mathcal { M } ( D ^ { \prime } ) \right) ( \varepsilon ) \leq \delta$ for all neighboring databases $D , D ^ { \prime }$ .
+
+Remark 2.2. Note that not all functions $\delta : \mathbb { R }  [ 0 , 1 ]$ are privacy curves. A characterization of privacy curves can be obtained using the $f$ -DP framework of [DRS19]. The notion of privacy curve $\delta ( X \vert \vert Y )$ and tradeoff function $T ( X \vert \vert Y )$ are dual to each other via convex duality [DRS19]. This implies a characterization of privacy curves as shown in [ZDW21].
+
+Definition 2.3 (Composition of privacy curves [DRS19]). Let $\delta _ { 1 } \equiv \delta ( X _ { 1 } | | Y _ { 1 } )$ and $\delta _ { 2 } \equiv \delta ( X _ { 2 } | | Y _ { 2 } )$ be any two privacy curves. The composition of the privacy curves, denoted by $\delta _ { 1 } \otimes \delta _ { 2 }$ , is defined as
+
+$$
+\delta _ { 1 } \otimes \delta _ { 2 } \equiv \delta \left( ( X _ { 1 } , X _ { 2 } ) | | ( Y _ { 1 } , Y _ { 2 } ) \right)
+$$
+
+where $X _ { 1 } , X _ { 2 }$ are independently sampled and $Y _ { 1 } , Y _ { 2 }$ are independently sampled.
+
+Note that there can be many pairs of random variables which have the same privacy curve, but the above operation is well-defined. If $\delta ( X _ { 1 } | | Y _ { 1 } ) \equiv \delta ( X _ { 1 } ^ { \prime } | | Y _ { 1 } ^ { \prime } )$ and $\delta ( X _ { 2 } | | Y _ { 2 } ) \equiv \delta ( X _ { 2 } ^ { \prime } | | Y _ { 2 } ^ { \prime } )$ , then it was shown by [DRS19] that
+
+$$
+\delta \left( ( X _ { 1 } , X _ { 2 } ) | | ( Y _ { 1 } , Y _ { 2 } ) \right) = \delta \left( ( X _ { 1 } ^ { \prime } , X _ { 2 } ^ { \prime } ) | | ( Y _ { 1 } ^ { \prime } , Y _ { 2 } ^ { \prime } ) \right) .
+$$
+
+[DRS19] also show that $\otimes$ is a commutative and associative operation.
+
+Given two DP algorithms $M _ { 1 }$ and $M _ { 2 }$ , the adaptive composition $( M _ { 2 } \circ M _ { 1 } ) ( D )$ is an algorithm which outputs $( M _ { 1 } ( D ) , M _ { 2 } ( D , M _ { 1 } ( D ) )$ , i.e., $M _ { 2 }$ can look at the database $D$ and also the output of the previous algorithm $M _ { 1 } ( D )$ . Adaptive composition of more than two algorithms is similarly defined. Suppose $M _ { 1 }$ has privacy curve $\delta _ { 1 }$ and $M _ { 2 }$ has privacy curve $\delta _ { 2 }$ (i.e., $M _ { 2 } ( \cdot , y )$ is a $\mathrm { D P }$ algorithm with privacy curve $\delta _ { 2 }$ for any fixed $y$ .). The following composition theorem shows how to get the privacy curve of $M _ { 2 } \circ M _ { 1 }$ .
+
+Theorem 2.4 (Composition theorem [DRS19]). Let $M _ { 1 } , M _ { 2 } , \ldots , M _ { k }$ be DP algorithms with privacy curves given by $\delta _ { 1 } , \delta _ { 2 } , \dots , \delta _ { k }$ respectively. The privacy curve of the adaptive composition $M _ { k } \circ$ $M _ { k - 1 } \circ \cdots \circ M _ { 1 }$ is given by $\delta _ { 1 } \otimes \delta _ { 2 } \otimes \cdot \cdot \cdot \otimes \delta _ { k }$ .
+
+# 3 Privacy loss random variables
+
+The notion of privacy loss random variables (PRVs) is a unique way to assign a pair $( X , Y )$ for any privacy curve $\delta$ such that $\delta \equiv \delta ( X | | Y )$ . PRVs allow us to compute composition of two algorithms via summing random variables (Theorem 3.5) (equivalently, convolving their distributions). Thus PRVs can be thought of as a reparametrization of privacy curves where composition becomes convolution. In this paper, we differ from the usual definition of PRVs given in [DR16, $\mathrm { K J H } ^ { + } 2 0 ]$ , which are tied to a specific algorithm. Instead we think of them as a reparametrization of privacy curves and study them directly. This allows us to succinctly prove many useful properties of PRVs.
+
+Let ${ \overline { { \mathbb { R } } } } = \mathbb { R } \cup \{ - \infty , \infty \}$ be the extended real line where we define $\infty + x = \infty$ and $- \infty + x = - \infty$ for $x \in \mathbb { R }$ .
+
+Definition 3.1 (Privacy loss random variables (PRVs)). Given a privacy curve $\delta : \mathbb { R }  [ 0 , 1 ]$ , we say that $( X , Y )$ are privacy loss random variables for $\delta$ , if they satisfy the following conditions:
+
+• $X , Y$ are supported on ${ \overline { { \mathbb { R } } } } ,$ , • $\delta ( X | | Y ) \equiv \delta$ , • $Y ( t ) = e ^ { t } X ( t )$ for every $t \in \mathbb { R }$ and • $Y ( - \infty ) = 0 a n d X ( \infty ) = 0$
+
+where $X ( t ) , Y ( t )$ are probability density functions of $X , Y$ respectively.
+
+Mathematically, the correct way to write the condition $Y ( t ) = e ^ { t } X ( t )$ is to say that $\mathbb { E } _ { Y } [ \phi ( Y ) ] =$ $\mathbb { E } _ { X } [ \phi ( X ) e ^ { X } ]$ for all test functions $\phi : \overline { { \mathbb { R } } }  [ 0 , 1 ]$ with $\phi ( \infty ) = \phi ( - \infty ) = 0$ . This will generalize to all situations where $X , Y$ are continuous or discrete or both. For ease of exposition, we ignore this complication and assume that $X ( t ) , Y ( t )$ represent the PDFs if $X , Y$ are continuous at $t$ , or the probability masses if they have point masses at $t$ .
+
+The following theorem shows that the PRVs for a privacy curve $\delta = \delta ( P | | Q )$ are given by the log-likelihood random variables of $P , Q$ .
+
+Theorem 3.2. Let $\delta : \mathbb { R }  [ 0 , 1 ]$ be a privacy curve given by $\delta \equiv \delta ( P | | Q )$ where $P , Q$ are two random variables supported on $\Omega$ . The PRVs $( X , Y )$ for the privacy curve $\delta$ are given $b y ^ { 5 }$ :
+
+$$
+\begin{array} { r } { X = \log \left( \frac { Q ( \omega ) } { P ( \omega ) } \right) ~ w h e r e ~ \omega \sim P , } \\ { Y = \log \left( \frac { Q ( \omega ) } { P ( \omega ) } \right) ~ w h e r e ~ \omega \sim Q . } \end{array}
+$$
+
+The following theorem provides a formula for computing the privacy curve $\delta$ in terms of the PRVs and conversely a formula for PRVs in terms of the privacy curve. A similar statement appears in [SMM19, $\mathrm { K J H } ^ { + } 2 0 ]$ ].
+
+Theorem 3.3. The privacy curve $\delta$ can be expressed in terms of PRVs $( X , Y )$ as:
+
+$$
+\delta ( \varepsilon ) = \operatorname* { P r } [ Y > \varepsilon ] - e ^ { \varepsilon } \operatorname* { P r } [ X > \varepsilon ] = \mathbb { E } _ { Y } [ ( 1 - e ^ { \varepsilon - Y } ) _ { + } ] = \operatorname* { P r } [ Y \geq \varepsilon + Z ] .
+$$
+
+where $Z$ is an exponential random variable.6 Conversely, given a privacy curve $\delta : \mathbb { R }  [ 0 , 1 ]$ , we can compute the PDFs of its $P R V s$ $( X , Y )$ as:
+
+$$
+Y ( t ) = \delta ^ { \prime \prime } ( t ) - \delta ^ { \prime } ( t ) a n d X ( t ) = e ^ { t } ( \delta ^ { \prime \prime } ( t ) - \delta ^ { \prime } ( t ) ) .
+$$
+
+Remark 3.4. Theorem 3.3 shows that the PRVs $X , Y$ do not depend on the particular $P , Q$ used to represent the privacy curve $\delta$ in Theorem 3.2. So we should think of the PDF of of the $P R V Y$ $( o r X )$ as an equivalent reparametrization of the privacy curve $\delta : \mathbb { R }  [ 0 , 1 ]$ , just as the notion of $f$ -DP [DRS19] is a reparametrization of the privacy curve $\delta$ .
+
+PRVs are useful in computing privacy curves because the composition of two privacy curves can be computed by adding the corresponding pairs of PRVs. A similar statement appears in [DR16].
+
+Theorem 3.5. Let $\delta _ { 1 } , \delta _ { 2 }$ be two privacy curves with PRVs $( X _ { 1 } , Y _ { 1 } )$ and $( X _ { 2 } , Y _ { 2 } )$ respectively. Then the PRVs for $\delta _ { 1 } \otimes \delta _ { 2 } = \delta ( X _ { 1 } , X _ { 2 } | | Y _ { 1 } , Y _ { 2 } )$ are given by $\left( X _ { 1 } + X _ { 2 } , Y _ { 1 } + Y _ { 2 } \right)$ . In particular,
+
+$$
+\delta _ { 1 } \otimes \delta _ { 2 } = \delta ( X _ { 1 } + X _ { 2 } | | Y _ { 1 } + Y _ { 2 } ) .
+$$
+
+Proof. Let $( X , Y )$ be the privacy random variables for $\delta ( X _ { 1 } , X _ { 2 } | | Y _ { 1 } , Y _ { 2 } )$ . By Theorem 3.2,
+
+$$
+\begin{array} { r l } & { X = \log \left( \frac { ( Y _ { 1 } , Y _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } { ( X _ { 1 } , X _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } \right) \mathrm { ~ w h e r e ~ } ( t _ { 1 } , t _ { 2 } ) \sim ( X _ { 1 } , X _ { 2 } ) } \\ & { \quad = \log \left( \frac { Y _ { 1 } ( t _ { 1 } ) Y _ { 2 } ( t _ { 2 } ) } { X _ { 1 } ( t _ { 1 } ) \cdot X _ { 2 } ( t _ { 2 } ) } \right) \mathrm { ~ w h e r e ~ } t _ { 1 } \sim X _ { 1 } , t _ { 2 } \sim X _ { 2 } } \\ & { \qquad \mathrm { ( B y ~ i n d e p e n d e n c e ~ o f ~ } X _ { 1 } , X _ { 2 } \mathrm { ~ a n d ~ i n d p e n d } } \\ & { \quad = \log \left( e ^ { t _ { 1 } } \cdot e ^ { t _ { 2 } } \right) \mathrm { ~ w h e r e ~ } t _ { 1 } \sim X _ { 1 } , t _ { 2 } \sim X _ { 2 } } \\ & { \quad = t _ { 1 } + t _ { 2 } \mathrm { ~ w h e r e ~ } t _ { 1 } \sim X _ { 1 } , t _ { 2 } \sim X _ { 2 } } \\ & { \quad = X _ { 1 } + X _ { 2 } . } \end{array}
+$$
+
+Similarly,
+
+$$
+{ \begin{array} { r l } & { Y = \log \left( { \frac { ( Y _ { 1 } , Y _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } { ( X _ { 1 } , X _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } } \right) { \mathrm { ~ w h e r e ~ } } ( t _ { 1 } , t _ { 2 } ) \sim ( Y _ { 1 } , Y _ { 2 } ) } \\ & { \quad = t _ { 1 } + t _ { 2 } { \mathrm { ~ w h e r e ~ } } t _ { 1 } \sim Y _ { 1 } , t _ { 2 } \sim Y _ { 2 } } \\ & { \quad = Y _ { 1 } + Y _ { 2 } . } \end{array} }
+$$
+
+In the supplementary material, we provide a proof of Theorems 3.2 and 3.3. We also discuss how to compute the PRVs for a subsampled mechanism given the PRVs for the original mechanism and give examples of PRVs for few standard mechanisms. These are used in our experiments to calculate the PRVs for DP-SGD.
+
+# 4 Numerical composition of privacy curves
+
+In this section, we present an efficient and numerically accurate method, ComposePRV (Algorithm 1), for composing privacy guarantees by utilizing the notion of PRVs.
+
+Input: CDFs of PRVs $Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }$ , mesh size $h$ , Truncation parameter $\textstyle L \in { \frac { h } { 2 } } + h \mathbb { Z } ^ { > 0 }$   
+Output: PDF of an approximation $\widetilde { Y }$ for $\begin{array} { r } { Y = \sum _ { i = 1 } ^ { k } Y _ { i } } \end{array}$ . $\widetilde { Y }$ will be supported on $\mu + ( h \mathbb { Z } \cap [ - L , L ] )$ for some $\mu \in [ 0 , \frac { h } { 2 } ]$ .   
+for $\ell = 1$ to $k$ do $\begin{array} { r l } { \big | } & { { } \widetilde { Y } _ { i } \gets \mathsf { D i s c r e t i z e P R V } ( Y _ { i } , L , h ) ; } \end{array}$   
+end   
+Compute PDF Of $\widetilde { Y } = \widetilde { Y } _ { 1 } \oplus _ { L } \widetilde { Y } _ { 2 } \oplus _ { L } \cdot \cdot \cdot \oplus _ { L } \widetilde { Y } _ { k }$ by convolving PDFs of $\widetilde { Y } _ { 1 } , \widetilde { Y } _ { 2 } , \ldots , \widetilde { Y } _ { k }$ using FFT;   
+Compu $\mathbf { \varepsilon } \mathbf { e } \delta _ { \widetilde { Y } } ( \varepsilon ) = \mathbb { E } _ { \widetilde { Y } } \left[ \left( 1 - e ^ { \varepsilon - \widetilde { Y } } \right) _ { + } \right] \mathrm { f o r } \mathrm { a l l } \varepsilon \in [ 0 , L ] ;$   
+Return Y , δ e Y (·)
+
+In the algorithm ComposePRV, we compute the circular convolution $\oplus _ { L }$ using Fast Fourier Transform (FFT). Fix some $L > 0$ . For $x \in \mathbb { R }$ , we define $x$ $( { \bmod { 2 } } L ) = x - 2 L n$ where $n \in \mathbb { Z }$ is chosen such that $x - 2 L n \in ( - L , L ]$ . Given $x , y$ , we define the circular addition $x \oplus _ { L } y = x + y$ (mod $2 L$ ). When we use FFT to compute the convolution of two discrete distributions $Y _ { 1 } , Y _ { 2 }$ supported on $h \mathbb { Z } \cap [ - L , L ]$ , we are implicitly calculating the the distribution of $Y _ { 1 } \oplus _ { L } Y _ { 2 }$ . We will later show that $\widetilde { Y } _ { 1 } \oplus _ { L } \widetilde { Y } _ { 2 } \oplus _ { L } \cdot \cdot \cdot \oplus _ { L } \widetilde { Y } _ { k }$ is a good approximation of $Y _ { 1 } + Y _ { 2 } + \cdots + Y _ { k }$ .
+
+The subroutine DiscretizePRV (Algorithm 2) is used to truncate and discretize PRVs. In this subroutine, we shift the discretized random variables such that it has the same mean as the original variables. This is one of main differences between our algorithm and the algorithm in [KJPH21, KH21]. We show that this significantly decreases the discretization error and allow us to use much coarser mesh $h \approx 1 / \sqrt { k }$ instead of $h \approx 1 / k$ .
+
+Input: $\mathrm { C D F } _ { Y } ( \cdot )$ of a PRV $Y$ , mesh size $h$ , Truncation parameter $\textstyle L \in { \frac { h } { 2 } } + h \mathbb { Z } ^ { > 0 }$   
+Output: PDF of an approximation $\widetilde { Y }$ supported on $\mu + ( h \mathbb { Z } \cap [ - L , L ] )$ for some $\mu \in [ 0 , \frac { h } { 2 } ]$ . $n  { \frac { L - { \frac { h } { 2 } } } { h } }$   
+for $i = - n$ to $n$ do   
+$| \quad q _ { i } \gets \mathrm { C D F } _ { Y } ( i h + h / 2 ) - \mathrm { C D F } _ { Y } ( i h - h / 2 ) ;$   
+end   
+$q  q / ( \sum _ { i = - n } ^ { n } q _ { i } )$ ; // Normalize $q$ to make it a probability distribution $Y ^ { L } \gets Y | _ { | Y | \leq L }$ (i.e., $Y$ conditioned on $| Y | \leq L )$ ;   
+$\begin{array} { r l } & { \mu  \mathbb { E } [ Y ^ { \dot { L } } ] \stackrel { - } { - } \textstyle \sum _ { i = - n } ^ { n } i h \cdot q _ { i } ; } \\ & { \widetilde { Y }  \{ i h + \mu \quad \mathrm { ~ w . p . ~ } q _ { i } \mathrm { ~ f o r ~ } - n \leq i \leq n ; } \end{array}$   
+Return $\widetilde { Y }$ ;
+
+For simplicity, throughout this paper, we will assume that the PRVs $Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }$ do not have any mass at $\infty$ . This is with out loss of generality. Suppose $\mathrm { P r } [ Y _ { i } = \infty ] = \delta _ { i }$ for each $i$ . Let $Y _ { i } ^ { \prime } = Y _ { i } | _ { Y _ { i } \neq \infty }$ . Then
+
+$$
+\begin{array} { r } { Y _ { 1 } + Y _ { 2 } + \cdots + Y _ { k } = \left\{ \begin{array} { l l } { Y _ { 1 } ^ { \prime } + Y _ { 2 } ^ { \prime } + \cdots + Y _ { k } ^ { \prime } } & { \mathrm { w . p . ~ } ( 1 - \delta _ { 1 } ) ( 1 - \delta _ { 2 } ) \cdots ( 1 - \delta _ { k } ) } \\ { \infty } & { \mathrm { w . p . ~ } 1 - ( 1 - \delta _ { 1 } ) ( 1 - \delta _ { 2 } ) \cdots ( 1 - \delta _ { k } ) . } \end{array} \right. } \end{array}
+$$
+
+Therefore we can use Algorithm 1 to approximate the distribution of $Y _ { 1 } ^ { \prime } + Y _ { 2 } ^ { \prime } + \cdots + Y _ { k } ^ { \prime }$ , and use it to approximate the distribution of $Y _ { 1 } + Y _ { 2 } + \cdots + Y _ { k }$ .
+
+# 5 Error analysis
+
+To analyze the discretization error, we introduce the notion of coupling approximation, a variant of Wasserstein distance. Intuitively, a good coupling approximation is a coupling where the two random variables are close to each other with high probability.
+
+Definition 5.1 (coupling approximation). Given two random variables $Y _ { 1 } , Y _ { 2 }$ , we write $| Y _ { 1 } - Y _ { 2 } | \leq _ { \eta }$ h if there exists a coupling between $Y _ { 1 } , Y _ { 2 }$ such that $\operatorname* { P r } [ | Y _ { 1 } - Y _ { 2 } | > h ] \leq \eta$ .
+
+The following lemma shows that if we have a good coupling approximation $\widetilde { Y }$ to a PRV $Y$ , then the privacy curves $\delta _ { Y } ( \varepsilon )$ and $\delta _ { \widetilde { Y } } ( \varepsilon )$ should be close.
+
+Lemma 5.2. If $Y$ and $\widetilde { Y }$ are two random variables such that $| Y - { \widetilde { Y } } | \leq _ { \eta } h ,$ , then for every $\varepsilon \in \mathbb { R } ,$
+
+$$
+\delta _ { \widetilde { Y } } ( \varepsilon + h ) - \eta \leq \delta _ { Y } ( \varepsilon ) \leq \delta _ { \widetilde { Y } } ( \varepsilon - h ) + \eta .
+$$
+
+Proof. By Theorem 3.2, $\delta _ { Y } ( \varepsilon ) = \operatorname* { P r } [ Y \geq \varepsilon + Z ]$ and hence
+
+$$
+\begin{array} { r l } & { \delta _ { Y } ( \varepsilon ) = \operatorname* { P r } [ Y - \widetilde { Y } + \widetilde { Y } \geq \varepsilon + Z ] } \\ & { \qquad \leq \operatorname* { P r } [ Y - \widetilde { Y } \geq h ] + \operatorname* { P r } [ \widetilde { Y } \geq \varepsilon - h + Z ] } \\ & { \qquad \leq \eta + \delta _ { \widetilde { Y } } ( \varepsilon - h ) . } \end{array}
+$$
+
+Similarly, we have $\delta _ { \widetilde { Y } } ( \varepsilon ) \leq \eta + \delta _ { Y } ( \varepsilon - h )$ for all $\varepsilon \in \mathbb { R }$
+
+Therefore the goal of our analysis is to show that the ComposePRV algorithm finds a good coupling approximation $\widetilde { Y }$ to $Y = { \dot { \sum _ { i = 1 } ^ { k } } } Y _ { i }$ . We first show that the DiscretizePRV algorithm computes a good coupling approximation to the PRVs and crucially, it preserves the expected value after truncation. Lemma D.5 shows that $| \widetilde { Y } - Y ^ { L } | \le _ { 0 } h$ where $\widetilde { Y }$ is the approximation of a PRV $Y$ output by Algorithm 2 and $Y ^ { L }$ is the truncation of $Y$ to $[ - L , L ]$ .
+
+We then use the following key lemma which shows that when we add independent coupling approximations (where expected values match), we get a much better coupling approximation than what the triangle inequality predicts.
+
+Lemma 5.3. Suppose $Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }$ and $\widetilde { Y } _ { 1 } , \widetilde { Y } _ { 2 } , \ldots , \widetilde { Y } _ { k }$ are two collections of independent random variables such that $| Y _ { i } - \widetilde { Y } _ { i } | \le _ { 0 } h$ and $\mathbb { E } [ Y _ { i } ] = \mathbb { E } [ \widetilde { Y } _ { i } ]$ for all $i$ , then
+
+$$
+\left| \sum _ { i = 1 } ^ { k } Y _ { i } - \sum _ { i = 1 } ^ { k } { \widetilde { Y } } _ { i } \right| \leq _ { \eta } h { \sqrt { 2 k \log { \frac { 2 } { \eta } } } } .
+$$
+
+Proof. Let $X _ { i } = Y _ { i } - \widetilde { Y } _ { i }$ where $( Y _ { i } , \widetilde { Y _ { i } } )$ are coupled such that $| Y _ { i } { - } \widetilde { Y } _ { i } | \leq h$ w.p. 1. Then $X _ { i } \in [ - h , h ]$ w.p. 1. Note that $X _ { 1 } , X _ { 2 } , \ldots , X _ { k }$ are independent of each other. By Hoeffding’s inequality,
+
+$$
+\operatorname* { P r } \left[ \left| \sum _ { i } X _ { i } \right| \geq t \right] \leq 2 \exp \left( - { \frac { 2 t ^ { 2 } } { k ( 2 h ) ^ { 2 } } } \right) = \eta
+$$
+
+if we set $t = h \sqrt { 2 k \log \frac { 2 } { \eta } }$
+
+This lemma shows that the error of $k$ times composition is around $\sqrt { k } \cdot h$ and hence setting $h \approx 1 / \sqrt { k }$ gives small enough error. Next, we bound the domain size $L$ . Naively, the domain size $L$ should be of the order of $\sqrt { k }$ because $Y$ is the sum of $k$ independent random variables with each bounded by a constant. In the supplementary material, we prove a tighter tail bound of $Y$ .
+
+Lemma 5.4. Let $( X , Y )$ be the privacy random variables for a $( \varepsilon , \delta )$ -DP algorithm, then for any $t \geq 0$ , we have
+
+$$
+\operatorname* { P r } [ | Y | \geq \varepsilon + t ] \leq { \frac { \delta \left( 1 + e ^ { - \varepsilon - t } \right) } { 1 - e ^ { - t } } } .
+$$
+
+This shows that $\begin{array} { r } { \operatorname* { P r } [ | Y | \ge \varepsilon + 2 ] \le \frac { 4 } { 3 } \delta } \end{array}$ and hence truncating the domain with $L = 2 + \varepsilon$ only introduces an additive $\delta$ error in the privacy curve. Therefore, if the composition satisfies a good privacy guarantee (namely $\varepsilon = O ( 1 )$ for small enough $\delta$ ), we can truncate the domain at $L = \Theta ( 1 )$ . Together with the fact that mesh size is $1 / \sqrt { k }$ , this gives a $O ( { \sqrt { k } } )$ -time algorithm for computing the privacy curve when we compose the same mechanism with itself $k$ times. The following theorem gives a formal statement of the error bounds of our algorithm, it is proved in the supplementary material.
+
+Theorem 5.5. Let $\varepsilon _ { \mathrm { e r r o r } }$ , $\delta _ { \mathrm { { e r r o r } } } > 0$ be some fixed error terms. Let $\mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \ldots , \mathcal { M } _ { k }$ be $D P$ algorithms with privacy curves $\delta _ { { \cal M } _ { i } } ( \varepsilon )$ . Let $Y _ { i }$ be the PRV corresponding to $\mathcal { M } _ { i }$ such that $\delta _ { { \cal M } _ { i } } ( \varepsilon ) =$ $\delta _ { Y _ { i } } ( \varepsilon )$ for $\varepsilon \geq 0$ . Let $\mathcal { M }$ be the (adaptive) composition of $\mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \ldots , \mathcal { M } _ { k }$ and let $\delta _ { \mathcal { M } } ( \varepsilon )$ be its privacy curve. Set $L \ge 2 + \varepsilon _ { \mathrm { e r r o r } }$ sufficiently large such that
+
+$$
+\sum _ { i = 1 } ^ { k } \delta _ { \mathcal { M } _ { i } } ( L - 2 ) \leq \frac { \delta _ { \mathrm { e r r o r } } } { 8 } a n d \delta _ { \mathcal { M } } ( L - 2 - \varepsilon _ { \mathrm { e r r o r } } ) \leq \frac { \delta _ { \mathrm { e r r o r } } } { 4 } .
+$$
+
+Let $\widetilde { Y }$ be the approximation of $\textstyle Y = \sum _ { i = 1 } ^ { k } Y _ { i }$ produced by ComposePRV algorithm with mesh size
+
+$$
+h = { \frac { \varepsilon _ { \mathrm { e r r o r } } } { \sqrt { { \frac { k } { 2 } } \log { \frac { 1 2 } { \delta _ { \mathrm { e r r o r } } } } } } } .
+$$
+
+Then
+
+$$
+\delta _ { \widetilde { Y } } ( \varepsilon + \varepsilon _ { \mathrm { e r r o r } } ) - \delta _ { \mathrm { e r r o r } } \leq \delta _ { Y } ( \varepsilon ) = \delta _ { \mathcal { M } } ( \varepsilon ) \leq \delta _ { \widetilde { Y } } ( \varepsilon - \varepsilon _ { \mathrm { e r r o r } } ) + \delta _ { \mathrm { e r r o r } } .
+$$
+
+Furthermore, our algorithm takes $O \left( b { \frac { L } { h } } \log \left( { \frac { L } { h } } \right) \right)$ time where $b$ is the number of distinct algorithms among $\mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \ldots , \mathcal { M } _ { k }$ .
+
+Remark 5.6. A simple way to set $L$ such that the condition (7) holds is by choosing an $L$ such that:
+
+$$
+L \geq 2 + \operatorname* { m a x } \left\{ \varepsilon _ { \mathrm { e r r o r } } + \varepsilon _ { \mathcal { M } } \left( \frac { \delta _ { \mathrm { e r r o r } } } { 4 } \right) , \operatorname* { m a x } _ { i \in [ k ] } \varepsilon _ { \mathcal { M } _ { i } } \left( \frac { \delta _ { \mathrm { e r r o r } } } { 8 k } \right) \right\}
+$$
+
+where $\varepsilon _ { \mathcal { A } } ( \delta )$ is the inverse of $\delta _ { \mathcal { A } } ( \varepsilon )$ . To set the value of $L$ , we do not need the exact value of $\varepsilon _ { \mathcal { M } }$ $( o r \varepsilon _ { \mathcal { M } _ { i } } ,$ ). We only need an upper bound on $\varepsilon _ { \mathcal { M } }$ , which can often by obtained by using the $R D P$ Accountant or any other method to derive upper bounds on privacy.
+
+# 6 Experiments
+
+We demonstrate the utility of our composition method by computing the privacy curves for the DP-SGD algorithm, which is one of the most important algorithms in differential privacy.
+
+The DP-SGD algorithm $[ \mathsf { A C G } ^ { + } 1 6 ]$ is a variant of stochastic gradient descent with $k$ steps. In each step, the algorithm selects a $p$ fraction of training examples uniformly at random. The algorithm adds a Gaussian vector with variance $\propto \sigma ^ { 2 }$ to the clipped gradient of the selected batch. Then it performs a gradient step (or any other iterative methods) using the noisy gradient computed. The privacy loss of DP-SGD involves composing the privacy curve of each iteration with itself $k$ times. The PRVs for each iteration have a closed form and depend only $p , \sigma$ (see supplementary material). Our algorithms use this closed form of PRVs.
+
+![](images/dd651af1c46cfcf948587466619d538bd872fe0785ba6d83d42653f689a3929f.jpg)  
+Figure 2: Setting $p = 1$ and comparing to the analytical solution (10).
+
+See Figure 1(b) for the comparison between our algorithm and the GDP and RDP Accountant. Our method provides a lower and upper bound of the privacy curve according to (8). In Figure 1(a), we compare our algorithm with [KJPH21] (implemented in [KP21]). Under the same mesh size, our algorithm computes a much closer upper and lower bound.
+
+We validate our program for the case $p = 1$ . When $p = 1$ , we have an exact formula for
+
+$$
+\delta ( \varepsilon ) = \Phi \left( - { \frac { \varepsilon } { \mu } } + { \frac { \mu } { 2 } } \right) - e ^ { \varepsilon } \Phi \left( - { \frac { \varepsilon } { \mu } } - { \frac { \mu } { 2 } } \right)
+$$
+
+where the bo $\textstyle \mu = { \frac { \sqrt { k } } { \sigma } }$ . In Figure 2, we show that the true privacy curve is indeed sandwicompute and that the vertical distance between our bounds is indeed etweenwith a $2 \varepsilon _ { \mathrm { e r r o r } }$ neglible δerror of 10−10.
+
+Floating point errors Note that our error analysis in Section 5 ignores floating point errors. This is because they are negligible compared to the discretization and truncation errors we analyzed in Section 5 for the range of $\delta$ we are interested in. See supplementary material for more details.
+
+# 6.1 Comparison with [KJPH21]
+
+![](images/ca0b62a1d9d64fe917e548bfcb7e7c09e98d280fde0ba10ba8c592a3576fb941.jpg)  
+Figure 3: (a) Comparison of error bounds of $\delta$ with varying number of discretisation points for $p = 4 \times 1 0 ^ { - 3 } , \sigma = 0 . 8 , \varepsilon = 1 . 5 , k = 1 0 0 0$ . (b) Comparing runtimes for our algorithm with that of [KJPH21] when aligned on accuracy for $\sigma = 0 . 8$ , $p = \dot { 4 } \times \dot { 1 } 0 ^ { - 3 }$ . We can see a significant reduction in runtime in particular for large number of DPSGD steps. We were not able to run the algorithm of [KJPH21] beyond 2,000 steps, since it becomes unstable beyond that point.7. We also plot the speed up directly on the secondary $y$ -axis.
+
+In this section, we provide more results demonstrating the practical use of our algorithm. We compare runtimes of our algorithm with [KJPH21], which is the state-of-the-art, for typical values of privacy parameters $\mathit { \check { \sigma } } = 0 . 8$ , $p = 4 \times 1 0 ^ { - 3 }$ , $\varepsilon = 1 . 5$ ). See Figure 3(a) for the effect of the number of discretisation points $n$ on the accuracy of $\delta$ . Our algorithm requires about a few orders of magnitude smaller number of discretization points to converge compared to the algorithm of [KJPH21].
+
+In order to compare runtimes, we align the accuracy of both FFT algorithms. We set numerical parameters (number of discretization bins and domain length) such that both algorithms give similarly accurate bounds and verify it visually. Figure 3(b) shows a significant speedup (100x) using our algorithms. We note that runtimes are directly proportional to the memory required by the algorithms and so a separate memory analysis is not required; the runtime and memory are dominated by the number of points in the discretization of PRV. All experiments are performed on a Intel Xeon W-2155 CPU with 3.30GHz with 128GB of memory.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank Janardhan Kulkarni and Sergey Yekhanin for several useful discussions and encouraging us to work on this problem. L.W. would like to thank Daniel Jones and Victor Rühle for fruitful discussions and helpful guidance.
+
+References   
+$[ \mathrm { A C G ^ { + } } 1 6 ]$ Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 308–318, 2016.   
+[BDLS19] Zhiqi Bu, Jinshuo Dong, Qi Long, and Weijie J. Su. Deep learning with gaussian differential privacy, 2019.   
+[BDRS18] Mark Bun, Cynthia Dwork, Guy N Rothblum, and Thomas Steinke. Composable and versatile privacy via truncated cdp. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 74–86, 2018.   
+$[ \mathrm { B G K } ^ { + } 2 1 ]$ Zhiqi Bu, Sivakanth Gopi, Janardhan Kulkarni, Yin Tat Lee, Judy Hanwen Shen, and Uthaipon Tantipongpipat. Fast and memory efficient differentially private-sgd via jl projections. arXiv preprint arXiv:2102.03013, 2021. [BS16] Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635–658. Springer, 2016. [BW18] Borja Balle and Yu-Xiang Wang. Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising. In International Conference on Machine Learning, pages 403–412, 2018.   
+$[ \mathrm { D K M ^ { + } } 0 6 ]$ Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni Naor. Our data, ourselves: Privacy via distributed noise generation. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 486– 503. Springer, 2006. [DL09] Cynthia Dwork and Jing Lei. Differential privacy and robust statistics. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 371–380, 2009.   
+[DMNS06] Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography Conference, pages 265–284. Springer, 2006. $[ \mathrm { D R } ^ { + } 1 4 ]$ Cynthia Dwork, Aaron Roth, et al. The algorithmic foundations of differential privacy. Foundations and Trends $\textsuperscript { \textregistered }$ in Theoretical Computer Science, 9(3–4):211–407, 2014. [DR16] Cynthia Dwork and Guy N Rothblum. Concentrated differential privacy. arXiv preprint arXiv:1603.01887, 2016. [DRS19] Jinshuo Dong, Aaron Roth, and Weijie J. Su. Gaussian differential privacy, 2019. [DRV10] Cynthia Dwork, Guy N Rothblum, and Salil Vadhan. Boosting and differential privacy. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 51–60. IEEE, 2010. [KH21] Antti Koskela and Antti Honkela. Computing differential privacy guarantees for heterogeneous compositions using fft. arXiv preprint arXiv:2102.12412, 2021.   
+$[ \mathrm { K J H } ^ { + } 2 0 ]$ Antti Koskela, Joonas Jälkö, Antti Honkela, et al. Computing tight differential privacy guarantees using fft. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. AISTATS, 2020.   
+[KJPH21] Antti Koskela, Joonas Jälkö, Lukas Prediger, and Antti Honkela. Tight differential privacy for discrete-valued mechanisms and for the subsampled gaussian mechanism using fft. In Arindam Banerjee and Kenji Fukumizu, editors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine Learning Research, pages 3358–3366. PMLR, 13–15 Apr 2021.   
+[KOV15] Peter Kairouz, Sewoong Oh, and Pramod Viswanath. The composition theorem for differential privacy. In International conference on machine learning, pages 1376–1385. PMLR, 2015. [KP21] Antti Koskela and Lukas Prediger. Github repository for fourier accountant. https: //github.com/DPBayes/PLD-Accountant, 2021. [Mir17] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 263–275. IEEE, 2017.   
+[MM18] Sebastian Meiser and Esfandiar Mohammadi. Tight on budget? tight bounds for r-fold approximate differential privacy. In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, pages 247–264, 2018. [MV16] Jack Murtagh and Salil Vadhan. The complexity of computing the optimal composition of differential privacy. In Theory of Cryptography Conference, pages 157–175. Springer, 2016.   
+[SMM19] David M Sommer, Sebastian Meiser, and Esfandiar Mohammadi. Privacy loss classes: The central limit theorem in differential privacy. Proceedings on privacy enhancing technologies, 2019(2):245–269, 2019.   
+[ZDW21] Yuqing Zhu, Jinshuo Dong, and Yu-Xiang Wang. Optimal accounting of differential privacy via characteristic function. arXiv preprint arXiv:2106.08567, 2021.

parse/train/GSXEx6iYd0/GSXEx6iYd0_content_list.json

@@ -0,0 +1,1650 @@
+[
+    {
+        "type": "text",
+        "text": "Numerical Composition of Differential Privacy∗ ",
+        "text_level": 1,
+        "bbox": [
+            214,
+            122,
+            792,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sivakanth Gopi Microsoft Research sigopi@microsoft.com ",
+        "bbox": [
+            184,
+            200,
+            357,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yin Tat Lee University of Washington yintat@uw.edu ",
+        "bbox": [
+            375,
+            200,
+            545,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lukas Wutschitz Microsoft lukas.wutschitz@microsoft.com ",
+        "bbox": [
+            562,
+            200,
+            813,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            279,
+            535,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We give a fast algorithm to optimally compose privacy guarantees of differentially private (DP) algorithms to arbitrary accuracy. Our method is based on the notion of privacy loss random variables to quantify the privacy loss of DP algorithms. The running time and memory needed for our algorithm to approximate the privacy√ curve of a DP algorithm composed with itself $k$ times is $\\bar { \\tilde { O } } ( { \\sqrt { k } } )$ . This improves over the best prior method by [KH21] which requires $\\tilde { \\Omega } ( k ^ { 1 . 5 } )$ running time. We demonstrate the utility of our algorithm by accurately computing the privacy loss of DP-SGD algorithm of Abadi et al. $[ \\mathsf { A C G } ^ { + } 1 6 ]$ and showing that our algorithm speeds up the privacy computations by a few orders of magnitude compared to prior work, while maintaining similar accuracy. ",
+        "bbox": [
+            232,
+            308,
+            766,
+            450
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            472,
+            310,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Differential privacy (DP) introduced by [DMNS06] provides a provable and quantifiable guarantee of privacy when the results of an algorithm run on private data are made public. Formally, we can define an $( \\varepsilon , \\delta )$ -differentially private algorithm as follows. ",
+        "bbox": [
+            176,
+            502,
+            823,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Definition 1.1 ( $( \\varepsilon , \\delta )$ -DP [DMNS06, $\\mathrm { D K M ^ { + } 0 6 ] }$ ). An algorithm $\\mathcal { M }$ is $( \\varepsilon , \\delta )$ -DP if for any two neighboring databases $D , D ^ { \\prime }$ differing in exactly one user and any subset $S$ of outputs, we have $\\operatorname* { P r } [ \\mathcal { M } ( D ) \\in S ] \\le e ^ { \\varepsilon } \\operatorname* { P r } [ \\mathcal { M } ( D ^ { \\prime } ) \\in S ] + \\delta$ . ",
+        "bbox": [
+            173,
+            546,
+            825,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, it says that looking at the outcome of $\\mathcal { M }$ , we cannot tell whether it was run on $D$ or $D ^ { \\prime }$ . Hence an adversary cannot infer the existence of any particular user in the input database, and therefore cannot learn any personal data of any particular user. ",
+        "bbox": [
+            174,
+            598,
+            825,
+            640
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DP algorithms have an important property called composition. Suppose $M _ { 1 }$ and $M _ { 2 }$ are DP algorithms and say $M ( D ) = ( M _ { 1 } ( D ) , M _ { 2 } ( D ) )$ , i.e., $M$ runs both the algorithms on $D$ and outputs their results. Then $M$ is also a DP algorithm. ",
+        "bbox": [
+            174,
+            646,
+            825,
+            689
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1.2 (Simple composition $\\mathrm { [ D K M ^ { + } 0 6 }$ , DL09]). If $M _ { 1 }$ is $( \\varepsilon _ { 1 } , \\delta _ { 1 } )$ -DP and $M _ { 2 }$ is $( \\varepsilon _ { 2 } , \\delta _ { 2 } )$ - $D P ,$ then $M ( D ) = ( \\bar { M } _ { 1 } ( D ) , \\bar { M } _ { 2 } ( D ) )$ is $( \\varepsilon _ { 1 } + \\varepsilon _ { 2 } , \\delta _ { 1 } + \\delta _ { 2 } ) – D P .$ . ",
+        "bbox": [
+            171,
+            690,
+            825,
+            720
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This also holds under adaptive composition (denoted by $M = M _ { 2 } \\circ M _ { 1 } )$ , where $M _ { 2 }$ can look at both the database and the output of $M _ { 1 }$ (here $M ( D ) = ( M _ { 1 } ( D ) , M _ { 2 } ( D , M _ { 1 } ( D ) ) ) )$ . It turns out that both compositions enjoy much better DP guarantees than this simple composition rule. Let $M$ be an $( \\varepsilon , \\delta )$ -DP algorithm and let $M ^ { \\circ k }$ denote the (adaptive) composition of $M$ with itself $k$ times. The naive composition rule shows that $M ^ { \\circ k }$ is $( k \\varepsilon , k \\delta )$ -DP. This was significantly improved in [DRV10]. ",
+        "bbox": [
+            173,
+            727,
+            826,
+            800
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1.3 (Advanced composition [DRV10, $\\mathrm { D R } ^ { + } 1 4 ]$ ). If $M$ is $( \\varepsilon , \\delta )$ -DP, then $M ^ { \\circ k }$ is $( \\varepsilon ^ { \\prime } , k \\delta +$ $\\delta ^ { \\prime }$ )-DP where ",
+        "bbox": [
+            171,
+            801,
+            826,
+            830
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/786915f476a1670946c693ff5ae90c703a0a78e10296dab60fc04ea4cdfa4f8b.jpg",
+        "text": "$$\n\\varepsilon ^ { \\prime } = \\varepsilon \\sqrt { 2 k \\log \\left( \\frac { 1 } { \\delta ^ { \\prime } } \\right) } + k \\varepsilon ( e ^ { \\varepsilon } - 1 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            829,
+            619,
+            872
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Note that if $\\begin{array} { r } { \\varepsilon = O \\left( \\frac { 1 } { \\sqrt { k } } \\right) } \\end{array}$ and $\\begin{array} { r } { \\delta = o \\left( \\frac { 1 } { k } \\right) } \\end{array}$ , then $M ^ { \\circ k }$ satisfies $( O _ { \\delta ^ { \\prime } } ( 1 ) , \\delta ^ { \\prime } )$ -DP. Using simple composition (Proposition 1.2), we can only claim that $M ^ { \\circ k }$ is $( O ( { \\sqrt { k } } ) , o ( 1 ) )$ -DP. Thus advanced composition often results in $\\sqrt { k }$ -factor savings in privacy which is significant in practice. The optimal DP guarantees for $k$ -fold composition of an $( \\varepsilon , \\delta )$ -DP algorithm were finally obtained by [KOV15]. For composing different algorithms, the situation is more complicated. If $M _ { 1 } , M _ { 2 } , \\ldots , M _ { k }$ are DP algorithms such that $M _ { i }$ is $( \\varepsilon _ { i } , \\delta _ { i } )$ -DP, then it is shown by [MV16] that computing the exact DP guarantees for $M = M _ { 1 } \\circ M _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ M _ { k }$ is #P-complete. They also give an algorithm to approximate the DP guarantees of $M$ to desired accuracy $\\eta$ which runs in ",
+        "bbox": [
+            173,
+            88,
+            826,
+            214
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe54608e491e192096fe4984b697023884f0947310bb9fc0b11a2f012027ed6d.jpg",
+        "text": "$$\n\\tilde { O } \\left( \\frac { k ^ { 3 } \\bar { \\varepsilon } ( 1 + \\bar { \\varepsilon } ) } { \\eta } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            219,
+            558,
+            257
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "time where $\\textstyle { \\bar { \\varepsilon } } = ( \\sum _ { i = 1 } ^ { k } \\varepsilon _ { i } ) / k$ .2 If each $\\begin{array} { r } { \\varepsilon _ { i } \\approx \\frac { 1 } { \\sqrt { k } } } \\end{array}$ (so that $M$ will satisfy reasonable privacy guarantees by advanced composition), then the running time is $\\tilde { O } ( k ^ { 2 . 5 } / \\eta )$ . ",
+        "bbox": [
+            174,
+            263,
+            825,
+            301
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In most situations, DP algorithms come with a collection of $( \\varepsilon , \\delta )$ -DP guarantees, i.e., for each value of $\\varepsilon$ , there exists $\\delta$ such that the algorithm is $( \\varepsilon , \\delta )$ -DP. ",
+        "bbox": [
+            174,
+            305,
+            823,
+            335
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Definition 1.4 (Privacy curve). A DP algorithm $M$ is said to have privacy curve $\\delta : \\mathbb { R }  [ 0 , 1 ] ,$ if for every $\\varepsilon \\in \\mathbb { R }$ , $M$ is $( \\varepsilon , \\delta ( \\varepsilon ) ) \\lrcorner D P .$ ",
+        "bbox": [
+            174,
+            338,
+            825,
+            368
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For example the privacy curve of a Gaussian mechanism (with sensitivity 1 and noise scale $\\sigma$ ) is given by $\\delta ( \\varepsilon ) = \\Phi \\left( - \\varepsilon \\sigma + 1 / 2 \\sigma \\right) - e ^ { \\varepsilon } \\Phi \\left( - \\varepsilon \\sigma - 1 / 2 \\sigma \\right)$ where $\\Phi ( \\cdot )$ is the Gaussian CDF [BW18]. Suppose we want to compose several Gaussian mechanisms, which $( \\varepsilon , \\delta )$ -DP guarantee should we choose for each mechanism? Any choice will lead to suboptimal DP guarantees for the final composition. Instead, we need a way to compose the privacy curves directly. This was suggested through the use of privacy region in [KOV15] and explicitly studied in the $f$ -DP framework of [DRS19]. $f$ -DP is a dual way (and equivalent) to look at the privacy curve $\\delta ( \\varepsilon )$ . ",
+        "bbox": [
+            173,
+            377,
+            825,
+            477
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Independently, an algorithm called Privacy Buckets for approximately composing privacy curves using the notion of was initiated in [MM18]. This algorithm depends on the notion of Privacy Loss Random Variable (PRV) [DR16], whose distribution is called Privacy Loss Distribution (PLD). For any DP-algorithm, one can associate a PRV and the privacy curve of that algorithm can be easily obtained from the PRV. The really useful property of PRVs is that under adaptive composition, they just add up; the PRV $Y$ of the composition $M = M _ { 1 } \\circ M _ { 2 } \\circ \\cdots \\circ M _ { k }$ is given by $\\begin{array} { r } { \\bar { Y } = \\sum _ { i = 1 } ^ { k } \\bar { Y _ { i } } } \\end{array}$ where $Y _ { i }$ is the PRV of $M _ { i }$ .3 Therefore, one can find the distribution of $Y$ by the convolution of the distributions of $Y _ { 1 } , Y _ { 2 } , \\ldots , Y _ { k }$ . In an important paper, $[ \\mathrm { K J H ^ { + } } 2 0 ]$ proposed that one can speed up the convolutions using Fast Fourier Transform (FFT). Explicit error bounds were obtained for the approximation obtained by their algorithm in $[ \\mathrm { K J H ^ { + } } 2 0 $ , KJPH21, KH21]. The running time of this algorithm was analyzed in [KH21] where it was shown that the privacy curve $\\delta _ { M } ( \\varepsilon )$ of $M = M _ { 1 } \\circ M _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ M _ { k }$ can be computed up to an additive error of $\\delta _ { \\mathrm { e r r o r } }$ in time ",
+        "bbox": [
+            173,
+            481,
+            825,
+            652
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92d1f198d0aa1f21006d24cdc838d47dd7e695dd1b43a7c893bbfce9d51a2417.jpg",
+        "text": "$$\n\\tilde { O } \\left( \\frac { k ^ { 3 } \\bar { \\varepsilon } } { \\delta _ { \\mathrm { e r r o r } } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            452,
+            659,
+            544,
+            695
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "if each algorithm $M _ { i }$ is satisfies $( \\varepsilon _ { i } , 0 )$ -DP and $\\begin{array} { r } { \\bar { \\varepsilon } = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\varepsilon _ { i } } \\end{array}$ . Assuming that each $\\begin{array} { r } { \\varepsilon _ { i } \\approx \\frac { 1 } { \\sqrt { k } } } \\end{array}$ , we get $\\tilde { O } ( k ^ { 2 . 5 } / \\delta _ { \\mathrm { e r r o r } } )$ running time. Note that this is slightly worse than (1), where the denominator $\\eta$ is the multiplicative error in $\\delta _ { M }$ . When composing the same algorithm with itself for $k$ times, the running time can be improved to $\\begin{array} { r } { \\tilde { O } \\left( \\frac { k ^ { 2 } \\bar { \\varepsilon } } { \\delta _ { \\mathrm { e r r o r } } } \\right) } \\end{array}$ , which is $\\begin{array} { r } { \\tilde { O } \\left( \\frac { k ^ { 1 . 5 } } { \\delta _ { \\mathrm { e r r o r } } } \\right) } \\end{array}$ when ε¯ = √1k . ",
+        "bbox": [
+            173,
+            702,
+            825,
+            777
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Moments Accountant and Renyi DP In an influential paper where they introduce Differentially Private Deep Learning, $[ \\mathsf { A C G } ^ { + } 1 6 ]$ proposed a method called the Moments Accountant (MA) for giving an upper bound the privacy curve of a composition of DP algorithms. They applied their method to bound the privacy loss of differentially private Stochastic Gradient Descent (DP-SGD) algorithm which they introduced. Analyzing the privacy loss of DP-SGD involves composing the privacy curve of each iteration of training with itself $k$ times, where $k$ is the total of number of training iterations. Typical values of $k$ range from 1000 to 300000 (such as when training large models like GPT3). The Moments Accountant was subsumed into the framework of Renyi Differential Privacy (RDP) introduced by [Mir17]. The running time of these accountants are independent of $k$ , but they only give an upper bound and cannot approximate the privacy curve to arbitrary accuracy. ",
+        "bbox": [
+            173,
+            790,
+            825,
+            861
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "DP-SGD is one of the most important DP algorithms in practice, because one can use it to train neural networks to achieve good privacy-vs-utility tradeoffs. Therefore obtaining accurate and tight privacy guarantees for DP-SGD is important. For example reducing $\\varepsilon$ from 2 to 1, can mean that one can train the network for 4 times more epochs while staying within the same privacy budget. Therefore DP-SGD is one of the main motivations for this work. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            237
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are also situations when the PRVs do not have bounded moments and so Moments Accountant or Renyi DP cannot be applied for analyzing privacy. An example of such an algorithm is the DP-SGD-JL algorithm of $[ \\mathrm { B G K } ^ { + } 2 1 ]$ which uses numerical composition of PRVs to analyze privacy. ",
+        "bbox": [
+            174,
+            242,
+            825,
+            285
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "GDP Accountant [DRS19, BDLS19] introduced the notion of Gaussian Differential Privacy (GDP) and used it to develop an accountant for DP-SGD. The accountant is based on central limit theorem and only gives an approximation to the true privacy curve, where the approximation gets better with $k$ . But as we show in Figure 1, GDP accountant can significantly underreport the true epsilon value. ",
+        "bbox": [
+            173,
+            297,
+            825,
+            356
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Several different notions of privacy were introduced for obtaining good upper bounds on the privacy curve of composition of DP algorithms such as Concentrated DP (CDP) [DR16, BS16], Truncated CDP [BDRS18] etc. None of these methods can approximate the privacy curve of compositions to arbitrary accuracy. The notion of $f$ -DP introduced by [DRS19], allows for a lossless composition theorem, but computing the privacy curve of composition seems computationally hard and they do not give any algorithms for doing it. ",
+        "bbox": [
+            173,
+            361,
+            825,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1.1 Our Contributions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            460,
+            344,
+            474
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main contribution of this work is a new algorithm with an improved analysis for computing the privacy curve of the composition of a large number of DP algorithms. ",
+        "bbox": [
+            174,
+            486,
+            821,
+            513
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1.5 (Informal version of Theorem 5.5). Suppose $M _ { 1 } , M _ { 2 } , \\ldots , M _ { k }$ are $D P$ algorithms. Then the privacy curve $\\delta _ { M } ( \\varepsilon )$ of adaptive composition $M = M _ { 1 } \\circ M _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ M _ { k }$ can be approximated in time ",
+        "bbox": [
+            173,
+            517,
+            823,
+            558
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a23daebc6a82f5e56b371deea0422bec77d6376c25750e49ae0d14fb50fd84a3.jpg",
+        "text": "$$\nO \\left( \\frac { \\varepsilon _ { \\mathrm { u p p e r } } k ^ { 1 . 5 } \\log k \\sqrt { \\log \\frac { 1 } { \\delta _ { \\mathrm { e r r o r } } } } } { \\varepsilon _ { \\mathrm { e r r o r } } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            555,
+            617,
+            606
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\varepsilon _ { \\mathrm { e r r o r } }$ is the additive error in $\\varepsilon$ , $\\delta _ { \\mathrm { e r r o r } }$ is the additive error in $\\delta$ and $\\varepsilon _ { \\mathrm { u p p e r } }$ is an upper bound on max $\\begin{array} { r } { \\left\\{ \\varepsilon _ { M } ( \\delta _ { \\mathrm { e r r o r } } ) , \\mathrm { m a x } _ { i } \\varepsilon _ { M _ { i } } \\left( \\frac { \\delta _ { \\mathrm { e r r o r } } } { k } \\right) \\right\\} } \\end{array}$ . 4 ",
+        "bbox": [
+            173,
+            608,
+            826,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If each $M _ { i }$ satisfies $\\scriptstyle \\left( { \\frac { 1 } { \\sqrt { k } } } , { \\frac { o ( 1 ) } { k } } \\right)$ -DP, then by advanced composition (Proposition 1.3), we can set $\\varepsilon _ { \\mathrm { u p p e r } } = O ( 1 )$ . Therefore the running time of our algorithm in this case is $\\tilde { O } \\left( \\frac { k ^ { 1 . 5 } \\sqrt { \\log \\frac { 1 } { \\delta _ { \\mathrm { e r r o r } } } } } { \\varepsilon _ { \\mathrm { e r r o r } } } \\right)$ . We can save a factor of $k$ , when we compose the same algorithm with itself $k$ times. ",
+        "bbox": [
+            173,
+            650,
+            826,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1.6. Suppose $M$ is a DP algorithm. Then the privacy curve $\\delta _ { M ^ { \\circ } k } ( \\varepsilon )$ of $M$ (adaptively) composed with itself $k$ times can be approximated in time ",
+        "bbox": [
+            171,
+            729,
+            825,
+            758
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82c6e1e703e748d79c0ecf5c52e660855da93a9e247debedfbde657537e0646c.jpg",
+        "text": "$$\nO \\left( \\frac { \\varepsilon _ { \\mathrm { u p p e r } } k ^ { \\frac { 1 } { 2 } } \\log k \\sqrt { \\log \\frac { 1 } { \\delta _ { \\mathrm { e r r o r } } } } } { \\varepsilon _ { \\mathrm { e r r o r } } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            763,
+            614,
+            815
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\varepsilon _ { \\mathrm { e r r o r } }$ is the additive error in $\\varepsilon$ , $\\delta _ { \\mathrm { e r r o r } }$ is the additive error in $\\delta$ and $\\varepsilon _ { \\mathrm { u p p e r } }$ is an upper bound on max $\\begin{array} { r } { \\left\\{ \\varepsilon _ { M ^ { \\circ k } } ( \\delta _ { \\mathrm { e r r o r } } ) , \\varepsilon _ { M } \\left( \\frac { \\delta _ { \\mathrm { e r r o r } } } { k } \\right) \\right\\} } \\end{array}$ . ",
+        "bbox": [
+            174,
+            819,
+            823,
+            853
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Thus we improve the state-of-the-art by at least a factor of $k$ in running time. We also note that our algorithm improves the memory required by a factor of $k$ . See Figure 1 for a comparison of our algorithm with that of [KJPH21]. Also note that RDP Accountant (equivalent to the Moments Accountant) significantly overestimates the true $\\varepsilon$ , while the GDP Accountant significantly underestimates the true $\\varepsilon$ . In contrast, the upper and lower bounds provided by our algorithm lie very close to each other. ",
+        "bbox": [
+            173,
+            859,
+            825,
+            888
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            826,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/8e9f81b58d5b20927335c64d5bde49592def1d97d29b397a5ab89e225d8706ef.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            159,
+            454,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d2af7302e29615a33af3e6ff3f26516bd66935578ca15397ae42aa3cec7bb70.jpg",
+        "image_caption": [
+            "(b) Our algorithm can improve significantly over the RDP Accountant. We also see that GDP Accountant can significantly underreport the true $\\varepsilon$ . We have set $\\varepsilon _ { \\mathrm { e r r o r } } = 0 . 1$ , $\\delta _ { \\mathrm { e r r o r } } = { \\bar { \\delta } } / 1 0 0 0$ here. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            508,
+            171,
+            789,
+            327
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(a) Our algorithm gives much closer upper and lower bounds on the true privacy curve compared to [KJPH21], under the same mesh size of $4 ~ \\times$ $1 0 ^ { - 5 }$ . Our upper and lower bounds are nearly coinciding. ",
+        "bbox": [
+            173,
+            321,
+            468,
+            385
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 1: Case study on DP-SGD. Sampling probability $p = 1 0 ^ { - 3 }$ , noise scale $\\sigma = 0 . 8$ , $\\delta = 1 0 ^ { - 7 }$ ",
+        "bbox": [
+            171,
+            392,
+            820,
+            409
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our Techniques Our algorithm (also the prior work of $\\mathrm { [ K J H ^ { + } 2 0 ] }$ ) proceeds by approximating the privacy loss random variables (PRVs) by truncating and discretizing them. We then use Fast Fourier Transform (FFT) to convolve the distributions efficiently. The main difference is in the approximation procedure and the error analysis. In the approximation procedure, we correct the approximation so that the expected value of the discretization matches with the expected value of the PRV. ",
+        "bbox": [
+            173,
+            433,
+            825,
+            502
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To analyze the approximation error, we introduce the concept of coupling approximation (Definition 5.1), which is a variant of Wasserstein (optimal transport) distance specifically tailored to this application. We first show that the approximation output by our algorithm to each privacy random variable is a good coupling approximation. We then show that when independent coupling approximations are added, cancellation happens between the errors due to Hoeffding bound, producing a much better coupling approximation than one naively expects from the triangle inequality. This allows us to choose the mesh size in our discretization to be $\\approx { \\frac { 1 } { \\sqrt { k } } }$ , whereas [KH21] choose a mesh size of $\\approx \\frac { 1 } { k }$ . The other improvement is the truncation procedure. We give a tight tail bound of the PRVs (Lemma 5.4). This allows us to choose the domain size for in truncation to be√ $\\approx \\tilde { O } ( 1 )$ , whereas [KH21] choose $\\approx \\tilde { O } ( \\sqrt { k } )$ . Both ideas together saves a factor of $k$ in the run time and memory. ",
+        "bbox": [
+            173,
+            508,
+            826,
+            672
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the analysis, the previous paper analyzes the discretization error by studying the stability of convolution. This leads to complicated calculations with the runtime linear in $1 / \\bar { \\delta } _ { \\mathrm { e r r o r } } \\bar { }$ (see (2)). Since $\\delta _ { \\mathrm { e r r o r } } \\ll \\delta \\ll 1 / N$ is required to give meaningful privacy guarantee $N$ is the number of users), this term $1 / \\delta _ { \\mathrm { e r r o r } }$ is huge. In this paper, we show various facts about how coupling approximation accumulates and use them to give a runtime depending only on $\\sqrt { \\log ( 1 / \\delta _ { \\mathrm { e r r o r } } ) }$ . ",
+        "bbox": [
+            174,
+            678,
+            825,
+            751
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2 DP Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
+            349,
+            785
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a DP algorithm $\\mathcal { M }$ , for each value of $\\varepsilon \\geq 0$ , there exists some $\\delta \\in [ 0 , 1 ]$ such that $\\mathcal { M }$ is $( \\varepsilon , \\delta )$ -DP. We can represent all these privacy guarantees by a function $\\delta _ { \\mathcal { M } } ( \\varepsilon ) : \\mathbb { R } ^ { \\geq 0 }  [ 0 , 1 ]$ and say that $\\delta _ { \\mathcal { M } } ( \\cdot )$ is the privacy curve of $\\mathcal { M }$ . This inspires the following definition of a privacy curve between two random variables. ",
+        "bbox": [
+            173,
+            799,
+            825,
+            854
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 (Privacy curve). Given two random variables $X , Y$ supported on some set $\\Omega$ , define $\\delta ( X \\vert \\vert Y ) : \\mathbb { R } \\to [ 0 , 1 ]$ as: ",
+        "bbox": [
+            173,
+            857,
+            823,
+            886
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56c3e027976d7e7080fbb80490a1b3a84ba66ba9a20a34b3da9fa3b3e134faec.jpg",
+        "text": "$$\n\\delta ( X | | Y ) ( \\varepsilon ) = \\operatorname* { s u p } _ { S \\subset \\Omega } \\operatorname* { P r } [ Y \\in S ] - e ^ { \\varepsilon } \\operatorname* { P r } [ X \\in S ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            888,
+            656,
+            915
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore an algorithm $\\mathcal { M }$ is $( \\varepsilon , \\delta )$ -DP iff $\\delta \\left( \\mathcal { M } ( D ) \\vert \\vert \\mathcal { M } ( D ^ { \\prime } ) \\right) ( \\varepsilon ) \\leq \\delta$ for all neighboring databases $D , D ^ { \\prime }$ . ",
+        "bbox": [
+            173,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 2.2. Note that not all functions $\\delta : \\mathbb { R }  [ 0 , 1 ]$ are privacy curves. A characterization of privacy curves can be obtained using the $f$ -DP framework of [DRS19]. The notion of privacy curve $\\delta ( X \\vert \\vert Y )$ and tradeoff function $T ( X \\vert \\vert Y )$ are dual to each other via convex duality [DRS19]. This implies a characterization of privacy curves as shown in [ZDW21]. ",
+        "bbox": [
+            173,
+            121,
+            826,
+            178
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.3 (Composition of privacy curves [DRS19]). Let $\\delta _ { 1 } \\equiv \\delta ( X _ { 1 } | | Y _ { 1 } )$ and $\\delta _ { 2 } \\equiv \\delta ( X _ { 2 } | | Y _ { 2 } )$ be any two privacy curves. The composition of the privacy curves, denoted by $\\delta _ { 1 } \\otimes \\delta _ { 2 }$ , is defined as ",
+        "bbox": [
+            173,
+            179,
+            823,
+            209
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf8f61bbadcb9f839c8e707419841f359c8c000d21171241639f1a8f6bceb9ed.jpg",
+        "text": "$$\n\\delta _ { 1 } \\otimes \\delta _ { 2 } \\equiv \\delta \\left( ( X _ { 1 } , X _ { 2 } ) | | ( Y _ { 1 } , Y _ { 2 } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            210,
+            609,
+            228
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $X _ { 1 } , X _ { 2 }$ are independently sampled and $Y _ { 1 } , Y _ { 2 }$ are independently sampled. ",
+        "bbox": [
+            174,
+            229,
+            702,
+            246
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that there can be many pairs of random variables which have the same privacy curve, but the above operation is well-defined. If $\\delta ( X _ { 1 } | | Y _ { 1 } ) \\equiv \\delta ( X _ { 1 } ^ { \\prime } | | Y _ { 1 } ^ { \\prime } )$ and $\\delta ( X _ { 2 } | | Y _ { 2 } ) \\equiv \\delta ( X _ { 2 } ^ { \\prime } | | Y _ { 2 } ^ { \\prime } )$ , then it was shown by [DRS19] that ",
+        "bbox": [
+            176,
+            253,
+            823,
+            295
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3fe5c38f12900b684c1d9088bb6cd1795a5ad6d0f87f7149e6a58e0159ba91c.jpg",
+        "text": "$$\n\\delta \\left( ( X _ { 1 } , X _ { 2 } ) | | ( Y _ { 1 } , Y _ { 2 } ) \\right) = \\delta \\left( ( X _ { 1 } ^ { \\prime } , X _ { 2 } ^ { \\prime } ) | | ( Y _ { 1 } ^ { \\prime } , Y _ { 2 } ^ { \\prime } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            297,
+            663,
+            315
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "[DRS19] also show that $\\otimes$ is a commutative and associative operation. ",
+        "bbox": [
+            174,
+            318,
+            635,
+            332
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given two DP algorithms $M _ { 1 }$ and $M _ { 2 }$ , the adaptive composition $( M _ { 2 } \\circ M _ { 1 } ) ( D )$ is an algorithm which outputs $( M _ { 1 } ( D ) , M _ { 2 } ( D , M _ { 1 } ( D ) )$ , i.e., $M _ { 2 }$ can look at the database $D$ and also the output of the previous algorithm $M _ { 1 } ( D )$ . Adaptive composition of more than two algorithms is similarly defined. Suppose $M _ { 1 }$ has privacy curve $\\delta _ { 1 }$ and $M _ { 2 }$ has privacy curve $\\delta _ { 2 }$ (i.e., $M _ { 2 } ( \\cdot , y )$ is a $\\mathrm { D P }$ algorithm with privacy curve $\\delta _ { 2 }$ for any fixed $y$ .). The following composition theorem shows how to get the privacy curve of $M _ { 2 } \\circ M _ { 1 }$ . ",
+        "bbox": [
+            174,
+            337,
+            825,
+            421
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2.4 (Composition theorem [DRS19]). Let $M _ { 1 } , M _ { 2 } , \\ldots , M _ { k }$ be DP algorithms with privacy curves given by $\\delta _ { 1 } , \\delta _ { 2 } , \\dots , \\delta _ { k }$ respectively. The privacy curve of the adaptive composition $M _ { k } \\circ$ $M _ { k - 1 } \\circ \\cdots \\circ M _ { 1 }$ is given by $\\delta _ { 1 } \\otimes \\delta _ { 2 } \\otimes \\cdot \\cdot \\cdot \\otimes \\delta _ { k }$ . ",
+        "bbox": [
+            174,
+            422,
+            825,
+            467
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Privacy loss random variables ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            483,
+            457,
+            501
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The notion of privacy loss random variables (PRVs) is a unique way to assign a pair $( X , Y )$ for any privacy curve $\\delta$ such that $\\delta \\equiv \\delta ( X | | Y )$ . PRVs allow us to compute composition of two algorithms via summing random variables (Theorem 3.5) (equivalently, convolving their distributions). Thus PRVs can be thought of as a reparametrization of privacy curves where composition becomes convolution. In this paper, we differ from the usual definition of PRVs given in [DR16, $\\mathrm { K J H } ^ { + } 2 0 ]$ , which are tied to a specific algorithm. Instead we think of them as a reparametrization of privacy curves and study them directly. This allows us to succinctly prove many useful properties of PRVs. ",
+        "bbox": [
+            173,
+            513,
+            825,
+            612
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let ${ \\overline { { \\mathbb { R } } } } = \\mathbb { R } \\cup \\{ - \\infty , \\infty \\}$ be the extended real line where we define $\\infty + x = \\infty$ and $- \\infty + x = - \\infty$ for $x \\in \\mathbb { R }$ . ",
+        "bbox": [
+            173,
+            618,
+            821,
+            647
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1 (Privacy loss random variables (PRVs)). Given a privacy curve $\\delta : \\mathbb { R }  [ 0 , 1 ]$ , we say that $( X , Y )$ are privacy loss random variables for $\\delta$ , if they satisfy the following conditions: ",
+        "bbox": [
+            173,
+            650,
+            825,
+            679
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• $X , Y$ are supported on ${ \\overline { { \\mathbb { R } } } } ,$ , • $\\delta ( X | | Y ) \\equiv \\delta$ , • $Y ( t ) = e ^ { t } X ( t )$ for every $t \\in \\mathbb { R }$ and • $Y ( - \\infty ) = 0 a n d X ( \\infty ) = 0$ ",
+        "bbox": [
+            214,
+            688,
+            468,
+            773
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $X ( t ) , Y ( t )$ are probability density functions of $X , Y$ respectively. ",
+        "bbox": [
+            174,
+            781,
+            645,
+            796
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Mathematically, the correct way to write the condition $Y ( t ) = e ^ { t } X ( t )$ is to say that $\\mathbb { E } _ { Y } [ \\phi ( Y ) ] =$ $\\mathbb { E } _ { X } [ \\phi ( X ) e ^ { X } ]$ for all test functions $\\phi : \\overline { { \\mathbb { R } } }  [ 0 , 1 ]$ with $\\phi ( \\infty ) = \\phi ( - \\infty ) = 0$ . This will generalize to all situations where $X , Y$ are continuous or discrete or both. For ease of exposition, we ignore this complication and assume that $X ( t ) , Y ( t )$ represent the PDFs if $X , Y$ are continuous at $t$ , or the probability masses if they have point masses at $t$ . ",
+        "bbox": [
+            173,
+            804,
+            825,
+            877
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following theorem shows that the PRVs for a privacy curve $\\delta = \\delta ( P | | Q )$ are given by the log-likelihood random variables of $P , Q$ . ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. Let $\\delta : \\mathbb { R }  [ 0 , 1 ]$ be a privacy curve given by $\\delta \\equiv \\delta ( P | | Q )$ where $P , Q$ are two random variables supported on $\\Omega$ . The PRVs $( X , Y )$ for the privacy curve $\\delta$ are given $b y ^ { 5 }$ : ",
+        "bbox": [
+            169,
+            89,
+            825,
+            121
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c48ee62f238b5e21988dfc5d16fbe76bb0e34ff16130dbb4d52f77ca988637b.jpg",
+        "text": "$$\n\\begin{array} { r } { X = \\log \\left( \\frac { Q ( \\omega ) } { P ( \\omega ) } \\right) ~ w h e r e ~ \\omega \\sim P , } \\\\ { Y = \\log \\left( \\frac { Q ( \\omega ) } { P ( \\omega ) } \\right) ~ w h e r e ~ \\omega \\sim Q . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            125,
+            612,
+            200
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The following theorem provides a formula for computing the privacy curve $\\delta$ in terms of the PRVs and conversely a formula for PRVs in terms of the privacy curve. A similar statement appears in [SMM19, $\\mathrm { K J H } ^ { + } 2 0 ]$ ]. ",
+        "bbox": [
+            173,
+            205,
+            825,
+            250
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3. The privacy curve $\\delta$ can be expressed in terms of PRVs $( X , Y )$ as: ",
+        "bbox": [
+            174,
+            251,
+            709,
+            267
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61863da84611495cce385648cec20163a1bd9388460206de9d29abf8bf3ae4b1.jpg",
+        "text": "$$\n\\delta ( \\varepsilon ) = \\operatorname* { P r } [ Y > \\varepsilon ] - e ^ { \\varepsilon } \\operatorname* { P r } [ X > \\varepsilon ] = \\mathbb { E } _ { Y } [ ( 1 - e ^ { \\varepsilon - Y } ) _ { + } ] = \\operatorname* { P r } [ Y \\geq \\varepsilon + Z ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            272,
+            748,
+            291
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $Z$ is an exponential random variable.6 Conversely, given a privacy curve $\\delta : \\mathbb { R }  [ 0 , 1 ]$ , we can compute the PDFs of its $P R V s$ $( X , Y )$ as: ",
+        "bbox": [
+            173,
+            295,
+            825,
+            325
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eec634282743c9c3732076d4c40db553dd8e8ce9c3aeba02f8dc976fa971af10.jpg",
+        "text": "$$\nY ( t ) = \\delta ^ { \\prime \\prime } ( t ) - \\delta ^ { \\prime } ( t ) a n d X ( t ) = e ^ { t } ( \\delta ^ { \\prime \\prime } ( t ) - \\delta ^ { \\prime } ( t ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            329,
+            673,
+            347
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 3.4. Theorem 3.3 shows that the PRVs $X , Y$ do not depend on the particular $P , Q$ used to represent the privacy curve $\\delta$ in Theorem 3.2. So we should think of the PDF of of the $P R V Y$ $( o r X )$ as an equivalent reparametrization of the privacy curve $\\delta : \\mathbb { R }  [ 0 , 1 ]$ , just as the notion of $f$ -DP [DRS19] is a reparametrization of the privacy curve $\\delta$ . ",
+        "bbox": [
+            173,
+            351,
+            826,
+            409
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "PRVs are useful in computing privacy curves because the composition of two privacy curves can be computed by adding the corresponding pairs of PRVs. A similar statement appears in [DR16]. ",
+        "bbox": [
+            171,
+            417,
+            823,
+            446
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.5. Let $\\delta _ { 1 } , \\delta _ { 2 }$ be two privacy curves with PRVs $( X _ { 1 } , Y _ { 1 } )$ and $( X _ { 2 } , Y _ { 2 } )$ respectively. Then the PRVs for $\\delta _ { 1 } \\otimes \\delta _ { 2 } = \\delta ( X _ { 1 } , X _ { 2 } | | Y _ { 1 } , Y _ { 2 } )$ are given by $\\left( X _ { 1 } + X _ { 2 } , Y _ { 1 } + Y _ { 2 } \\right)$ . In particular, ",
+        "bbox": [
+            173,
+            449,
+            825,
+            479
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aca1b4fa25cd9470e1d5eb0f18b7c6b212a195ee9da9063962d3665c0fee69eb.jpg",
+        "text": "$$\n\\delta _ { 1 } \\otimes \\delta _ { 2 } = \\delta ( X _ { 1 } + X _ { 2 } | | Y _ { 1 } + Y _ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            483,
+            607,
+            501
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $( X , Y )$ be the privacy random variables for $\\delta ( X _ { 1 } , X _ { 2 } | | Y _ { 1 } , Y _ { 2 } )$ . By Theorem 3.2, ",
+        "bbox": [
+            169,
+            513,
+            774,
+            530
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd407302da6bf7eb8cc43098fcb923abf24f8d1819fdfdb300b0d831180413d8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { X = \\log \\left( \\frac { ( Y _ { 1 } , Y _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } { ( X _ { 1 } , X _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } \\right) \\mathrm { ~ w h e r e ~ } ( t _ { 1 } , t _ { 2 } ) \\sim ( X _ { 1 } , X _ { 2 } ) } \\\\ & { \\quad = \\log \\left( \\frac { Y _ { 1 } ( t _ { 1 } ) Y _ { 2 } ( t _ { 2 } ) } { X _ { 1 } ( t _ { 1 } ) \\cdot X _ { 2 } ( t _ { 2 } ) } \\right) \\mathrm { ~ w h e r e ~ } t _ { 1 } \\sim X _ { 1 } , t _ { 2 } \\sim X _ { 2 } } \\\\ & { \\qquad \\mathrm { ( B y ~ i n d e p e n d e n c e ~ o f ~ } X _ { 1 } , X _ { 2 } \\mathrm { ~ a n d ~ i n d p e n d } } \\\\ & { \\quad = \\log \\left( e ^ { t _ { 1 } } \\cdot e ^ { t _ { 2 } } \\right) \\mathrm { ~ w h e r e ~ } t _ { 1 } \\sim X _ { 1 } , t _ { 2 } \\sim X _ { 2 } } \\\\ & { \\quad = t _ { 1 } + t _ { 2 } \\mathrm { ~ w h e r e ~ } t _ { 1 } \\sim X _ { 1 } , t _ { 2 } \\sim X _ { 2 } } \\\\ & { \\quad = X _ { 1 } + X _ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            535,
+            723,
+            676
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly, ",
+        "bbox": [
+            173,
+            679,
+            240,
+            695
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90290373446198852d03f6c56680a073601d587d110b0dbf4ae1e7b0673769dd.jpg",
+        "text": "$$\n{ \\begin{array} { r l } & { Y = \\log \\left( { \\frac { ( Y _ { 1 } , Y _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } { ( X _ { 1 } , X _ { 2 } ) ( t _ { 1 } , t _ { 2 } ) } } \\right) { \\mathrm { ~ w h e r e ~ } } ( t _ { 1 } , t _ { 2 } ) \\sim ( Y _ { 1 } , Y _ { 2 } ) } \\\\ & { \\quad = t _ { 1 } + t _ { 2 } { \\mathrm { ~ w h e r e ~ } } t _ { 1 } \\sim Y _ { 1 } , t _ { 2 } \\sim Y _ { 2 } } \\\\ & { \\quad = Y _ { 1 } + Y _ { 2 } . } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            699,
+            684,
+            770
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the supplementary material, we provide a proof of Theorems 3.2 and 3.3. We also discuss how to compute the PRVs for a subsampled mechanism given the PRVs for the original mechanism and give examples of PRVs for few standard mechanisms. These are used in our experiments to calculate the PRVs for DP-SGD. ",
+        "bbox": [
+            173,
+            803,
+            825,
+            859
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Numerical composition of privacy curves ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            89,
+            547,
+            107
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present an efficient and numerically accurate method, ComposePRV (Algorithm 1), for composing privacy guarantees by utilizing the notion of PRVs. ",
+        "bbox": [
+            174,
+            119,
+            825,
+            148
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: CDFs of PRVs $Y _ { 1 } , Y _ { 2 } , \\ldots , Y _ { k }$ , mesh size $h$ , Truncation parameter $\\textstyle L \\in { \\frac { h } { 2 } } + h \\mathbb { Z } ^ { > 0 }$   \nOutput: PDF of an approximation $\\widetilde { Y }$ for $\\begin{array} { r } { Y = \\sum _ { i = 1 } ^ { k } Y _ { i } } \\end{array}$ . $\\widetilde { Y }$ will be supported on $\\mu + ( h \\mathbb { Z } \\cap [ - L , L ] )$ for some $\\mu \\in [ 0 , \\frac { h } { 2 } ]$ .   \nfor $\\ell = 1$ to $k$ do $\\begin{array} { r l } { \\big | } & { { } \\widetilde { Y } _ { i } \\gets \\mathsf { D i s c r e t i z e P R V } ( Y _ { i } , L , h ) ; } \\end{array}$   \nend   \nCompute PDF Of $\\widetilde { Y } = \\widetilde { Y } _ { 1 } \\oplus _ { L } \\widetilde { Y } _ { 2 } \\oplus _ { L } \\cdot \\cdot \\cdot \\oplus _ { L } \\widetilde { Y } _ { k }$ by convolving PDFs of $\\widetilde { Y } _ { 1 } , \\widetilde { Y } _ { 2 } , \\ldots , \\widetilde { Y } _ { k }$ using FFT;   \nCompu $\\mathbf { \\varepsilon } \\mathbf { e } \\delta _ { \\widetilde { Y } } ( \\varepsilon ) = \\mathbb { E } _ { \\widetilde { Y } } \\left[ \\left( 1 - e ^ { \\varepsilon - \\widetilde { Y } } \\right) _ { + } \\right] \\mathrm { f o r } \\mathrm { a l l } \\varepsilon \\in [ 0 , L ] ;$   \nReturn Y , δ e Y (·) ",
+        "bbox": [
+            171,
+            184,
+            785,
+            366
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the algorithm ComposePRV, we compute the circular convolution $\\oplus _ { L }$ using Fast Fourier Transform (FFT). Fix some $L > 0$ . For $x \\in \\mathbb { R }$ , we define $x$ $( { \\bmod { 2 } } L ) = x - 2 L n$ where $n \\in \\mathbb { Z }$ is chosen such that $x - 2 L n \\in ( - L , L ]$ . Given $x , y$ , we define the circular addition $x \\oplus _ { L } y = x + y$ (mod $2 L$ ). When we use FFT to compute the convolution of two discrete distributions $Y _ { 1 } , Y _ { 2 }$ supported on $h \\mathbb { Z } \\cap [ - L , L ]$ , we are implicitly calculating the the distribution of $Y _ { 1 } \\oplus _ { L } Y _ { 2 }$ . We will later show that $\\widetilde { Y } _ { 1 } \\oplus _ { L } \\widetilde { Y } _ { 2 } \\oplus _ { L } \\cdot \\cdot \\cdot \\oplus _ { L } \\widetilde { Y } _ { k }$ is a good approximation of $Y _ { 1 } + Y _ { 2 } + \\cdots + Y _ { k }$ . ",
+        "bbox": [
+            173,
+            383,
+            825,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The subroutine DiscretizePRV (Algorithm 2) is used to truncate and discretize PRVs. In this subroutine, we shift the discretized random variables such that it has the same mean as the original variables. This is one of main differences between our algorithm and the algorithm in [KJPH21, KH21]. We show that this significantly decreases the discretization error and allow us to use much coarser mesh $h \\approx 1 / \\sqrt { k }$ instead of $h \\approx 1 / k$ . ",
+        "bbox": [
+            173,
+            476,
+            826,
+            549
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: $\\mathrm { C D F } _ { Y } ( \\cdot )$ of a PRV $Y$ , mesh size $h$ , Truncation parameter $\\textstyle L \\in { \\frac { h } { 2 } } + h \\mathbb { Z } ^ { > 0 }$   \nOutput: PDF of an approximation $\\widetilde { Y }$ supported on $\\mu + ( h \\mathbb { Z } \\cap [ - L , L ] )$ for some $\\mu \\in [ 0 , \\frac { h } { 2 } ]$ . $n  { \\frac { L - { \\frac { h } { 2 } } } { h } }$   \nfor $i = - n$ to $n$ do   \n$| \\quad q _ { i } \\gets \\mathrm { C D F } _ { Y } ( i h + h / 2 ) - \\mathrm { C D F } _ { Y } ( i h - h / 2 ) ;$   \nend   \n$q  q / ( \\sum _ { i = - n } ^ { n } q _ { i } )$ ; // Normalize $q$ to make it a probability distribution $Y ^ { L } \\gets Y | _ { | Y | \\leq L }$ (i.e., $Y$ conditioned on $| Y | \\leq L )$ ;   \n$\\begin{array} { r l } & { \\mu  \\mathbb { E } [ Y ^ { \\dot { L } } ] \\stackrel { - } { - } \\textstyle \\sum _ { i = - n } ^ { n } i h \\cdot q _ { i } ; } \\\\ & { \\widetilde { Y }  \\{ i h + \\mu \\quad \\mathrm { ~ w . p . ~ } q _ { i } \\mathrm { ~ f o r ~ } - n \\leq i \\leq n ; } \\end{array}$   \nReturn $\\widetilde { Y }$ ; ",
+        "bbox": [
+            171,
+            585,
+            799,
+            775
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For simplicity, throughout this paper, we will assume that the PRVs $Y _ { 1 } , Y _ { 2 } , \\ldots , Y _ { k }$ do not have any mass at $\\infty$ . This is with out loss of generality. Suppose $\\mathrm { P r } [ Y _ { i } = \\infty ] = \\delta _ { i }$ for each $i$ . Let $Y _ { i } ^ { \\prime } = Y _ { i } | _ { Y _ { i } \\neq \\infty }$ . Then ",
+        "bbox": [
+            173,
+            792,
+            825,
+            837
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9e5c4d37a572293b934807bce1f2a0ded96a29eca5dbb4c8eaf67d8609296d5.jpg",
+        "text": "$$\n\\begin{array} { r } { Y _ { 1 } + Y _ { 2 } + \\cdots + Y _ { k } = \\left\\{ \\begin{array} { l l } { Y _ { 1 } ^ { \\prime } + Y _ { 2 } ^ { \\prime } + \\cdots + Y _ { k } ^ { \\prime } } & { \\mathrm { w . p . ~ } ( 1 - \\delta _ { 1 } ) ( 1 - \\delta _ { 2 } ) \\cdots ( 1 - \\delta _ { k } ) } \\\\ { \\infty } & { \\mathrm { w . p . ~ } 1 - ( 1 - \\delta _ { 1 } ) ( 1 - \\delta _ { 2 } ) \\cdots ( 1 - \\delta _ { k } ) . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            842,
+            784,
+            877
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Therefore we can use Algorithm 1 to approximate the distribution of $Y _ { 1 } ^ { \\prime } + Y _ { 2 } ^ { \\prime } + \\cdots + Y _ { k } ^ { \\prime }$ , and use it to approximate the distribution of $Y _ { 1 } + Y _ { 2 } + \\cdots + Y _ { k }$ . ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Error analysis ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            89,
+            326,
+            107
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To analyze the discretization error, we introduce the notion of coupling approximation, a variant of Wasserstein distance. Intuitively, a good coupling approximation is a coupling where the two random variables are close to each other with high probability. ",
+        "bbox": [
+            173,
+            121,
+            825,
+            164
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Definition 5.1 (coupling approximation). Given two random variables $Y _ { 1 } , Y _ { 2 }$ , we write $| Y _ { 1 } - Y _ { 2 } | \\leq _ { \\eta }$ h if there exists a coupling between $Y _ { 1 } , Y _ { 2 }$ such that $\\operatorname* { P r } [ | Y _ { 1 } - Y _ { 2 } | > h ] \\leq \\eta$ . ",
+        "bbox": [
+            174,
+            167,
+            823,
+            198
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The following lemma shows that if we have a good coupling approximation $\\widetilde { Y }$ to a PRV $Y$ , then the privacy curves $\\delta _ { Y } ( \\varepsilon )$ and $\\delta _ { \\widetilde { Y } } ( \\varepsilon )$ should be close. ",
+        "bbox": [
+            173,
+            210,
+            821,
+            241
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.2. If $Y$ and $\\widetilde { Y }$ are two random variables such that $| Y - { \\widetilde { Y } } | \\leq _ { \\eta } h ,$ , then for every $\\varepsilon \\in \\mathbb { R } ,$ ",
+        "bbox": [
+            173,
+            246,
+            816,
+            263
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e12b759a6b4ac9150a71e749222cfb39017fadb7cad294c13440457da63a6f6a.jpg",
+        "text": "$$\n\\delta _ { \\widetilde { Y } } ( \\varepsilon + h ) - \\eta \\leq \\delta _ { Y } ( \\varepsilon ) \\leq \\delta _ { \\widetilde { Y } } ( \\varepsilon - h ) + \\eta .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            270,
+            640,
+            289
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proof. By Theorem 3.2, $\\delta _ { Y } ( \\varepsilon ) = \\operatorname* { P r } [ Y \\geq \\varepsilon + Z ]$ and hence ",
+        "bbox": [
+            173,
+            304,
+            573,
+            321
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f620c90b39e14745244fbc37b89fcecccb67792e05a2588435344eace24af2bf.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\delta _ { Y } ( \\varepsilon ) = \\operatorname* { P r } [ Y - \\widetilde { Y } + \\widetilde { Y } \\geq \\varepsilon + Z ] } \\\\ & { \\qquad \\leq \\operatorname* { P r } [ Y - \\widetilde { Y } \\geq h ] + \\operatorname* { P r } [ \\widetilde { Y } \\geq \\varepsilon - h + Z ] } \\\\ & { \\qquad \\leq \\eta + \\delta _ { \\widetilde { Y } } ( \\varepsilon - h ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            327,
+            658,
+            390
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we have $\\delta _ { \\widetilde { Y } } ( \\varepsilon ) \\leq \\eta + \\delta _ { Y } ( \\varepsilon - h )$ for all $\\varepsilon \\in \\mathbb { R }$ ",
+        "bbox": [
+            173,
+            393,
+            549,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Therefore the goal of our analysis is to show that the ComposePRV algorithm finds a good coupling approximation $\\widetilde { Y }$ to $Y = { \\dot { \\sum _ { i = 1 } ^ { k } } } Y _ { i }$ . We first show that the DiscretizePRV algorithm computes a good coupling approximation to the PRVs and crucially, it preserves the expected value after truncation. Lemma D.5 shows that $| \\widetilde { Y } - Y ^ { L } | \\le _ { 0 } h$ where $\\widetilde { Y }$ is the approximation of a PRV $Y$ output by Algorithm 2 and $Y ^ { L }$ is the truncation of $Y$ to $[ - L , L ]$ . ",
+        "bbox": [
+            173,
+            426,
+            826,
+            503
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We then use the following key lemma which shows that when we add independent coupling approximations (where expected values match), we get a much better coupling approximation than what the triangle inequality predicts. ",
+        "bbox": [
+            174,
+            508,
+            825,
+            551
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.3. Suppose $Y _ { 1 } , Y _ { 2 } , \\ldots , Y _ { k }$ and $\\widetilde { Y } _ { 1 } , \\widetilde { Y } _ { 2 } , \\ldots , \\widetilde { Y } _ { k }$ are two collections of independent random variables such that $| Y _ { i } - \\widetilde { Y } _ { i } | \\le _ { 0 } h$ and $\\mathbb { E } [ Y _ { i } ] = \\mathbb { E } [ \\widetilde { Y } _ { i } ]$ for all $i$ , then ",
+        "bbox": [
+            171,
+            555,
+            821,
+            590
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b90306cbcfff5a33e611ee05782f2621a7c5260a45d06b02ec4a150a81dc749.jpg",
+        "text": "$$\n\\left| \\sum _ { i = 1 } ^ { k } Y _ { i } - \\sum _ { i = 1 } ^ { k } { \\widetilde { Y } } _ { i } \\right| \\leq _ { \\eta } h { \\sqrt { 2 k \\log { \\frac { 2 } { \\eta } } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            597,
+            616,
+            641
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $X _ { i } = Y _ { i } - \\widetilde { Y } _ { i }$ where $( Y _ { i } , \\widetilde { Y _ { i } } )$ are coupled such that $| Y _ { i } { - } \\widetilde { Y } _ { i } | \\leq h$ w.p. 1. Then $X _ { i } \\in [ - h , h ]$ w.p. 1. Note that $X _ { 1 } , X _ { 2 } , \\ldots , X _ { k }$ are independent of each other. By Hoeffding’s inequality, ",
+        "bbox": [
+            173,
+            659,
+            825,
+            690
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5a776d5b4358b3ba1364cf2b53780c7883366c2ceed67f296081f20b3f42761.jpg",
+        "text": "$$\n\\operatorname* { P r } \\left[ \\left| \\sum _ { i } X _ { i } \\right| \\geq t \\right] \\leq 2 \\exp \\left( - { \\frac { 2 t ^ { 2 } } { k ( 2 h ) ^ { 2 } } } \\right) = \\eta\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            695,
+            651,
+            739
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "if we set $t = h \\sqrt { 2 k \\log \\frac { 2 } { \\eta } }$ ",
+        "bbox": [
+            173,
+            747,
+            348,
+            772
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This lemma shows that the error of $k$ times composition is around $\\sqrt { k } \\cdot h$ and hence setting $h \\approx 1 / \\sqrt { k }$ gives small enough error. Next, we bound the domain size $L$ . Naively, the domain size $L$ should be of the order of $\\sqrt { k }$ because $Y$ is the sum of $k$ independent random variables with each bounded by a constant. In the supplementary material, we prove a tighter tail bound of $Y$ . ",
+        "bbox": [
+            173,
+            792,
+            826,
+            852
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.4. Let $( X , Y )$ be the privacy random variables for a $( \\varepsilon , \\delta )$ -DP algorithm, then for any $t \\geq 0$ , we have ",
+        "bbox": [
+            166,
+            856,
+            823,
+            883
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ff97cefeb66e12f3e9c657e8741e3fa94e20d0c33c4924370406b122baa542b.jpg",
+        "text": "$$\n\\operatorname* { P r } [ | Y | \\geq \\varepsilon + t ] \\leq { \\frac { \\delta \\left( 1 + e ^ { - \\varepsilon - t } \\right) } { 1 - e ^ { - t } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            883,
+            612,
+            916
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This shows that $\\begin{array} { r } { \\operatorname* { P r } [ | Y | \\ge \\varepsilon + 2 ] \\le \\frac { 4 } { 3 } \\delta } \\end{array}$ and hence truncating the domain with $L = 2 + \\varepsilon$ only introduces an additive $\\delta$ error in the privacy curve. Therefore, if the composition satisfies a good privacy guarantee (namely $\\varepsilon = O ( 1 )$ for small enough $\\delta$ ), we can truncate the domain at $L = \\Theta ( 1 )$ . Together with the fact that mesh size is $1 / \\sqrt { k }$ , this gives a $O ( { \\sqrt { k } } )$ -time algorithm for computing the privacy curve when we compose the same mechanism with itself $k$ times. The following theorem gives a formal statement of the error bounds of our algorithm, it is proved in the supplementary material. ",
+        "bbox": [
+            173,
+            89,
+            826,
+            190
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.5. Let $\\varepsilon _ { \\mathrm { e r r o r } }$ , $\\delta _ { \\mathrm { { e r r o r } } } > 0$ be some fixed error terms. Let $\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\ldots , \\mathcal { M } _ { k }$ be $D P$ algorithms with privacy curves $\\delta _ { { \\cal M } _ { i } } ( \\varepsilon )$ . Let $Y _ { i }$ be the PRV corresponding to $\\mathcal { M } _ { i }$ such that $\\delta _ { { \\cal M } _ { i } } ( \\varepsilon ) =$ $\\delta _ { Y _ { i } } ( \\varepsilon )$ for $\\varepsilon \\geq 0$ . Let $\\mathcal { M }$ be the (adaptive) composition of $\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\ldots , \\mathcal { M } _ { k }$ and let $\\delta _ { \\mathcal { M } } ( \\varepsilon )$ be its privacy curve. Set $L \\ge 2 + \\varepsilon _ { \\mathrm { e r r o r } }$ sufficiently large such that ",
+        "bbox": [
+            173,
+            195,
+            825,
+            252
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9df3e2db0c8d4df6b0066978de70f5b09afa32bd3c9c00cf4a8aea2027bd3df9.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { k } \\delta _ { \\mathcal { M } _ { i } } ( L - 2 ) \\leq \\frac { \\delta _ { \\mathrm { e r r o r } } } { 8 } a n d \\delta _ { \\mathcal { M } } ( L - 2 - \\varepsilon _ { \\mathrm { e r r o r } } ) \\leq \\frac { \\delta _ { \\mathrm { e r r o r } } } { 4 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            258,
+            699,
+            301
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Let $\\widetilde { Y }$ be the approximation of $\\textstyle Y = \\sum _ { i = 1 } ^ { k } Y _ { i }$ produced by ComposePRV algorithm with mesh size ",
+        "bbox": [
+            176,
+            309,
+            823,
+            328
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20375c04877704332bbf536bfceb15f92dd6540daf009030ddaf009124d06d06.jpg",
+        "text": "$$\nh = { \\frac { \\varepsilon _ { \\mathrm { e r r o r } } } { \\sqrt { { \\frac { k } { 2 } } \\log { \\frac { 1 2 } { \\delta _ { \\mathrm { e r r o r } } } } } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            333,
+            565,
+            375
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Then ",
+        "bbox": [
+            173,
+            381,
+            209,
+            395
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc507513aa9a69e347f364a6cdd802f29e8df65c3ade95c43a493de155635c64.jpg",
+        "text": "$$\n\\delta _ { \\widetilde { Y } } ( \\varepsilon + \\varepsilon _ { \\mathrm { e r r o r } } ) - \\delta _ { \\mathrm { e r r o r } } \\leq \\delta _ { Y } ( \\varepsilon ) = \\delta _ { \\mathcal { M } } ( \\varepsilon ) \\leq \\delta _ { \\widetilde { Y } } ( \\varepsilon - \\varepsilon _ { \\mathrm { e r r o r } } ) + \\delta _ { \\mathrm { e r r o r } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            393,
+            727,
+            412
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, our algorithm takes $O \\left( b { \\frac { L } { h } } \\log \\left( { \\frac { L } { h } } \\right) \\right)$ time where $b$ is the number of distinct algorithms among $\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\ldots , \\mathcal { M } _ { k }$ . ",
+        "bbox": [
+            176,
+            416,
+            823,
+            446
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Remark 5.6. A simple way to set $L$ such that the condition (7) holds is by choosing an $L$ such that: ",
+        "bbox": [
+            173,
+            449,
+            818,
+            465
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a7353099e28c77a626403976c6969e4d704175ddb5a30630218607188da7b37.jpg",
+        "text": "$$\nL \\geq 2 + \\operatorname* { m a x } \\left\\{ \\varepsilon _ { \\mathrm { e r r o r } } + \\varepsilon _ { \\mathcal { M } } \\left( \\frac { \\delta _ { \\mathrm { e r r o r } } } { 4 } \\right) , \\operatorname* { m a x } _ { i \\in [ k ] } \\varepsilon _ { \\mathcal { M } _ { i } } \\left( \\frac { \\delta _ { \\mathrm { e r r o r } } } { 8 k } \\right) \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            472,
+            704,
+            507
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\varepsilon _ { \\mathcal { A } } ( \\delta )$ is the inverse of $\\delta _ { \\mathcal { A } } ( \\varepsilon )$ . To set the value of $L$ , we do not need the exact value of $\\varepsilon _ { \\mathcal { M } }$ $( o r \\varepsilon _ { \\mathcal { M } _ { i } } ,$ ). We only need an upper bound on $\\varepsilon _ { \\mathcal { M } }$ , which can often by obtained by using the $R D P$ Accountant or any other method to derive upper bounds on privacy. ",
+        "bbox": [
+            173,
+            513,
+            826,
+            558
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            575,
+            312,
+            593
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate the utility of our composition method by computing the privacy curves for the DP-SGD algorithm, which is one of the most important algorithms in differential privacy. ",
+        "bbox": [
+            173,
+            607,
+            485,
+            662
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The DP-SGD algorithm $[ \\mathsf { A C G } ^ { + } 1 6 ]$ is a variant of stochastic gradient descent with $k$ steps. In each step, the algorithm selects a $p$ fraction of training examples uniformly at random. The algorithm adds a Gaussian vector with variance $\\propto \\sigma ^ { 2 }$ to the clipped gradient of the selected batch. Then it performs a gradient step (or any other iterative methods) using the noisy gradient computed. The privacy loss of DP-SGD involves composing the privacy curve of each iteration with itself $k$ times. The PRVs for each iteration have a closed form and depend only $p , \\sigma$ (see supplementary material). Our algorithms use this closed form of PRVs. ",
+        "bbox": [
+            174,
+            669,
+            485,
+            863
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/dd651af1c46cfcf948587466619d538bd872fe0785ba6d83d42653f689a3929f.jpg",
+        "image_caption": [
+            "Figure 2: Setting $p = 1$ and comparing to the analytical solution (10). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            503,
+            626,
+            849,
+            823
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "See Figure 1(b) for the comparison between our algorithm and the GDP and RDP Accountant. Our method provides a lower and upper bound of the privacy curve according to (8). In Figure 1(a), we compare our algorithm with [KJPH21] (implemented in [KP21]). Under the same mesh size, our algorithm computes a much closer upper and lower bound. ",
+        "bbox": [
+            171,
+            869,
+            826,
+            912
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We validate our program for the case $p = 1$ . When $p = 1$ , we have an exact formula for ",
+        "bbox": [
+            173,
+            126,
+            748,
+            141
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1f102a28fdb38c6c3c7063e71f28f903136ea395de468f46495f179039f9c93.jpg",
+        "text": "$$\n\\delta ( \\varepsilon ) = \\Phi \\left( - { \\frac { \\varepsilon } { \\mu } } + { \\frac { \\mu } { 2 } } \\right) - e ^ { \\varepsilon } \\Phi \\left( - { \\frac { \\varepsilon } { \\mu } } - { \\frac { \\mu } { 2 } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            156,
+            642,
+            190
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where the bo $\\textstyle \\mu = { \\frac { \\sqrt { k } } { \\sigma } }$ . In Figure 2, we show that the true privacy curve is indeed sandwicompute and that the vertical distance between our bounds is indeed etweenwith a $2 \\varepsilon _ { \\mathrm { e r r o r } }$ neglible δerror of 10−10. ",
+        "bbox": [
+            174,
+            195,
+            825,
+            239
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Floating point errors Note that our error analysis in Section 5 ignores floating point errors. This is because they are negligible compared to the discretization and truncation errors we analyzed in Section 5 for the range of $\\delta$ we are interested in. See supplementary material for more details. ",
+        "bbox": [
+            176,
+            253,
+            825,
+            296
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1 Comparison with [KJPH21] ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            313,
+            408,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca0b62a1d9d64fe917e548bfcb7e7c09e98d280fde0ba10ba8c592a3576fb941.jpg",
+        "image_caption": [
+            "Figure 3: (a) Comparison of error bounds of $\\delta$ with varying number of discretisation points for $p = 4 \\times 1 0 ^ { - 3 } , \\sigma = 0 . 8 , \\varepsilon = 1 . 5 , k = 1 0 0 0$ . (b) Comparing runtimes for our algorithm with that of [KJPH21] when aligned on accuracy for $\\sigma = 0 . 8$ , $p = \\dot { 4 } \\times \\dot { 1 } 0 ^ { - 3 }$ . We can see a significant reduction in runtime in particular for large number of DPSGD steps. We were not able to run the algorithm of [KJPH21] beyond 2,000 steps, since it becomes unstable beyond that point.7. We also plot the speed up directly on the secondary $y$ -axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            347,
+            838,
+            516
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide more results demonstrating the practical use of our algorithm. We compare runtimes of our algorithm with [KJPH21], which is the state-of-the-art, for typical values of privacy parameters $\\mathit { \\check { \\sigma } } = 0 . 8$ , $p = 4 \\times 1 0 ^ { - 3 }$ , $\\varepsilon = 1 . 5$ ). See Figure 3(a) for the effect of the number of discretisation points $n$ on the accuracy of $\\delta$ . Our algorithm requires about a few orders of magnitude smaller number of discretization points to converge compared to the algorithm of [KJPH21]. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            694
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In order to compare runtimes, we align the accuracy of both FFT algorithms. We set numerical parameters (number of discretization bins and domain length) such that both algorithms give similarly accurate bounds and verify it visually. Figure 3(b) shows a significant speedup (100x) using our algorithms. We note that runtimes are directly proportional to the memory required by the algorithms and so a separate memory analysis is not required; the runtime and memory are dominated by the number of points in the discretization of PRV. All experiments are performed on a Intel Xeon W-2155 CPU with 3.30GHz with 128GB of memory. ",
+        "bbox": [
+            174,
+            699,
+            825,
+            796
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            815,
+            553,
+            833
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Janardhan Kulkarni and Sergey Yekhanin for several useful discussions and encouraging us to work on this problem. L.W. would like to thank Daniel Jones and Victor Rühle for fruitful discussions and helpful guidance. ",
+        "bbox": [
+            174,
+            848,
+            825,
+            888
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n$[ \\mathrm { A C G ^ { + } } 1 6 ]$ Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 308–318, 2016.   \n[BDLS19] Zhiqi Bu, Jinshuo Dong, Qi Long, and Weijie J. Su. Deep learning with gaussian differential privacy, 2019.   \n[BDRS18] Mark Bun, Cynthia Dwork, Guy N Rothblum, and Thomas Steinke. Composable and versatile privacy via truncated cdp. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 74–86, 2018.   \n$[ \\mathrm { B G K } ^ { + } 2 1 ]$ Zhiqi Bu, Sivakanth Gopi, Janardhan Kulkarni, Yin Tat Lee, Judy Hanwen Shen, and Uthaipon Tantipongpipat. Fast and memory efficient differentially private-sgd via jl projections. arXiv preprint arXiv:2102.03013, 2021. [BS16] Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635–658. Springer, 2016. [BW18] Borja Balle and Yu-Xiang Wang. Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising. In International Conference on Machine Learning, pages 403–412, 2018.   \n$[ \\mathrm { D K M ^ { + } } 0 6 ]$ Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni Naor. Our data, ourselves: Privacy via distributed noise generation. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 486– 503. Springer, 2006. [DL09] Cynthia Dwork and Jing Lei. Differential privacy and robust statistics. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 371–380, 2009.   \n[DMNS06] Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography Conference, pages 265–284. Springer, 2006. $[ \\mathrm { D R } ^ { + } 1 4 ]$ Cynthia Dwork, Aaron Roth, et al. The algorithmic foundations of differential privacy. Foundations and Trends $\\textsuperscript { \\textregistered }$ in Theoretical Computer Science, 9(3–4):211–407, 2014. [DR16] Cynthia Dwork and Guy N Rothblum. Concentrated differential privacy. arXiv preprint arXiv:1603.01887, 2016. [DRS19] Jinshuo Dong, Aaron Roth, and Weijie J. Su. Gaussian differential privacy, 2019. [DRV10] Cynthia Dwork, Guy N Rothblum, and Salil Vadhan. Boosting and differential privacy. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 51–60. IEEE, 2010. [KH21] Antti Koskela and Antti Honkela. Computing differential privacy guarantees for heterogeneous compositions using fft. arXiv preprint arXiv:2102.12412, 2021.   \n$[ \\mathrm { K J H } ^ { + } 2 0 ]$ Antti Koskela, Joonas Jälkö, Antti Honkela, et al. Computing tight differential privacy guarantees using fft. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. AISTATS, 2020.   \n[KJPH21] Antti Koskela, Joonas Jälkö, Lukas Prediger, and Antti Honkela. Tight differential privacy for discrete-valued mechanisms and for the subsampled gaussian mechanism using fft. In Arindam Banerjee and Kenji Fukumizu, editors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine Learning Research, pages 3358–3366. PMLR, 13–15 Apr 2021.   \n[KOV15] Peter Kairouz, Sewoong Oh, and Pramod Viswanath. The composition theorem for differential privacy. In International conference on machine learning, pages 1376–1385. PMLR, 2015. [KP21] Antti Koskela and Lukas Prediger. Github repository for fourier accountant. https: //github.com/DPBayes/PLD-Accountant, 2021. [Mir17] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 263–275. IEEE, 2017.   \n[MM18] Sebastian Meiser and Esfandiar Mohammadi. Tight on budget? tight bounds for r-fold approximate differential privacy. In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, pages 247–264, 2018. [MV16] Jack Murtagh and Salil Vadhan. The complexity of computing the optimal composition of differential privacy. In Theory of Cryptography Conference, pages 157–175. Springer, 2016.   \n[SMM19] David M Sommer, Sebastian Meiser, and Esfandiar Mohammadi. Privacy loss classes: The central limit theorem in differential privacy. Proceedings on privacy enhancing technologies, 2019(2):245–269, 2019.   \n[ZDW21] Yuqing Zhu, Jinshuo Dong, and Yu-Xiang Wang. Optimal accounting of differential privacy via characteristic function. arXiv preprint arXiv:2106.08567, 2021. ",
+        "bbox": [
+            174,
+            70,
+            828,
+            929
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            184,
+            90,
+            828,
+            404
+        ],
+        "page_idx": 11
+    }
+]

parse/train/Hk6a8N5xe/Hk6a8N5xe.md

@@ -0,0 +1,245 @@
+# CLASSIFY OR SELECT: NEURAL ARCHITECTURES FOR EXTRACTIVE DOCUMENT SUMMARIZATION
+
+Ramesh Nallapati, Bowen Zhou IBM Watson Yorktown Heights, NY 10598 USA {nallapati,zhou}@us.ibm.com
+
+Mingbo Ma   
+Oregon State University   
+Kelley Engineering Center, Corvallis, OR, 97331   
+mam@oregonstate.edu
+
+# ABSTRACT
+
+We present two novel and contrasting Recurrent Neural Network (RNN) based architectures for extractive summarization of documents. The Classifier based architecture sequentially accepts or rejects each sentence in the original document order for its membership in the final summary. The Selector architecture, on the other hand, is free to pick one sentence at a time in any arbitrary order to piece together the summary.
+
+Our models under both architectures jointly capture the notions of salience and redundancy of sentences. In addition, these models have the advantage of being very interpretable, since they allow visualization of their predictions broken up by abstract features such as information content, salience and redundancy.
+
+We show that our models reach or outperform state-of-the-art supervised models on two different corpora. We also recommend the conditions under which one architecture is superior to the other based on experimental evidence.
+
+# 1 INTRODUCTION
+
+Document summarization is an important problem that has many applications in information retrieval and natural language understanding. Summarization techniques are mainly classified into two categories: extractive and abstractive. Extractive methods aim to select salient snippets, sentences or passages from documents, while abstractive summarization techniques aim to concisely paraphrase the information content in the documents.
+
+A vast majority of the literature on document summarization is devoted to extractive summarization. Traditional methods for extractive summarization can be broadly classified into greedy approaches (e.g., Carbonell & Goldstein (1998)), graph based approaches (e.g., Radev & Erkan (2004)) and constraint optimization based approaches (e.g., McDonald (2007)).
+
+Recently, neural network based approaches have become popular for extractive summarization. For example, Kageback et al. (2014) employed the recursive autoencoder (Socher et al. (2011)) to summarize documents, producing best performance on the Opinosis dataset (Ganesan et al. (2010)). Yin & Pei (2015) applied Convolutional Neural Networks (CNN) to project sentences to continuous vector space and then select sentences by minimizing the cost based on their ‘prestige’ and ‘diverseness’, on the task of multi-document extractive summarization. Another related work is that of Cao et al. (2016), who address the problem of query-focused multi-document summarization using query-attention-weighted CNNs.
+
+Recently, with the emergence of strong generative neural models for text Bahdanau et al. (2014), abstractive techniques are also becoming increasingly popular (Rush et al. (2015), Nallapati et al. (2016b) and Nallapati et al. (2016a)). Despite the emergence of abstractive techniques, extractive techniques are still attractive as they are less complex, less expensive, and generate grammatically and semantically correct summaries most of the time. In a very recent work, Cheng & Lapata (2016) proposed an attentional encoder-decoder for extractive single-document summarization and trained it on Daily Mail corpus, a large news data set, achieving state-of-the-art performance. Like Cheng & Lapata (2016), our work also focuses only on sentential extractive summarization of single documents using neural networks.
+
+# 2 TWO ARCHITECTURES
+
+Our architectures are motivated by two intuitive strategies that humans tend to adopt when they are tasked with extracting salient sentences in a document. The first strategy, which we call Classify, involves reading the whole document once to understand its contents, and then traversing through the sentences in the original document order and deciding whether or not each sentence belongs to the summary. The other strategy that we call Select involves memorizing the whole document once as before, and then picking sentences that should belong to the summary one at a time, in any order of one’s choosing. Qualitatively, the latter strategy appears to be a better one since it allows us to make globally optimal decisions at each step. While it may be harder for humans to follow this strategy since we are forgetful by nature, one may expect that the Select strategy could deliver an advantage for the machines, since ’forgetfulness’ is not a real ‘concern’ for them. In this work, we will explore both the strategies empirically and make a recommendation on which strategy is optimal under what conditions.
+
+Broadly, our Classify architecture involves an RNN based sequence classification model that sequentially classifies each sentence into 0/1 binary labels, while the Select architecture involves a generative model that sequentially generates the indices of the sentences that should belong to the summary. We will first discuss the components shared by both the architectures and then we will present each architecture separately.
+
+Shared Building Blocks: Both architectures begin with word-level bidirectional Gated Recurrent Unit (GRU) based RNNs (Chung et al. (2014)) run independently over each sentence in the document, where each time-step of the RNN corresponds to a word index in the sentence. The average pooling of the concatenated hidden states of this bidirectional RNN is then used as an input to another bidirectional RNN whose time steps correspond to sentence indices in the document. The concatenated hidden states $\mathbf { \hat { h } } \mathbf { \Phi } $ from the forward and backward layers of this second layer of bidirectional RNN at each time step are used as corresponding sentence representations. We also use the average pooling of the sentence representations as the document representation $\mathbf { \cdot } _ { \mathbf { d } } ,$ . Both architectures also maintain a dynamic summary representation ‘s’ whose estimation is architecture dependent. Models under each architecture compute a score for each sentence towards its summary membership. Motivated by the need to build humanly interpretable models, we compute this score by explicitly modeling abstract features such as salience, novelty and information content as shown below:
+
+$$
+\begin{array} { r l r l } { \mathord { \mathrm { s c o r e } } ( \mathbf { h } _ { j } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { j } ) = w _ { c } \sigma ( \mathbf { W } _ { c } ^ { T } \mathbf { h } _ { j } ) } & { } & { \textnormal { \# } ( \mathrm { c o n t e n t ~ r i c h n e s s } ) } \\ { + w _ { s } \sigma ( \cos ( \mathbf { h } _ { j } , \mathbf { d } ) ) } & { } & { \textnormal { \# } ( \mathrm { s a l ~ i e n c e ~ w . r . t . ~ d o c u m e n t } ) } \\ { + w _ { p } \sigma ( \mathbf { W } _ { p } ^ { T } \mathbf { p } _ { j } ) } & { } & { \textnormal { \# } ( \mathrm { p o s i t i o n a l ~ i m p o r t a n c e } ) } \\ { - w _ { r } \sigma ( \cos ( \mathbf { h } _ { j } , \mathbf { s } _ { j } ) ) } & { } & { \textnormal { \# } ( \mathrm { r e d u n d a n c y ~ w . r . t . ~ s u m m a r y } ) } \\ { + b ) , } & { } & { \textnormal { \# } ( \mathrm { b i a s ~ t e r m } ) } \end{array}
+$$
+
+where $j$ is the index of the sentence in the document, $\mathbf { p } _ { j }$ is the positional embedding of the sentence computed by concatenation of embeddings corresponding to forward and backward position indices of the sentence in the document; $\cos ( \mathbf { a } , \mathbf { b } )$ is the standard cosine similarity between the two vectors $\mathbf { a }$ and $\mathbf { b }$ ; $\mathbf { W } _ { c }$ and $\mathbf { W } _ { p }$ are parameter vectors to model content richness and positional importance of sentences respectively; and $w _ { c } , w _ { s } , w _ { p }$ and $w _ { r }$ are scalar weights to model relative importance of various abstract features, and are learned automatically. In the equation above, the abstract feature that each term represents is printed against the term in comments. In other words, assuming the importance weights are positive, in order for a sentence to score high for summary membership, it needs to be highly salient, content rich and occupy important positions in the document, while being least redundant with respect to the summary generated till that point. Note that our formulation of the scoring function simultaneously captures both salience of the sentence $\mathbf { h } _ { j }$ with respect to the document $\mathbf { d }$ as well as its redundancy with respect to the current summary representation ${ \bf s } _ { j }$ . In the next subsection, we will describe the estimation of dynamic summary representation ${ \bf s } _ { j }$ and the formulation of the cost function for training in each architecture. We will also present shallow and deep models under each architecture.
+
+# 2.1 CLASSIFIER ARCHITECTURE
+
+In this architecture, we sequentially visit each sentence in the original document order and binaryclassify the sentence in terms of whether it belongs to the summary. The probability of the sentence belonging to the summary, $P ( y _ { j } = 1 )$ is given as follows:
+
+$$
+P ( y _ { j } = 1 | \mathbf { h } _ { j } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { j } ) = \sigma ( { \mathsf { s c o r e } } ( \mathbf { h } _ { j } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { j } )
+$$
+
+The objective function to minimize at training is the negative log-likelihood of the training data labels:
+
+$$
+\ell ( \mathbf { W } , \mathbf { w } , \mathbf { b } ) = - \sum _ { d = 1 } ^ { N } \sum _ { j = 1 } ^ { N _ { d } } ( y _ { j } ^ { d } \log P ( y _ { j } ^ { d } = 1 | \mathbf { h } _ { j } ^ { d } , \mathbf { s } _ { j } ^ { d } , \mathbf { d } _ { d } ) + ( 1 - y _ { j } ^ { d } ) \log ( 1 - P ( y _ { j } ^ { d } = 1 | \mathbf { h } _ { j } ^ { d } , \mathbf { s } _ { j } ^ { d } , \mathbf { d } _ { d } ) )
+$$
+
+where $N$ is the size of the training corpus and $N _ { d }$ is the number of sentences in the document $d$ . Now the only detail that remains is how the dynamic summary representation ${ \bf s } _ { j }$ is estimated. This is where the shallow and deep models under this architecture differ, and we describe them below.
+
+![](images/22dfb52afa32430a634e87ff92dc82078c1f5d3c0928962112dea681948f3f26.jpg)  
+Figure 1: The shallow and deep versions of the Classifier architecture for extractive summarization.
+
+Shallow Model: In the shallow model, we estimate the dynamic summary representation as the running sum of the representations of the sentences visited so far weighted by their probability of being in the summary.
+
+$$
+\begin{array}{c} \begin{array} { r c l } { { { \bf { s } } _ { j } } } & { { = } } & { { \displaystyle \sum _ { i = 1 } ^ { j - 1 } { { \bf { h } } _ { i } y _ { i } \qquad } } } \\ { { { \bf { s } } _ { j } } } & { { = } } & { { \displaystyle \sum _ { i = 1 } ^ { j - 1 } { { \bf { h } } _ { i } y _ { i } \qquad } } } \end{array} \textstyle { \Psi } { \mathrm { ~ ( t r a i n i n g ~ t ~ t i m e ) } }  \\ { { \bf { s } } _ { j } } & { { = } } & { { \displaystyle \sum _ { i = 1 } ^ { j - 1 } { { \bf { h } } _ { i } { \cal P } \big ( { y _ { i } = 1 \big | { { \bf { h } } _ { i } , { \bf { s } } _ { i } , { \bf { d } } } \big ) \mathrm { ~ } } \mathrm { ~ \# ~ ( t e s t ~ t i m e ) } } } } \end{array}
+$$
+
+In other words, at training time, since the summary membership of sentences is known, the probabilities are binary, whereas at test time we use a weighted pooling based on the estimated probability that each sentence belongs to the summary. There is no need to normalize the summary representations since the cosine similarity metric we use in the scoring function of Eq. (1) automatically normalizes them.
+
+Deep Model: In the deep model, we introduce an additional layer of unidirectional sentence-level GRU-RNN that takes as input the sentence representations $\mathbf { h } _ { j }$ at each time-step. The hidden state of the new GRU $\hat { \mathbf { h } } _ { j } = G R U ( \mathbf { h } _ { j } )$ is used as a replacement for sentence representation $\mathbf { h } _ { j }$ in computing summary membership scores using Eq. (1) as well as in computing the dynamic summary representation using Eq. (3). The main idea behind using this additional layer of GRU is to allow a greater degree of non-linearity in computing the summary representation.
+
+The graphical representations of the shallow and deep models under the Classifier architecture are displayed in Figure 1 with their full set of dependencies.
+
+# 2.2 SELECTOR ARCHITECTURE
+
+In this architecture, the models do not make decisions in the sequence of sentence ordering; instead, they pick one sentence at a time in an order that they deem fit. The act of picking a sentence is cast as a sequential generative model in which one sentence-index is emitted at each time step that maximizes the score in Eq. 1. Accordingly, the probability of picking a sentence with index $I ( j ) = k \in \{ 1 , \dots , N _ { d } \}$ at time-step $j$ is given by the softmax over the scoring function:
+
+$$
+P ( I ( j ) = k | \mathbf { s } _ { j } , \mathbf { h } _ { k } , \mathbf { d } ) = \frac { \exp ( \operatorname { s c o r e } ( \mathbf { h } _ { k } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { k } ) ) } { \sum _ { l \in \{ 1 , \dots , N _ { d } \} } \exp ( \operatorname { s c o r e } ( \mathbf { h } _ { l } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { l } ) ) }
+$$
+
+The loss function in this case is the negative log-likelihood of the selected sentences in the ground truth data as shown below.
+
+$$
+\begin{array} { r l r } { \ell ( { \mathbf W } , { \mathbf w } , { \mathbf b } ) } & { = } & { - \displaystyle \sum _ { d = 1 } ^ { N } \sum _ { j = 1 } ^ { M _ { d } } \log P ( I ( j ) ^ { ( d ) } | { \mathbf h } _ { I ( j ) ^ { ( d ) } } , { \mathbf s } _ { j } ^ { d } , { \mathbf d } _ { d } ) } \end{array}
+$$
+
+where $M _ { d }$ is the number of sentences selected in the ground truth of document $d$ , $\{ I ( 1 ) ^ { ( d ) } , \dots , I ( M _ { d } ) ^ { ( d ) } \}$ is the ordered list of selected sentence indices in the ground truth of document $d$ . The dependence of the loss function on the order of the selected sentences can be gauged by the fact that the probability of selecting a sentence at time step $j$ depends on the dynamic summary representation ${ \bf s } _ { j }$ , which is estimated based on the all sentences selected up to time step $j - 1$ .
+
+At test time, at each time-step, the model emits the index of the sentence that has the best score given the current summary representation as shown below.
+
+$$
+I ( j ) = \arg \operatorname* { m a x } _ { k \in \{ 1 , \dots , N _ { d } \} } \mathtt { s c o r e } ( \mathbf { h } _ { k } , \mathbf { s } _ { j } , \mathbf { d } , \mathbf { p } _ { k } )
+$$
+
+The estimation of dynamic summary representation is done differently for the shallow and deep selector models as described below.
+
+Shallow Model: In this model, we sum the representations of the selected sentences until the time step $j$ as the dynamic summary representation. This is true for both training time and test time.
+
+$$
+\mathbf { s } _ { j } = \sum _ { i = 1 } ^ { j - 1 } \mathbf { h } _ { I ( i ) } .
+$$
+
+Deep Model: In the deep model, we introduce an additional GRU-RNN whose time steps correspond to the sentence index emission events. At each time-step, it takes as input the representation of the previously selected sentence $\mathbf { h } _ { I ( j - 1 ) }$ , and computes a new hidden state $\hat { \mathbf { h } } _ { j } = G R U ( \mathbf { h } _ { I ( j - 1 ) } )$ . Unlike the shallow model that maintains a separate vector for summary representation ${ \bf s } _ { j }$ , we use $\hat { \mathbf { h } } _ { j }$ as the summary representation ${ \bf s } _ { j }$ at time step $j$ . This makes sense for the case of the Selector architecture since both at training and test time we make hard decisions of sentence selection, with the effect that the hidden state of the new GRU can capture a non-linear aggregation of the sentences selected until time step $j - 1$ .
+
+![](images/eb9e93e233925f834a6b17c74e15ea081d07fe5886a372e0868ee6806320d105.jpg)  
+Figure 2: Selector architecture for extractive summarization. The shallow and deep versions are identical except for the fact that the simple vector representation for summary representation in the shallow version is replaced with a gated recurrent unit in the deep version.
+
+Fig. 2 shows the graphical representation of the Selector architecture with all the dependencies between the nodes. The architecture is the same for both shallow and deep models with the only difference being that the simple summary representation in the former is replaced with a gated recurrent unit in the latter.
+
+# 3 RELATED WORK
+
+Previous researcher such as Shen et al. (2007) have proposed modeling extractive document summarization as a sequence classification problem using Conditional Random Fields. Our approach is different from theirs in the sense that we use RNNs in our model that do not require any handcrafted features for representing sentences and documents.
+
+The Selector architecture broadly involves ranking of sentences by some criterion, therefore does correspond to traditional methods for extractive summarization such as TextRank (Mihalcea & Tarau (2004)) that also involve ranking of sentences by salience and novelty. However, to the best of our knowledge, our Selector framework is a novel deep learning framework for extractive summarization. Broader efforts are being made in the deep learning community to build more sophisticated sequence to sequence models towards the objective of automatically learning complex tasks such as sorting sequences (Oriol Vinyals (2015); Graves et al. (2014)), but their utility for extractive summarization remains to be explored.
+
+In the deep learning framework, the extractive summarization work of Cheng & Lapata (2016) is the closest to our work. Their model is based on an encoder-decoder approach where the encoder learns the representation of sentences and documents while the decoder classifies each sentence using an attention mechanism. Broadly, their model is also in the Classifier framework, but architecturally, our approaches are different. While their approach can be termed as a multi-pass approach where both the encoder and decoder consume the same sentence representations, our approach is a deep one where the representations learned by the bidirectional GRU encoder are in turn consumed by the Classifier or Selector models. Another key difference between our work and theirs is that unlike our unsupervised greedy approach to convert abstractive summaries to extractive labels, Cheng & Lapata (2016) chose to train a separate supervised classifier using manually created labels on a subset of the data. This may yield more accurate gold extractive labels which may help boost the performance of their models, but incurs additional annotation costs.
+
+# 4 EXPERIMENTS AND RESULTS
+
+Pseudo ground-truth generation: In order to train our extractive Classifier and Selector models, for each document we need ground truth in the form of sentence-level binary labels and ordered list of selected sentences respectively. However, most summarization corpora only contain human written abstractive summaries as ground truth. To solve this problem, we use an unsupervised approach to convert the abstractive summaries to extractive labels. Our approach is based on the idea that the selected sentences from the document should be the ones that maximize the Rouge score with respect to gold abstractive summaries. Since it is computationally expensive to find a globally optimal subset of sentences that maximizes the Rouge score, we employ a greedy approach, where we add one sentence at a time incrementally to the summary, such that the Rouge score of the current set of selected sentences is maximized with respect to the entire gold summary. We stop adding sentences when either none of the remaining candidate sentences improves the Rouge score upon addition to the current summary set or when the maximum summary length is reached. We return this ordered list of sentences as the ground-truth for the Selector architecture. The ordered list is converted into binary summary-membership labels that are consumed by the Classifier architecture for training.
+
+We note that similar approaches have been employed by other researchers such as Svore et al. (2007) to handle the problem of converting abstractive summaries to extractive ground truth. We would also like to point readers to a recent work by Cao et al. (2015) that proposes an ILP based approach to solve this problem optimally. Since this is not the focus of this work, we chose a simple greedy algorithm.
+
+Corpora: For our experiments, we used the Daily Mail corpus originally constructed by Hermann et al. (2015) for the task of passage-based question answering, and re-purposed for the task of document summarization as proposed in Cheng & Lapata (2016) for extractive summarization and Nallapati et al. (2016a) for abstractive summarization. Overall, we have 196,557 training documents, 12,147 validation documents and 10,396 test documents from the Daily Mail corpus. On average, there are about 28 sentences per document in the training set, and an average of 3-4 sentences in the reference summaries. The average word count per document in the training set is 802.
+
+We also used the DUC 2002 single-document summarization dataset1 consisting of 567 documents as an additional out-of-domain test set to evaluate our models.
+
+Evaluation: In our experiments below, we evaluate the performance of our models using different variants of the Rouge metric2 computed with respect to the gold abstractive summaries. Following Cheng & Lapata (2016), we use limited length Rouge recall at 75 bytes of summary as well as 275 bytes on the Daily Mail corpus. On DUC 2002 corpus, following the official guidelines, we use limited length Rouge recall at 75 words. We report the scores from Rouge-1, Rouge-2 and RougeL, which are computed using matches of unigrams, bigrams and longest common subsequences respectively, with the ground truth summaries.
+
+Baselines: On all datasets, we use Lead-3 model, which simply produces the leading three sentences of the document as the summary, as a baseline. On the Daily Mail and DUC 2002 corpora, we also report performance of $L R e g$ , a feature-rich logistic classifier used as a baseline by Cheng & Lapata (2016). On DUC 2002 corpus, we report several baselines such as Integer Linear Programming based approach (Woodsend & Lapata (2010)), and graph based approaches such as TGRAPH (Parveen et al. (2015)) and URANK (Wan (2010)) which achieve very high performance on this corpus. In addition, we also compare with the state-of-the art deep learning supervised extractive model from Cheng & Lapata (2016).
+
+Experimental Settings: We used 100-dimensional word2vec (Mikolov et al. (2013)) embeddings trained on the Daily Mail corpus as our embedding initialization. We limited the vocabulary size to 150K and the maximum sentence length to 50 words, to speed up computation. We fixed the model hidden state size at 200. We used a batch size of 32 at training time, and employed adadelta (Zeiler (2012)) to train our model. We employed gradient clipping and L-2 regularization to prevent overfitting and an early stopping criterion based on validation cost.
+
+At test time, for the Classifier models we pick sentences sorted by the predicted probabilities until we exceed the length limit as determined by the Rouge metric. Likewise, we allow the Selector models to emit sentence indices until the desired summary length is reached. For the Selector model, we also make sure the emitted sentence ids are not repeated across time steps by traversing down the sorted predicted probabilities of the softmax layer at each time step until we reach a sentence-id that was not emitted before.
+
+We note that it is possible to optimize the Classifier performance at test time using the Viterbi algorithm to compute the best sequence of labels, subject to the Markovian assumptions of the architecture and model. Similarly, it is also possible to further boost the Selector’s performance by using beam search at test time. However, in this work we used greedy classification/selection for inference since our primary interest is in comparing the two architectures, and our choice allows us to make a fair apples-to-apples comparison.
+
+Results on Daily Mail corpus: Table 1 shows the performance comparison of our models with state-of-the-art model of Cheng & Lapata (2016) and other baselines on the DailyMail corpus using Rouge recall at two different summary lengths.
+
+Table 1: Performance of various models on the entire Daily Mail test set using the limited length recall variants of Rouge with respect to the abstractive ground truth at 75 bytes and 275 bytes. Entries with asterisk are statistically significant using $9 5 \%$ confidence interval with respect to the nearest state-of-the-art model, as estimated by the Rouge script.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Recall at 75 bytes</td><td colspan="3">Recall at 275 bytes</td></tr><tr><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>Lead-3</td><td>21.9</td><td>7.2</td><td>11.6</td><td>40.5</td><td>14.9</td><td>32.6</td></tr><tr><td>LReg(500)</td><td>18.5</td><td>6.9</td><td>10.2</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td>Cheng &#x27;16</td><td>22.7</td><td>8.5</td><td>12.5</td><td>42.2</td><td>17.3*</td><td>34.8</td></tr><tr><td>Shal.-Select</td><td>25.6</td><td>10.3</td><td>14.0</td><td>41.3</td><td>16.8</td><td>34.9</td></tr><tr><td>Deep-Select</td><td>26.1</td><td>10.7</td><td>14.4</td><td>41.3</td><td>15.3</td><td>33.5</td></tr><tr><td>Shal.-Cls.</td><td>26.0</td><td>10.5</td><td>14.23</td><td>42.1</td><td>16.8</td><td>34.8</td></tr><tr><td>Deep-Cls.</td><td>26.2* ±0.4</td><td>10.7* ±0.4</td><td>14.4* ±0.4</td><td>42.2 ±0.2</td><td>16.8 ±0.2</td><td>35.0 ±0.2</td></tr></table>
+
+The results show that contrary to our initial expectation, the Classifier architecture is superior to the Selector architecture. Within each architecture, the deeper models are better performing than the shallower ones. Our deep classifier model outperforms Cheng & Lapata (2016) with a statistically significant margin at 75 bytes, while matching their model at 275 bytes. One potential reason our models do not consistently outperform the extractive model of Cheng & Lapata (2016) is the additional supervised training they used to create sentence-level extractive labels to train their model. Our models instead use an unsupervised greedy approximation to create extractive labels from abstractive summaries, and as a result, may generate noisier ground truth than theirs.
+
+Results on the Out-of-Domain DUC 2002 corpus: We also evaluated the models trained on the DailyMail corpus on the out-of-domain DUC 2002 set as shown in Table 2. The performance trend is similar to that on Daily Mail. Our best model, Deep Classifier is again statistically on par with the model of Cheng & Lapata (2016). However, both models perform worse than graph-based TGRAPH (Parveen et al. (2015)) and URANK (Wan (2010)) algorithms, which are the state-of-the-art models on this corpus. Deep learning based supervised models such as ours and that of Cheng & Lapata (2016) perform very well on the domain they are trained on, but may suffer from domain adaptation issues when tested on a different corpus such as DUC 2002.
+
+# 5 DISCUSSION
+
+Impact of Document Structure: In all our experiments thus far, the classifier architecture has proven superior to the selector architecture. We conjecture that decision making in the same sequence as the original sentence ordering is perhaps advantageous in document summarization since there is a smooth sequential discourse structure in news stories starting with the main highlights of the story in the beginning, more elaborate description in the middle and ending with conclusive remarks. If this is true, then in scenarios where sentence ordering is less structured, the selector architecture should be superior since it has freedom to select salient sentences in any arbitrary order. Such scenarios actually do occur in practice, e.g., summarization of a cluster of tweets on a topic where there is no specific discourse structure between individual tweets, or in multi-document summarization where a pair of sentences across document boundaries have no specific ordering. In order to test this hypothesis, we simulated such data in the Daily Mail corpus by randomly shuffling the sentences in each document in the training set and retraining models under both the architectures, and evaluating them on the original test sets. The results, summarized in Table 3, show that the Classifier architecture suffers bigger losses than the Selector architecture when the document structure is destroyed. In fact, the Selector architecture performs slightly better than the Classifier architecture when trained on the shuffled data, indicating that our hypothesis may indeed be true.
+
+Table 2: Performance of various models on the DUC 2002 set using the limited length recall variants of Rouge at 75 words. Our Deep Classifier is statistically within the margin of error at $9 5 \%$ C.I. with respect to the model of Cheng & Lapata (2016), but both are lower than state-of-the-art results due to out-of-domain training.   
+
+<table><tr><td></td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>Lead-3</td><td>43.6</td><td>21.0</td><td>40.2</td></tr><tr><td>LReg</td><td>43.8</td><td>20.7</td><td>40.3</td></tr><tr><td>ILP</td><td>45.4</td><td>21.3</td><td>42.8</td></tr><tr><td>TGRAPH</td><td>48.1</td><td>24.3*</td><td></td></tr><tr><td>URANK</td><td>48.5*</td><td>21.5</td><td>=</td></tr><tr><td>Cheng et al &#x27;16</td><td>47.4</td><td>23.0</td><td>43.5</td></tr><tr><td>Shallow-Selector</td><td>44.6</td><td>20.0</td><td>41.1</td></tr><tr><td>Deep-Selector</td><td>45.9</td><td>21.5</td><td>42.4</td></tr><tr><td>Shallow-Classifier</td><td>45.9</td><td>21.5</td><td>42.3</td></tr><tr><td>Deep-Classifier</td><td>46.8 ±0.9</td><td>22.6 ±0.9</td><td>43.1 ±0.9</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>Trained on original data</td><td rowspan=1 colspan=3>Trained on shuffled sentences</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Rouge-1</td><td rowspan=1 colspan=1>Rouge-2</td><td rowspan=1 colspan=1>Rouge-L</td><td rowspan=1 colspan=1>Rouge-1</td><td rowspan=1 colspan=1>Rouge-2</td><td rowspan=1 colspan=1>Rouge-L</td></tr><tr><td rowspan=2 colspan=1>Shallow-SelectorShallow-Classifier</td><td rowspan=2 colspan=1>41.342.1</td><td rowspan=2 colspan=1>16.816.8</td><td rowspan=1 colspan=1>34.9</td><td rowspan=1 colspan=1>40.6</td><td rowspan=2 colspan=1>15.615.3</td><td rowspan=2 colspan=1>33.032.9</td></tr><tr><td rowspan=1 colspan=1>35.0</td><td rowspan=1 colspan=1>40.1</td></tr><tr><td rowspan=1 colspan=1>Deep-SelectorDeep-Classifier</td><td rowspan=1 colspan=1>41.342.2</td><td rowspan=1 colspan=1>15.316.8</td><td rowspan=1 colspan=1>33.535.0</td><td rowspan=1 colspan=1>40.540.1</td><td rowspan=1 colspan=1>15.315.1</td><td rowspan=1 colspan=1>32.532.9</td></tr></table>
+
+Table 3: Simulated experiment to demonstrate the impact of document discourse structure on model performance. Evaluation is done using Rouge limited length recall at 275 bytes. The Selector architecture exhibits superior performance when the discourse structure of the document is destroyed.
+
+Qualitative Analysis: One of the advantages of our model design is teasing out various abstract features for the sake of interpretability of system predictions. In the appendix, we present a visualization (see Fig. 3 in the Appendix) of the system predictions based on the scores for various abstract features listed in Eq. (1). We also present the learned importance weights of these features in Table 4. A few representative documents are also presented in the appendix highlighting the sentences chosen by our models for summarization.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+In this work, we propose two neural architectures for extractive summarization. Our proposed models under these architectures are not only very interpretable, but also achieve state-of-the-art performance on two different data sets. We also empirically compare our two frameworks and suggest conditions under which each of them can deliver optimal performance.
+
+As part of our future work, we plan to further investigate the applicability of the novel Selector architecture to relatively less structured summarization problems such as summarization of multiple documents or topical clusters of tweets. In addition, we also intend to perform additional experiments on the Daily Mail dataset such as incorporating beam search in both model inference as well in pseudo ground truth generation that may result in further performance improvements.
+
+# REFERENCES
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.
+
+Ziqiang Cao, Chengyao Chen, Wenjie Li, Sujian Li, Furu Wei, and Ming Zhou. Tgsum: Build tweet guided multi-document summarization dataset. CoRR, abs/1511.08417, 2015. URL http: //arxiv.org/abs/1511.08417.
+
+Ziqiang Cao, Wenjie Li, Sujian Li, and Furu Wei. Attsum: Joint learning of focusing and summarization with neural attention. arXiv preprint arXiv:1604.00125, 2016.
+
+Jaime Carbonell and Jade Goldstein. The use of mmr, diversity-based reranking for reordering documents and producing summaries. In Proceedings of the 21st annual international ACM SIGIR conference on Research and development in information retrieval, pp. 335–336. ACM, 1998.
+
+Jianpeng Cheng and Mirella Lapata. Neural summarization by extracting sentences and words. 54th Annual Meeting of the Association for Computational Linguistics, 2016. URL http://arxiv. org/abs/1603.07252.
+
+Junyoung Chung, C¸ aglar Gulc¸ehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation ¨ of gated recurrent neural networks on sequence modeling. CoRR, abs/1412.3555, 2014. URL http://arxiv.org/abs/1412.3555.
+
+Kavita Ganesan, ChengXiang Zhai, and Jiawei Han. Opinosis: a graph-based approach to abstractive summarization of highly redundant opinions. In Proceedings of the 23rd international conference on computational linguistics, pp. 340–348. Association for Computational Linguistics, 2010.
+
+Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. CoRR, abs/1410.5401, 2014. URL http://arxiv.org/abs/1410.5401.
+
+Karl Moritz Hermann, Tomas Kocisk ´ y, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Su-´ leyman, and Phil Blunsom. Teaching machines to read and comprehend. CoRR, abs/1506.03340, 2015. URL http://arxiv.org/abs/1506.03340.
+
+Mikael Kageback, Olof Mogren, Nina Tahmasebi, and Devdatt Dubhashi. Extractive summarization using continuous vector space models. pp. 31–39. 2014.
+
+Ryan McDonald. A study of global inference algorithms in multi-document summarization. pp. 557–564. 2007.
+
+Rada Mihalcea and Paul Tarau. Textrank: Bringing order into texts. Empirical Methods in Natural Language Processing, 2004.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013.
+
+Ramesh Nallapati, Bowen Zhou, Cicero dos Santos, Caglar Gulcehre, and Bing Xiang. Abstractive text summarization using sequence-to-sequence rnns and beyond. The SIGNLL Conference on Computational Natural Language Learning, 2016a.
+
+Ramesh Nallapati, Bowen Zhou, and Bing Xiang. Sequence-to-sequence rnns for text summarization. International Conference on Learning Representations, Workshop track, 2016b.
+
+Manjunath Kudlur Oriol Vinyals, Samy Bengio. Order matters: Sequence to sequence for sets. International Conference on Learning Representations, 2015.
+
+Daraksha Parveen, Hans-Martin Ramsl, and Michael Strube. Topical coherence for graph-based extractive summarization. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pp. 19491954, 2015.
+
+Dragomir Radev and Gunes Erkan. Lexrank: Graph-based lexical centrality as salience in text ¨ summarization. Journal of Artificial Intelligence Research, pp. 457–479, 2004.
+
+Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. arXiv preprint arXiv:1509.00685, 2015.   
+Dou Shen, Jian-Tao Sun, Hua Li, Qiang Yang, and Zheng Chen. Document summarization using conditional random fields. In Proceedings of IJCAI, 2007.   
+Richard Socher, Eric H. Huang, Jeffrey Pennin, Christopher D. Manning, and Andrew Y. Ng. Dynamic pooling and unfolding recursive autoencoders for paraphrase detection. pp. 801–809. 2011.   
+Krysta M. Svore, Lucy Vanderwende, and Christopher J.C. Burges. Enhancing single-document summarization by combining ranknet and third-party sources. In Proceedings of the Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, pp. 448–457, 2007.   
+Xiaojun Wan. Towards a unified approach to simultaneous single-document and multidocument summarizations. In In Proceedings of the 23rd COLING, pp. 11371145, 2010.   
+Kristian Woodsend and Mirella Lapata. Automatic generation of story highlights. In In Proceedings of the 48th ACL, pp. 565574, 2010.   
+Wenpeng Yin and Yulong Pei. Optimizing sentence modeling and selection for document summarization. In Proceedings of the 24th International Conference on Artificial Intelligence, pp. 1383–1389. AAAI Press, 2015.   
+Matthew D. Zeiler. ADADELTA: an adaptive learning rate method. CoRR, abs/1212.5701, 2012. URL http://arxiv.org/abs/1212.5701.
+
+# 7 APPENDIX
+
+In this section, we will present some additional qualitative and quantitative analysis of our models that we hope will shed some light on their behavior.
+
+# 7.1 VISUALIZATION OF MODEL OUTPUT
+
+In addition to being state-of-the-art performers, our models have the additional advantage of being very interpretable. The clearly separated terms in the scoring function (see Eqn. 1) allow us to tease out various factors responsible for the classification/selection of each sentence. This is illustrated in Figure 3, where we display a representative document from our validation set along with normalized scores from each abstract feature from the deep classifier model. Such visualization is especially useful in explaining to the end-user the decisions made by the system.
+
+# 7.2 LEARNED IMPORTANCE WEIGHTS
+
+We display in Table 4 the learned importance weights corresponding to various abstract features for deep sentence selector. Confirming our intuition, the model learns that salience and redundancy are the most important predictive features for summary membership of a sentence, followed by positional feature and content based feature. Further, when the same model is trained on documents with randomly shuffled sentences, it learns very small weight for the positional features, which is exactly what one expects.
+
+<table><tr><td rowspan=1 colspan=1>Trainingcondition</td><td rowspan=1 colspan=1>Salience</td><td rowspan=1 colspan=1>Content</td><td rowspan=1 colspan=1>Position</td><td rowspan=1 colspan=1>Redundancy</td></tr><tr><td rowspan=1 colspan=1>Original dataShuffled data</td><td rowspan=1 colspan=1>42.759.69</td><td rowspan=1 colspan=1>14.832.85</td><td rowspan=1 colspan=1>-31.090.20</td><td rowspan=1 colspan=1>40.9916.08</td></tr></table>
+
+Table 4: Learned weights of various abstract features from the deep sentence selector model. Salience and redundancy are the most important features as learned by the model, followed by position and content. The negative sign for position weights has no particular significance. The positional feature gets very low weight when the document structure is destroyed by randomly shuffling sentences in each document the training data.
+
+<table><tr><td rowspan=1 colspan=1>Gold Summary:Redpath has ended his eight-year association with SaleSharks. Redpath spent five years as a player and three as acoach at sale. He has thanked the owners, coaches and players for their support.</td><td rowspan=1 colspan=1>Salience</td><td rowspan=1 colspan=1>Content</td><td rowspan=1 colspan=1>Novelty</td><td rowspan=1 colspan=1>Position Prob.</td><td rowspan=1 colspan=1>Position Prob.</td></tr><tr><td rowspan=1 colspan=1>Bryan Redpath has left his coaching role at Sale Sharks withimmediate effect.</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.3</td></tr><tr><td rowspan=1 colspan=1>The 43- year- old Scot ends an eight-year association withthe Aviva Premiership side, having spent five years with them as a player and three as a coach.</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.7</td></tr><tr><td rowspan=1 colspan=1> Redpath returned to Sale in June 2012 as director of rugbyafter starting a coaching career at Gloucester and progressing to the top job at Kingsholm .</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.6</td></tr><tr><td rowspan=2 colspan=1>Redpath spent five years with Sale Sharks as a player and afurther three as a coach but with Sale Sharks struggling fourmonths into Redpath&#x27;s tenure, he was removed from thedirector of rugby role at the Salford-based side and has sincebeen operating as head coach .</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>0.9</td><td rowspan=2 colspan=1>0.7</td><td rowspan=2 colspan=1>0.8</td><td rowspan=2 colspan=1>0.9</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>‘I would like to thank the owners,coaches,players and stafffor all their helpand support since I returned to the club in2012.</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.7</td><td rowspan=2 colspan=1>0.2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1> Also to the supporters who have been great with me both asa player and as a coach,&#x27; Redpath said.</td><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>0.3</td><td rowspan=1 colspan=1>0.2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr></table>
+
+# 7.3 ABLATION EXPERIMENTS
+
+We evaluated the performance of the deep selector and deep classifier models on the validation set by deleting one abstract feature at a time from the model, with replacement. The performance numbers, displayed in Table 5, show that removing any of the features results in a small loss in performance. Note that the priority of features in the ablation experiments need not correspond to their priority in terms of learned weights in Table 4, since feature correlations may affect the two metrics differently. For the deep classifier, content and redundancy seem to matter the most while for the deep selector, dropping positional features hurts the most. Based on this analysis, we plan to investigate more thoroughly the reasons behind the poor ablation performance of salience and redundancy in the classifier and selector models respectively.
+
+<table><tr><td></td><td colspan="3">Deep Classifier</td><td colspan="3">Deep Selector</td></tr><tr><td>Features</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>All</td><td>42.43</td><td>17.32</td><td>34.07</td><td>41.55</td><td>16.52</td><td>32.41</td></tr><tr><td>-Salience</td><td>42.40</td><td>17.27</td><td>34.09</td><td>40.82</td><td>15.99</td><td>31.45</td></tr><tr><td>-Position</td><td>41.78</td><td>16.76</td><td>33.58</td><td>39.06</td><td>14.32</td><td>29.85</td></tr><tr><td>-Content</td><td>41.12</td><td>15.78</td><td>33.23</td><td>40.68</td><td>15.83</td><td>31.13</td></tr><tr><td>-Redundancy</td><td>41.67</td><td>16.86</td><td>32.93</td><td>41.46</td><td>16.50</td><td>32.31</td></tr></table>
+
+Table 5: Ablation experiments on the validation set to gauge the relative importance of each abstract feature. The top row is where all four abstract features are present. The following rows corresponding to removal of one feature at a time with replacement. Evaluation is done using Rouge limited length recall at 275 bytes. Bold faced entries correspond to largest reduction in performance when the corresponding features are dropped.
+
+# 7.4 REPRESENTATIVE DOCUMENTS AND EXTRACTIVE SUMMARIES
+
+We display a couple of representative documents, one each from the Daily Mail and DUC corpora, highlighting the sentences chosen by deep classifier and comparing them with the gold summaries in Table 6. The examples demonstrate qualitatively that the model performs a reasonably good job in identifying the key messages of a document.
+
+<table><tr><td>Document:@entityO have an interest in @entity3 defender @entity2 but are unlikely to make a move until january . the 00 - year - old @entity6 captain has yet to open talks over a new contract at @entity3 and his current deal runs out in 0000 .@entity3 defender @entity2 could be targeted by @entityO in the january transfer window @entityO like @entity2 but do n&#x27;t expect @ entity3 to sell yet they know he will be free to talk to foreign clubs from january．@entity12 will make a Omillion offer for @entity3 goalkeeper @entity14 this summer．the OO - year - old is poised to leave @entityl6 and wants to play for a @entity18 contender . @entity12 are set to make a Omillion bid for @entity2’s @en-</td></tr><tr><td>tity3 team - mate @entity14 in the summer Gold Summary:@entity2&#x27;s contract at @entity3 expires at the end of next season .OO - year-old has yet to open talks over a new deal at @entityl6 ．@entity14 is poised to leave @entity3 at the end of</td></tr><tr><td>the season Document: today, the foreign ministry said that control opera- tions carried out by the corvette spiro against a korean-flagged asreceived ship fishing illegally in argentine waters were car- ried out “ in accordance with international law and in coordi- nation with the foreign ministry ”.the foreign ministry thus ap- proved the intervention by the argentine corvette when it discovered the korean ship chin yuan hsing violating argentine jurisdictional waters on OO may.... the korean ship，which had been fishing illegally in argentine waters，was sunk by its own crew after failing to answer to the argentine ship &#x27;s warnings .the crew was transferred to the chin chuan hsing，which was sailing nearby and</td></tr><tr><td>approached to rescue the crew of the sinking ship... Gold Summary:the korean-flagged fishing vessel chin yuan hs- ing was scuttled in waters off argentina on OO may OOoO ．adverse weather conditions prevailed when the argentine corvette spiro spot- ted the korean ship fishing illegally in restricted argentine waters . the korean vessel did not respond to the corvette &#x27;s warning .instead ,the korean crew sank their ship,and transferred to another korean ship sailing nearby.in accordance with a uk-argentine agreement, the argentine navy turned the surveillance of the second korean ves- sel over to the british when it approached within OO nautical miles of the malvinas（falkland）islands .</td></tr></table>

parse/train/Hk6a8N5xe/Hk6a8N5xe_content_list.json

@@ -0,0 +1,1288 @@
+[
+    {
+        "type": "text",
+        "text": "CLASSIFY OR SELECT: NEURAL ARCHITECTURES FOR EXTRACTIVE DOCUMENT SUMMARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ramesh Nallapati, Bowen Zhou IBM Watson Yorktown Heights, NY 10598 USA {nallapati,zhou}@us.ibm.com ",
+        "bbox": [
+            183,
+            170,
+            447,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mingbo Ma   \nOregon State University   \nKelley Engineering Center, Corvallis, OR, 97331   \nmam@oregonstate.edu ",
+        "bbox": [
+            183,
+            247,
+            504,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            340,
+            544,
+            354
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present two novel and contrasting Recurrent Neural Network (RNN) based architectures for extractive summarization of documents. The Classifier based architecture sequentially accepts or rejects each sentence in the original document order for its membership in the final summary. The Selector architecture, on the other hand, is free to pick one sentence at a time in any arbitrary order to piece together the summary. ",
+        "bbox": [
+            233,
+            369,
+            764,
+            454
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our models under both architectures jointly capture the notions of salience and redundancy of sentences. In addition, these models have the advantage of being very interpretable, since they allow visualization of their predictions broken up by abstract features such as information content, salience and redundancy. ",
+        "bbox": [
+            232,
+            455,
+            766,
+            511
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We show that our models reach or outperform state-of-the-art supervised models on two different corpora. We also recommend the conditions under which one architecture is superior to the other based on experimental evidence. ",
+        "bbox": [
+            232,
+            512,
+            764,
+            554
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            579,
+            336,
+            594
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Document summarization is an important problem that has many applications in information retrieval and natural language understanding. Summarization techniques are mainly classified into two categories: extractive and abstractive. Extractive methods aim to select salient snippets, sentences or passages from documents, while abstractive summarization techniques aim to concisely paraphrase the information content in the documents. ",
+        "bbox": [
+            174,
+            609,
+            823,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A vast majority of the literature on document summarization is devoted to extractive summarization. Traditional methods for extractive summarization can be broadly classified into greedy approaches (e.g., Carbonell & Goldstein (1998)), graph based approaches (e.g., Radev & Erkan (2004)) and constraint optimization based approaches (e.g., McDonald (2007)). ",
+        "bbox": [
+            174,
+            686,
+            823,
+            742
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, neural network based approaches have become popular for extractive summarization. For example, Kageback et al. (2014) employed the recursive autoencoder (Socher et al. (2011)) to summarize documents, producing best performance on the Opinosis dataset (Ganesan et al. (2010)). Yin & Pei (2015) applied Convolutional Neural Networks (CNN) to project sentences to continuous vector space and then select sentences by minimizing the cost based on their ‘prestige’ and ‘diverseness’, on the task of multi-document extractive summarization. Another related work is that of Cao et al. (2016), who address the problem of query-focused multi-document summarization using query-attention-weighted CNNs. ",
+        "bbox": [
+            174,
+            750,
+            825,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, with the emergence of strong generative neural models for text Bahdanau et al. (2014), abstractive techniques are also becoming increasingly popular (Rush et al. (2015), Nallapati et al. (2016b) and Nallapati et al. (2016a)). Despite the emergence of abstractive techniques, extractive techniques are still attractive as they are less complex, less expensive, and generate grammatically and semantically correct summaries most of the time. In a very recent work, Cheng & Lapata (2016) proposed an attentional encoder-decoder for extractive single-document summarization and trained it on Daily Mail corpus, a large news data set, achieving state-of-the-art performance. Like Cheng & Lapata (2016), our work also focuses only on sentential extractive summarization of single documents using neural networks. ",
+        "bbox": [
+            174,
+            868,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 TWO ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            193,
+            395,
+            209
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our architectures are motivated by two intuitive strategies that humans tend to adopt when they are tasked with extracting salient sentences in a document. The first strategy, which we call Classify, involves reading the whole document once to understand its contents, and then traversing through the sentences in the original document order and deciding whether or not each sentence belongs to the summary. The other strategy that we call Select involves memorizing the whole document once as before, and then picking sentences that should belong to the summary one at a time, in any order of one’s choosing. Qualitatively, the latter strategy appears to be a better one since it allows us to make globally optimal decisions at each step. While it may be harder for humans to follow this strategy since we are forgetful by nature, one may expect that the Select strategy could deliver an advantage for the machines, since ’forgetfulness’ is not a real ‘concern’ for them. In this work, we will explore both the strategies empirically and make a recommendation on which strategy is optimal under what conditions. ",
+        "bbox": [
+            174,
+            224,
+            825,
+            391
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Broadly, our Classify architecture involves an RNN based sequence classification model that sequentially classifies each sentence into 0/1 binary labels, while the Select architecture involves a generative model that sequentially generates the indices of the sentences that should belong to the summary. We will first discuss the components shared by both the architectures and then we will present each architecture separately. ",
+        "bbox": [
+            174,
+            397,
+            825,
+            468
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Shared Building Blocks: Both architectures begin with word-level bidirectional Gated Recurrent Unit (GRU) based RNNs (Chung et al. (2014)) run independently over each sentence in the document, where each time-step of the RNN corresponds to a word index in the sentence. The average pooling of the concatenated hidden states of this bidirectional RNN is then used as an input to another bidirectional RNN whose time steps correspond to sentence indices in the document. The concatenated hidden states $\\mathbf { \\hat { h } } \\mathbf { \\Phi } $ from the forward and backward layers of this second layer of bidirectional RNN at each time step are used as corresponding sentence representations. We also use the average pooling of the sentence representations as the document representation $\\mathbf { \\cdot } _ { \\mathbf { d } } ,$ . Both architectures also maintain a dynamic summary representation ‘s’ whose estimation is architecture dependent. Models under each architecture compute a score for each sentence towards its summary membership. Motivated by the need to build humanly interpretable models, we compute this score by explicitly modeling abstract features such as salience, novelty and information content as shown below: ",
+        "bbox": [
+            174,
+            474,
+            825,
+            655
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c7e6710a5c007487cc3620f9b1389ff122cd46abf63d01d530bc2fb4a02ad338.jpg",
+        "text": "$$\n\\begin{array} { r l r l } { \\mathord { \\mathrm { s c o r e } } ( \\mathbf { h } _ { j } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { j } ) = w _ { c } \\sigma ( \\mathbf { W } _ { c } ^ { T } \\mathbf { h } _ { j } ) } & { } & { \\textnormal { \\# } ( \\mathrm { c o n t e n t ~ r i c h n e s s } ) } \\\\ { + w _ { s } \\sigma ( \\cos ( \\mathbf { h } _ { j } , \\mathbf { d } ) ) } & { } & { \\textnormal { \\# } ( \\mathrm { s a l ~ i e n c e ~ w . r . t . ~ d o c u m e n t } ) } \\\\ { + w _ { p } \\sigma ( \\mathbf { W } _ { p } ^ { T } \\mathbf { p } _ { j } ) } & { } & { \\textnormal { \\# } ( \\mathrm { p o s i t i o n a l ~ i m p o r t a n c e } ) } \\\\ { - w _ { r } \\sigma ( \\cos ( \\mathbf { h } _ { j } , \\mathbf { s } _ { j } ) ) } & { } & { \\textnormal { \\# } ( \\mathrm { r e d u n d a n c y ~ w . r . t . ~ s u m m a r y } ) } \\\\ { + b ) , } & { } & { \\textnormal { \\# } ( \\mathrm { b i a s ~ t e r m } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            657,
+            785,
+            753
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $j$ is the index of the sentence in the document, $\\mathbf { p } _ { j }$ is the positional embedding of the sentence computed by concatenation of embeddings corresponding to forward and backward position indices of the sentence in the document; $\\cos ( \\mathbf { a } , \\mathbf { b } )$ is the standard cosine similarity between the two vectors $\\mathbf { a }$ and $\\mathbf { b }$ ; $\\mathbf { W } _ { c }$ and $\\mathbf { W } _ { p }$ are parameter vectors to model content richness and positional importance of sentences respectively; and $w _ { c } , w _ { s } , w _ { p }$ and $w _ { r }$ are scalar weights to model relative importance of various abstract features, and are learned automatically. In the equation above, the abstract feature that each term represents is printed against the term in comments. In other words, assuming the importance weights are positive, in order for a sentence to score high for summary membership, it needs to be highly salient, content rich and occupy important positions in the document, while being least redundant with respect to the summary generated till that point. Note that our formulation of the scoring function simultaneously captures both salience of the sentence $\\mathbf { h } _ { j }$ with respect to the document $\\mathbf { d }$ as well as its redundancy with respect to the current summary representation ${ \\bf s } _ { j }$ . In the next subsection, we will describe the estimation of dynamic summary representation ${ \\bf s } _ { j }$ and the formulation of the cost function for training in each architecture. We will also present shallow and deep models under each architecture. ",
+        "bbox": [
+            173,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 CLASSIFIER ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            161,
+            413,
+            176
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this architecture, we sequentially visit each sentence in the original document order and binaryclassify the sentence in terms of whether it belongs to the summary. The probability of the sentence belonging to the summary, $P ( y _ { j } = 1 )$ is given as follows: ",
+        "bbox": [
+            173,
+            188,
+            825,
+            231
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf639cb5e09c35be2bed554e89629a8fcec3a69221bf950d74103991f9d28a6b.jpg",
+        "text": "$$\nP ( y _ { j } = 1 | \\mathbf { h } _ { j } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { j } ) = \\sigma ( { \\mathsf { s c o r e } } ( \\mathbf { h } _ { j } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            234,
+            669,
+            252
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The objective function to minimize at training is the negative log-likelihood of the training data labels: ",
+        "bbox": [
+            174,
+            256,
+            821,
+            285
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf5817724e3e5450e54caecd6a746a0469aa79cc2f456b07d8c1ff1eb821c552.jpg",
+        "text": "$$\n\\ell ( \\mathbf { W } , \\mathbf { w } , \\mathbf { b } ) = - \\sum _ { d = 1 } ^ { N } \\sum _ { j = 1 } ^ { N _ { d } } ( y _ { j } ^ { d } \\log P ( y _ { j } ^ { d } = 1 | \\mathbf { h } _ { j } ^ { d } , \\mathbf { s } _ { j } ^ { d } , \\mathbf { d } _ { d } ) + ( 1 - y _ { j } ^ { d } ) \\log ( 1 - P ( y _ { j } ^ { d } = 1 | \\mathbf { h } _ { j } ^ { d } , \\mathbf { s } _ { j } ^ { d } , \\mathbf { d } _ { d } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            285,
+            820,
+            332
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $N$ is the size of the training corpus and $N _ { d }$ is the number of sentences in the document $d$ . Now the only detail that remains is how the dynamic summary representation ${ \\bf s } _ { j }$ is estimated. This is where the shallow and deep models under this architecture differ, and we describe them below. ",
+        "bbox": [
+            174,
+            334,
+            825,
+            377
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/22dfb52afa32430a634e87ff92dc82078c1f5d3c0928962112dea681948f3f26.jpg",
+        "image_caption": [
+            "Figure 1: The shallow and deep versions of the Classifier architecture for extractive summarization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            387,
+            767,
+            703
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Shallow Model: In the shallow model, we estimate the dynamic summary representation as the running sum of the representations of the sentences visited so far weighted by their probability of being in the summary. ",
+        "bbox": [
+            174,
+            739,
+            825,
+            782
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/47ffb7d6698d4e3efd7472eb5951b5bd73f0f888eca720366762305560e45b08.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { r c l } { { { \\bf { s } } _ { j } } } & { { = } } & { { \\displaystyle \\sum _ { i = 1 } ^ { j - 1 } { { \\bf { h } } _ { i } y _ { i } \\qquad } } } \\\\ { { { \\bf { s } } _ { j } } } & { { = } } & { { \\displaystyle \\sum _ { i = 1 } ^ { j - 1 } { { \\bf { h } } _ { i } y _ { i } \\qquad } } } \\end{array} \\textstyle { \\Psi } { \\mathrm { ~ ( t r a i n i n g ~ t ~ t i m e ) } }  \\\\ { { \\bf { s } } _ { j } } & { { = } } & { { \\displaystyle \\sum _ { i = 1 } ^ { j - 1 } { { \\bf { h } } _ { i } { \\cal P } \\big ( { y _ { i } = 1 \\big | { { \\bf { h } } _ { i } , { \\bf { s } } _ { i } , { \\bf { d } } } \\big ) \\mathrm { ~ } } \\mathrm { ~ \\# ~ ( t e s t ~ t i m e ) } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            785,
+            681,
+            876
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In other words, at training time, since the summary membership of sentences is known, the probabilities are binary, whereas at test time we use a weighted pooling based on the estimated probability that each sentence belongs to the summary. There is no need to normalize the summary representations since the cosine similarity metric we use in the scoring function of Eq. (1) automatically normalizes them. ",
+        "bbox": [
+            173,
+            895,
+            826,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Deep Model: In the deep model, we introduce an additional layer of unidirectional sentence-level GRU-RNN that takes as input the sentence representations $\\mathbf { h } _ { j }$ at each time-step. The hidden state of the new GRU $\\hat { \\mathbf { h } } _ { j } = G R U ( \\mathbf { h } _ { j } )$ is used as a replacement for sentence representation $\\mathbf { h } _ { j }$ in computing summary membership scores using Eq. (1) as well as in computing the dynamic summary representation using Eq. (3). The main idea behind using this additional layer of GRU is to allow a greater degree of non-linearity in computing the summary representation. ",
+        "bbox": [
+            173,
+            151,
+            825,
+            239
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The graphical representations of the shallow and deep models under the Classifier architecture are displayed in Figure 1 with their full set of dependencies. ",
+        "bbox": [
+            174,
+            246,
+            821,
+            275
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 SELECTOR ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            291,
+            403,
+            306
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this architecture, the models do not make decisions in the sequence of sentence ordering; instead, they pick one sentence at a time in an order that they deem fit. The act of picking a sentence is cast as a sequential generative model in which one sentence-index is emitted at each time step that maximizes the score in Eq. 1. Accordingly, the probability of picking a sentence with index $I ( j ) = k \\in \\{ 1 , \\dots , N _ { d } \\}$ at time-step $j$ is given by the softmax over the scoring function: ",
+        "bbox": [
+            173,
+            318,
+            825,
+            388
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b0d9c4387598c9b9954720e0905f8350b985fb7868d46178e2cd3d6f11e80d6.jpg",
+        "text": "$$\nP ( I ( j ) = k | \\mathbf { s } _ { j } , \\mathbf { h } _ { k } , \\mathbf { d } ) = \\frac { \\exp ( \\operatorname { s c o r e } ( \\mathbf { h } _ { k } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { k } ) ) } { \\sum _ { l \\in \\{ 1 , \\dots , N _ { d } \\} } \\exp ( \\operatorname { s c o r e } ( \\mathbf { h } _ { l } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { l } ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            395,
+            720,
+            433
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The loss function in this case is the negative log-likelihood of the selected sentences in the ground truth data as shown below. ",
+        "bbox": [
+            173,
+            439,
+            823,
+            468
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ec3bd4e94b4860a9d6ce678e780a77416d6387435a913826ed5ac00cd0a7f00.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\ell ( { \\mathbf W } , { \\mathbf w } , { \\mathbf b } ) } & { = } & { - \\displaystyle \\sum _ { d = 1 } ^ { N } \\sum _ { j = 1 } ^ { M _ { d } } \\log P ( I ( j ) ^ { ( d ) } | { \\mathbf h } _ { I ( j ) ^ { ( d ) } } , { \\mathbf s } _ { j } ^ { d } , { \\mathbf d } _ { d } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            472,
+            696,
+            518
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $M _ { d }$ is the number of sentences selected in the ground truth of document $d$ , $\\{ I ( 1 ) ^ { ( d ) } , \\dots , I ( M _ { d } ) ^ { ( d ) } \\}$ is the ordered list of selected sentence indices in the ground truth of document $d$ . The dependence of the loss function on the order of the selected sentences can be gauged by the fact that the probability of selecting a sentence at time step $j$ depends on the dynamic summary representation ${ \\bf s } _ { j }$ , which is estimated based on the all sentences selected up to time step $j - 1$ . ",
+        "bbox": [
+            174,
+            523,
+            825,
+            595
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At test time, at each time-step, the model emits the index of the sentence that has the best score given the current summary representation as shown below. ",
+        "bbox": [
+            173,
+            602,
+            821,
+            631
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96fc557beda6ad47d02757ae82abe22e1a34bc6c9492e1b395b85e8593cd4af4.jpg",
+        "text": "$$\nI ( j ) = \\arg \\operatorname* { m a x } _ { k \\in \\{ 1 , \\dots , N _ { d } \\} } \\mathtt { s c o r e } ( \\mathbf { h } _ { k } , \\mathbf { s } _ { j } , \\mathbf { d } , \\mathbf { p } _ { k } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            637,
+            648,
+            664
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The estimation of dynamic summary representation is done differently for the shallow and deep selector models as described below. ",
+        "bbox": [
+            173,
+            678,
+            823,
+            707
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Shallow Model: In this model, we sum the representations of the selected sentences until the time step $j$ as the dynamic summary representation. This is true for both training time and test time. ",
+        "bbox": [
+            171,
+            713,
+            823,
+            742
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/404ea9edf7be6c6e8a616daac75b8348b88549298d776ea286b85355b31becf9.jpg",
+        "text": "$$\n\\mathbf { s } _ { j } = \\sum _ { i = 1 } ^ { j - 1 } \\mathbf { h } _ { I ( i ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            748,
+            550,
+            792
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Deep Model: In the deep model, we introduce an additional GRU-RNN whose time steps correspond to the sentence index emission events. At each time-step, it takes as input the representation of the previously selected sentence $\\mathbf { h } _ { I ( j - 1 ) }$ , and computes a new hidden state $\\hat { \\mathbf { h } } _ { j } = G R U ( \\mathbf { h } _ { I ( j - 1 ) } )$ . Unlike the shallow model that maintains a separate vector for summary representation ${ \\bf s } _ { j }$ , we use $\\hat { \\mathbf { h } } _ { j }$ as the summary representation ${ \\bf s } _ { j }$ at time step $j$ . This makes sense for the case of the Selector architecture since both at training and test time we make hard decisions of sentence selection, with the effect that the hidden state of the new GRU can capture a non-linear aggregation of the sentences selected until time step $j - 1$ . ",
+        "bbox": [
+            173,
+            806,
+            825,
+            925
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb9e93e233925f834a6b17c74e15ea081d07fe5886a372e0868ee6806320d105.jpg",
+        "image_caption": [
+            "Figure 2: Selector architecture for extractive summarization. The shallow and deep versions are identical except for the fact that the simple vector representation for summary representation in the shallow version is replaced with a gated recurrent unit in the deep version. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            246,
+            111,
+            750,
+            362
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 shows the graphical representation of the Selector architecture with all the dependencies between the nodes. The architecture is the same for both shallow and deep models with the only difference being that the simple summary representation in the former is replaced with a gated recurrent unit in the latter. ",
+        "bbox": [
+            173,
+            448,
+            825,
+            505
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
+            343,
+            553
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Previous researcher such as Shen et al. (2007) have proposed modeling extractive document summarization as a sequence classification problem using Conditional Random Fields. Our approach is different from theirs in the sense that we use RNNs in our model that do not require any handcrafted features for representing sentences and documents. ",
+        "bbox": [
+            174,
+            575,
+            825,
+            631
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Selector architecture broadly involves ranking of sentences by some criterion, therefore does correspond to traditional methods for extractive summarization such as TextRank (Mihalcea & Tarau (2004)) that also involve ranking of sentences by salience and novelty. However, to the best of our knowledge, our Selector framework is a novel deep learning framework for extractive summarization. Broader efforts are being made in the deep learning community to build more sophisticated sequence to sequence models towards the objective of automatically learning complex tasks such as sorting sequences (Oriol Vinyals (2015); Graves et al. (2014)), but their utility for extractive summarization remains to be explored. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            750
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the deep learning framework, the extractive summarization work of Cheng & Lapata (2016) is the closest to our work. Their model is based on an encoder-decoder approach where the encoder learns the representation of sentences and documents while the decoder classifies each sentence using an attention mechanism. Broadly, their model is also in the Classifier framework, but architecturally, our approaches are different. While their approach can be termed as a multi-pass approach where both the encoder and decoder consume the same sentence representations, our approach is a deep one where the representations learned by the bidirectional GRU encoder are in turn consumed by the Classifier or Selector models. Another key difference between our work and theirs is that unlike our unsupervised greedy approach to convert abstractive summaries to extractive labels, Cheng & Lapata (2016) chose to train a separate supervised classifier using manually created labels on a subset of the data. This may yield more accurate gold extractive labels which may help boost the performance of their models, but incurs additional annotation costs. ",
+        "bbox": [
+            173,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            447,
+            117
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Pseudo ground-truth generation: In order to train our extractive Classifier and Selector models, for each document we need ground truth in the form of sentence-level binary labels and ordered list of selected sentences respectively. However, most summarization corpora only contain human written abstractive summaries as ground truth. To solve this problem, we use an unsupervised approach to convert the abstractive summaries to extractive labels. Our approach is based on the idea that the selected sentences from the document should be the ones that maximize the Rouge score with respect to gold abstractive summaries. Since it is computationally expensive to find a globally optimal subset of sentences that maximizes the Rouge score, we employ a greedy approach, where we add one sentence at a time incrementally to the summary, such that the Rouge score of the current set of selected sentences is maximized with respect to the entire gold summary. We stop adding sentences when either none of the remaining candidate sentences improves the Rouge score upon addition to the current summary set or when the maximum summary length is reached. We return this ordered list of sentences as the ground-truth for the Selector architecture. The ordered list is converted into binary summary-membership labels that are consumed by the Classifier architecture for training. ",
+        "bbox": [
+            174,
+            141,
+            825,
+            334
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We note that similar approaches have been employed by other researchers such as Svore et al. (2007) to handle the problem of converting abstractive summaries to extractive ground truth. We would also like to point readers to a recent work by Cao et al. (2015) that proposes an ILP based approach to solve this problem optimally. Since this is not the focus of this work, we chose a simple greedy algorithm. ",
+        "bbox": [
+            174,
+            342,
+            823,
+            411
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corpora: For our experiments, we used the Daily Mail corpus originally constructed by Hermann et al. (2015) for the task of passage-based question answering, and re-purposed for the task of document summarization as proposed in Cheng & Lapata (2016) for extractive summarization and Nallapati et al. (2016a) for abstractive summarization. Overall, we have 196,557 training documents, 12,147 validation documents and 10,396 test documents from the Daily Mail corpus. On average, there are about 28 sentences per document in the training set, and an average of 3-4 sentences in the reference summaries. The average word count per document in the training set is 802. ",
+        "bbox": [
+            174,
+            419,
+            825,
+            516
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also used the DUC 2002 single-document summarization dataset1 consisting of 567 documents as an additional out-of-domain test set to evaluate our models. ",
+        "bbox": [
+            174,
+            523,
+            821,
+            551
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation: In our experiments below, we evaluate the performance of our models using different variants of the Rouge metric2 computed with respect to the gold abstractive summaries. Following Cheng & Lapata (2016), we use limited length Rouge recall at 75 bytes of summary as well as 275 bytes on the Daily Mail corpus. On DUC 2002 corpus, following the official guidelines, we use limited length Rouge recall at 75 words. We report the scores from Rouge-1, Rouge-2 and RougeL, which are computed using matches of unigrams, bigrams and longest common subsequences respectively, with the ground truth summaries. ",
+        "bbox": [
+            174,
+            558,
+            825,
+            655
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines: On all datasets, we use Lead-3 model, which simply produces the leading three sentences of the document as the summary, as a baseline. On the Daily Mail and DUC 2002 corpora, we also report performance of $L R e g$ , a feature-rich logistic classifier used as a baseline by Cheng & Lapata (2016). On DUC 2002 corpus, we report several baselines such as Integer Linear Programming based approach (Woodsend & Lapata (2010)), and graph based approaches such as TGRAPH (Parveen et al. (2015)) and URANK (Wan (2010)) which achieve very high performance on this corpus. In addition, we also compare with the state-of-the art deep learning supervised extractive model from Cheng & Lapata (2016). ",
+        "bbox": [
+            174,
+            662,
+            825,
+            773
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Experimental Settings: We used 100-dimensional word2vec (Mikolov et al. (2013)) embeddings trained on the Daily Mail corpus as our embedding initialization. We limited the vocabulary size to 150K and the maximum sentence length to 50 words, to speed up computation. We fixed the model hidden state size at 200. We used a batch size of 32 at training time, and employed adadelta (Zeiler (2012)) to train our model. We employed gradient clipping and L-2 regularization to prevent overfitting and an early stopping criterion based on validation cost. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            864
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "At test time, for the Classifier models we pick sentences sorted by the predicted probabilities until we exceed the length limit as determined by the Rouge metric. Likewise, we allow the Selector models to emit sentence indices until the desired summary length is reached. For the Selector model, we also make sure the emitted sentence ids are not repeated across time steps by traversing down the sorted predicted probabilities of the softmax layer at each time step until we reach a sentence-id that was not emitted before. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            186
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We note that it is possible to optimize the Classifier performance at test time using the Viterbi algorithm to compute the best sequence of labels, subject to the Markovian assumptions of the architecture and model. Similarly, it is also possible to further boost the Selector’s performance by using beam search at test time. However, in this work we used greedy classification/selection for inference since our primary interest is in comparing the two architectures, and our choice allows us to make a fair apples-to-apples comparison. ",
+        "bbox": [
+            174,
+            194,
+            825,
+            279
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results on Daily Mail corpus: Table 1 shows the performance comparison of our models with state-of-the-art model of Cheng & Lapata (2016) and other baselines on the DailyMail corpus using Rouge recall at two different summary lengths. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            327
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c674ff98fa115f3adc20cf2f0c05d137bc81b9262da7e86af0e1c9942f9d4caa.jpg",
+        "table_caption": [
+            "Table 1: Performance of various models on the entire Daily Mail test set using the limited length recall variants of Rouge with respect to the abstractive ground truth at 75 bytes and 275 bytes. Entries with asterisk are statistically significant using $9 5 \\%$ confidence interval with respect to the nearest state-of-the-art model, as estimated by the Rouge script. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"3\">Recall at 75 bytes</td><td colspan=\"3\">Recall at 275 bytes</td></tr><tr><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>Lead-3</td><td>21.9</td><td>7.2</td><td>11.6</td><td>40.5</td><td>14.9</td><td>32.6</td></tr><tr><td>LReg(500)</td><td>18.5</td><td>6.9</td><td>10.2</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td>Cheng &#x27;16</td><td>22.7</td><td>8.5</td><td>12.5</td><td>42.2</td><td>17.3*</td><td>34.8</td></tr><tr><td>Shal.-Select</td><td>25.6</td><td>10.3</td><td>14.0</td><td>41.3</td><td>16.8</td><td>34.9</td></tr><tr><td>Deep-Select</td><td>26.1</td><td>10.7</td><td>14.4</td><td>41.3</td><td>15.3</td><td>33.5</td></tr><tr><td>Shal.-Cls.</td><td>26.0</td><td>10.5</td><td>14.23</td><td>42.1</td><td>16.8</td><td>34.8</td></tr><tr><td>Deep-Cls.</td><td>26.2* ±0.4</td><td>10.7* ±0.4</td><td>14.4* ±0.4</td><td>42.2 ±0.2</td><td>16.8 ±0.2</td><td>35.0 ±0.2</td></tr></table>",
+        "bbox": [
+            178,
+            342,
+            818,
+            470
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results show that contrary to our initial expectation, the Classifier architecture is superior to the Selector architecture. Within each architecture, the deeper models are better performing than the shallower ones. Our deep classifier model outperforms Cheng & Lapata (2016) with a statistically significant margin at 75 bytes, while matching their model at 275 bytes. One potential reason our models do not consistently outperform the extractive model of Cheng & Lapata (2016) is the additional supervised training they used to create sentence-level extractive labels to train their model. Our models instead use an unsupervised greedy approximation to create extractive labels from abstractive summaries, and as a result, may generate noisier ground truth than theirs. ",
+        "bbox": [
+            174,
+            554,
+            825,
+            666
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results on the Out-of-Domain DUC 2002 corpus: We also evaluated the models trained on the DailyMail corpus on the out-of-domain DUC 2002 set as shown in Table 2. The performance trend is similar to that on Daily Mail. Our best model, Deep Classifier is again statistically on par with the model of Cheng & Lapata (2016). However, both models perform worse than graph-based TGRAPH (Parveen et al. (2015)) and URANK (Wan (2010)) algorithms, which are the state-of-the-art models on this corpus. Deep learning based supervised models such as ours and that of Cheng & Lapata (2016) perform very well on the domain they are trained on, but may suffer from domain adaptation issues when tested on a different corpus such as DUC 2002. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            785
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
+            310,
+            823
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Impact of Document Structure: In all our experiments thus far, the classifier architecture has proven superior to the selector architecture. We conjecture that decision making in the same sequence as the original sentence ordering is perhaps advantageous in document summarization since there is a smooth sequential discourse structure in news stories starting with the main highlights of the story in the beginning, more elaborate description in the middle and ending with conclusive remarks. If this is true, then in scenarios where sentence ordering is less structured, the selector architecture should be superior since it has freedom to select salient sentences in any arbitrary order. Such scenarios actually do occur in practice, e.g., summarization of a cluster of tweets on a topic where there is no specific discourse structure between individual tweets, or in multi-document summarization where a pair of sentences across document boundaries have no specific ordering. In order to test this hypothesis, we simulated such data in the Daily Mail corpus by randomly shuffling the sentences in each document in the training set and retraining models under both the architectures, and evaluating them on the original test sets. The results, summarized in Table 3, show that the Classifier architecture suffers bigger losses than the Selector architecture when the document structure is destroyed. In fact, the Selector architecture performs slightly better than the Classifier architecture when trained on the shuffled data, indicating that our hypothesis may indeed be true. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/da1efe1b550f6b75950576215403f8c42ab702ad038ed7167d510f3e4b2a22ff.jpg",
+        "table_caption": [
+            "Table 2: Performance of various models on the DUC 2002 set using the limited length recall variants of Rouge at 75 words. Our Deep Classifier is statistically within the margin of error at $9 5 \\%$ C.I. with respect to the model of Cheng & Lapata (2016), but both are lower than state-of-the-art results due to out-of-domain training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>Lead-3</td><td>43.6</td><td>21.0</td><td>40.2</td></tr><tr><td>LReg</td><td>43.8</td><td>20.7</td><td>40.3</td></tr><tr><td>ILP</td><td>45.4</td><td>21.3</td><td>42.8</td></tr><tr><td>TGRAPH</td><td>48.1</td><td>24.3*</td><td></td></tr><tr><td>URANK</td><td>48.5*</td><td>21.5</td><td>=</td></tr><tr><td>Cheng et al &#x27;16</td><td>47.4</td><td>23.0</td><td>43.5</td></tr><tr><td>Shallow-Selector</td><td>44.6</td><td>20.0</td><td>41.1</td></tr><tr><td>Deep-Selector</td><td>45.9</td><td>21.5</td><td>42.4</td></tr><tr><td>Shallow-Classifier</td><td>45.9</td><td>21.5</td><td>42.3</td></tr><tr><td>Deep-Classifier</td><td>46.8 ±0.9</td><td>22.6 ±0.9</td><td>43.1 ±0.9</td></tr></table>",
+        "bbox": [
+            299,
+            101,
+            697,
+            256
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            348,
+            825,
+            488
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ad4ede34543583c50ab82344e5946fad61694ea338b6414a34d5bee74ee96656.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>Trained on original data</td><td rowspan=1 colspan=3>Trained on shuffled sentences</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Rouge-1</td><td rowspan=1 colspan=1>Rouge-2</td><td rowspan=1 colspan=1>Rouge-L</td><td rowspan=1 colspan=1>Rouge-1</td><td rowspan=1 colspan=1>Rouge-2</td><td rowspan=1 colspan=1>Rouge-L</td></tr><tr><td rowspan=2 colspan=1>Shallow-SelectorShallow-Classifier</td><td rowspan=2 colspan=1>41.342.1</td><td rowspan=2 colspan=1>16.816.8</td><td rowspan=1 colspan=1>34.9</td><td rowspan=1 colspan=1>40.6</td><td rowspan=2 colspan=1>15.615.3</td><td rowspan=2 colspan=1>33.032.9</td></tr><tr><td rowspan=1 colspan=1>35.0</td><td rowspan=1 colspan=1>40.1</td></tr><tr><td rowspan=1 colspan=1>Deep-SelectorDeep-Classifier</td><td rowspan=1 colspan=1>41.342.2</td><td rowspan=1 colspan=1>15.316.8</td><td rowspan=1 colspan=1>33.535.0</td><td rowspan=1 colspan=1>40.540.1</td><td rowspan=1 colspan=1>15.315.1</td><td rowspan=1 colspan=1>32.532.9</td></tr></table>",
+        "bbox": [
+            197,
+            501,
+            799,
+            589
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Simulated experiment to demonstrate the impact of document discourse structure on model performance. Evaluation is done using Rouge limited length recall at 275 bytes. The Selector architecture exhibits superior performance when the discourse structure of the document is destroyed. ",
+        "bbox": [
+            176,
+            599,
+            823,
+            641
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Qualitative Analysis: One of the advantages of our model design is teasing out various abstract features for the sake of interpretability of system predictions. In the appendix, we present a visualization (see Fig. 3 in the Appendix) of the system predictions based on the scores for various abstract features listed in Eq. (1). We also present the learned importance weights of these features in Table 4. A few representative documents are also presented in the appendix highlighting the sentences chosen by our models for summarization. ",
+        "bbox": [
+            173,
+            655,
+            825,
+            739
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            758,
+            495,
+            776
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose two neural architectures for extractive summarization. Our proposed models under these architectures are not only very interpretable, but also achieve state-of-the-art performance on two different data sets. We also empirically compare our two frameworks and suggest conditions under which each of them can deliver optimal performance. ",
+        "bbox": [
+            174,
+            791,
+            825,
+            847
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As part of our future work, we plan to further investigate the applicability of the novel Selector architecture to relatively less structured summarization problems such as summarization of multiple documents or topical clusters of tweets. In addition, we also intend to perform additional experiments on the Daily Mail dataset such as incorporating beam search in both model inference as well in pseudo ground truth generation that may result in further performance improvements. ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. ",
+        "bbox": [
+            174,
+            126,
+            823,
+            155
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ziqiang Cao, Chengyao Chen, Wenjie Li, Sujian Li, Furu Wei, and Ming Zhou. Tgsum: Build tweet guided multi-document summarization dataset. CoRR, abs/1511.08417, 2015. URL http: //arxiv.org/abs/1511.08417. ",
+        "bbox": [
+            174,
+            161,
+            821,
+            205
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ziqiang Cao, Wenjie Li, Sujian Li, and Furu Wei. Attsum: Joint learning of focusing and summarization with neural attention. arXiv preprint arXiv:1604.00125, 2016. ",
+        "bbox": [
+            171,
+            213,
+            823,
+            242
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jaime Carbonell and Jade Goldstein. The use of mmr, diversity-based reranking for reordering documents and producing summaries. In Proceedings of the 21st annual international ACM SIGIR conference on Research and development in information retrieval, pp. 335–336. ACM, 1998. ",
+        "bbox": [
+            173,
+            251,
+            825,
+            308
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jianpeng Cheng and Mirella Lapata. Neural summarization by extracting sentences and words. 54th Annual Meeting of the Association for Computational Linguistics, 2016. URL http://arxiv. org/abs/1603.07252. ",
+        "bbox": [
+            171,
+            315,
+            823,
+            358
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Junyoung Chung, C¸ aglar Gulc¸ehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation ¨ of gated recurrent neural networks on sequence modeling. CoRR, abs/1412.3555, 2014. URL http://arxiv.org/abs/1412.3555. ",
+        "bbox": [
+            173,
+            366,
+            823,
+            410
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kavita Ganesan, ChengXiang Zhai, and Jiawei Han. Opinosis: a graph-based approach to abstractive summarization of highly redundant opinions. In Proceedings of the 23rd international conference on computational linguistics, pp. 340–348. Association for Computational Linguistics, 2010. ",
+        "bbox": [
+            174,
+            416,
+            823,
+            460
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. CoRR, abs/1410.5401, 2014. URL http://arxiv.org/abs/1410.5401. ",
+        "bbox": [
+            169,
+            468,
+            823,
+            497
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Karl Moritz Hermann, Tomas Kocisk ´ y, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Su-´ leyman, and Phil Blunsom. Teaching machines to read and comprehend. CoRR, abs/1506.03340, 2015. URL http://arxiv.org/abs/1506.03340. ",
+        "bbox": [
+            174,
+            505,
+            825,
+            549
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mikael Kageback, Olof Mogren, Nina Tahmasebi, and Devdatt Dubhashi. Extractive summarization using continuous vector space models. pp. 31–39. 2014. ",
+        "bbox": [
+            173,
+            556,
+            823,
+            585
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ryan McDonald. A study of global inference algorithms in multi-document summarization. pp. 557–564. 2007. ",
+        "bbox": [
+            173,
+            593,
+            823,
+            622
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Rada Mihalcea and Paul Tarau. Textrank: Bringing order into texts. Empirical Methods in Natural Language Processing, 2004. ",
+        "bbox": [
+            171,
+            631,
+            823,
+            660
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013. ",
+        "bbox": [
+            173,
+            667,
+            825,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ramesh Nallapati, Bowen Zhou, Cicero dos Santos, Caglar Gulcehre, and Bing Xiang. Abstractive text summarization using sequence-to-sequence rnns and beyond. The SIGNLL Conference on Computational Natural Language Learning, 2016a. ",
+        "bbox": [
+            173,
+            718,
+            825,
+            762
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ramesh Nallapati, Bowen Zhou, and Bing Xiang. Sequence-to-sequence rnns for text summarization. International Conference on Learning Representations, Workshop track, 2016b. ",
+        "bbox": [
+            173,
+            768,
+            823,
+            799
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Manjunath Kudlur Oriol Vinyals, Samy Bengio. Order matters: Sequence to sequence for sets. International Conference on Learning Representations, 2015. ",
+        "bbox": [
+            173,
+            806,
+            821,
+            837
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Daraksha Parveen, Hans-Martin Ramsl, and Michael Strube. Topical coherence for graph-based extractive summarization. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pp. 19491954, 2015. ",
+        "bbox": [
+            173,
+            843,
+            823,
+            887
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dragomir Radev and Gunes Erkan. Lexrank: Graph-based lexical centrality as salience in text ¨ summarization. Journal of Artificial Intelligence Research, pp. 457–479, 2004. ",
+        "bbox": [
+            173,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. arXiv preprint arXiv:1509.00685, 2015.   \nDou Shen, Jian-Tao Sun, Hua Li, Qiang Yang, and Zheng Chen. Document summarization using conditional random fields. In Proceedings of IJCAI, 2007.   \nRichard Socher, Eric H. Huang, Jeffrey Pennin, Christopher D. Manning, and Andrew Y. Ng. Dynamic pooling and unfolding recursive autoencoders for paraphrase detection. pp. 801–809. 2011.   \nKrysta M. Svore, Lucy Vanderwende, and Christopher J.C. Burges. Enhancing single-document summarization by combining ranknet and third-party sources. In Proceedings of the Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, pp. 448–457, 2007.   \nXiaojun Wan. Towards a unified approach to simultaneous single-document and multidocument summarizations. In In Proceedings of the 23rd COLING, pp. 11371145, 2010.   \nKristian Woodsend and Mirella Lapata. Automatic generation of story highlights. In In Proceedings of the 48th ACL, pp. 565574, 2010.   \nWenpeng Yin and Yulong Pei. Optimizing sentence modeling and selection for document summarization. In Proceedings of the 24th International Conference on Artificial Intelligence, pp. 1383–1389. AAAI Press, 2015.   \nMatthew D. Zeiler. ADADELTA: an adaptive learning rate method. CoRR, abs/1212.5701, 2012. URL http://arxiv.org/abs/1212.5701. ",
+        "bbox": [
+            171,
+            102,
+            826,
+            440
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            465,
+            294,
+            482
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will present some additional qualitative and quantitative analysis of our models that we hope will shed some light on their behavior. ",
+        "bbox": [
+            173,
+            497,
+            825,
+            526
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7.1 VISUALIZATION OF MODEL OUTPUT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            542,
+            472,
+            558
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In addition to being state-of-the-art performers, our models have the additional advantage of being very interpretable. The clearly separated terms in the scoring function (see Eqn. 1) allow us to tease out various factors responsible for the classification/selection of each sentence. This is illustrated in Figure 3, where we display a representative document from our validation set along with normalized scores from each abstract feature from the deep classifier model. Such visualization is especially useful in explaining to the end-user the decisions made by the system. ",
+        "bbox": [
+            174,
+            569,
+            825,
+            652
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7.2 LEARNED IMPORTANCE WEIGHTS",
+        "text_level": 1,
+        "bbox": [
+            178,
+            670,
+            454,
+            684
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We display in Table 4 the learned importance weights corresponding to various abstract features for deep sentence selector. Confirming our intuition, the model learns that salience and redundancy are the most important predictive features for summary membership of a sentence, followed by positional feature and content based feature. Further, when the same model is trained on documents with randomly shuffled sentences, it learns very small weight for the positional features, which is exactly what one expects. ",
+        "bbox": [
+            173,
+            695,
+            825,
+            780
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/e85a3606233ca3d28c839bfe465c4bd7505ccec1ff2fd0e5e9413c9161db2268.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Trainingcondition</td><td rowspan=1 colspan=1>Salience</td><td rowspan=1 colspan=1>Content</td><td rowspan=1 colspan=1>Position</td><td rowspan=1 colspan=1>Redundancy</td></tr><tr><td rowspan=1 colspan=1>Original dataShuffled data</td><td rowspan=1 colspan=1>42.759.69</td><td rowspan=1 colspan=1>14.832.85</td><td rowspan=1 colspan=1>-31.090.20</td><td rowspan=1 colspan=1>40.9916.08</td></tr></table>",
+        "bbox": [
+            264,
+            792,
+            730,
+            838
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Learned weights of various abstract features from the deep sentence selector model. Salience and redundancy are the most important features as learned by the model, followed by position and content. The negative sign for position weights has no particular significance. The positional feature gets very low weight when the document structure is destroyed by randomly shuffling sentences in each document the training data. ",
+        "bbox": [
+            173,
+            849,
+            826,
+            905
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/244190df16aecc023b9ca11dcff4d50d95b5a2c0d6ba85f9fb8f6245c7aeb136.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Gold Summary:Redpath has ended his eight-year association with SaleSharks. Redpath spent five years as a player and three as acoach at sale. He has thanked the owners, coaches and players for their support.</td><td rowspan=1 colspan=1>Salience</td><td rowspan=1 colspan=1>Content</td><td rowspan=1 colspan=1>Novelty</td><td rowspan=1 colspan=1>Position Prob.</td><td rowspan=1 colspan=1>Position Prob.</td></tr><tr><td rowspan=1 colspan=1>Bryan Redpath has left his coaching role at Sale Sharks withimmediate effect.</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.3</td></tr><tr><td rowspan=1 colspan=1>The 43- year- old Scot ends an eight-year association withthe Aviva Premiership side, having spent five years with them as a player and three as a coach.</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.7</td></tr><tr><td rowspan=1 colspan=1> Redpath returned to Sale in June 2012 as director of rugbyafter starting a coaching career at Gloucester and progressing to the top job at Kingsholm .</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.6</td></tr><tr><td rowspan=2 colspan=1>Redpath spent five years with Sale Sharks as a player and afurther three as a coach but with Sale Sharks struggling fourmonths into Redpath&#x27;s tenure, he was removed from thedirector of rugby role at the Salford-based side and has sincebeen operating as head coach .</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>0.9</td><td rowspan=2 colspan=1>0.7</td><td rowspan=2 colspan=1>0.8</td><td rowspan=2 colspan=1>0.9</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>‘I would like to thank the owners,coaches,players and stafffor all their helpand support since I returned to the club in2012.</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.7</td><td rowspan=2 colspan=1>0.2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1> Also to the supporters who have been great with me both asa player and as a coach,&#x27; Redpath said.</td><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>0.3</td><td rowspan=1 colspan=1>0.2</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr></table>",
+        "bbox": [
+            214,
+            300,
+            784,
+            598
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7.3 ABLATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            393,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the performance of the deep selector and deep classifier models on the validation set by deleting one abstract feature at a time from the model, with replacement. The performance numbers, displayed in Table 5, show that removing any of the features results in a small loss in performance. Note that the priority of features in the ablation experiments need not correspond to their priority in terms of learned weights in Table 4, since feature correlations may affect the two metrics differently. For the deep classifier, content and redundancy seem to matter the most while for the deep selector, dropping positional features hurts the most. Based on this analysis, we plan to investigate more thoroughly the reasons behind the poor ablation performance of salience and redundancy in the classifier and selector models respectively. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            256
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f81d97cdf9cc32027fac174a7a130fd80d78bbd4fe82de4fccf1182628f2d274.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Deep Classifier</td><td colspan=\"3\">Deep Selector</td></tr><tr><td>Features</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td><td>Rouge-1</td><td>Rouge-2</td><td>Rouge-L</td></tr><tr><td>All</td><td>42.43</td><td>17.32</td><td>34.07</td><td>41.55</td><td>16.52</td><td>32.41</td></tr><tr><td>-Salience</td><td>42.40</td><td>17.27</td><td>34.09</td><td>40.82</td><td>15.99</td><td>31.45</td></tr><tr><td>-Position</td><td>41.78</td><td>16.76</td><td>33.58</td><td>39.06</td><td>14.32</td><td>29.85</td></tr><tr><td>-Content</td><td>41.12</td><td>15.78</td><td>33.23</td><td>40.68</td><td>15.83</td><td>31.13</td></tr><tr><td>-Redundancy</td><td>41.67</td><td>16.86</td><td>32.93</td><td>41.46</td><td>16.50</td><td>32.31</td></tr></table>",
+        "bbox": [
+            215,
+            267,
+            782,
+            369
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Ablation experiments on the validation set to gauge the relative importance of each abstract feature. The top row is where all four abstract features are present. The following rows corresponding to removal of one feature at a time with replacement. Evaluation is done using Rouge limited length recall at 275 bytes. Bold faced entries correspond to largest reduction in performance when the corresponding features are dropped. ",
+        "bbox": [
+            173,
+            380,
+            825,
+            435
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7.4 REPRESENTATIVE DOCUMENTS AND EXTRACTIVE SUMMARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            653,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We display a couple of representative documents, one each from the Daily Mail and DUC corpora, highlighting the sentences chosen by deep classifier and comparing them with the gold summaries in Table 6. The examples demonstrate qualitatively that the model performs a reasonably good job in identifying the key messages of a document. ",
+        "bbox": [
+            174,
+            487,
+            825,
+            542
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/ee63e3b22c5dfda7cd16fe7bbf12cbd235e06179061d481426db0d9ce4fae6e1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Document:@entityO have an interest in @entity3 defender @entity2 but are unlikely to make a move until january . the 00 - year - old @entity6 captain has yet to open talks over a new contract at @entity3 and his current deal runs out in 0000 .@entity3 defender @entity2 could be targeted by @entityO in the january transfer window @entityO like @entity2 but do n&#x27;t expect @ entity3 to sell yet they know he will be free to talk to foreign clubs from january．@entity12 will make a Omillion offer for @entity3 goalkeeper @entity14 this summer．the OO - year - old is poised to leave @entityl6 and wants to play for a @entity18 contender . @entity12 are set to make a Omillion bid for @entity2’s @en-</td></tr><tr><td>tity3 team - mate @entity14 in the summer Gold Summary:@entity2&#x27;s contract at @entity3 expires at the end of next season .OO - year-old has yet to open talks over a new deal at @entityl6 ．@entity14 is poised to leave @entity3 at the end of</td></tr><tr><td>the season Document: today, the foreign ministry said that control opera- tions carried out by the corvette spiro against a korean-flagged asreceived ship fishing illegally in argentine waters were car- ried out “ in accordance with international law and in coordi- nation with the foreign ministry ”.the foreign ministry thus ap- proved the intervention by the argentine corvette when it discovered the korean ship chin yuan hsing violating argentine jurisdictional waters on OO may.... the korean ship，which had been fishing illegally in argentine waters，was sunk by its own crew after failing to answer to the argentine ship &#x27;s warnings .the crew was transferred to the chin chuan hsing，which was sailing nearby and</td></tr><tr><td>approached to rescue the crew of the sinking ship... Gold Summary:the korean-flagged fishing vessel chin yuan hs- ing was scuttled in waters off argentina on OO may OOoO ．adverse weather conditions prevailed when the argentine corvette spiro spot- ted the korean ship fishing illegally in restricted argentine waters . the korean vessel did not respond to the corvette &#x27;s warning .instead ,the korean crew sank their ship,and transferred to another korean ship sailing nearby.in accordance with a uk-argentine agreement, the argentine navy turned the surveillance of the second korean ves- sel over to the british when it approached within OO nautical miles of the malvinas（falkland）islands .</td></tr></table>",
+        "bbox": [
+            289,
+            248,
+            707,
+            733
+        ],
+        "page_idx": 12
+    }
+]

parse/train/_IY3_4psXuf/_IY3_4psXuf.md

@@ -0,0 +1,307 @@
+# Decoupling the Depth and Scope of Graph Neural Networks
+
+Hanqing Zeng USC zengh@usc.edu
+
+Muhan Zhang Peking University, BIGAI muhan@pku.edu.cn
+
+Yinglong Xia Facebook AI yxia@fb.com
+
+Ajitesh Srivastava USC ajiteshs@usc.edu
+
+# Rajgopal Kannan
+
+Andrey Malevich Facebook AI amalevich@fb.com
+
+US ARL
+
+Viktor Prasanna USC prasanna@usc.edu
+
+rajgopal.kannan.civ@mail.mil
+
+Long Jin   
+Facebook AI   
+longjin@fb.com   
+Ren Chen   
+Facebook AI   
+renchen@fb.com
+
+# Abstract
+
+State-of-the-art Graph Neural Networks (GNNs) have limited scalability with respect to the graph and model sizes. On large graphs, increasing the model depth often means exponential expansion of the scope (i.e., receptive field). Beyond just a few layers, two fundamental challenges emerge: 1. degraded expressivity due to oversmoothing, and 2. expensive computation due to neighborhood explosion. We propose a design principle to decouple the depth and scope of GNNs – to generate representation of a target entity (i.e., a node or an edge), we first extract a localized subgraph as the bounded-size scope, and then apply a GNN of arbitrary depth on top of the subgraph. A properly extracted subgraph consists of a small number of critical neighbors, while excluding irrelevant ones. The GNN, no matter how deep it is, smooths the local neighborhood into informative representation rather than oversmoothing the global graph into “white noise”. Theoretically, decoupling improves the GNN expressive power from the perspectives of graph signal processing (GCN), function approximation (GraphSAGE) and topological learning (GIN). Empirically, on seven graphs (with up to 110M nodes) and six backbone GNN architectures, our design achieves significant accuracy improvement with orders of magnitude reduction in computation and hardware cost.
+
+# 1 Introduction
+
+Graph Neural Networks (GNNs) have now become the state-of-the-art models for graph mining [48, 13, 58], facilitating applications such as social recommendation [35, 52, 37], knowledge understanding [40, 38, 59] and drug discovery [43, 32]. With the numerous architectures proposed [22, 12, 44, 49], it still remains an open question how to effectively scale up GNNs with respect to both the model size and graph size. There are two fundamental obstacles when we increase the number of GNN layers:
+
+• Expressivity challenge (i.e., oversmoothing [30, 36, 39, 17]): iterative mixing of neighbor features collapses embedding vectors of different nodes into a fixed, low-dimensional subspace.
+
+• Scalability challenge (i.e., neighbor explosion [7, 8, 9, 55]): recursive expansion of multi-hop neighborhood results in exponentially growing receptive field size (and thus computation cost).
+
+To address the expressivity challenge, most remedies focus on neural architecture exploration: [44, 12, 49, 29] propose more expressive aggregation functions when propagating neighbor features. [50, 28, 18, 34, 1, 33, 31] use residue-style design components to construct flexible and dynamic receptive fields. Among them, [50, 28, 18] use skip-connection across multiple GNN layers, and [34, 1, 33, 31] encourage multi-hop message passing within each single layer. As for the scalability challenge, sampling methods have been explored to improve the training speed and efficiency. Importance based layer-wise sampling [8, 7, 61] and subgraph-based sampling [54, 9, 55] alleviate neighbor explosion, while preserving training accuracy. Unfortunately, such sampling methods cannot be naturally generalized to inference without accuracy loss (see also Section 4).
+
+The above lines of research have only guided us to partial solutions. Yet what is the root cause of both the expressivity and scalability challenges? Setting aside the design of GNN architectures or sampling schemes, we provide an alternative perspective by interpretting the data in a different way.
+
+Two views on the graph. Given an input graph $\mathcal { G }$ with node set $\nu$ , the most straightforward way to understand $\mathcal { G }$ is by viewing it as a single global graph. So any two nodes $u$ and $v$ belong to the same $\mathcal { G }$ , and if $u$ and $v$ lie in the same connected component, they will ultimately see each other in their own neighborhood no matter how far away $u$ and $v$ are. Alternative to the above global view, we can take a local view on $\mathcal { G }$ . For each node $v$ , there is a latent $\mathcal { G } _ { [ v ] }$ surrounding it which captures the characteristics of just $v$ itself. The full $\mathcal { G }$ is observed (by the data collection process) as the union of all such $\mathcal { G } _ { [ v ] }$ . Consequently, $\nu _ { [ v ] }$ rather than $\nu$ defines $v$ ’s neighborhood: if $u \notin \mathcal { V } _ { [ v ] }$ , $v$ will never consider $u$ as a neighbor. Our “decoupling” design is based on the local view.
+
+Scope of GNNs. Both the expressivity and scalability challenges are closely related to the enlargement of the GNN’s scope (i.e., receptive field). More importantly, how we define the scope is determined by how we view $\mathcal { G }$ . With the global view above, an $L$ -layer GNN has the scope of the full $L$ -hop neighborhood. With the local view, the GNN scope is simply $\nu _ { [ v ] }$ regardless of the GNN depth. The two existing lines of research, one on architectural exploration and the other on sampling, both take the global view. Consequently, the depth (i.e., number of layers) and scope of such GNNs are tightly coupled. Such coupling significantly limits the design space exploration of GNNs with various depths [53]. Consider the example of ogbn-products, a medium-scale graph in Open Graph Benchmark [16]. The average number of 4-hop neighbors is around $0 . 6 \mathbf { M }$ , corresponding to $2 5 \%$ of the full graph size. To generate representation of a single target node, a 4-layer coupled GNN needs to propagate features from the $0 . 6 \mathbf { M }$ neighbors. Such propagation can be inefficient or even harmful since most nodes in the huge neighborhood would be barely relevant to the target node.
+
+Decoupling the GNN depth and scope. Taking the local view on $\mathcal { G }$ , we propose a general design principle to decouple the GNN depth and scope. To generate the representation of the target node $v$ , we first extract from $\mathcal { G }$ a small subgraph $\mathcal { G } _ { [ v ] }$ surrounding $v$ . On top of $\mathcal { G } _ { [ v ] }$ , we apply a GNN whose number of layers and message passing functions can be flexibly chosen. “Decoupling” means we treat the scope extraction function and GNN depth as two independently tuned parameters – effectively we introduce a new dimension in the GNN design space. We intuitively illustrate the benefits of decoupling by an example GNN construction, where the scope is the $L$ -hop neighborhood and depth is $L ^ { \prime }$ $L ^ { \prime } > L  )$ ). When we use more layers $( L ^ { \prime } )$ than hops $( L )$ , each pair of subgraph nodes may exchange messages multiple times. The extra message passing helps the GNN better absorb and embed the information within scope, and thus leads to higher expressivity. We further justify the above intuition with multifaceted theoretical analysis. From the graph signal processing perspective, we prove that decoupled-GCN performs local-smoothing rather than oversmoothing, as long as the scopes of different target nodes are different. From the function approximation perspective, we construct a linear target function on neighbor features and show that decoupling the GraphSAGE model reduces the function approximation error. From the topological learning perspective, we apply deep GIN-style message passing to differentiate non-regular subgraphs of a regular graph. As a result, our model is more powerful than the 1-dimensional Weisfeiler-Lehman test [41].
+
+Practical implementation: SHADOW-GNN. The decoupling principle leads to a practical implementation, SHADOW-GNN: Decoupled GNN on a shallow subgraph. In SHADOW-GNN, the scope is a shallow yet informative subgraph, only containing a fraction of the 2- or 3-hop neighbors of $\mathcal { G }$ (see Section 5). On the other hand, the model of SHADOW-GNN is deeper (e.g., $L ^ { \prime } = 5$ ). To efficiently construct the shallow scope on commodity hardware, we propose various subgraph extraction functions. To better utilize the subgraph node embeddings after deep message passing, we propose neural architecture extensions such as pooling and ensemble. Empirically, our “decoupling” design improves both the accuracy and scalability. On seven benchmarks (including the largest ogbn-papers100M graph with 111M nodes) and across two graph learning tasks, SHADOW-GNNs achieve significant accuracy gains compared to the original models. Meanwhile, the computation and hardware costs are reduced by orders of magnitude. Our code is available at https://github.com/facebookresearch/shaDow_GNN
+
+# 2 Preliminaries
+
+Let $\mathcal { G } \left( \mathcal { V } , \mathcal { E } , \boldsymbol { X } \right)$ be an undirected graph, with node set $\nu$ , edge set $\mathcal { E } \subseteq \mathcal { V } \times \mathcal { V }$ and node feature matrix $\pmb { X } \in \mathbb { R } ^ { | \mathcal { V } | \times d }$ . Let $\mathcal { N } _ { v }$ denote the set of $v$ ’s direct neighbors in $\mathcal { G }$ . The $u ^ { \mathrm { t h } }$ row of $\boldsymbol { X }$ corresponds to the length- $d$ feature of node $u$ . Let $\pmb { A }$ be the adjacency matrix of $\mathcal { G }$ where $A _ { u , v } = 1$ if edge $( u , v ) \in \mathcal { E }$ and $A _ { u , v } = 0$ otherwise. Let $_ { D }$ be the diagonal degree matrix of $\pmb { A }$ . Denote $\widetilde { A } = D _ { * } ^ { - \frac 1 2 } A _ { * } D _ { * } ^ { - \frac 1 2 }$ as the adjacency matrix after symmetric normalization ( $\cdot _ { * } , *$ means augmented with self-edges), and $\stackrel { \frown } { A } = D ^ { \stackrel { - 1 } { - } } A$ (or $D _ { * } ^ { - 1 } A _ { * , }$ ) as the one after random walk normalization. Let subscript “ $[ u ] ^ { \dag }$ mark the quantities corresponding to a subgraph surrounding node $u$ . For example, the subgraph itself is $\mathcal { G } _ { [ u ] }$ . Subgraph matrices $X _ { [ v ] }$ and $\boldsymbol { A } _ { [ v ] }$ have the same dimension as the original $\boldsymbol { X }$ and $\pmb { A }$ . Yet, row vector $\left[ \boldsymbol { X } _ { [ v ] } \right] _ { u } = \mathbf { 0 }$ for $u \notin \mathcal { V } _ { [ v ] }$ . Element $\left[ { \pmb A } _ { [ v ] } \right] _ { u , w } = 0$ if either $u \notin \mathcal { V } _ { [ v ] }$ or $w \notin \mathcal { V } _ { [ v ] }$ For an $L$ -layer GNN, let superscript “ $( \ell ) ^ { \dag }$ denote the layer- $\ell$ quantities. Let $d ^ { ( \ell ) }$ be the number of channels for layer $\ell$ ; $\pmb { H } ^ { ( \ell - 1 ) } \in \mathbb { R } ^ { | \mathcal { V } | \times d ^ { ( \ell - 1 ) } }$ and $H ^ { ( \ell ) } \in \mathbb { R } ^ { | \mathcal { V } | \times d ^ { ( \ell ) } }$ be the input and output features. So $H ^ { ( 0 ) } = X$ and $d ^ { ( 0 ) } = d$ . Further, let $\sigma$ be the activation and $\mathbf { \boldsymbol { W } } ^ { ( \ell ) }$ be the learnable weight. For example, a GCN layer performs $\pmb { H } ^ { ( \ell ) } = \sigma \left( \widetilde { A } \pmb { H } ^ { ( \ell - 1 ) } \pmb { W } ^ { ( \ell ) } \right)$ . A GraphSAGE layer performs $\begin{array} { r } { \pmb { H } ^ { ( \ell ) } = \sigma \left( \pmb { H } ^ { ( \ell - 1 ) } \pmb { W } _ { 1 } ^ { ( \ell ) } + \widehat { \pmb { A } } \pmb { H } ^ { ( \ell - 1 ) } \pmb { W } _ { 2 } ^ { ( \ell ) } \right) . } \end{array}$
+
+Our analysis in Section 3 mostly focuses on the node classification task. Yet our design can be generalized to the link prediction task, as demonstrated by our experiments in Section 5.
+
+Definition 2.1. (Depth of subgraph) Assume the subgraph $\mathcal { G } _ { [ v ] }$ is connected. The depth of $\mathcal { G } _ { [ v ] }$ is defined as $\mathrm { m a x } _ { u \in \mathcal { V } _ { [ v ] } } d ( u , v ) ;$ , where $d ( u , v )$ denotes the shortest path distance from u to v.
+
+The above definition enables us to make comparison such as “the GNN is deeper than the subgraph”.   
+For “decoupling the depth and scope”, we refer to the model depth rather than the subgraph depth.
+
+# 3 Decoupling the Depth and Scope of GNNs
+
+“Decoupling the depth and scope of GNNs” is a design principle to improve the expressivity and scalability of GNNs without modifying the layer architecture. We name a GNN after decoupling a SHADOW-GNN (see Section 3.6 for explanation of the name). Compared with a normal GNN, SHADOW-GNN contains an additional component: the subgraph extractor EXTRACT. To generate embedding of a target node $v$ , SHADOW-GNN proceeds as follows: 1. We use EXTRACT $( v , { \mathcal { G } } )$ to return a connected $\mathcal { G } _ { [ v ] }$ , where $\mathcal { G } _ { [ v ] }$ is a subgraph containing $v$ , and the depth of $\mathcal { G } _ { [ v ] }$ is $L$ . 2. We build an $L ^ { \prime }$ -layer GNN on $\mathcal { G } _ { [ v ] }$ by treating $\mathcal { G } _ { [ v ] }$ as the new full graph and by ignoring all nodes / edges not in $\mathcal { G } _ { [ v ] }$ . So $\mathcal { G } _ { [ v ] }$ is the scope of SHADOW-GNN. The key point reflecting “decoupling” is that $L ^ { \prime } > L$
+
+A normal GNN is closely related to a SHADOW-GNN. Under the normal setup, an $L$ -layer GNN operates on the full $\mathcal { G }$ and propagates the influence from all the neighbors up to $L$ hops away from $v$ Such a GNN is equivalent to a model where EXTRACT returns the full $L$ -hop subgraph and $L ^ { \prime } = L$ .
+
+We theoretical demonstrate how SHADOW-GNN improves expressivity from three different angles. On SHADOW-GCN (Section 3.1), we come from the graph signal processing perspective. The GCN propagation can be interpreted as applying filtering on the node signals [47]. Deep models correspond to high-pass filters. Filtering the local graph $\mathcal { G } _ { [ v ] }$ preserves richer information than the global $\mathcal { G }$ . On SHADOW-SAGE (Section 3.2), we view the GNN as a function approximator. We construct a target function and study how decoupling reduces the approximation error. On SHADOW-GIN (Section 3.3), we focus on learning topological information. We show that decoupling helps capture local graph structure which the 1D Weisfeiler-Lehman test fails to capture.
+
+# 3.1 Expressivity Analysis on SHADOW-GCN: Graph Signal Processing Perspective
+
+GCNs [22] suffer from “oversmoothing” [30] – Each GCN layer smooths the features of the direct (i.e., 1-hop) neighbors, and many GCN layers smooths the features of the full graph. Eventually, such repeated smoothing process propagates to any target node just the averaged feature of all $\nu$ . “Oversmoothing” thus incurs significant information loss by wiping out all local information.
+
+Formally, suppose the original features $\boldsymbol { X }$ reside in a high-dimensional space $\mathbb { R } ^ { | \nu | \times d }$ . Oversmoothing pushes $\boldsymbol { X }$ towards a low-dimensional subspace $\mathbb { R } ^ { | \nu | \times d ^ { \prime } }$ , where $d ^ { \prime } < d$ . Corresponding analysis comes from two perspectives: oversmoothing by a deep $G C N$ , and oversmoothing by repeated GCNstyle propagation. The former considers the full neural network with non-linear activation, weight and bias. The later characterizes the aggregation matrix $M = \operatorname* { l i m } _ { L \to \infty } \widetilde { A } ^ { L } X$ . It is shown that even with the vanilla architecture, a deep GCN with bias parameters does not oversmooth [17]. In addition, various tricks [60, 39, 34] can prevent oversmoothing from the neural network perspective. However, a deep GCN still suffers from accuracy drop, indicating that the GCN-style propagation (rather than other GCN components like activation and bias) may be the fundamental reason causing difficulty in learning. Therefore, we study the asymptotic behavior of the aggregation matrix $M$ under the normal and SHADOW design. In other words, here in Section 3.1, we ignore the non-linear activation and bias parameters. Such setup is consistent with many existing literature such as [30, 33, 34, 60].
+
+Proposition 3.1. $\infty$ number of feature propagation by SHADOW-GCN leads to
+
+$$
+\pmb { m } _ { [ v ] } = \left[ e _ { [ v ] } \right] _ { v } \cdot \left( e _ { [ v ] } ^ { \top } \pmb { X } _ { [ v ] } \right)
+$$
+
+where $e _ { [ v ] }$ is defined by $\begin{array} { r } { \left[ e _ { [ v ] } \right] _ { u } = \sqrt { \frac { \delta _ { [ v ] } ( u ) } { \sum _ { w \in \mathcal { V } _ { [ v ] } } \delta _ { [ v ] } ( w ) } } } \end{array}$ ; $\delta _ { [ v ] } \left( u \right)$ returns the degree of u in $\mathcal { G } _ { [ v ] } p l u s ~ l$
+
+Oversmoothing by normal GCN propagation. With a large enough $L$ , the full $L$ -hop neighborhood becomes $\nu$ (assuming connected $\mathcal { G }$ ). So $\forall ~ u , v$ , we have $\mathcal { G } _ { [ u ] } = \mathcal { G } _ { [ v ] } = \mathcal { G }$ , implying $e _ { [ u ] } = e _ { [ v ] }$ and $X _ { [ u ] } = X _ { [ v ] } = X$ . From Proposition 3.1, the aggregation converges to a point where $n o$ feature and little structural information of the target is preserved. The only information in $m _ { [ v ] }$ is $v$ ’s degree.
+
+Local-smoothing by SHADOW-GCN propagation. With a fixed subgraph, no matter how many times we aggregate using $\widetilde { A } _ { [ v ] }$ , the layers will not include the faraway irrelevant nodes. From Proposition 3.1, $m _ { [ v ] }$ is a linear combination of the neighbor features $X _ { [ v ] }$ . Increasing the number of layers only pushes the coefficients of each neighbor features to the stationary values. The domain $X _ { [ v ] }$ of such linear transformation is solely determined by EXTRACT and is independent of the model depth. Intuitively, if EXTRACT picks non-identical subgraphs for two nodes $u$ and $v$ , the aggregations should be different due to the different domains of the linear transformation. Therefore, SHADOW-GCN preserves local feature information whereas normal GCN preserves none. For structural information in $m _ { [ v ] }$ , note that $e _ { [ v ] }$ is a normalized degree distribution of the subgraph around $v$ , and -e[v]v indicates the role of the target node in the subgraph. By simply letting EXTRACT return the 1-hop subgraph, $\left[ e _ { [ v ] } \right] _ { v }$ alone already contains all the information preserved by a normal GCN, which is $v$ ’s degree in $\bar { \mathcal { G } }$ . For the general EXTRACT, $e _ { [ v ] }$ additionally reflects $v$ ’s ego-net structure. Thus, a deep SHADOW-GCN preserves more structural information than a deep GCN.
+
+Theorem 3.2. Let $\overline { { \pmb { m } } } _ { [ v ] } = \phi _ { \mathcal { G } } \left( v \right) \cdot \pmb { m } _ { [ v ] }$ where $\phi _ { \mathcal { G } }$ is any non-zero function only depending on the structural property of $v$ . Let $\mathcal { M } = \mathrm { \large ~ \left\{ ~ \overline { { ~ } } m _ { [ v ] } ~ \lvert ~ v ~ \in ~ \mathcal { V } \right\} ~ }$ . Given $\mathcal { G }$ , EXTRACT and some continuous probability distribution in $\mathbb { R } ^ { | \nu | \times d }$ to generate $\boldsymbol { X }$ , then $\overline { { \pmb { m } } } _ { [ v ] } \neq \overline { { \pmb { m } } } _ { [ u ] }$ if $\mathcal { V } _ { [ u ] } \neq \mathcal { V } _ { [ v ] }$ , almost surely.
+
+Corollary 3.2.1. Consider EXTRA $C T _ { 1 }$ , where $\forall v \in \mathcal { V }$ , $| \nu _ { [ v ] } | \leq n$ . Then $\begin{array} { r } { | \mathcal { M } | \geq \left\lceil \frac { | \mathcal { V } | } { n } \right\rceil } \end{array}$ a.s.
+
+Corollary 3.2.2. Consider EXTRA $C T _ { 2 }$ , where $\forall u , v \in \mathcal { V }$ , $\mathcal { V } _ { [ v ] } \neq \mathcal { V } _ { [ u ] }$ . Then $| \mathcal { M } | = | \mathcal { V } | a . s$
+
+Theorem 3.2 proves SHADOW-GCN does not oversmooth: 1. A normal GCN pushes the aggregation of same-degree nodes to the same point, while SHADOW-GCN with $\mathtt { E X T R A C T } _ { 2 }$ ensures any two nodes (even with the same degree) have different aggregation. 2. A normal GCN wipes out all information in $\boldsymbol { X }$ after many times of aggregation, while SHADOW-GCN always preserves feature information. Particularly, with φG (v) = δ[v](v)
−1/2, a normal GCN generates only one unique value of for all $v$ . By contrast, SHADOW-GNN generates $| \nu |$ different values for any $\phi _ { \mathcal { G } }$ function.
+
+We compare the expressivity by showing 1. SHADOW-SAGE can express all functions GraphSAGE can, and 2. SHADOW-SAGE can express some function GraphSAGE cannot. Recall, a GraphSAGE layer performs the following: $\begin{array} { r } { \dot { \overline { { h _ { v } ^ { ( \ell ) } } } } = \sigma \left( \left( W _ { 1 } ^ { ( \ell ) } \right) ^ { \top } h _ { v } ^ { ( \ell - 1 ) } + \left( W _ { 2 } ^ { ( \ell ) } \right) ^ { \top } \left( \frac { 1 } { | \mathcal { N } _ { v } | } \sum _ { u \in \mathcal { N } _ { v } } \dot { h _ { u } ^ { ( \ell - 1 ) } } \right) \right) . } \end{array}$ We can prove Point 1 by making an $L ^ { \prime }$ -layer SHADOW-SAGE identical to an $L$ -layer GraphSAGE with the following steps: 1. let EXTRACT return the full $L$ -hop neighborhood, and 2. set $\bar { \boldsymbol { W } } _ { 1 } ^ { ( \ell ) } = \boldsymbol { I }$ , $W _ { 2 } ^ { ( \ell ) } = { \bf 0 }$ for $L + 1 \leq \ell \leq L ^ { \prime }$ . For point 2, we consider a target function: $\tau \left( \boldsymbol { X } , \boldsymbol { \mathcal { G } } _ { [ v ] } \right) =$ $\begin{array} { r } { C \cdot \sum _ { u \in \mathcal { V } _ { [ v ] } } \delta _ { [ v ] } \left( u \right) \cdot \pmb { x } _ { u } } \end{array}$ for some neighborhood $\mathcal { G } _ { [ v ] }$ , scaling constant $C$ and $\delta _ { [ v ] } \left( u \right)$ as defined in Proposition 3.1. An expressive model should be able to learn well this simple linear function $\tau$ .
+
+GraphSAGE cannot learn $\tau$ accurately, while SHADOW-SAGE can. We first show the GraphSAGE case. Let the depth of $\mathcal { G } _ { [ v ] }$ be $L$ . Firstly, we need GraphSAGE to perform message passing for exactly $L$ times (where such a model can be implemented by, e.g., $L$ layers or $L ^ { \prime }$ layers with $W _ { 2 } = \mathbf { 0 }$ for $L ^ { \prime } - L$ layers). Otherwise, the extra $L ^ { \prime } - L$ message passings will propagate influence from nodes $v ^ { \prime } \notin \mathcal { V } _ { [ v ] }$ , violating the condition that $\tau$ is independent of $v ^ { \prime }$ . Next, suppose GraphSAGE can learn a function $\zeta$ such that on some $\mathcal { G } _ { [ v ] } ^ { \prime }$ , we have $\zeta \left( \mathcal { G } _ { [ v ] } ^ { \prime } \right) = \tau \left( \mathcal { G } _ { [ v ] } ^ { \prime } \right)$ . We construct another $\mathcal { G } _ { [ v ] } ^ { \prime \prime }$ by adding an extra edge $e$ connecting two depth- $L$ nodes in $\mathcal { G } _ { [ v ] } ^ { \prime }$ . Edge $e$ changes the degree distribution $\delta _ { [ v ] } \left( \cdot \right)$ , and thus $\tau \left( \mathcal { G } _ { [ v ] } ^ { \prime } \right) \neq \tau \left( \mathcal { G } _ { [ v ] } ^ { \prime \prime } \right)$ . On the other hand, there is no way for GraphSAGE to propagate the influence of edge $e$ to the target $v$ , unless the model performs at least $L + 1$ message passings. So $\zeta \left( \mathcal { G } _ { [ v ] } ^ { \prime } \right) = \zeta \left( \bar { \mathcal { G } } _ { [ v ] } ^ { \prime \prime } \right)$ regardless of the activation function and weight parameters. Therefore, $\zeta \neq \tau$ . For SHADOW-SAGE, let EXTRACT return $\mathcal { G } _ { [ v ] }$ . Then the model can output $\zeta ^ { \prime } = \left[ \widehat { \cal A } _ { [ v ] } ^ { L ^ { \prime } } { \cal X } \right] _ { v , }$ after we 1. set $W _ { 1 } ^ { ( \ell ) } = \mathbf { 0 }$ and $W _ { 2 } ^ { ( \ell ) } = I$ for all layers, and 2. either remove the non-linear activation or bypass ReLU by shifting $\boldsymbol { X }$ with bias. With known results in Markov chain convergence theorem [27], we derive the following theorem by analyzing the convergence of $\widehat { A } _ { [ v ] } ^ { L ^ { \prime } }$ when $L ^ { \prime } \to \infty$ .
+
+Theorem 3.3. SHADOW-SAGE can approximate $\tau$ with error decaying exponentially with depth.
+
+We have the following conclusions from above: 1. SHADOW-SAGE is more expressive than GraphSAGE. 2. appropriate EXTRACT function improves SHADOW-GNN expressivity, 3. There exists cases where it may be desirable to set the SHADOW-GNN depth much larger than the subgraph depth.
+
+# 3.3 Expressivity Analysis on SHADOW-GIN: Topological Learning Perspective
+
+While GCN and GraphSAGE are popular architectures in practice, they are not the theoretically most discriminative ones. The work in [49] establishes the relation in discriminativeness between GNNs and 1-dimensional Weisfeiler-Lehman test (i.e., 1-WL). And GIN [49] is an example architecture achieving the same discriminativeness as 1-WL. We show that applying the decoupling principle can further improve the discriminativeness of such GNNs, making them more powerful than 1-WL.
+
+1-WL is a graph isomorphism test aiming at distinguishing graphs of different structures. A GNN as expressive as 1-WL thus well captures the topological property of the target node. While 1-WL is already very powerful, it may still fail in some cases. e.g., it cannot distinguish certain non-isomorphic, regular graphs. To understand why SHADOW-GNN works, we first need to un
+
+![](images/0fd819a2beeeba3fca87a30aa3012f69f2b800a51d7754d5d403f4ebe7ed94bb.jpg)  
+Figure 1: Example 3-regular graph and the 1-hop subgraphs of the target nodes $u$ and $v$ .
+
+derstand why 1-WL fails. In a regular graph, all nodes have the same degree, and thus the “regular” property describes a global topological symmetry among nodes. Unfortunately, 1-WL (and the corresponding normal GNN) also operates globally on $\mathcal { G }$ . Intuitively, on two different regular graphs, there is no way for 1-WL (and the normal GNN) to assign different labels by breaking such symmetry.
+
+On the other hand, SHADOW-GNN can break such symmetry by applying decoupling. In Section 1, we have discussed how SHADOW-GNN is built from the local perspective on the full graph. The key property benefiting SHADOW-GNN is that a subgraph of a regular graph may not be regular. Thus, SHADOW-GNN can distinguish nodes in a regular graph with the non-regular subgraphs as the scope. We illustrate the intuition with the example in Figure 1. The graph $\mathcal { G }$ is 3-regular and we assume all nodes have identical features. Our goal is to discriminate nodes $u$ and $v$ since their neighborhood structures are different. No matter how many iterations 1-WL runs, or how many layers the normal GNN has, they cannot distinguish $u$ and $v$ . On the other hand, a SHADOW-GNN with 1-hop EXTRACT and at least 2 layers can discriminate $u$ and $v$ .
+
+Theorem 3.4. Consider GNNs whose layer function is defined by
+
+$$
+\pmb { h } _ { v } ^ { ( \ell ) } = f _ { 1 } ^ { ( \ell ) } \left( \pmb { h } _ { v } ^ { ( \ell - 1 ) } , \ \sum _ { u \in \mathcal { N } _ { v } } f _ { 2 } ^ { ( \ell ) } \left( \pmb { h } _ { v } ^ { ( \ell - 1 ) } , \pmb { h } _ { u } ^ { ( \ell - 1 ) } \right) \right) ,
+$$
+
+where $f _ { 1 } ^ { ( \ell ) }$ and $f _ { 2 } ^ { ( \ell ) }$ are the update and message functions of layer- $\ell ,$ , implemented as MLPs. Then, such SHADOW-GNN is more discriminative than the -dimensional Weisfeiler-Lehman test.
+
+The theorem also implies that SHADOW-GIN is more discriminative than a normal GIN due to the correspondence between GIN and 1-WL. See Appendix A for the proof of all theorems in Section 3.
+
+# 3.4 Subgraph Extraction Algorithms
+
+Our decoupling principle does not rely on specific subgraph extraction algorithms. Appropriate EXTRACT can be customized given the characteristic of $\mathcal { G }$ , and different EXTRACT leads to different implementation of our decoupling principle. In general, we summarize three approaches to design EXTRACT: 1. heuristic based, where we pick graph metrics that reflect neighbor importance and then design EXTRACT by such metrics; 2. model based, where we assume a generation process on $\mathcal { G }$ and set EXTRACT as the reverse process, and 3. learning based, where we integrate the design of EXTRACT as part of the GNN training. In the following, we present several examples on heuristic based EXTRACT, which we also empirically evaluate in Section 5. We leave detailed evaluation on the model based and learning based EXTRACT as future work. See also Appendix C for details.
+
+Example heuristic based EXTRACT. The algorithm is derived from the selected graph metrics. For example, with the metric being shortest path distance, we design a $L$ -hop extractor. i.e., EXTRACT returns the full set or randomly selected subset of the target node’s $L$ -hop neighbors in $\mathcal { G }$ . Picking the random walk landing probability as the metric, we can design a PPR-based extractor. i.e., we first run the Personalized PageRank (PPR) algorithm on $\mathcal { G }$ to derive the PPR score of other nodes relative to the target node. Then EXTRACT define $\nu _ { [ v ] }$ by picking the top- $K$ nodes with the highest PPR scores. The subgraph $\mathcal { G } _ { [ v ] }$ is the node-induced subgraph1 of $\mathcal { G }$ from $\nu _ { [ v ] }$ . One can easily extend this approach by using other metrics such as Katz index [20], SimRank [19] and feature similarity.
+
+# 3.5 Architecture
+
+Subgraph pooling. For a normal GNN performing node classification, the multi-layer message passing follows a “tree structure”. The nodes at level $L$ of the tree correspond to the $L$ -hop neighborhood. And the tree root outputs the final embedding of the target node. Thus, there is no way to apply subgraph pooling or READOUT on the final layer output, since the “pool” only contains a single vector. For a SHADOW-GNN, since we decouple the $\bar { L } ^ { \mathrm { t h } }$ layer from the $L$ -hop neighborhood, it is natural to let each layer (including the final layer) output embeddings for all subgraph nodes. This leads to the design to READOUT all the subgraph node embeddings as the target node embedding.
+
+We can understand the pooling for SHADOW-GNN from another perspective. In a normal GNN, the target node at the final layer receives messages from all neighbors, but two neighbor nodes may not have a chance to exchange any message to each other (e.g., two nodes $L$ -hop away from the target may be $2 L$ -hop away from each other). In our design, a SHADOW-GNN can pass messages between any pair of neighbors when the model depth is large enough. Therefore, all the subgraph node embeddings at the final layer capture meaningful information of the neighborhood.
+
+In summary, the power of the decoupling principle lies in that it establishes the connection between the node- / link-level task and the graph-level task. e.g., to classify a node is seen as to classify the subgraph surrounding the node. From the neural architecture perspective, we can apply any subgraph pooling / READOUT operation originally designed for graph classification (e.g., [57, 24, 4]) to enhance the node classification / link prediction of SHADOW-GNN. In particular, in the vanilla SHADOW-GNN, we can implement a trivial READOUT as “discarding all neighbor embeddings”, corresponding to performing center pooling. See Appendix D and F.3 for algorithm and experiments.
+
+Subgraph ensemble. It may be challenging in practice to design a single EXTRACT capturing all meaningful characteristics of the neighborhood. We can use multiple EXTRACT to jointly define the receptive field, and then ensemble multiple SHADOW-GNN at the subgraph level. Consider $R$ candidates $\{ \mathtt { E X T R A C T } _ { i } \}$ , each returning $\mathcal { G } _ { [ v ] } ^ { i }$ . To generate $v$ ’s embedding, we first use $R$ branches of $L ^ { \prime }$ -layer GNN to obtain intermediate embeddings for each $\mathcal { G } _ { v } ^ { i }$ , and then aggregate the $R$ embeddings by some learnable function $g$ . In practice, we design $g$ as an attention based aggregation function (see Appendix D.2). Subgraph ensemble is useful both when $\{ \mathtt { E X T R A C T } _ { i } \}$ consists of different algorithms and when each EXTRA $\Omega  { \mathrm { T } } _ { i }$ performs the same algorithm under different parameters.
+
+CASE STUDY Consider PPR-based $\mathtt { E X T R A C T } _ { i }$ with different threshold $\theta _ { i }$ on the neighbor PPR score. A SHADOW-GNN-ensemble can approximate PPRGo [6]. PPRGo generates embedding as: ${ \pmb { \xi } } _ { v } =$ $\sum _ { u \in \mathcal { V } _ { [ v ] } } \pi _ { u } h _ { v }$ , where $\pi _ { u }$ is $u$ ’s PPR score and $\begin{array} { r } { h _ { v } = \mathtt { M L P } \left( \pmb { x } _ { v } \right) } \end{array}$ . We can partition $\begin{array} { r } { \mathcal { V } _ { [ v ] } = \bigcup _ { i = 1 } ^ { R } \mathcal { V } _ { [ v ] } ^ { i } } \end{array}$ s.t. nodes in $\mathcal { V } _ { [ v ] } ^ { i }$ have similar PPR scores denoted by $\widetilde { \pi } _ { i }$ , and $\widetilde { \pi } _ { i } \leq \widetilde { \pi } _ { i + 1 }$ . $\begin{array} { r } { \mathrm { S o } \xi _ { v } = \sum _ { i = 1 } ^ { R } \rho _ { i } \left( \sum _ { u \in \mathcal { V } _ { i } ^ { ' } } \pmb { h } _ { u } \right) } \end{array}$ , where $\begin{array} { r } { \rho _ { i } = \widetilde { \pi } _ { i } - \sum _ { j < i } \widetilde { \pi } _ { j } } \end{array}$ and $\begin{array} { r } { \mathcal { V } _ { i } ^ { \prime } = \bigcup _ { k = i } ^ { R } \mathcal { V } _ { [ v ] } ^ { k } } \end{array}$ . Now for each branch of SHADOW-GNN-ensemble, elet parameter $\theta _ { i } = \widetilde { \pi } _ { i }$ eso that $\mathtt { E X T R A C T } _ { i }$ returns $\mathcal { V } _ { i } ^ { \prime }$ . The GNN on $\mathcal { V } _ { i } ^ { \prime }$ can then learn $\sum _ { u \in \mathcal { V } _ { i } ^ { \prime } } \boldsymbol { h } _ { u }$ (e.g., by a simple “mean” READOUT). Finally, set the ensemble weight as $\rho _ { i }$ . SHADOW-GNN-ensemble learns $\xi _ { v }$ . As EXTRACT also preserves graph topology, our model can be more expressive than PPRGo.
+
+# 3.6 Practical Design: SHADOW-GNN
+
+We now discuss the practical implementation of decoupled GNN – SHADOW-GNN. As the name suggests, in SHADOW-GNN, the scope is a shallow subgraph (i.e., with depth often set to 2 or 3).
+
+In many realistic scenarios (e.g., citation networks, social networks, product recommendation graphs), a shallow neighborhood is both necessary and sufficient for the GNN to learn well. On “sufficiency”, we consider the social network example: the friend of a friend of a friend may share little commonality with you, and close friends may be at most 2 hops away. Formally, by the $\gamma$ -decaying theorem [56], a shallow neighborhood is sufficient to accurately estimate various graph metrics. On “necessity”, since the neighborhood size may grow exponentially with hops, a deep neighborhood would be dominated by nodes irrelevant to the target. The corresponding GNN would first need to differentiate the many useless nodes from the very few useful ones, before it can extract meaningful features from the useful nodes. Finally, a shallow subgraph ensures scalability by avoiding ���neighborhood explosion”.
+
+Remark on decoupling. So far we have defined a decoupled model as having the model depth $L ^ { \prime }$ larger than the subgraph depth $L$ . Strictly speaking, a decoupled model also admits $L ^ { \prime } = \bar { L }$ . For example, suppose in the full $L$ -hop neighborhood, there are $70 \%$ nodes $L$ hops away. Applying decoupling, the EXTRACT excludes most of the $L$ -hop neighbors, and the resulting subgraph $\mathcal { G } _ { [ v ] }$ contains only $20 \%$ nodes $L$ hops away. Then it is reasonable to consider an $L$ -layer model on such a depth- $L$ subgraph as also a decouple model. Compared with an $L$ -layer model on the full $L$ -hop neighborhood, an $L$ -layer model on such a depth- $L$ $\mathcal { G } _ { [ v ] }$ propagates much less information from nodes $L$ hops away. So the $L$ message passings are indeed decoupled from the full $L$ -hop neighborhood.
+
+Remark on neighborhood. The “sufficiency” and “necessity” in shallow neighborhood are not universal. In many other applications, long-range dependencies can be critical, as studied in [2]. In such cases, our practical implementation of SHADOW-GNN would incur accuracy loss. However, our decoupling principle in general may still be beneficial – “shallow subgraph” is a practical guideline rather than a theoretical requirement. We leave the study on such applications as future work.
+
+# 4 Related Work
+
+Deep GNNs. To improve the GNN performance while increasing the model depth, various layer architectures have been proposed. AS-GCN [18], DeepGCN [28], JK-net [50], MixHop [1], Snowball [33], DAGNN [31] and GCNII [34] all include some variants of residue connection, either across multiple layers or within a single layer. In principle, such architectures can also benefit the feature propagation of a deep SHADOW-GNN, since their design does not rely on a specific neighborhood (e.g., $L$ -hop). In addition to architectures, DropEdge [39] and Bayesian-GDC [14] propose regularization techniques by adapting dropout [42] to graphs. Such techniques are only applied during training, and inference may still suffer from issues such as oversmoothing.
+
+Sampling based methods. Neighbor or subgraph sampling techniques have been proposed to improve training efficiency. FastGCN [8], VR-GCN [7], AS-GCN [18], LADIES [61] and MVSGNN [10] sample neighbor nodes per GNN layer. Cluster-GCN [9] and GraphSAINT [55] sample a subgraph as the training minibatch. While sampling also changes the receptive field, all the above methods are fundamentally different from ours. The training samplers aim at estimating the quantities related to the full graph (e.g., the aggregation of the full $L$ -hop neighborhood), and so the inference model still operates on the full neighborhood to avoid accuracy loss. For SHADOW-GNN, since the decoupling principle is derived from a local view on $\mathcal { G }$ , our EXTRACT does not estimate any full neighborhood quantities. Consequently, the sampling based methods only improve the training efficiency, while SHADOW-GNN addresses the computation challenge for both training and inference.
+
+Re-defining the neighborhood. Various works reconstruct the original graph and apply the GNN on the re-defined neighborhood. GDC [23] views the reconstructed adjacency matrix as the diffusion matrix. SIGN [11] applies reconstruction for customized graph operators. AM-GCN [46] utilizes the reconstruction to separate feature and structure information. The above work re-define the neighborhood. However, they still have tightly coupled depth and scope. SHADOW-GNN can also work with the reconstructed graph $\mathcal { G } ^ { \prime }$ by simply applying EXTRACT on $\mathcal { G } ^ { \prime }$ .
+
+# 5 Experiments
+
+Setup. We evaluate SHADOW-GNN on seven graphs. Six of them are for the node classification task: Flickr [55], Reddit [12], Yelp [55], ogbn-arxiv, ogbn-products and ogbn-papers100M [16]. The remaining is for the link prediction task: ogbl-collab [16]. The sizes of the seven graphs range from 9K nodes (Flickr) to 110M nodes (ogbn-papers100M). Flickr, Reddit and Yelp are under the inductive setting. ogbn-arxiv, ogbn-products and ogbn-papers100M are transductive. Consistent with the original papers, for the graphs on node classification, we measure “F1-micro” score for Yelp and “accuracy” for the remaining five graphs. For the link prediction task, we use “Hits $@ 5 0 ^ { \circ }$ as the metric. See Appendix E.1 for details. We use W&B [5] for experiment tracking.
+
+We construct SHADOW with six backbone models: GCN [22], GraphSAGE [12], GAT [44], JK-Net [50], GIN [49], SGC [47]. The first five are representatives of the state-of-the-art GNN architectures, jointly covering various message aggregation functions as well as the skip connection design. SGC simplifies normal GCN by moving all the neighbor propagation to the pre-processing step. Therefore, SGC is suitable for evaluating oversmoothing. The non-SHADOW models are trained with both full-batch and GraphSAINT-minibatch [55]. Due to the massive size of the full $L$ -hop neighborhood, we need to perform sampling when training normal GNNs in order to make the computation time tractable. GraphSAINT is suitable for our purpose since 1. it is the state-of-the-art minibatch method which achieves high accuracy, and 2. it supports various GNN architectures and scales well to large graphs. On the other hand, for SHADOW-GNN, both training and inference are always executed in minibatches. One advantage of SHADOW-GNN is that the decoupling enables straightforward minibatch construction: each target just independently extracts the small subgraph on its own.
+
+We implement two EXTRACT described in Section 3.4: 1. “PPR”, where we set the node budget $K$ as $\{ 2 0 0 , 4 0 0 \}$ for the largest ogbn-papers100M and $K \leq 2 0 0$ for all other graphs; 2. “ $L$ -hop”, where we set the depth as $\{ 1 , 2 \}$ . We implement various subgraph pooling functions: “mean” and “max” evaluated in this section and others evaluated in Appendix F.3. For the model depth, since $L ^ { \prime } = 3$ is the standard setting in the literature (e.g., see the benchmarking in OGB [16]), we start from $L ^ { \prime } = 3$ and further evaluate a deeper model of $L ^ { \prime } = 5$ . All accuracy are measured by 5 runs without fixing random seeds. Hyperparameter tuning and architecture configurations are in Appendix E.4.
+
+SHADOW-GNN neighborhood. For both normal and SHADOW GNNs, Figure 2 shows on average how many neighbors are $L$ hops away from the target. For a normal GNN, the size of the neighborhood grows rapidly with respect to $L$ , and the nodes 4 hops away dominate the neighborhood. For SHADOW-GNN using the Table 1 EXTRACT, most neighbors concentrate within 2 hops. A small number of nodes are 3 hops away. Almost no nodes are 4 or more hops away. Importantly, not only does the composition of the two kinds of neighborhood differ significantly, but also the size of SHADOW-GNN scope is much smaller (see also Table 1). Such small subgraphs are essential to high computation efficiency. Finally, we can ignore the very few distant neighbors $ { \left( L \geq 4 \right) }$ , and regard the (effective) depth of SHADOW-GNN subgraph as $L = 2$ (or at most 3). Under such practical value of $L$ , a model with $L ^ { \prime } \geq 3$ is indeed a SHADOW-GNN (see “Remark on decoupling” in Section 3.6).
+
+![](images/3ff0d72768a180365177f62556e21fbfcee94cfc5f872b8f2a83346e32c2987c.jpg)  
+Figure 2: Neighborhood composition
+
+![](images/0d3040806271ed912e8436ec3211d9ba63a3ac476cee9100cccc0c4712716703.jpg)  
+Figure 3: SGC oversmoothing
+
+Table 1: Comparison on test accuracy / F1-micro score and inference cost (tuned with DropEdge)   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Layers</td><td colspan="2">Flickr</td><td colspan="2">Reddit</td><td colspan="2">Yelp</td><td colspan="2">ogbn-arxiv</td><td colspan="2">ogbn-products</td></tr><tr><td>Accuracy</td><td>Cost</td><td>Accuracy</td><td>Cost</td><td>F1-micro</td><td>Cost</td><td>Accuracy</td><td>Cost</td><td>Accuracy</td><td>Cost</td></tr><tr><td rowspan="3">GCN</td><td>3 5</td><td>0.5159±0.0017</td><td>2E0</td><td>0.9532±0.0003</td><td>6E1</td><td>0.4028±0.0019</td><td>2E1</td><td>0.7170±0.0026</td><td>1E1</td><td>0.7567±0.0018</td><td>5E0</td></tr><tr><td></td><td>0.5217±0.0016</td><td>2E2</td><td>0.9495±0.0012</td><td>1E3</td><td>OOM</td><td>1E3</td><td>0.7186±0.0017</td><td>1E3</td><td>OOM</td><td>9E2</td></tr><tr><td>3</td><td>0.5155±0.0027</td><td>2E0</td><td>0.9523±0.0003</td><td>6E1</td><td>0.5110±0.0012</td><td>2E1</td><td>0.7093±0.0003</td><td>1E1</td><td>0.8003±0.0024</td><td>5E0</td></tr><tr><td>GCN-SAINT</td><td>5</td><td>0.5165±0.0026</td><td>2E2</td><td>0.9523±0.0012</td><td>1E3</td><td>0.5012±0.0021</td><td>1E3</td><td>0.7039±0.0020</td><td>1E3</td><td>0.7992±0.0021</td><td>9E2</td></tr><tr><td>SHADOW-GCN</td><td>3</td><td>0.5234±0.0009</td><td>(1</td><td>0.9576±0.0005</td><td>1</td><td>0.5291±0.0020</td><td>(1</td><td>0.7180±0.0024</td><td>1</td><td>0.7742±0.0037</td><td>1)</td></tr><tr><td>(PPR)</td><td>5</td><td>0.5268±0.0008</td><td>1E0</td><td>0.9564±0.0004</td><td>1E0</td><td>0.5323±0.0020</td><td>2E0</td><td>0.7206±0.0025</td><td>2E0</td><td>0.7821±0.0043</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5286±0.0013</td><td>1E0</td><td>0.9624±0.0002</td><td>1E0</td><td>0.5393±0.0036</td><td>2E0</td><td>0.7223±0.0018</td><td>2E0</td><td>0.7914±0.0044</td><td>2E0</td></tr><tr><td rowspan="3">GraphSAGE</td><td>3</td><td>0.5140±0.0014</td><td>3E0</td><td>0.9653±0.0002</td><td>5E1</td><td>0.6178±0.0033</td><td>2E1</td><td>0.7192±0.0027</td><td>1E1</td><td>0.7858±0.0013</td><td>4E0</td></tr><tr><td>5</td><td>0.5154±0.0052</td><td>2E2</td><td>0.9626±0.0004</td><td>1E3</td><td>0OM</td><td>2E3</td><td>0.7193±0.0037</td><td>1E3</td><td>0OM</td><td>1E3</td></tr><tr><td>3</td><td>0.5176±0.0032</td><td>3E0</td><td>0.9671±0.0003</td><td>5E1</td><td>0.6453±0.0011</td><td>2E1</td><td>0.7107±0.0003</td><td>1E1</td><td>0.7923±0.0023</td><td>4E0</td></tr><tr><td>SAGE-SAINT</td><td>5</td><td>0.5201±0.0032</td><td>2E2</td><td>0.9670±0.0010</td><td>1E3</td><td>0.6394±0.0003</td><td>2E3</td><td>0.7013±0.0021</td><td>1E3</td><td>0.7964±0.0022</td><td>1E3</td></tr><tr><td>SHADOW-SAGE</td><td>3</td><td>0.5312±0.0019</td><td>1E0</td><td>0.9672±0.0003</td><td>1E0</td><td>0.6542±0.0002</td><td>1E0</td><td>0.7163±0.0028</td><td>1E0</td><td>0.7935±0.0031</td><td>1E0</td></tr><tr><td>(2-hop)</td><td>5</td><td>0.5335±0.0015</td><td>2E0</td><td>0.9675±0.0005</td><td>2E0</td><td>0.6525±0.0003</td><td>2E0</td><td>0.7180±0.0030</td><td>2E0</td><td>0.7995±0.0022</td><td>2E0</td></tr><tr><td> SHADOW-SAGE</td><td>3</td><td>0.5356±0.0013</td><td>(1)</td><td>0.9688±0.0002</td><td>(1)</td><td>0.6538±0.0003</td><td>(1)</td><td>0.7227±0.0012</td><td>(1)</td><td>0.7905±0.0026</td><td>(1)</td></tr><tr><td>(PPR)</td><td>5</td><td>0.5417±0.0006</td><td>2E0</td><td>0.9692±0.0007</td><td>2E0</td><td>0.6518±0.0002</td><td>2E0</td><td>0.7238±0.0007</td><td>2E0</td><td>0.8005±0.0040</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5395±0.0013</td><td>2E0</td><td>0.9703±0.0003</td><td>2E0</td><td>0.6564±0.0004</td><td>1E0</td><td>0.7258±0.0017</td><td>2E0</td><td>0.8067±0.0037</td><td>1E0</td></tr><tr><td rowspan="3">GAT</td><td>3</td><td>0.5070±0.0032</td><td>2E1</td><td>0OM</td><td>3E2</td><td>0OM</td><td>2E2</td><td>0.7201±0.0011</td><td>1E2</td><td>0OM</td><td>3E1</td></tr><tr><td>5</td><td>0.5164±0.0033</td><td>2E2</td><td>OOM</td><td>2E3</td><td>OOM</td><td>2E3</td><td>OOM</td><td>3E3</td><td>OOM</td><td>2E3</td></tr><tr><td>3</td><td>0.5225±0.0053</td><td>2E1</td><td>0.9671±0.0003</td><td>3E2</td><td>0.6459±0.0002</td><td>2E2</td><td>0.6977±0.0003</td><td>1E2</td><td>0.8027±0.0028</td><td>3E1</td></tr><tr><td>GAT-SAINT</td><td>5</td><td>0.5153±0.0034</td><td>2E2</td><td>0.9651±0.0024</td><td>2E3</td><td>0.6478±0.0012</td><td>2E3</td><td>0.6954±0.0013</td><td>3E3</td><td>0.7990±0.0072</td><td>2E3</td></tr><tr><td>SHADOW-GAT</td><td></td><td>0.5349±0.0023</td><td>(1)</td><td>0.9707±0.0004</td><td>(1)</td><td>0.6575±0.0004</td><td>(1)</td><td>0.7235±0.0020</td><td>(1)</td><td>0.8006±0.0014</td><td>(1)</td></tr><tr><td>(PPR)</td><td>3</td><td>0.5352±0.0028</td><td>2E0</td><td>0.9713±0.0004</td><td>2E0</td><td>0.6559±0.0002</td><td>2E0</td><td>0.7274±0.0022</td><td>2E0</td><td>0.8071±0.0004</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5364±0.0026</td><td>1E0</td><td>0.9710±0.0004</td><td>2E0</td><td>0.6566±0.0005</td><td>1E0</td><td>0.7265±0.0028</td><td>2E0</td><td>0.8142±0.0031</td><td>1E0</td></tr></table>
+
+Comparison with baselines. Table 1 shows the performance comparison of SHADOW-GNN with the normal GNNs. All models on all datasets have uniform hidden dimension of 256. We define the metric “inference cost” as the average amount of computation to generate prediction for one test node. Inference cost is a measure of computation complexity (see Appendix B for the calculation) and is independent of hardware / implementation factors such as parallelization strategy, batch processing, etc. For cost of SHADOW-GNN, we do not include the overhead on computing EXTRACT, since it is hard to calculate such cost analytically. Empirically, subgraph extraction is much cheaper than model computation (see Figure 8, 9 for time evaluation on CPU and GPU). During training, we apply DropEdge [39] to both the baseline and SHADOW models. DropEdge helps improve the baseline accuracy by alleviating oversmoothing, and benefits SHADOW-GNN due to its regularization effects. See Appendix F.2 for results on other architectures including GIN and JK-Net.
+
+ACCURACY We aim at answering the following questions: 1. Can we improve accuracy by decoupling a baseline model? 2. What architecture components can we tune to improve accuracy of a decoupled model? 3. What EXTRACT can we tune to improve the accuracy of a decoupled model? To answer Q1, we fix the backbone architecture and remove pooling. Then we inspect “3-layer normal GNN vs. 3-layer SHADOW-GNN” and “5-layer normal GNN vs. 5-layer SHADOW-GNN”. Clearly, SHADOW-GNNs (with scope size no more than 200) in general achieve significantly higher accuracy than the normal GNNs. This indicates that a shallow neighborhood contains sufficient information, and customizing the scope can benefit accuracy even without architecture changes (from Figure 2, a depth-3 $\mathcal { G } _ { [ v ] }$ differs significantly from the 3-hop neighborhood). To answer Q1, we focus on the PPR EXTRACT and thus compare the rows in blue background. We use the 3-layer SHADOW-GNN without pooling as the baseline and analyze the effects of 1. increasing the GNN depth without expanding scope, and 2. adding subgraph pooling. Comparing among the rows in light blue background, we observe that in many cases, simply increasing the depth from 3 to 5 leads to significant accuracy gain. Comparing the ligh blue rows with the dark blue rows, we observe that sometimes pooling can further improve the accuracy of a SHADOW-GNN. In conclusion, both types of architecture tuning are effective ways of optimizing a SHADOW-GNN. Finally, to answer Q3, we compare the light blue rows with the light yellow rows. In general, PPR EXTRACT leads to higher accuracy than 2-hop EXTRACT, demonstrating the importance of designing a good EXTRACT.
+
+INFERENCE COST Inference cost of SHADOW-GNN is orders of magnitude lower than the normal GNNs (a 5-layer SHADOW-GNN is still much cheaper than a 3-layer normal GNN). The high cost of the baselines is due to the “neighborhood explosion” with respect to more layers. SHADOW-GNN is efficient and scalable as the cost only grows linearly with the model depth. Note that GraphSAINT only improves efficiency during training since its inference operates on the full $L$ -hop neighborhood.
+
+Table 2: Leaderboard comparison on papers100M   
+
+<table><tr><td>Method</td><td>Test accuracy</td><td>Val accuracy</td><td>Neigh size</td></tr><tr><td>GraphSAGE+incep</td><td>0.6706±0.0017</td><td>0.7032±0.0011</td><td>4E5</td></tr><tr><td>SIGN-XL</td><td>0.6606±0.0019</td><td>0.6984±0.0006</td><td>&gt;4E5</td></tr><tr><td>SGC</td><td>0.6329±0.0019</td><td>0.6648±0.0020</td><td>&gt;4E5</td></tr><tr><td>SHADOW-GAT200</td><td>0.6681±0.0016</td><td>0.7019±0.0011</td><td>2E2</td></tr><tr><td>SHADOW-GAT400</td><td>0.6708±0.0017</td><td>0.7073±0.0011</td><td>3E2</td></tr></table>
+
+Scaling to 100 million nodes. We further scale SHADOW-GNN to ogbn-papers100M, one of the largest public dataset. Even through the full graph size is at least two orders of magnitude larger than the graphs in Table 1, the localized scope of SHADOW-GNN barely needs to increase. Since SHADOW-GNN performs
+
+minibatch computation, a low-end GPU with limited memory capacity can compute SHADOW-GNN on ogbn-papers100M efficiently. We show in Appendix F.1 that we can train and inference our model with as little as 4GB GPU memory consumption. This is infeasible using normal GNNs. Table 2 summarizes our comparison with the top leaderboard methods [45, 11, 47]. We only include those methods that do not use node labels as the model input (i.e., the most standard setup). We achieve at least 3 orders of magnitude reduction in neighborhood size without sacrificing accuracy. For SIGN-XL and SGC, their neighborhood is too large to count the exact size. Also, their preprocessing consumes $5 \times$ more CPU memory than SHADOW-GNN (Appendix F.1).
+
+Table 3: Leaderboard comparison on collab   
+
+<table><tr><td>Method</td><td>Test Hits @50</td><td>Val Hits @50</td></tr><tr><td>SEAL</td><td>0.5371±0.0047</td><td>0.6389±0.0049</td></tr><tr><td>DeeperGCN</td><td>0.5273±0.0047</td><td>0.6187±0.0045</td></tr><tr><td>LRGA+GCN</td><td>0.5221±0.0072</td><td>0.6088±0.0059</td></tr><tr><td>SHADOW-SAGE</td><td>0.5492±0.0022</td><td>0.6524±0.0017</td></tr></table>
+
+Extending to link-level task. We further show that SHADOW-GNN is general and can be extended to the link prediction task. There are two settings of ogbl-collab. We follow the one where validation edges cannot be used in training updates. This is the setting which most leaderboard methods follow. Table 3 shows the comparison with the top GNN models under the same setting. SHADOW-SAGE outperforms the rank-1 model with significant margin.
+
+Oversmoothing. To validate Theorem 3.2, we pick SGC as the backbone architecture. SGC with power $L$ is equivalent to $L$ -layer GCN without activation. Performance comparison between SGC and SHADOW-SGC thus reveals the effect of oversmoothing without introducing other factors due to optimizing deep neural networks (e.g., vanishing gradients). In Figure 3, we vary the power of SGC and SHADOW-SGC from 1 to 40 (see Appendix E.5 for details). While SGC gradually collapses local information into global “white noise”, accuracy of SHADOW-SGC does not degrade. This validates our theory that extracting local subgraphs prevents oversmoothing.
+
+# 6 Conclusion
+
+We have presented a design principle to decouple the depth and scope of GNNs. Applying such a principle on various GNN architectures simultaneously improves expressivity and computation scalability of the corresponding models. We have presented thorough theoretical analysis on expressivity from three different perspectives, and also rich design components (e.g., subgraph extraction functions, architecture extensions) to implement such design principle. Experiments show significant performance improvement over a wide range of graphs, GNN architectures and learning tasks.
+
+References   
+[1] Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. arXiv preprint arXiv:1905.00067, 2019.   
+[2] Uri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In International Conference on Learning Representations, 2021.   
+[3] Reid Andersen, Fan Chung, and Kevin Lang. Local graph partitioning using pagerank vectors. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pages 475–486. IEEE, 2006.   
+[4] Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Spectral clustering with graph neural networks for graph pooling. In International Conference on Machine Learning, pages 874–883. PMLR, 2020.   
+[5] Lukas Biewald. Experiment tracking with weights and biases, 2020. Software available from wandb.com.   
+[6] Aleksandar Bojchevski, Johannes Klicpera, Bryan Perozzi, Amol Kapoor, Martin Blais, Benedek Rózemberczki, Michal Lukasik, and Stephan Günnemann. Scaling graph neural networks with approximate pagerank. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 2464–2473, New York, NY, USA, 2020. Association for Computing Machinery.   
+[7] Jianfei Chen, Jun Zhu, and Le Song. Stochastic training of graph convolutional networks with variance reduction, 2017.   
+[8] Jie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast learning with graph convolutional networks via importance sampling. In International Conference on Learning Representations, 2018.   
+[9] Wei-Lin Chiang, Xuanqing Liu, Si Si, Yang Li, Samy Bengio, and Cho-Jui Hsieh. Cluster-gcn: An efficient algorithm for training deep and large graph convolutional networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’19, page 257–266, New York, NY, USA, 2019. Association for Computing Machinery.   
+[10] Weilin Cong, Rana Forsati, Mahmut Kandemir, and Mehrdad Mahdavi. Minimal variance sampling with provable guarantees for fast training of graph neural networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 1393–1403, New York, NY, USA, 2020. Association for Computing Machinery.   
+[11] Fabrizio Frasca, Emanuele Rossi, Davide Eynard, Ben Chamberlain, Michael Bronstein, and Federico Monti. SIGN: Scalable inception graph neural networks, 2020.   
+[12] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in neural information processing systems, pages 1024–1034, 2017.   
+[13] William L Hamilton, Rex Ying, and Jure Leskovec. Representation learning on graphs: Methods and applications. arXiv preprint arXiv:1709.05584, 2017.   
+[14] Arman Hasanzadeh, Ehsan Hajiramezanali, Shahin Boluki, Mingyuan Zhou, Nick Duffield, Krishna Narayanan, and Xiaoning Qian. Bayesian graph neural networks with adaptive connection sampling, 2020.   
+[15] Kurt Hornik, Maxwell Stinchcombe, Halbert White, et al. Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366, 1989.   
+[16] Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs, 2020.   
+[17] Wenbing Huang, Yu Rong, Tingyang Xu, Fuchun Sun, and Junzhou Huang. Tackling oversmoothing for general Graph Convolutional Networks, 2020.   
+[18] Wenbing Huang, Tong Zhang, Yu Rong, and Junzhou Huang. Adaptive sampling towards fast graph representation learning. In Advances in neural information processing systems, pages 4558–4567, 2018.   
+[19] Glen Jeh and Jennifer Widom. Simrank: a measure of structural-context similarity. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 538–543, 2002.   
+[20] Leo Katz. A new status index derived from sociometric analysis. Psychometrika, 18(1):39–43, 1953.   
+[21] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   
+[22] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   
+[23] Johannes Klicpera, Stefan Weißenberger, and Stephan Günnemann. Diffusion improves graph learning, 2019.   
+[24] Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In International Conference on Machine Learning, pages 3734–3743. PMLR, 2019.   
+[25] Jure Leskovec and Christos Faloutsos. Sampling from large graphs. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’06, page 631–636, New York, NY, USA, 2006. Association for Computing Machinery.   
+[26] Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graph evolution: Densification and shrinking diameters. ACM Trans. Knowl. Discov. Data, 1(1):2–es, March 2007.   
+[27] David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017.   
+[28] Guohao Li, Matthias Müller, Ali Thabet, and Bernard Ghanem. DeepGCNs: Can GCNs go as deep as CNNs? In The IEEE International Conference on Computer Vision (ICCV), 2019.   
+[29] Guohao Li, Chenxin Xiong, Ali Thabet, and Bernard Ghanem. DeeperGCN: All you need to train deeper gcns, 2020.   
+[30] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into Graph Convolutional Networks for semi-supervised learning. arXiv preprint arXiv:1801.07606, 2018.   
+[31] Meng Liu, Hongyang Gao, and Shuiwang Ji. Towards deeper graph neural networks. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Aug 2020.   
+[32] Yu-Chen Lo, Stefano E Rensi, Wen Torng, and Russ B Altman. Machine learning in chemoinformatics and drug discovery. Drug discovery today, 23(8):1538–1546, 2018.   
+[33] Sitao Luan, Mingde Zhao, Xiao-Wen Chang, and Doina Precup. Break the ceiling: Stronger multi-scale deep graph convolutional networks. In Advances in Neural Information Processing Systems 32, pages 10945–10955. Curran Associates, Inc., 2019.   
+[34] Zhewei Wei Ming Chen, Bolin Ding Zengfeng Huang, and Yaliang Li. Simple and deep graph convolutional networks. 2020.   
+[35] Federico Monti, Michael Bronstein, and Xavier Bresson. Geometric matrix completion with recurrent multi-graph neural networks. In Advances in Neural Information Processing Systems, pages 3697–3707, 2017.   
+[36] Kenta Oono and Taiji Suzuki. Graph Neural Networks exponentially lose expressive power for node classification. In International Conference on Learning Representations, 2020.
+
+[37] Aditya Pal, Chantat Eksombatchai, Yitong Zhou, Bo Zhao, Charles Rosenberg, and Jure Leskovec. PinnerSage: Multi-modal user embedding framework for recommendations at Pinterest. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Aug 2020.
+
+[38] Namyong Park, Andrey Kan, Xin Luna Dong, Tong Zhao, and Christos Faloutsos. Estimating node importance in knowledge graphs using graph neural networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 596–606, 2019.   
+[39] Yu Rong, Wenbing Huang, Tingyang Xu, and Junzhou Huang. DropEdge: Towards deep Graph Convolutional Networks on node classification. In International Conference on Learning Representations, 2020.   
+[40] Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional networks. In European Semantic Web Conference, pages 593–607. Springer, 2018.   
+[41] Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research, 12(9), 2011.   
+[42] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1):1929–1958, 2014.   
+[43] Jonathan M Stokes, Kevin Yang, Kyle Swanson, Wengong Jin, Andres Cubillos-Ruiz, Nina M Donghia, Craig R MacNair, Shawn French, Lindsey A Carfrae, Zohar Bloom-Ackerman, et al. A deep learning approach to antibiotic discovery. Cell, 180(4):688–702, 2020.   
+[44] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua ´ Bengio. Graph Attention Networks. In International Conference on Learning Representations, 2018.   
+[45] Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou, Chao Ma, Lingfan Yu, Yu Gai, Tianjun Xiao, Tong He, George Karypis, Jinyang Li, and Zheng Zhang. Deep graph library: A graph-centric, highly-performant package for graph neural networks, 2020.   
+[46] Xiao Wang, Meiqi Zhu, Deyu Bo, Peng Cui, Chuan Shi, and Jian Pei. AM-GCN: Adaptive multichannel graph convolutional networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 1243–1253, New York, NY, USA, 2020. Association for Computing Machinery.   
+[47] Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr, Christopher Fifty, Tao Yu, and Kilian Q Weinberger. Simplifying graph convolutional networks. arXiv preprint arXiv:1902.07153, 2019.   
+[48] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2020.   
+[49] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018.   
+[50] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks, 2018.   
+[51] Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. GNNExplainer: Generating explanations for graph neural networks, 2019.   
+[52] Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 974–983, 2018.   
+[53] Jiaxuan You, Zhitao Ying, and Jure Leskovec. Design space for graph neural networks. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 17009–17021. Curran Associates, Inc., 2020.   
+[54] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. Accurate, efficient and scalable graph embedding. In 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 462–471, 2019.   
+[55] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. GraphSAINT: Graph sampling based inductive learning method. In International Conference on Learning Representations, 2020.   
+[56] Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pages 5165–5175, 2018.   
+[57] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   
+[58] Shuai Zhang, Lina Yao, Aixin Sun, and Yi Tay. Deep learning based recommender system: A survey and new perspectives. ACM Computing Surveys (CSUR), 52(1):1–38, 2019.   
+[59] Yuyu Zhang, Xinshi Chen, Yuan Yang, Arun Ramamurthy, Bo Li, Yuan Qi, and Le Song. Efficient probabilistic logic reasoning with graph neural networks, 2020.   
+[60] Lingxiao Zhao and Leman Akoglu. PairNorm: Tackling oversmoothing in gnns. In International Conference on Learning Representations, 2020.   
+[61] Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. Layerdependent importance sampling for training deep and large graph convolutional networks. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See append
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See the released code.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See the released code. It contains all configurations.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See citation after each dataset.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/_IY3_4psXuf/_IY3_4psXuf_content_list.json

@@ -0,0 +1,1357 @@
+[
+    {
+        "type": "text",
+        "text": "Decoupling the Depth and Scope of Graph Neural Networks ",
+        "text_level": 1,
+        "bbox": [
+            285,
+            122,
+            712,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hanqing Zeng USC zengh@usc.edu ",
+        "bbox": [
+            196,
+            227,
+            308,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Muhan Zhang Peking University, BIGAI muhan@pku.edu.cn ",
+        "bbox": [
+            336,
+            226,
+            509,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yinglong Xia Facebook AI yxia@fb.com ",
+        "bbox": [
+            539,
+            227,
+            635,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ajitesh Srivastava USC ajiteshs@usc.edu ",
+        "bbox": [
+            665,
+            227,
+            800,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Rajgopal Kannan ",
+        "text_level": 1,
+        "bbox": [
+            436,
+            290,
+            560,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andrey Malevich Facebook AI amalevich@fb.com ",
+        "bbox": [
+            202,
+            289,
+            343,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "US ARL ",
+        "bbox": [
+            470,
+            304,
+            526,
+            315
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Viktor Prasanna USC prasanna@usc.edu ",
+        "bbox": [
+            656,
+            290,
+            794,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "rajgopal.kannan.civ@mail.mil ",
+        "bbox": [
+            379,
+            318,
+            617,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Long Jin   \nFacebook AI   \nlongjin@fb.com   \nRen Chen   \nFacebook AI   \nrenchen@fb.com ",
+        "bbox": [
+            307,
+            353,
+            429,
+            395
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            568,
+            352,
+            691,
+            393
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            430,
+            535,
+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "State-of-the-art Graph Neural Networks (GNNs) have limited scalability with respect to the graph and model sizes. On large graphs, increasing the model depth often means exponential expansion of the scope (i.e., receptive field). Beyond just a few layers, two fundamental challenges emerge: 1. degraded expressivity due to oversmoothing, and 2. expensive computation due to neighborhood explosion. We propose a design principle to decouple the depth and scope of GNNs – to generate representation of a target entity (i.e., a node or an edge), we first extract a localized subgraph as the bounded-size scope, and then apply a GNN of arbitrary depth on top of the subgraph. A properly extracted subgraph consists of a small number of critical neighbors, while excluding irrelevant ones. The GNN, no matter how deep it is, smooths the local neighborhood into informative representation rather than oversmoothing the global graph into “white noise”. Theoretically, decoupling improves the GNN expressive power from the perspectives of graph signal processing (GCN), function approximation (GraphSAGE) and topological learning (GIN). Empirically, on seven graphs (with up to 110M nodes) and six backbone GNN architectures, our design achieves significant accuracy improvement with orders of magnitude reduction in computation and hardware cost. ",
+        "bbox": [
+            232,
+            463,
+            766,
+            698
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            727,
+            310,
+            744
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph Neural Networks (GNNs) have now become the state-of-the-art models for graph mining [48, 13, 58], facilitating applications such as social recommendation [35, 52, 37], knowledge understanding [40, 38, 59] and drug discovery [43, 32]. With the numerous architectures proposed [22, 12, 44, 49], it still remains an open question how to effectively scale up GNNs with respect to both the model size and graph size. There are two fundamental obstacles when we increase the number of GNN layers: ",
+        "bbox": [
+            174,
+            760,
+            825,
+            829
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Expressivity challenge (i.e., oversmoothing [30, 36, 39, 17]): iterative mixing of neighbor features collapses embedding vectors of different nodes into a fixed, low-dimensional subspace. ",
+        "bbox": [
+            178,
+            843,
+            823,
+            872
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Scalability challenge (i.e., neighbor explosion [7, 8, 9, 55]): recursive expansion of multi-hop neighborhood results in exponentially growing receptive field size (and thus computation cost). ",
+        "bbox": [
+            181,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To address the expressivity challenge, most remedies focus on neural architecture exploration: [44, 12, 49, 29] propose more expressive aggregation functions when propagating neighbor features. [50, 28, 18, 34, 1, 33, 31] use residue-style design components to construct flexible and dynamic receptive fields. Among them, [50, 28, 18] use skip-connection across multiple GNN layers, and [34, 1, 33, 31] encourage multi-hop message passing within each single layer. As for the scalability challenge, sampling methods have been explored to improve the training speed and efficiency. Importance based layer-wise sampling [8, 7, 61] and subgraph-based sampling [54, 9, 55] alleviate neighbor explosion, while preserving training accuracy. Unfortunately, such sampling methods cannot be naturally generalized to inference without accuracy loss (see also Section 4). ",
+        "bbox": [
+            174,
+            132,
+            825,
+            257
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The above lines of research have only guided us to partial solutions. Yet what is the root cause of both the expressivity and scalability challenges? Setting aside the design of GNN architectures or sampling schemes, we provide an alternative perspective by interpretting the data in a different way. ",
+        "bbox": [
+            176,
+            263,
+            823,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Two views on the graph. Given an input graph $\\mathcal { G }$ with node set $\\nu$ , the most straightforward way to understand $\\mathcal { G }$ is by viewing it as a single global graph. So any two nodes $u$ and $v$ belong to the same $\\mathcal { G }$ , and if $u$ and $v$ lie in the same connected component, they will ultimately see each other in their own neighborhood no matter how far away $u$ and $v$ are. Alternative to the above global view, we can take a local view on $\\mathcal { G }$ . For each node $v$ , there is a latent $\\mathcal { G } _ { [ v ] }$ surrounding it which captures the characteristics of just $v$ itself. The full $\\mathcal { G }$ is observed (by the data collection process) as the union of all such $\\mathcal { G } _ { [ v ] }$ . Consequently, $\\nu _ { [ v ] }$ rather than $\\nu$ defines $v$ ’s neighborhood: if $u \\notin \\mathcal { V } _ { [ v ] }$ , $v$ will never consider $u$ as a neighbor. Our “decoupling” design is based on the local view. ",
+        "bbox": [
+            174,
+            310,
+            825,
+            422
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Scope of GNNs. Both the expressivity and scalability challenges are closely related to the enlargement of the GNN’s scope (i.e., receptive field). More importantly, how we define the scope is determined by how we view $\\mathcal { G }$ . With the global view above, an $L$ -layer GNN has the scope of the full $L$ -hop neighborhood. With the local view, the GNN scope is simply $\\nu _ { [ v ] }$ regardless of the GNN depth. The two existing lines of research, one on architectural exploration and the other on sampling, both take the global view. Consequently, the depth (i.e., number of layers) and scope of such GNNs are tightly coupled. Such coupling significantly limits the design space exploration of GNNs with various depths [53]. Consider the example of ogbn-products, a medium-scale graph in Open Graph Benchmark [16]. The average number of 4-hop neighbors is around $0 . 6 \\mathbf { M }$ , corresponding to $2 5 \\%$ of the full graph size. To generate representation of a single target node, a 4-layer coupled GNN needs to propagate features from the $0 . 6 \\mathbf { M }$ neighbors. Such propagation can be inefficient or even harmful since most nodes in the huge neighborhood would be barely relevant to the target node. ",
+        "bbox": [
+            173,
+            429,
+            825,
+            594
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Decoupling the GNN depth and scope. Taking the local view on $\\mathcal { G }$ , we propose a general design principle to decouple the GNN depth and scope. To generate the representation of the target node $v$ , we first extract from $\\mathcal { G }$ a small subgraph $\\mathcal { G } _ { [ v ] }$ surrounding $v$ . On top of $\\mathcal { G } _ { [ v ] }$ , we apply a GNN whose number of layers and message passing functions can be flexibly chosen. “Decoupling” means we treat the scope extraction function and GNN depth as two independently tuned parameters – effectively we introduce a new dimension in the GNN design space. We intuitively illustrate the benefits of decoupling by an example GNN construction, where the scope is the $L$ -hop neighborhood and depth is $L ^ { \\prime }$ $L ^ { \\prime } > L  )$ ). When we use more layers $( L ^ { \\prime } )$ than hops $( L )$ , each pair of subgraph nodes may exchange messages multiple times. The extra message passing helps the GNN better absorb and embed the information within scope, and thus leads to higher expressivity. We further justify the above intuition with multifaceted theoretical analysis. From the graph signal processing perspective, we prove that decoupled-GCN performs local-smoothing rather than oversmoothing, as long as the scopes of different target nodes are different. From the function approximation perspective, we construct a linear target function on neighbor features and show that decoupling the GraphSAGE model reduces the function approximation error. From the topological learning perspective, we apply deep GIN-style message passing to differentiate non-regular subgraphs of a regular graph. As a result, our model is more powerful than the 1-dimensional Weisfeiler-Lehman test [41]. ",
+        "bbox": [
+            174,
+            601,
+            825,
+            835
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Practical implementation: SHADOW-GNN. The decoupling principle leads to a practical implementation, SHADOW-GNN: Decoupled GNN on a shallow subgraph. In SHADOW-GNN, the scope is a shallow yet informative subgraph, only containing a fraction of the 2- or 3-hop neighbors of $\\mathcal { G }$ (see Section 5). On the other hand, the model of SHADOW-GNN is deeper (e.g., $L ^ { \\prime } = 5$ ). To efficiently construct the shallow scope on commodity hardware, we propose various subgraph extraction functions. To better utilize the subgraph node embeddings after deep message passing, we propose neural architecture extensions such as pooling and ensemble. Empirically, our “decoupling” design improves both the accuracy and scalability. On seven benchmarks (including the largest ogbn-papers100M graph with 111M nodes) and across two graph learning tasks, SHADOW-GNNs achieve significant accuracy gains compared to the original models. Meanwhile, the computation and hardware costs are reduced by orders of magnitude. Our code is available at https://github.com/facebookresearch/shaDow_GNN ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            826,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            207,
+            318,
+            224
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { G } \\left( \\mathcal { V } , \\mathcal { E } , \\boldsymbol { X } \\right)$ be an undirected graph, with node set $\\nu$ , edge set $\\mathcal { E } \\subseteq \\mathcal { V } \\times \\mathcal { V }$ and node feature matrix $\\pmb { X } \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times d }$ . Let $\\mathcal { N } _ { v }$ denote the set of $v$ ’s direct neighbors in $\\mathcal { G }$ . The $u ^ { \\mathrm { t h } }$ row of $\\boldsymbol { X }$ corresponds to the length- $d$ feature of node $u$ . Let $\\pmb { A }$ be the adjacency matrix of $\\mathcal { G }$ where $A _ { u , v } = 1$ if edge $( u , v ) \\in \\mathcal { E }$ and $A _ { u , v } = 0$ otherwise. Let $_ { D }$ be the diagonal degree matrix of $\\pmb { A }$ . Denote $\\widetilde { A } = D _ { * } ^ { - \\frac 1 2 } A _ { * } D _ { * } ^ { - \\frac 1 2 }$ as the adjacency matrix after symmetric normalization ( $\\cdot _ { * } , *$ means augmented with self-edges), and $\\stackrel { \\frown } { A } = D ^ { \\stackrel { - 1 } { - } } A$ (or $D _ { * } ^ { - 1 } A _ { * , }$ ) as the one after random walk normalization. Let subscript “ $[ u ] ^ { \\dag }$ mark the quantities corresponding to a subgraph surrounding node $u$ . For example, the subgraph itself is $\\mathcal { G } _ { [ u ] }$ . Subgraph matrices $X _ { [ v ] }$ and $\\boldsymbol { A } _ { [ v ] }$ have the same dimension as the original $\\boldsymbol { X }$ and $\\pmb { A }$ . Yet, row vector $\\left[ \\boldsymbol { X } _ { [ v ] } \\right] _ { u } = \\mathbf { 0 }$ for $u \\notin \\mathcal { V } _ { [ v ] }$ . Element $\\left[ { \\pmb A } _ { [ v ] } \\right] _ { u , w } = 0$ if either $u \\notin \\mathcal { V } _ { [ v ] }$ or $w \\notin \\mathcal { V } _ { [ v ] }$ For an $L$ -layer GNN, let superscript “ $( \\ell ) ^ { \\dag }$ denote the layer- $\\ell$ quantities. Let $d ^ { ( \\ell ) }$ be the number of channels for layer $\\ell$ ; $\\pmb { H } ^ { ( \\ell - 1 ) } \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times d ^ { ( \\ell - 1 ) } }$ and $H ^ { ( \\ell ) } \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times d ^ { ( \\ell ) } }$ be the input and output features. So $H ^ { ( 0 ) } = X$ and $d ^ { ( 0 ) } = d$ . Further, let $\\sigma$ be the activation and $\\mathbf { \\boldsymbol { W } } ^ { ( \\ell ) }$ be the learnable weight. For example, a GCN layer performs $\\pmb { H } ^ { ( \\ell ) } = \\sigma \\left( \\widetilde { A } \\pmb { H } ^ { ( \\ell - 1 ) } \\pmb { W } ^ { ( \\ell ) } \\right)$ . A GraphSAGE layer performs $\\begin{array} { r } { \\pmb { H } ^ { ( \\ell ) } = \\sigma \\left( \\pmb { H } ^ { ( \\ell - 1 ) } \\pmb { W } _ { 1 } ^ { ( \\ell ) } + \\widehat { \\pmb { A } } \\pmb { H } ^ { ( \\ell - 1 ) } \\pmb { W } _ { 2 } ^ { ( \\ell ) } \\right) . } \\end{array}$ ",
+        "bbox": [
+            173,
+            238,
+            826,
+            477
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our analysis in Section 3 mostly focuses on the node classification task. Yet our design can be generalized to the link prediction task, as demonstrated by our experiments in Section 5. ",
+        "bbox": [
+            173,
+            481,
+            823,
+            508
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1. (Depth of subgraph) Assume the subgraph $\\mathcal { G } _ { [ v ] }$ is connected. The depth of $\\mathcal { G } _ { [ v ] }$ is defined as $\\mathrm { m a x } _ { u \\in \\mathcal { V } _ { [ v ] } } d ( u , v ) ;$ , where $d ( u , v )$ denotes the shortest path distance from u to v. ",
+        "bbox": [
+            173,
+            512,
+            823,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The above definition enables us to make comparison such as “the GNN is deeper than the subgraph”.   \nFor “decoupling the depth and scope”, we refer to the model depth rather than the subgraph depth. ",
+        "bbox": [
+            173,
+            553,
+            823,
+            582
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Decoupling the Depth and Scope of GNNs ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            601,
+            555,
+            619
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "“Decoupling the depth and scope of GNNs” is a design principle to improve the expressivity and scalability of GNNs without modifying the layer architecture. We name a GNN after decoupling a SHADOW-GNN (see Section 3.6 for explanation of the name). Compared with a normal GNN, SHADOW-GNN contains an additional component: the subgraph extractor EXTRACT. To generate embedding of a target node $v$ , SHADOW-GNN proceeds as follows: 1. We use EXTRACT $( v , { \\mathcal { G } } )$ to return a connected $\\mathcal { G } _ { [ v ] }$ , where $\\mathcal { G } _ { [ v ] }$ is a subgraph containing $v$ , and the depth of $\\mathcal { G } _ { [ v ] }$ is $L$ . 2. We build an $L ^ { \\prime }$ -layer GNN on $\\mathcal { G } _ { [ v ] }$ by treating $\\mathcal { G } _ { [ v ] }$ as the new full graph and by ignoring all nodes / edges not in $\\mathcal { G } _ { [ v ] }$ . So $\\mathcal { G } _ { [ v ] }$ is the scope of SHADOW-GNN. The key point reflecting “decoupling” is that $L ^ { \\prime } > L$ ",
+        "bbox": [
+            173,
+            631,
+            825,
+            747
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A normal GNN is closely related to a SHADOW-GNN. Under the normal setup, an $L$ -layer GNN operates on the full $\\mathcal { G }$ and propagates the influence from all the neighbors up to $L$ hops away from $v$ Such a GNN is equivalent to a model where EXTRACT returns the full $L$ -hop subgraph and $L ^ { \\prime } = L$ . ",
+        "bbox": [
+            176,
+            752,
+            823,
+            794
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We theoretical demonstrate how SHADOW-GNN improves expressivity from three different angles. On SHADOW-GCN (Section 3.1), we come from the graph signal processing perspective. The GCN propagation can be interpreted as applying filtering on the node signals [47]. Deep models correspond to high-pass filters. Filtering the local graph $\\mathcal { G } _ { [ v ] }$ preserves richer information than the global $\\mathcal { G }$ . On SHADOW-SAGE (Section 3.2), we view the GNN as a function approximator. We construct a target function and study how decoupling reduces the approximation error. On SHADOW-GIN (Section 3.3), we focus on learning topological information. We show that decoupling helps capture local graph structure which the 1D Weisfeiler-Lehman test fails to capture. ",
+        "bbox": [
+            173,
+            799,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Expressivity Analysis on SHADOW-GCN: Graph Signal Processing Perspective ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            90,
+            761,
+            107
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "GCNs [22] suffer from “oversmoothing” [30] – Each GCN layer smooths the features of the direct (i.e., 1-hop) neighbors, and many GCN layers smooths the features of the full graph. Eventually, such repeated smoothing process propagates to any target node just the averaged feature of all $\\nu$ . “Oversmoothing” thus incurs significant information loss by wiping out all local information. ",
+        "bbox": [
+            173,
+            117,
+            826,
+            174
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Formally, suppose the original features $\\boldsymbol { X }$ reside in a high-dimensional space $\\mathbb { R } ^ { | \\nu | \\times d }$ . Oversmoothing pushes $\\boldsymbol { X }$ towards a low-dimensional subspace $\\mathbb { R } ^ { | \\nu | \\times d ^ { \\prime } }$ , where $d ^ { \\prime } < d$ . Corresponding analysis comes from two perspectives: oversmoothing by a deep $G C N$ , and oversmoothing by repeated GCNstyle propagation. The former considers the full neural network with non-linear activation, weight and bias. The later characterizes the aggregation matrix $M = \\operatorname* { l i m } _ { L \\to \\infty } \\widetilde { A } ^ { L } X$ . It is shown that even with the vanilla architecture, a deep GCN with bias parameters does not oversmooth [17]. In addition, various tricks [60, 39, 34] can prevent oversmoothing from the neural network perspective. However, a deep GCN still suffers from accuracy drop, indicating that the GCN-style propagation (rather than other GCN components like activation and bias) may be the fundamental reason causing difficulty in learning. Therefore, we study the asymptotic behavior of the aggregation matrix $M$ under the normal and SHADOW design. In other words, here in Section 3.1, we ignore the non-linear activation and bias parameters. Such setup is consistent with many existing literature such as [30, 33, 34, 60]. ",
+        "bbox": [
+            173,
+            179,
+            826,
+            352
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.1. $\\infty$ number of feature propagation by SHADOW-GCN leads to ",
+        "bbox": [
+            173,
+            357,
+            699,
+            372
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f18b14966c3feacd2a4fac99bf5459fe265f8f2def6e9ff7aa5fcbbe503511b.jpg",
+        "text": "$$\n\\pmb { m } _ { [ v ] } = \\left[ e _ { [ v ] } \\right] _ { v } \\cdot \\left( e _ { [ v ] } ^ { \\top } \\pmb { X } _ { [ v ] } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            381,
+            593,
+            407
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $e _ { [ v ] }$ is defined by $\\begin{array} { r } { \\left[ e _ { [ v ] } \\right] _ { u } = \\sqrt { \\frac { \\delta _ { [ v ] } ( u ) } { \\sum _ { w \\in \\mathcal { V } _ { [ v ] } } \\delta _ { [ v ] } ( w ) } } } \\end{array}$ ; $\\delta _ { [ v ] } \\left( u \\right)$ returns the degree of u in $\\mathcal { G } _ { [ v ] } p l u s ~ l$ ",
+        "bbox": [
+            176,
+            417,
+            812,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Oversmoothing by normal GCN propagation. With a large enough $L$ , the full $L$ -hop neighborhood becomes $\\nu$ (assuming connected $\\mathcal { G }$ ). So $\\forall ~ u , v$ , we have $\\mathcal { G } _ { [ u ] } = \\mathcal { G } _ { [ v ] } = \\mathcal { G }$ , implying $e _ { [ u ] } = e _ { [ v ] }$ and $X _ { [ u ] } = X _ { [ v ] } = X$ . From Proposition 3.1, the aggregation converges to a point where $n o$ feature and little structural information of the target is preserved. The only information in $m _ { [ v ] }$ is $v$ ’s degree. ",
+        "bbox": [
+            174,
+            460,
+            825,
+            517
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Local-smoothing by SHADOW-GCN propagation. With a fixed subgraph, no matter how many times we aggregate using $\\widetilde { A } _ { [ v ] }$ , the layers will not include the faraway irrelevant nodes. From Proposition 3.1, $m _ { [ v ] }$ is a linear combination of the neighbor features $X _ { [ v ] }$ . Increasing the number of layers only pushes the coefficients of each neighbor features to the stationary values. The domain $X _ { [ v ] }$ of such linear transformation is solely determined by EXTRACT and is independent of the model depth. Intuitively, if EXTRACT picks non-identical subgraphs for two nodes $u$ and $v$ , the aggregations should be different due to the different domains of the linear transformation. Therefore, SHADOW-GCN preserves local feature information whereas normal GCN preserves none. For structural information in $m _ { [ v ] }$ , note that $e _ { [ v ] }$ is a normalized degree distribution of the subgraph around $v$ , and -e[v]\u0003v indicates the role of the target node in the subgraph. By simply letting EXTRACT return the 1-hop subgraph, $\\left[ e _ { [ v ] } \\right] _ { v }$ alone already contains all the information preserved by a normal GCN, which is $v$ ’s degree in $\\bar { \\mathcal { G } }$ . For the general EXTRACT, $e _ { [ v ] }$ additionally reflects $v$ ’s ego-net structure. Thus, a deep SHADOW-GCN preserves more structural information than a deep GCN. ",
+        "bbox": [
+            173,
+            522,
+            826,
+            705
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. Let $\\overline { { \\pmb { m } } } _ { [ v ] } = \\phi _ { \\mathcal { G } } \\left( v \\right) \\cdot \\pmb { m } _ { [ v ] }$ where $\\phi _ { \\mathcal { G } }$ is any non-zero function only depending on the structural property of $v$ . Let $\\mathcal { M } = \\mathrm { \\large ~ \\left\\{ ~ \\overline { { ~ } } m _ { [ v ] } ~ \\lvert ~ v ~ \\in ~ \\mathcal { V } \\right\\} ~ }$ . Given $\\mathcal { G }$ , EXTRACT and some continuous probability distribution in $\\mathbb { R } ^ { | \\nu | \\times d }$ to generate $\\boldsymbol { X }$ , then $\\overline { { \\pmb { m } } } _ { [ v ] } \\neq \\overline { { \\pmb { m } } } _ { [ u ] }$ if $\\mathcal { V } _ { [ u ] } \\neq \\mathcal { V } _ { [ v ] }$ , almost surely. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            760
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3.2.1. Consider EXTRA $C T _ { 1 }$ , where $\\forall v \\in \\mathcal { V }$ , $| \\nu _ { [ v ] } | \\leq n$ . Then $\\begin{array} { r } { | \\mathcal { M } | \\geq \\left\\lceil \\frac { | \\mathcal { V } | } { n } \\right\\rceil } \\end{array}$ a.s. ",
+        "bbox": [
+            171,
+            765,
+            763,
+            787
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3.2.2. Consider EXTRA $C T _ { 2 }$ , where $\\forall u , v \\in \\mathcal { V }$ , $\\mathcal { V } _ { [ v ] } \\neq \\mathcal { V } _ { [ u ] }$ . Then $| \\mathcal { M } | = | \\mathcal { V } | a . s$ ",
+        "bbox": [
+            171,
+            794,
+            769,
+            813
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 proves SHADOW-GCN does not oversmooth: 1. A normal GCN pushes the aggregation of same-degree nodes to the same point, while SHADOW-GCN with $\\mathtt { E X T R A C T } _ { 2 }$ ensures any two nodes (even with the same degree) have different aggregation. 2. A normal GCN wipes out all information in $\\boldsymbol { X }$ after many times of aggregation, while SHADOW-GCN always preserves feature information. Particularly, with φG (v) = \u0000δ[v](v)\u0001−1/2, a normal GCN generates only one unique value of for all $v$ . By contrast, SHADOW-GNN generates $| \\nu |$ different values for any $\\phi _ { \\mathcal { G } }$ function. ",
+        "bbox": [
+            173,
+            821,
+            825,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We compare the expressivity by showing 1. SHADOW-SAGE can express all functions GraphSAGE can, and 2. SHADOW-SAGE can express some function GraphSAGE cannot. Recall, a GraphSAGE layer performs the following: $\\begin{array} { r } { \\dot { \\overline { { h _ { v } ^ { ( \\ell ) } } } } = \\sigma \\left( \\left( W _ { 1 } ^ { ( \\ell ) } \\right) ^ { \\top } h _ { v } ^ { ( \\ell - 1 ) } + \\left( W _ { 2 } ^ { ( \\ell ) } \\right) ^ { \\top } \\left( \\frac { 1 } { | \\mathcal { N } _ { v } | } \\sum _ { u \\in \\mathcal { N } _ { v } } \\dot { h _ { u } ^ { ( \\ell - 1 ) } } \\right) \\right) . } \\end{array}$ We can prove Point 1 by making an $L ^ { \\prime }$ -layer SHADOW-SAGE identical to an $L$ -layer GraphSAGE with the following steps: 1. let EXTRACT return the full $L$ -hop neighborhood, and 2. set $\\bar { \\boldsymbol { W } } _ { 1 } ^ { ( \\ell ) } = \\boldsymbol { I }$ , $W _ { 2 } ^ { ( \\ell ) } = { \\bf 0 }$ for $L + 1 \\leq \\ell \\leq L ^ { \\prime }$ . For point 2, we consider a target function: $\\tau \\left( \\boldsymbol { X } , \\boldsymbol { \\mathcal { G } } _ { [ v ] } \\right) =$ $\\begin{array} { r } { C \\cdot \\sum _ { u \\in \\mathcal { V } _ { [ v ] } } \\delta _ { [ v ] } \\left( u \\right) \\cdot \\pmb { x } _ { u } } \\end{array}$ for some neighborhood $\\mathcal { G } _ { [ v ] }$ , scaling constant $C$ and $\\delta _ { [ v ] } \\left( u \\right)$ as defined in Proposition 3.1. An expressive model should be able to learn well this simple linear function $\\tau$ . ",
+        "bbox": [
+            173,
+            116,
+            826,
+            257
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "GraphSAGE cannot learn $\\tau$ accurately, while SHADOW-SAGE can. We first show the GraphSAGE case. Let the depth of $\\mathcal { G } _ { [ v ] }$ be $L$ . Firstly, we need GraphSAGE to perform message passing for exactly $L$ times (where such a model can be implemented by, e.g., $L$ layers or $L ^ { \\prime }$ layers with $W _ { 2 } = \\mathbf { 0 }$ for $L ^ { \\prime } - L$ layers). Otherwise, the extra $L ^ { \\prime } - L$ message passings will propagate influence from nodes $v ^ { \\prime } \\notin \\mathcal { V } _ { [ v ] }$ , violating the condition that $\\tau$ is independent of $v ^ { \\prime }$ . Next, suppose GraphSAGE can learn a function $\\zeta$ such that on some $\\mathcal { G } _ { [ v ] } ^ { \\prime }$ , we have $\\zeta \\left( \\mathcal { G } _ { [ v ] } ^ { \\prime } \\right) = \\tau \\left( \\mathcal { G } _ { [ v ] } ^ { \\prime } \\right)$ . We construct another $\\mathcal { G } _ { [ v ] } ^ { \\prime \\prime }$ by adding an extra edge $e$ connecting two depth- $L$ nodes in $\\mathcal { G } _ { [ v ] } ^ { \\prime }$ . Edge $e$ changes the degree distribution $\\delta _ { [ v ] } \\left( \\cdot \\right)$ , and thus $\\tau \\left( \\mathcal { G } _ { [ v ] } ^ { \\prime } \\right) \\neq \\tau \\left( \\mathcal { G } _ { [ v ] } ^ { \\prime \\prime } \\right)$ . On the other hand, there is no way for GraphSAGE to propagate the influence of edge $e$ to the target $v$ , unless the model performs at least $L + 1$ message passings. So $\\zeta \\left( \\mathcal { G } _ { [ v ] } ^ { \\prime } \\right) = \\zeta \\left( \\bar { \\mathcal { G } } _ { [ v ] } ^ { \\prime \\prime } \\right)$ regardless of the activation function and weight parameters. Therefore, $\\zeta \\neq \\tau$ . For SHADOW-SAGE, let EXTRACT return $\\mathcal { G } _ { [ v ] }$ . Then the model can output $\\zeta ^ { \\prime } = \\left[ \\widehat { \\cal A } _ { [ v ] } ^ { L ^ { \\prime } } { \\cal X } \\right] _ { v , }$ after we 1. set $W _ { 1 } ^ { ( \\ell ) } = \\mathbf { 0 }$ and $W _ { 2 } ^ { ( \\ell ) } = I$ for all layers, and 2. either remove the non-linear activation or bypass ReLU by shifting $\\boldsymbol { X }$ with bias. With known results in Markov chain convergence theorem [27], we derive the following theorem by analyzing the convergence of $\\widehat { A } _ { [ v ] } ^ { L ^ { \\prime } }$ when $L ^ { \\prime } \\to \\infty$ . ",
+        "bbox": [
+            173,
+            261,
+            826,
+            520
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3. SHADOW-SAGE can approximate $\\tau$ with error decaying exponentially with depth. ",
+        "bbox": [
+            171,
+            523,
+            816,
+            539
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We have the following conclusions from above: 1. SHADOW-SAGE is more expressive than GraphSAGE. 2. appropriate EXTRACT function improves SHADOW-GNN expressivity, 3. There exists cases where it may be desirable to set the SHADOW-GNN depth much larger than the subgraph depth. ",
+        "bbox": [
+            174,
+            550,
+            826,
+            593
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Expressivity Analysis on SHADOW-GIN: Topological Learning Perspective ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            608,
+            732,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While GCN and GraphSAGE are popular architectures in practice, they are not the theoretically most discriminative ones. The work in [49] establishes the relation in discriminativeness between GNNs and 1-dimensional Weisfeiler-Lehman test (i.e., 1-WL). And GIN [49] is an example architecture achieving the same discriminativeness as 1-WL. We show that applying the decoupling principle can further improve the discriminativeness of such GNNs, making them more powerful than 1-WL. ",
+        "bbox": [
+            173,
+            635,
+            826,
+            705
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1-WL is a graph isomorphism test aiming at distinguishing graphs of different structures. A GNN as expressive as 1-WL thus well captures the topological property of the target node. While 1-WL is already very powerful, it may still fail in some cases. e.g., it cannot distinguish certain non-isomorphic, regular graphs. To understand why SHADOW-GNN works, we first need to un",
+        "bbox": [
+            174,
+            710,
+            491,
+            820
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fd819a2beeeba3fca87a30aa3012f69f2b800a51d7754d5d403f4ebe7ed94bb.jpg",
+        "image_caption": [
+            "Figure 1: Example 3-regular graph and the 1-hop subgraphs of the target nodes $u$ and $v$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            509,
+            713,
+            808,
+            782
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "derstand why 1-WL fails. In a regular graph, all nodes have the same degree, and thus the “regular” property describes a global topological symmetry among nodes. Unfortunately, 1-WL (and the corresponding normal GNN) also operates globally on $\\mathcal { G }$ . Intuitively, on two different regular graphs, there is no way for 1-WL (and the normal GNN) to assign different labels by breaking such symmetry. ",
+        "bbox": [
+            174,
+            821,
+            826,
+            877
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, SHADOW-GNN can break such symmetry by applying decoupling. In Section 1, we have discussed how SHADOW-GNN is built from the local perspective on the full graph. The key property benefiting SHADOW-GNN is that a subgraph of a regular graph may not be regular. Thus, SHADOW-GNN can distinguish nodes in a regular graph with the non-regular subgraphs as the scope. We illustrate the intuition with the example in Figure 1. The graph $\\mathcal { G }$ is 3-regular and we assume all nodes have identical features. Our goal is to discriminate nodes $u$ and $v$ since their neighborhood structures are different. No matter how many iterations 1-WL runs, or how many layers the normal GNN has, they cannot distinguish $u$ and $v$ . On the other hand, a SHADOW-GNN with 1-hop EXTRACT and at least 2 layers can discriminate $u$ and $v$ . ",
+        "bbox": [
+            173,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            188
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.4. Consider GNNs whose layer function is defined by ",
+        "bbox": [
+            173,
+            190,
+            606,
+            207
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78c78712f53087d85da2fdecc5a7b531a5fabe4e3fc2e2cd08324d0241da86af.jpg",
+        "text": "$$\n\\pmb { h } _ { v } ^ { ( \\ell ) } = f _ { 1 } ^ { ( \\ell ) } \\left( \\pmb { h } _ { v } ^ { ( \\ell - 1 ) } , \\ \\sum _ { u \\in \\mathcal { N } _ { v } } f _ { 2 } ^ { ( \\ell ) } \\left( \\pmb { h } _ { v } ^ { ( \\ell - 1 ) } , \\pmb { h } _ { u } ^ { ( \\ell - 1 ) } \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            209,
+            673,
+            253
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { 1 } ^ { ( \\ell ) }$ and $f _ { 2 } ^ { ( \\ell ) }$ are the update and message functions of layer- $\\ell ,$ , implemented as MLPs. Then, such SHADOW-GNN is more discriminative than the -dimensional Weisfeiler-Lehman test. ",
+        "bbox": [
+            176,
+            260,
+            825,
+            290
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The theorem also implies that SHADOW-GIN is more discriminative than a normal GIN due to the correspondence between GIN and 1-WL. See Appendix A for the proof of all theorems in Section 3. ",
+        "bbox": [
+            174,
+            299,
+            825,
+            328
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 Subgraph Extraction Algorithms ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            442,
+            359
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our decoupling principle does not rely on specific subgraph extraction algorithms. Appropriate EXTRACT can be customized given the characteristic of $\\mathcal { G }$ , and different EXTRACT leads to different implementation of our decoupling principle. In general, we summarize three approaches to design EXTRACT: 1. heuristic based, where we pick graph metrics that reflect neighbor importance and then design EXTRACT by such metrics; 2. model based, where we assume a generation process on $\\mathcal { G }$ and set EXTRACT as the reverse process, and 3. learning based, where we integrate the design of EXTRACT as part of the GNN training. In the following, we present several examples on heuristic based EXTRACT, which we also empirically evaluate in Section 5. We leave detailed evaluation on the model based and learning based EXTRACT as future work. See also Appendix C for details. ",
+        "bbox": [
+            173,
+            368,
+            825,
+            494
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Example heuristic based EXTRACT. The algorithm is derived from the selected graph metrics. For example, with the metric being shortest path distance, we design a $L$ -hop extractor. i.e., EXTRACT returns the full set or randomly selected subset of the target node’s $L$ -hop neighbors in $\\mathcal { G }$ . Picking the random walk landing probability as the metric, we can design a PPR-based extractor. i.e., we first run the Personalized PageRank (PPR) algorithm on $\\mathcal { G }$ to derive the PPR score of other nodes relative to the target node. Then EXTRACT define $\\nu _ { [ v ] }$ by picking the top- $K$ nodes with the highest PPR scores. The subgraph $\\mathcal { G } _ { [ v ] }$ is the node-induced subgraph1 of $\\mathcal { G }$ from $\\nu _ { [ v ] }$ . One can easily extend this approach by using other metrics such as Katz index [20], SimRank [19] and feature similarity. ",
+        "bbox": [
+            173,
+            500,
+            825,
+            614
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 Architecture ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            628,
+            300,
+            643
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Subgraph pooling. For a normal GNN performing node classification, the multi-layer message passing follows a “tree structure”. The nodes at level $L$ of the tree correspond to the $L$ -hop neighborhood. And the tree root outputs the final embedding of the target node. Thus, there is no way to apply subgraph pooling or READOUT on the final layer output, since the “pool” only contains a single vector. For a SHADOW-GNN, since we decouple the $\\bar { L } ^ { \\mathrm { t h } }$ layer from the $L$ -hop neighborhood, it is natural to let each layer (including the final layer) output embeddings for all subgraph nodes. This leads to the design to READOUT all the subgraph node embeddings as the target node embedding. ",
+        "bbox": [
+            173,
+            654,
+            825,
+            752
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We can understand the pooling for SHADOW-GNN from another perspective. In a normal GNN, the target node at the final layer receives messages from all neighbors, but two neighbor nodes may not have a chance to exchange any message to each other (e.g., two nodes $L$ -hop away from the target may be $2 L$ -hop away from each other). In our design, a SHADOW-GNN can pass messages between any pair of neighbors when the model depth is large enough. Therefore, all the subgraph node embeddings at the final layer capture meaningful information of the neighborhood. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            842
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, the power of the decoupling principle lies in that it establishes the connection between the node- / link-level task and the graph-level task. e.g., to classify a node is seen as to classify the subgraph surrounding the node. From the neural architecture perspective, we can apply any subgraph pooling / READOUT operation originally designed for graph classification (e.g., [57, 24, 4]) to enhance the node classification / link prediction of SHADOW-GNN. In particular, in the vanilla SHADOW-GNN, we can implement a trivial READOUT as “discarding all neighbor embeddings”, corresponding to performing center pooling. See Appendix D and F.3 for algorithm and experiments. ",
+        "bbox": [
+            176,
+            848,
+            823,
+            876
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            161
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Subgraph ensemble. It may be challenging in practice to design a single EXTRACT capturing all meaningful characteristics of the neighborhood. We can use multiple EXTRACT to jointly define the receptive field, and then ensemble multiple SHADOW-GNN at the subgraph level. Consider $R$ candidates $\\{ \\mathtt { E X T R A C T } _ { i } \\}$ , each returning $\\mathcal { G } _ { [ v ] } ^ { i }$ . To generate $v$ ’s embedding, we first use $R$ branches of $L ^ { \\prime }$ -layer GNN to obtain intermediate embeddings for each $\\mathcal { G } _ { v } ^ { i }$ , and then aggregate the $R$ embeddings by some learnable function $g$ . In practice, we design $g$ as an attention based aggregation function (see Appendix D.2). Subgraph ensemble is useful both when $\\{ \\mathtt { E X T R A C T } _ { i } \\}$ consists of different algorithms and when each EXTRA $\\Omega  { \\mathrm { T } } _ { i }$ performs the same algorithm under different parameters. ",
+        "bbox": [
+            173,
+            166,
+            826,
+            282
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "CASE STUDY Consider PPR-based $\\mathtt { E X T R A C T } _ { i }$ with different threshold $\\theta _ { i }$ on the neighbor PPR score. A SHADOW-GNN-ensemble can approximate PPRGo [6]. PPRGo generates embedding as: ${ \\pmb { \\xi } } _ { v } =$ $\\sum _ { u \\in \\mathcal { V } _ { [ v ] } } \\pi _ { u } h _ { v }$ , where $\\pi _ { u }$ is $u$ ’s PPR score and $\\begin{array} { r } { h _ { v } = \\mathtt { M L P } \\left( \\pmb { x } _ { v } \\right) } \\end{array}$ . We can partition $\\begin{array} { r } { \\mathcal { V } _ { [ v ] } = \\bigcup _ { i = 1 } ^ { R } \\mathcal { V } _ { [ v ] } ^ { i } } \\end{array}$ s.t. nodes in $\\mathcal { V } _ { [ v ] } ^ { i }$ have similar PPR scores denoted by $\\widetilde { \\pi } _ { i }$ , and $\\widetilde { \\pi } _ { i } \\leq \\widetilde { \\pi } _ { i + 1 }$ . $\\begin{array} { r } { \\mathrm { S o } \\xi _ { v } = \\sum _ { i = 1 } ^ { R } \\rho _ { i } \\left( \\sum _ { u \\in \\mathcal { V } _ { i } ^ { ' } } \\pmb { h } _ { u } \\right) } \\end{array}$ , where $\\begin{array} { r } { \\rho _ { i } = \\widetilde { \\pi } _ { i } - \\sum _ { j < i } \\widetilde { \\pi } _ { j } } \\end{array}$ and $\\begin{array} { r } { \\mathcal { V } _ { i } ^ { \\prime } = \\bigcup _ { k = i } ^ { R } \\mathcal { V } _ { [ v ] } ^ { k } } \\end{array}$ . Now for each branch of SHADOW-GNN-ensemble, elet parameter $\\theta _ { i } = \\widetilde { \\pi } _ { i }$ eso that $\\mathtt { E X T R A C T } _ { i }$ returns $\\mathcal { V } _ { i } ^ { \\prime }$ . The GNN on $\\mathcal { V } _ { i } ^ { \\prime }$ can then learn $\\sum _ { u \\in \\mathcal { V } _ { i } ^ { \\prime } } \\boldsymbol { h } _ { u }$ (e.g., by a simple “mean” READOUT). Finally, set the ensemble weight as $\\rho _ { i }$ . SHADOW-GNN-ensemble learns $\\xi _ { v }$ . As EXTRACT also preserves graph topology, our model can be more expressive than PPRGo. ",
+        "bbox": [
+            173,
+            287,
+            826,
+            425
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.6 Practical Design: SHADOW-GNN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            440,
+            446,
+            455
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now discuss the practical implementation of decoupled GNN – SHADOW-GNN. As the name suggests, in SHADOW-GNN, the scope is a shallow subgraph (i.e., with depth often set to 2 or 3). ",
+        "bbox": [
+            174,
+            465,
+            823,
+            494
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In many realistic scenarios (e.g., citation networks, social networks, product recommendation graphs), a shallow neighborhood is both necessary and sufficient for the GNN to learn well. On “sufficiency”, we consider the social network example: the friend of a friend of a friend may share little commonality with you, and close friends may be at most 2 hops away. Formally, by the $\\gamma$ -decaying theorem [56], a shallow neighborhood is sufficient to accurately estimate various graph metrics. On “necessity”, since the neighborhood size may grow exponentially with hops, a deep neighborhood would be dominated by nodes irrelevant to the target. The corresponding GNN would first need to differentiate the many useless nodes from the very few useful ones, before it can extract meaningful features from the useful nodes. Finally, a shallow subgraph ensures scalability by avoiding “neighborhood explosion”. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            626
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark on decoupling. So far we have defined a decoupled model as having the model depth $L ^ { \\prime }$ larger than the subgraph depth $L$ . Strictly speaking, a decoupled model also admits $L ^ { \\prime } = \\bar { L }$ . For example, suppose in the full $L$ -hop neighborhood, there are $70 \\%$ nodes $L$ hops away. Applying decoupling, the EXTRACT excludes most of the $L$ -hop neighbors, and the resulting subgraph $\\mathcal { G } _ { [ v ] }$ contains only $20 \\%$ nodes $L$ hops away. Then it is reasonable to consider an $L$ -layer model on such a depth- $L$ subgraph as also a decouple model. Compared with an $L$ -layer model on the full $L$ -hop neighborhood, an $L$ -layer model on such a depth- $L$ $\\mathcal { G } _ { [ v ] }$ propagates much less information from nodes $L$ hops away. So the $L$ message passings are indeed decoupled from the full $L$ -hop neighborhood. ",
+        "bbox": [
+            174,
+            631,
+            825,
+            742
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remark on neighborhood. The “sufficiency” and “necessity” in shallow neighborhood are not universal. In many other applications, long-range dependencies can be critical, as studied in [2]. In such cases, our practical implementation of SHADOW-GNN would incur accuracy loss. However, our decoupling principle in general may still be beneficial – “shallow subgraph” is a practical guideline rather than a theoretical requirement. We leave the study on such applications as future work. ",
+        "bbox": [
+            174,
+            748,
+            825,
+            819
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            837,
+            321,
+            854
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Deep GNNs. To improve the GNN performance while increasing the model depth, various layer architectures have been proposed. AS-GCN [18], DeepGCN [28], JK-net [50], MixHop [1], Snowball [33], DAGNN [31] and GCNII [34] all include some variants of residue connection, either across multiple layers or within a single layer. In principle, such architectures can also benefit the feature propagation of a deep SHADOW-GNN, since their design does not rely on a specific neighborhood (e.g., $L$ -hop). In addition to architectures, DropEdge [39] and Bayesian-GDC [14] propose regularization techniques by adapting dropout [42] to graphs. Such techniques are only applied during training, and inference may still suffer from issues such as oversmoothing. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            823,
+            160
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sampling based methods. Neighbor or subgraph sampling techniques have been proposed to improve training efficiency. FastGCN [8], VR-GCN [7], AS-GCN [18], LADIES [61] and MVSGNN [10] sample neighbor nodes per GNN layer. Cluster-GCN [9] and GraphSAINT [55] sample a subgraph as the training minibatch. While sampling also changes the receptive field, all the above methods are fundamentally different from ours. The training samplers aim at estimating the quantities related to the full graph (e.g., the aggregation of the full $L$ -hop neighborhood), and so the inference model still operates on the full neighborhood to avoid accuracy loss. For SHADOW-GNN, since the decoupling principle is derived from a local view on $\\mathcal { G }$ , our EXTRACT does not estimate any full neighborhood quantities. Consequently, the sampling based methods only improve the training efficiency, while SHADOW-GNN addresses the computation challenge for both training and inference. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            305
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Re-defining the neighborhood. Various works reconstruct the original graph and apply the GNN on the re-defined neighborhood. GDC [23] views the reconstructed adjacency matrix as the diffusion matrix. SIGN [11] applies reconstruction for customized graph operators. AM-GCN [46] utilizes the reconstruction to separate feature and structure information. The above work re-define the neighborhood. However, they still have tightly coupled depth and scope. SHADOW-GNN can also work with the reconstructed graph $\\mathcal { G } ^ { \\prime }$ by simply applying EXTRACT on $\\mathcal { G } ^ { \\prime }$ . ",
+        "bbox": [
+            174,
+            311,
+            825,
+            395
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            416,
+            312,
+            434
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Setup. We evaluate SHADOW-GNN on seven graphs. Six of them are for the node classification task: Flickr [55], Reddit [12], Yelp [55], ogbn-arxiv, ogbn-products and ogbn-papers100M [16]. The remaining is for the link prediction task: ogbl-collab [16]. The sizes of the seven graphs range from 9K nodes (Flickr) to 110M nodes (ogbn-papers100M). Flickr, Reddit and Yelp are under the inductive setting. ogbn-arxiv, ogbn-products and ogbn-papers100M are transductive. Consistent with the original papers, for the graphs on node classification, we measure “F1-micro” score for Yelp and “accuracy” for the remaining five graphs. For the link prediction task, we use “Hits $@ 5 0 ^ { \\circ }$ as the metric. See Appendix E.1 for details. We use W&B [5] for experiment tracking. ",
+        "bbox": [
+            173,
+            449,
+            826,
+            560
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We construct SHADOW with six backbone models: GCN [22], GraphSAGE [12], GAT [44], JK-Net [50], GIN [49], SGC [47]. The first five are representatives of the state-of-the-art GNN architectures, jointly covering various message aggregation functions as well as the skip connection design. SGC simplifies normal GCN by moving all the neighbor propagation to the pre-processing step. Therefore, SGC is suitable for evaluating oversmoothing. The non-SHADOW models are trained with both full-batch and GraphSAINT-minibatch [55]. Due to the massive size of the full $L$ -hop neighborhood, we need to perform sampling when training normal GNNs in order to make the computation time tractable. GraphSAINT is suitable for our purpose since 1. it is the state-of-the-art minibatch method which achieves high accuracy, and 2. it supports various GNN architectures and scales well to large graphs. On the other hand, for SHADOW-GNN, both training and inference are always executed in minibatches. One advantage of SHADOW-GNN is that the decoupling enables straightforward minibatch construction: each target just independently extracts the small subgraph on its own. ",
+        "bbox": [
+            174,
+            565,
+            826,
+            732
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We implement two EXTRACT described in Section 3.4: 1. “PPR”, where we set the node budget $K$ as $\\{ 2 0 0 , 4 0 0 \\}$ for the largest ogbn-papers100M and $K \\leq 2 0 0$ for all other graphs; 2. “ $L$ -hop”, where we set the depth as $\\{ 1 , 2 \\}$ . We implement various subgraph pooling functions: “mean” and “max” evaluated in this section and others evaluated in Appendix F.3. For the model depth, since $L ^ { \\prime } = 3$ is the standard setting in the literature (e.g., see the benchmarking in OGB [16]), we start from $L ^ { \\prime } = 3$ and further evaluate a deeper model of $L ^ { \\prime } = 5$ . All accuracy are measured by 5 runs without fixing random seeds. Hyperparameter tuning and architecture configurations are in Appendix E.4. ",
+        "bbox": [
+            174,
+            738,
+            825,
+            835
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "SHADOW-GNN neighborhood. For both normal and SHADOW GNNs, Figure 2 shows on average how many neighbors are $L$ hops away from the target. For a normal GNN, the size of the neighborhood grows rapidly with respect to $L$ , and the nodes 4 hops away dominate the neighborhood. For SHADOW-GNN using the Table 1 EXTRACT, most neighbors concentrate within 2 hops. A small number of nodes are 3 hops away. Almost no nodes are 4 or more hops away. Importantly, not only does the composition of the two kinds of neighborhood differ significantly, but also the size of SHADOW-GNN scope is much smaller (see also Table 1). Such small subgraphs are essential to high computation efficiency. Finally, we can ignore the very few distant neighbors $ { \\left( L \\geq 4 \\right) }$ , and regard the (effective) depth of SHADOW-GNN subgraph as $L = 2$ (or at most 3). Under such practical value of $L$ , a model with $L ^ { \\prime } \\geq 3$ is indeed a SHADOW-GNN (see “Remark on decoupling” in Section 3.6). ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ff0d72768a180365177f62556e21fbfcee94cfc5f872b8f2a83346e32c2987c.jpg",
+        "image_caption": [
+            "Figure 2: Neighborhood composition "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            93,
+            540,
+            243
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0d3040806271ed912e8436ec3211d9ba63a3ac476cee9100cccc0c4712716703.jpg",
+        "image_caption": [
+            "Figure 3: SGC oversmoothing "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            594,
+            90,
+            812,
+            234
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            273,
+            825,
+            343
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/68b6276e9d4bb1c7d79dc8a30a7c1a9954eef416ca50dfae9050189b51d9129a.jpg",
+        "table_caption": [
+            "Table 1: Comparison on test accuracy / F1-micro score and inference cost (tuned with DropEdge) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Layers</td><td colspan=\"2\">Flickr</td><td colspan=\"2\">Reddit</td><td colspan=\"2\">Yelp</td><td colspan=\"2\">ogbn-arxiv</td><td colspan=\"2\">ogbn-products</td></tr><tr><td>Accuracy</td><td>Cost</td><td>Accuracy</td><td>Cost</td><td>F1-micro</td><td>Cost</td><td>Accuracy</td><td>Cost</td><td>Accuracy</td><td>Cost</td></tr><tr><td rowspan=\"3\">GCN</td><td>3 5</td><td>0.5159±0.0017</td><td>2E0</td><td>0.9532±0.0003</td><td>6E1</td><td>0.4028±0.0019</td><td>2E1</td><td>0.7170±0.0026</td><td>1E1</td><td>0.7567±0.0018</td><td>5E0</td></tr><tr><td></td><td>0.5217±0.0016</td><td>2E2</td><td>0.9495±0.0012</td><td>1E3</td><td>OOM</td><td>1E3</td><td>0.7186±0.0017</td><td>1E3</td><td>OOM</td><td>9E2</td></tr><tr><td>3</td><td>0.5155±0.0027</td><td>2E0</td><td>0.9523±0.0003</td><td>6E1</td><td>0.5110±0.0012</td><td>2E1</td><td>0.7093±0.0003</td><td>1E1</td><td>0.8003±0.0024</td><td>5E0</td></tr><tr><td>GCN-SAINT</td><td>5</td><td>0.5165±0.0026</td><td>2E2</td><td>0.9523±0.0012</td><td>1E3</td><td>0.5012±0.0021</td><td>1E3</td><td>0.7039±0.0020</td><td>1E3</td><td>0.7992±0.0021</td><td>9E2</td></tr><tr><td>SHADOW-GCN</td><td>3</td><td>0.5234±0.0009</td><td>(1</td><td>0.9576±0.0005</td><td>1</td><td>0.5291±0.0020</td><td>(1</td><td>0.7180±0.0024</td><td>1</td><td>0.7742±0.0037</td><td>1)</td></tr><tr><td>(PPR)</td><td>5</td><td>0.5268±0.0008</td><td>1E0</td><td>0.9564±0.0004</td><td>1E0</td><td>0.5323±0.0020</td><td>2E0</td><td>0.7206±0.0025</td><td>2E0</td><td>0.7821±0.0043</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5286±0.0013</td><td>1E0</td><td>0.9624±0.0002</td><td>1E0</td><td>0.5393±0.0036</td><td>2E0</td><td>0.7223±0.0018</td><td>2E0</td><td>0.7914±0.0044</td><td>2E0</td></tr><tr><td rowspan=\"3\">GraphSAGE</td><td>3</td><td>0.5140±0.0014</td><td>3E0</td><td>0.9653±0.0002</td><td>5E1</td><td>0.6178±0.0033</td><td>2E1</td><td>0.7192±0.0027</td><td>1E1</td><td>0.7858±0.0013</td><td>4E0</td></tr><tr><td>5</td><td>0.5154±0.0052</td><td>2E2</td><td>0.9626±0.0004</td><td>1E3</td><td>0OM</td><td>2E3</td><td>0.7193±0.0037</td><td>1E3</td><td>0OM</td><td>1E3</td></tr><tr><td>3</td><td>0.5176±0.0032</td><td>3E0</td><td>0.9671±0.0003</td><td>5E1</td><td>0.6453±0.0011</td><td>2E1</td><td>0.7107±0.0003</td><td>1E1</td><td>0.7923±0.0023</td><td>4E0</td></tr><tr><td>SAGE-SAINT</td><td>5</td><td>0.5201±0.0032</td><td>2E2</td><td>0.9670±0.0010</td><td>1E3</td><td>0.6394±0.0003</td><td>2E3</td><td>0.7013±0.0021</td><td>1E3</td><td>0.7964±0.0022</td><td>1E3</td></tr><tr><td>SHADOW-SAGE</td><td>3</td><td>0.5312±0.0019</td><td>1E0</td><td>0.9672±0.0003</td><td>1E0</td><td>0.6542±0.0002</td><td>1E0</td><td>0.7163±0.0028</td><td>1E0</td><td>0.7935±0.0031</td><td>1E0</td></tr><tr><td>(2-hop)</td><td>5</td><td>0.5335±0.0015</td><td>2E0</td><td>0.9675±0.0005</td><td>2E0</td><td>0.6525±0.0003</td><td>2E0</td><td>0.7180±0.0030</td><td>2E0</td><td>0.7995±0.0022</td><td>2E0</td></tr><tr><td> SHADOW-SAGE</td><td>3</td><td>0.5356±0.0013</td><td>(1)</td><td>0.9688±0.0002</td><td>(1)</td><td>0.6538±0.0003</td><td>(1)</td><td>0.7227±0.0012</td><td>(1)</td><td>0.7905±0.0026</td><td>(1)</td></tr><tr><td>(PPR)</td><td>5</td><td>0.5417±0.0006</td><td>2E0</td><td>0.9692±0.0007</td><td>2E0</td><td>0.6518±0.0002</td><td>2E0</td><td>0.7238±0.0007</td><td>2E0</td><td>0.8005±0.0040</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5395±0.0013</td><td>2E0</td><td>0.9703±0.0003</td><td>2E0</td><td>0.6564±0.0004</td><td>1E0</td><td>0.7258±0.0017</td><td>2E0</td><td>0.8067±0.0037</td><td>1E0</td></tr><tr><td rowspan=\"3\">GAT</td><td>3</td><td>0.5070±0.0032</td><td>2E1</td><td>0OM</td><td>3E2</td><td>0OM</td><td>2E2</td><td>0.7201±0.0011</td><td>1E2</td><td>0OM</td><td>3E1</td></tr><tr><td>5</td><td>0.5164±0.0033</td><td>2E2</td><td>OOM</td><td>2E3</td><td>OOM</td><td>2E3</td><td>OOM</td><td>3E3</td><td>OOM</td><td>2E3</td></tr><tr><td>3</td><td>0.5225±0.0053</td><td>2E1</td><td>0.9671±0.0003</td><td>3E2</td><td>0.6459±0.0002</td><td>2E2</td><td>0.6977±0.0003</td><td>1E2</td><td>0.8027±0.0028</td><td>3E1</td></tr><tr><td>GAT-SAINT</td><td>5</td><td>0.5153±0.0034</td><td>2E2</td><td>0.9651±0.0024</td><td>2E3</td><td>0.6478±0.0012</td><td>2E3</td><td>0.6954±0.0013</td><td>3E3</td><td>0.7990±0.0072</td><td>2E3</td></tr><tr><td>SHADOW-GAT</td><td></td><td>0.5349±0.0023</td><td>(1)</td><td>0.9707±0.0004</td><td>(1)</td><td>0.6575±0.0004</td><td>(1)</td><td>0.7235±0.0020</td><td>(1)</td><td>0.8006±0.0014</td><td>(1)</td></tr><tr><td>(PPR)</td><td>3</td><td>0.5352±0.0028</td><td>2E0</td><td>0.9713±0.0004</td><td>2E0</td><td>0.6559±0.0002</td><td>2E0</td><td>0.7274±0.0022</td><td>2E0</td><td>0.8071±0.0004</td><td>2E0</td></tr><tr><td>+Pooling</td><td>3/5</td><td>0.5364±0.0026</td><td>1E0</td><td>0.9710±0.0004</td><td>2E0</td><td>0.6566±0.0005</td><td>1E0</td><td>0.7265±0.0028</td><td>2E0</td><td>0.8142±0.0031</td><td>1E0</td></tr></table>",
+        "bbox": [
+            176,
+            381,
+            823,
+            638
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Comparison with baselines. Table 1 shows the performance comparison of SHADOW-GNN with the normal GNNs. All models on all datasets have uniform hidden dimension of 256. We define the metric “inference cost” as the average amount of computation to generate prediction for one test node. Inference cost is a measure of computation complexity (see Appendix B for the calculation) and is independent of hardware / implementation factors such as parallelization strategy, batch processing, etc. For cost of SHADOW-GNN, we do not include the overhead on computing EXTRACT, since it is hard to calculate such cost analytically. Empirically, subgraph extraction is much cheaper than model computation (see Figure 8, 9 for time evaluation on CPU and GPU). During training, we apply DropEdge [39] to both the baseline and SHADOW models. DropEdge helps improve the baseline accuracy by alleviating oversmoothing, and benefits SHADOW-GNN due to its regularization effects. See Appendix F.2 for results on other architectures including GIN and JK-Net. ",
+        "bbox": [
+            173,
+            656,
+            825,
+            808
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACCURACY We aim at answering the following questions: 1. Can we improve accuracy by decoupling a baseline model? 2. What architecture components can we tune to improve accuracy of a decoupled model? 3. What EXTRACT can we tune to improve the accuracy of a decoupled model? To answer Q1, we fix the backbone architecture and remove pooling. Then we inspect “3-layer normal GNN vs. 3-layer SHADOW-GNN” and “5-layer normal GNN vs. 5-layer SHADOW-GNN”. Clearly, SHADOW-GNNs (with scope size no more than 200) in general achieve significantly higher accuracy than the normal GNNs. This indicates that a shallow neighborhood contains sufficient information, and customizing the scope can benefit accuracy even without architecture changes (from Figure 2, a depth-3 $\\mathcal { G } _ { [ v ] }$ differs significantly from the 3-hop neighborhood). To answer Q1, we focus on the PPR EXTRACT and thus compare the rows in blue background. We use the 3-layer SHADOW-GNN without pooling as the baseline and analyze the effects of 1. increasing the GNN depth without expanding scope, and 2. adding subgraph pooling. Comparing among the rows in light blue background, we observe that in many cases, simply increasing the depth from 3 to 5 leads to significant accuracy gain. Comparing the ligh blue rows with the dark blue rows, we observe that sometimes pooling can further improve the accuracy of a SHADOW-GNN. In conclusion, both types of architecture tuning are effective ways of optimizing a SHADOW-GNN. Finally, to answer Q3, we compare the light blue rows with the light yellow rows. In general, PPR EXTRACT leads to higher accuracy than 2-hop EXTRACT, demonstrating the importance of designing a good EXTRACT. ",
+        "bbox": [
+            174,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            258
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "INFERENCE COST Inference cost of SHADOW-GNN is orders of magnitude lower than the normal GNNs (a 5-layer SHADOW-GNN is still much cheaper than a 3-layer normal GNN). The high cost of the baselines is due to the “neighborhood explosion” with respect to more layers. SHADOW-GNN is efficient and scalable as the cost only grows linearly with the model depth. Note that GraphSAINT only improves efficiency during training since its inference operates on the full $L$ -hop neighborhood. ",
+        "bbox": [
+            173,
+            263,
+            825,
+            335
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/11a6941c95351465d2299be4671f3d428265b59d9e5d5fb9cb547b22094f9e52.jpg",
+        "table_caption": [
+            "Table 2: Leaderboard comparison on papers100M "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Test accuracy</td><td>Val accuracy</td><td>Neigh size</td></tr><tr><td>GraphSAGE+incep</td><td>0.6706±0.0017</td><td>0.7032±0.0011</td><td>4E5</td></tr><tr><td>SIGN-XL</td><td>0.6606±0.0019</td><td>0.6984±0.0006</td><td>&gt;4E5</td></tr><tr><td>SGC</td><td>0.6329±0.0019</td><td>0.6648±0.0020</td><td>&gt;4E5</td></tr><tr><td>SHADOW-GAT200</td><td>0.6681±0.0016</td><td>0.7019±0.0011</td><td>2E2</td></tr><tr><td>SHADOW-GAT400</td><td>0.6708±0.0017</td><td>0.7073±0.0011</td><td>3E2</td></tr></table>",
+        "bbox": [
+            485,
+            352,
+            818,
+            428
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Scaling to 100 million nodes. We further scale SHADOW-GNN to ogbn-papers100M, one of the largest public dataset. Even through the full graph size is at least two orders of magnitude larger than the graphs in Table 1, the localized scope of SHADOW-GNN barely needs to increase. Since SHADOW-GNN performs ",
+        "bbox": [
+            174,
+            340,
+            470,
+            438
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "minibatch computation, a low-end GPU with limited memory capacity can compute SHADOW-GNN on ogbn-papers100M efficiently. We show in Appendix F.1 that we can train and inference our model with as little as 4GB GPU memory consumption. This is infeasible using normal GNNs. Table 2 summarizes our comparison with the top leaderboard methods [45, 11, 47]. We only include those methods that do not use node labels as the model input (i.e., the most standard setup). We achieve at least 3 orders of magnitude reduction in neighborhood size without sacrificing accuracy. For SIGN-XL and SGC, their neighborhood is too large to count the exact size. Also, their preprocessing consumes $5 \\times$ more CPU memory than SHADOW-GNN (Appendix F.1). ",
+        "bbox": [
+            173,
+            438,
+            825,
+            547
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/d4dbf7e06ebf934ed91b38cd6929e66e87e16f57dc0619724dc2320235aef67f.jpg",
+        "table_caption": [
+            "Table 3: Leaderboard comparison on collab "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Test Hits @50</td><td>Val Hits @50</td></tr><tr><td>SEAL</td><td>0.5371±0.0047</td><td>0.6389±0.0049</td></tr><tr><td>DeeperGCN</td><td>0.5273±0.0047</td><td>0.6187±0.0045</td></tr><tr><td>LRGA+GCN</td><td>0.5221±0.0072</td><td>0.6088±0.0059</td></tr><tr><td>SHADOW-SAGE</td><td>0.5492±0.0022</td><td>0.6524±0.0017</td></tr></table>",
+        "bbox": [
+            535,
+            565,
+            823,
+            638
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Extending to link-level task. We further show that SHADOW-GNN is general and can be extended to the link prediction task. There are two settings of ogbl-collab. We follow the one where validation edges cannot be used in training updates. This is the setting which most leaderboard methods follow. Table 3 shows the comparison with the top GNN models under the same setting. SHADOW-SAGE outperforms the rank-1 model with significant margin. ",
+        "bbox": [
+            174,
+            554,
+            519,
+            659
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            178,
+            651,
+            803,
+            665
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Oversmoothing. To validate Theorem 3.2, we pick SGC as the backbone architecture. SGC with power $L$ is equivalent to $L$ -layer GCN without activation. Performance comparison between SGC and SHADOW-SGC thus reveals the effect of oversmoothing without introducing other factors due to optimizing deep neural networks (e.g., vanishing gradients). In Figure 3, we vary the power of SGC and SHADOW-SGC from 1 to 40 (see Appendix E.5 for details). While SGC gradually collapses local information into global “white noise”, accuracy of SHADOW-SGC does not degrade. This validates our theory that extracting local subgraphs prevents oversmoothing. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            768
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            787,
+            299,
+            804
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have presented a design principle to decouple the depth and scope of GNNs. Applying such a principle on various GNN architectures simultaneously improves expressivity and computation scalability of the corresponding models. We have presented thorough theoretical analysis on expressivity from three different perspectives, and also rich design components (e.g., subgraph extraction functions, architecture extensions) to implement such design principle. Experiments show significant performance improvement over a wide range of graphs, GNN architectures and learning tasks. ",
+        "bbox": [
+            174,
+            819,
+            825,
+            902
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. arXiv preprint arXiv:1905.00067, 2019.   \n[2] Uri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In International Conference on Learning Representations, 2021.   \n[3] Reid Andersen, Fan Chung, and Kevin Lang. Local graph partitioning using pagerank vectors. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pages 475–486. IEEE, 2006.   \n[4] Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Spectral clustering with graph neural networks for graph pooling. In International Conference on Machine Learning, pages 874–883. PMLR, 2020.   \n[5] Lukas Biewald. Experiment tracking with weights and biases, 2020. Software available from wandb.com.   \n[6] Aleksandar Bojchevski, Johannes Klicpera, Bryan Perozzi, Amol Kapoor, Martin Blais, Benedek Rózemberczki, Michal Lukasik, and Stephan Günnemann. Scaling graph neural networks with approximate pagerank. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 2464–2473, New York, NY, USA, 2020. Association for Computing Machinery.   \n[7] Jianfei Chen, Jun Zhu, and Le Song. Stochastic training of graph convolutional networks with variance reduction, 2017.   \n[8] Jie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast learning with graph convolutional networks via importance sampling. In International Conference on Learning Representations, 2018.   \n[9] Wei-Lin Chiang, Xuanqing Liu, Si Si, Yang Li, Samy Bengio, and Cho-Jui Hsieh. Cluster-gcn: An efficient algorithm for training deep and large graph convolutional networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’19, page 257–266, New York, NY, USA, 2019. Association for Computing Machinery.   \n[10] Weilin Cong, Rana Forsati, Mahmut Kandemir, and Mehrdad Mahdavi. Minimal variance sampling with provable guarantees for fast training of graph neural networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 1393–1403, New York, NY, USA, 2020. Association for Computing Machinery.   \n[11] Fabrizio Frasca, Emanuele Rossi, Davide Eynard, Ben Chamberlain, Michael Bronstein, and Federico Monti. SIGN: Scalable inception graph neural networks, 2020.   \n[12] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in neural information processing systems, pages 1024–1034, 2017.   \n[13] William L Hamilton, Rex Ying, and Jure Leskovec. Representation learning on graphs: Methods and applications. arXiv preprint arXiv:1709.05584, 2017.   \n[14] Arman Hasanzadeh, Ehsan Hajiramezanali, Shahin Boluki, Mingyuan Zhou, Nick Duffield, Krishna Narayanan, and Xiaoning Qian. Bayesian graph neural networks with adaptive connection sampling, 2020.   \n[15] Kurt Hornik, Maxwell Stinchcombe, Halbert White, et al. Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366, 1989.   \n[16] Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs, 2020.   \n[17] Wenbing Huang, Yu Rong, Tingyang Xu, Fuchun Sun, and Junzhou Huang. Tackling oversmoothing for general Graph Convolutional Networks, 2020.   \n[18] Wenbing Huang, Tong Zhang, Yu Rong, and Junzhou Huang. Adaptive sampling towards fast graph representation learning. In Advances in neural information processing systems, pages 4558–4567, 2018.   \n[19] Glen Jeh and Jennifer Widom. Simrank: a measure of structural-context similarity. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 538–543, 2002.   \n[20] Leo Katz. A new status index derived from sociometric analysis. Psychometrika, 18(1):39–43, 1953.   \n[21] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \n[22] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   \n[23] Johannes Klicpera, Stefan Weißenberger, and Stephan Günnemann. Diffusion improves graph learning, 2019.   \n[24] Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In International Conference on Machine Learning, pages 3734–3743. PMLR, 2019.   \n[25] Jure Leskovec and Christos Faloutsos. Sampling from large graphs. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’06, page 631–636, New York, NY, USA, 2006. Association for Computing Machinery.   \n[26] Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graph evolution: Densification and shrinking diameters. ACM Trans. Knowl. Discov. Data, 1(1):2–es, March 2007.   \n[27] David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017.   \n[28] Guohao Li, Matthias Müller, Ali Thabet, and Bernard Ghanem. DeepGCNs: Can GCNs go as deep as CNNs? In The IEEE International Conference on Computer Vision (ICCV), 2019.   \n[29] Guohao Li, Chenxin Xiong, Ali Thabet, and Bernard Ghanem. DeeperGCN: All you need to train deeper gcns, 2020.   \n[30] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into Graph Convolutional Networks for semi-supervised learning. arXiv preprint arXiv:1801.07606, 2018.   \n[31] Meng Liu, Hongyang Gao, and Shuiwang Ji. Towards deeper graph neural networks. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Aug 2020.   \n[32] Yu-Chen Lo, Stefano E Rensi, Wen Torng, and Russ B Altman. Machine learning in chemoinformatics and drug discovery. Drug discovery today, 23(8):1538–1546, 2018.   \n[33] Sitao Luan, Mingde Zhao, Xiao-Wen Chang, and Doina Precup. Break the ceiling: Stronger multi-scale deep graph convolutional networks. In Advances in Neural Information Processing Systems 32, pages 10945–10955. Curran Associates, Inc., 2019.   \n[34] Zhewei Wei Ming Chen, Bolin Ding Zengfeng Huang, and Yaliang Li. Simple and deep graph convolutional networks. 2020.   \n[35] Federico Monti, Michael Bronstein, and Xavier Bresson. Geometric matrix completion with recurrent multi-graph neural networks. In Advances in Neural Information Processing Systems, pages 3697–3707, 2017.   \n[36] Kenta Oono and Taiji Suzuki. Graph Neural Networks exponentially lose expressive power for node classification. In International Conference on Learning Representations, 2020. ",
+        "bbox": [
+            173,
+            77,
+            828,
+            921
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            77,
+            828,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[37] Aditya Pal, Chantat Eksombatchai, Yitong Zhou, Bo Zhao, Charles Rosenberg, and Jure Leskovec. PinnerSage: Multi-modal user embedding framework for recommendations at Pinterest. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Aug 2020. ",
+        "bbox": [
+            173,
+            90,
+            826,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[38] Namyong Park, Andrey Kan, Xin Luna Dong, Tong Zhao, and Christos Faloutsos. Estimating node importance in knowledge graphs using graph neural networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 596–606, 2019.   \n[39] Yu Rong, Wenbing Huang, Tingyang Xu, and Junzhou Huang. DropEdge: Towards deep Graph Convolutional Networks on node classification. In International Conference on Learning Representations, 2020.   \n[40] Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional networks. In European Semantic Web Conference, pages 593–607. Springer, 2018.   \n[41] Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research, 12(9), 2011.   \n[42] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1):1929–1958, 2014.   \n[43] Jonathan M Stokes, Kevin Yang, Kyle Swanson, Wengong Jin, Andres Cubillos-Ruiz, Nina M Donghia, Craig R MacNair, Shawn French, Lindsey A Carfrae, Zohar Bloom-Ackerman, et al. A deep learning approach to antibiotic discovery. Cell, 180(4):688–702, 2020.   \n[44] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua ´ Bengio. Graph Attention Networks. In International Conference on Learning Representations, 2018.   \n[45] Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou, Chao Ma, Lingfan Yu, Yu Gai, Tianjun Xiao, Tong He, George Karypis, Jinyang Li, and Zheng Zhang. Deep graph library: A graph-centric, highly-performant package for graph neural networks, 2020.   \n[46] Xiao Wang, Meiqi Zhu, Deyu Bo, Peng Cui, Chuan Shi, and Jian Pei. AM-GCN: Adaptive multichannel graph convolutional networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’20, page 1243–1253, New York, NY, USA, 2020. Association for Computing Machinery.   \n[47] Felix Wu, Tianyi Zhang, Amauri Holanda de Souza Jr, Christopher Fifty, Tao Yu, and Kilian Q Weinberger. Simplifying graph convolutional networks. arXiv preprint arXiv:1902.07153, 2019.   \n[48] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2020.   \n[49] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018.   \n[50] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks, 2018.   \n[51] Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. GNNExplainer: Generating explanations for graph neural networks, 2019.   \n[52] Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 974–983, 2018.   \n[53] Jiaxuan You, Zhitao Ying, and Jure Leskovec. Design space for graph neural networks. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 17009–17021. Curran Associates, Inc., 2020.   \n[54] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. Accurate, efficient and scalable graph embedding. In 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 462–471, 2019.   \n[55] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. GraphSAINT: Graph sampling based inductive learning method. In International Conference on Learning Representations, 2020.   \n[56] Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pages 5165–5175, 2018.   \n[57] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   \n[58] Shuai Zhang, Lina Yao, Aixin Sun, and Yi Tay. Deep learning based recommender system: A survey and new perspectives. ACM Computing Surveys (CSUR), 52(1):1–38, 2019.   \n[59] Yuyu Zhang, Xinshi Chen, Yuan Yang, Arun Ramamurthy, Bo Li, Yuan Qi, and Le Song. Efficient probabilistic logic reasoning with graph neural networks, 2020.   \n[60] Lingxiao Zhao and Leman Akoglu. PairNorm: Tackling oversmoothing in gnns. In International Conference on Learning Representations, 2020.   \n[61] Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. Layerdependent importance sampling for training deep and large graph convolutional networks. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. ",
+        "bbox": [
+            169,
+            103,
+            828,
+            915
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            92,
+            826,
+            521
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            545,
+            254,
+            561
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            573,
+            339,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            592,
+            825,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            688,
+            493,
+            702
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See append ",
+        "bbox": [
+            238,
+            705,
+            758,
+            737
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            742,
+            393,
+            756
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See the released code.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See the released code. It contains all configurations.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] ",
+        "bbox": [
+            238,
+            760,
+            825,
+            892
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            207,
+            897,
+            823,
+            911
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] See citation after each dataset.   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] ",
+        "bbox": [
+            238,
+            90,
+            825,
+            213
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            215,
+            217,
+            705,
+            231
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            236,
+            825,
+            324
+        ],
+        "page_idx": 14
+    }
+]

parse/train/b8Kl8mcK6tb/b8Kl8mcK6tb.md

@@ -0,0 +1,357 @@
+# Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms
+
+Chi Jin Princeton University chij@princeton.edu
+
+Qinghua Liu Princeton University qinghual@princeton.edu
+
+Sobhan Miryoosefi Princeton University miryoosefi@cs.princeton.edu
+
+# Abstract
+
+Finding the minimal structural assumptions that empower sample-efficient learning is one of the most important research directions in Reinforcement Learning (RL). This paper advances our understanding of this fundamental question by introducing a new complexity measure—Bellman Eluder (BE) dimension. We show that the family of RL problems of low BE dimension is remarkably rich, which subsumes a vast majority of existing tractable RL problems including but not limited to tabular MDPs, linear MDPs, reactive POMDPs, low Bellman rank problems as well as low Eluder dimension problems. This paper further designs a new optimization-based algorithm—GOLF, and reanalyzes a hypothesis eliminationbased algorithm—OLIVE [proposed in 1]. We prove that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret and sample complexity results match or improve the best existing results for several well-known subclasses of low BE dimension problems.
+
+# 1 Introduction
+
+Modern Reinforcement Learning (RL) commonly engages practical problems with an enormous number of states, where function approximation must be deployed to approximate the true value function using functions from a prespecified function class. Function approximation, especially based on deep neural networks, lies at the heart of the recent practical successes of RL in domains such as Atari [2], Go [3], robotics [4], and dialogue systems [5].
+
+Despite its empirical success, RL with function approximation raises a new series of theoretical challenges when comparing to the classic tabular RL: (1) generalization, to generalize knowledge from the visited states to the unvisited states due to the enormous state space. (2) limited expressiveness, to handle the complicated issues where true value functions or intermediate steps computed in the algorithm can be functions outside the prespecified function class. (3) exploration, to address the tradeoff between exploration and exploitation when above challenges are present.
+
+Consequently, most existing theoretical results on efficient RL with function approximation rely on relatively strong structural assumptions. For instance, many require that the MDP admits a linear approximation [6–8], or that the model is precisely Linear Quadratic Regulator (LQR) [9–11]. Most of these structural assumptions rarely hold in practical applications. This naturally leads to one of the most fundamental questions in RL.
+
+![](images/77c926de0e00917b5f40e684448ed4d632520e6ebd7f236362d3c5c22bd1676f.jpg)  
+Figure 1: A schematic summarizing relations among families of RL problems1
+
+# What are the minimal structural assumptions that empower sample-efficient RL?
+
+We advance our understanding of this grand question via the following two steps: (1) identify a rich class of RL problems (with weak structural assumptions) that cover many practical applications of interests; (2) design sample-efficient algorithms that provably learn any RL problem in this class.
+
+The attempts to find weak or minimal structural assumptions that allow statistical learning can be traced in supervised learning where VC dimension [12] or Rademacher complexity [13] is proposed, or in online learning where Littlestone dimension [14] or sequential Rademacher complexity [15] is developed.
+
+In the area of reinforcement learning, there are two intriguing lines of recent works that have made significant progress in this direction. To begin with, Jiang et al. [1] introduces a generic complexity notion—Bellman rank, which can be proved small for many RL problems including linear MDPs [7], reactive POMDPs [16], etc. [1] further propose an hypothesis elimination-based algorithm— OLIVE for sample-efficient learning of problems with low Bellman rank. On the other hand, recent work by Wang et al. [17] considers general function approximation with low Eluder dimension [18], and designs a UCB-style algorithm with regret guarantee. Noticeably, generalized linear MDPs [6] and kernel MDPs (see Appendix C) are subclasses of low Eluder dimension problems, but not low Bellman rank.
+
+In this paper, we make the following three contributions.
+
+• We introduce a new complexity measure for RL—Bellman Eluder (BE) dimension. We prove that the family of RL problems of low BE dimension is remarkably rich, which subsumes both low Bellman rank problems and low Eluder dimension problems—two arguably most generic tractable function classes so far in the literature (see Figure 1). The family of low BE dimension further includes new problems such as kernel reactive POMDPs (see Appendix C) which were not known to be sample-efficiently learnable. We design a new optimization-based algorithm—GOLF, which provably learns nearoptimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret or sample complexity guarantees match [8] which is minimax optimal when specified to the linear setting. Our rates further improve upon [1, 17] in low Bellman rank and low Eluder dimension settings, respectively. We reanalyze the hypothesis elimination based algorithm—OLIVE proposed in [1]. We show it can also learn RL problems with low BE dimension sample-efficiently, under slightly weaker assumptions but with worse sample complexity comparing to GOLF.
+
+# 1.1 Related works
+
+This section reviews prior theoretical works on RL, under Markov Decision Process (MDP) models.
+
+We remark that there has been a long line of research on function approximation in the batch RL setting [see, e.g., 19–22]. In this setting, agents are provided with exploratory data or simulator, so that they do not need to explicitly address the challenge of exploration. In this paper, we do not make such assumption, and attack the exploration problem directly. In the following we focus exclusively on the RL results in the general setting where exploration is required.
+
+Tabular RL. Tabular RL concerns MDPs with a small number of states and actions, which has been thoroughly studied in recent years [see, e.g., 23–30]. In the episodic setting with non-stationary dynamics, the best regret bound $\tilde { \mathcal { O } } ( \sqrt { H ^ { 2 } | \boldsymbol { S } | | \boldsymbol { A } | T } )$ is achieved by both model-based [27] and modelfree [30] algorithms. Moreover, the bound is proved to be minimax-optimal [29, 31]. This minimax bound suggests that when the state-action space is enormous, RL is information-theoretically hard without further structural assumptions.
+
+RL with linear function approximation. A recent line of work studies RL with linear function approximation [see, e.g., 7, 6, 32, 8, 33–36] These papers assume certain completeness conditions, as well as the optimal value function can be well approximated by linear functions. Under one√ formulation of linear approximation, the minimax regret bound $\tilde { \mathcal { O } } ( \dot { d } \sqrt { T } )$ is achieved by algorithm ELEANOR [8], where $d$ is the ambient dimension of the feature space.
+
+RL with general function approximation. Beyond the linear setting, there is a flurry line of research studying RL with general function approximation [see, e.g., 37, 1, 36, 38, 17, 39, 40]. Among them, [1] and [17] are the closest to our work.
+
+Jiang et al. [1] propose a complexity measure named Bellman rank and design an algorithm OLIVE with PAC guarantees for problems with low Bellman rank. We note that low Bellman rank is a special case of low BE dimension. When specialized to the low Bellman rank setting, our result for OLIVE exactly matches the guarantee in [1]. Our result for GOLF requires an additional completeness assumption, but provides sharper sample complexity guarantee.
+
+Wang et al. [17] propose a UCB-type algorithm with a regret guarantee under the assumption that the function class has a low eluder dimension. Again, we will show that low Eluder dimension is a special case of low BE dimension. Comparing to [17], our algorithm GOLF works under a weaker completeness assumption, with a better regret guarantee.
+
+Finally, we remark that the algorithms proposed in [1, 33, 41] and this paper are all computationally inefficient in general. We notice several existing works [e.g., 7, 17] can be computationally efficient given suitable regression oracles but they require stronger representation conditions and also achieve worse regret guarantees.
+
+Relation to bilinear classes Concurrent to this work, Du et al. [41] propose a new general tractable class of RL problems—bilinear class with low effective dimension (also known as low critical information gain in Du et al. [41]). We comment on the similarities and differences between two works as follows.
+
+In terms of algorithms, both Algorithm 2 in this paper and the algorithm proposed in Du et al. [41] are based on OLIVE originally proposed in Jiang et al. [1]. The two algorithms share similar guarantees in terms of assumptions and complexity results. More importantly, our work further develops a new type of algorithm for general function approximation—GOLF, a natural and clean algorithm which can be viewed as an optimistic version of classical algorithm—Fitted Q-Iteration [42]. GOLF gives much sharper sample complexity guarantees compared to [41] for various settings, and is minimaxoptimal when applied to the linear setting [8].
+
+In terms of richness of new classes identified, it depends on (a) what structure of MDP the complexity measures are applied to, and (b) what complexity measures are used. For (a), BE dimension applies to the Bellman error, while the bilinear class allows general surrogate losses of the Bellman error. For (b), this paper uses Eluder dimension while Du et al. [41] uses effective dimension. It can be shown that low effective dimension always implies low Eluder dimension (see Appendix C.2). In short, Du et al. [41] is more general in (a), while our work is more general in (b). As a result, neither work fully captures the other.
+
+In particular, our BE framework covers a majority of the examples identified in Du et al. [41] including low occupancy complexity, linear $Q ^ { \star } / V ^ { \star }$ , $Q ^ { \star }$ state aggregation, feature selection/FLAMBE.
+
+Nevertheless, our work can not address examples with model-based function approximation (e.g., low witness rank [36]) while [41] can. On the other hand, Du et al. [41] can not address the class of RL problems with low Eluder dimension [17] while our work can. Moreover, for several classes of RL problems that both works cover, our complexity measure is sharper. For example, in the setting of function approximation with generalized linear functions, the BE dimension is ${ \tilde { O } } ( d )$ where $d$ is the ambient dimension of the feature vectors, while the effective dimension under the generalized bilinear framework of Du et al. [41] is at least $\tilde { \Omega } ( d ^ { 2 } )$ .
+
+# 1.2 Paper organization
+
+We present preliminaries in Section 2, the definition of Bellman-Eluder dimension as well as its relations to existing complexity notions in Section 3. We present the results for algorithm GOLF in Section 4, and conclude in Section 5. Due to space limit, we postpone the results for algorithm OLIVE to Appendix A. Further discussions on Q-type versus V-type variants of BE dimension, as well as the practical examples will be provided in Appendix B and C. All the proofs are postponed to the appendix.
+
+# 2 Preliminaries
+
+We consider episodic Markov Decision Process (MDP), denoted by $\mathcal { M } = ( \mathcal { S } , \mathcal { A } , H , \mathbb { P } , r )$ , where $s$   
+is the state space, $\mathcal { A }$ is the action space, $H$ is the number of steps in each episode, $\mathbb { P } = \left\{ \mathbb { P } _ { h } \right\} _ { h \in [ H ] }$   
+is the collection of transition measures with $\mathbb { P } _ { h } ( s ^ { \prime } \mid s , a )$ equal to the probability of transiting to   
+$s ^ { \prime }$ after taking action $a$ at state $s$ at the $h ^ { \mathrm { t h } }$ step, and $r ~ { = } ~ \bar { \{ } r _ { h } \} _ { h \in \{ H \} }$ is the collection of reward   
+funthe for ns with step. 2 possible $r _ { h } ( s , a )$ eoue stic reward received after takinme reward is non-negative, and . $a$ $s$ $h ^ { \mathrm { t h } }$ $\begin{array} { r } { \sum _ { h = 1 } ^ { H } r _ { h } ( s _ { h } , a _ { h } ) \le 1 } \end{array}$ $( s _ { 1 } , a _ { 1 } , \dotsc , s _ { H } , a _ { H } )$
+
+In each episode, the agent starts at a fixed initial state $s _ { 1 }$ . Then, at each step $h \in [ H ]$ , the agent observes its current state $s _ { h }$ , takes action $a _ { h }$ , receives reward $r _ { h } ( s _ { h } , a _ { h } )$ , and causes the environment to transit to $s _ { h + 1 } \sim \mathbb { P } _ { h } ( \cdot \mid s _ { h } , a _ { h } )$ . Without loss of generality, we assume there is a terminating state $s _ { \mathrm { e n d } }$ which the environment will always transit to at step $H + 1$ , and the episode terminates when $s _ { \mathrm { e n d } }$ is reached.
+
+Policy and value functions A (deterministic) policy $\mathcal { A } \} _ { h = 1 } ^ { H }$ . We denote ed sum of th $V _ { h } ^ { \pi } : { \cal S }  \mathbb { R }$ as the value function at step ewards received under policy is a collection of $h$ for policy , starting f $\pi$ $H$ , sm functions $V _ { h } ^ { \pi } ( s )$ $\{ \pi _ { h } : { \mathcal { S } } \to$ gives thehe end of $\pi$ $s _ { h } = s$ the episode. In symbol,
+
+$$
+V _ { h } ^ { \pi } ( s ) : = \mathbb { E } _ { \pi } [ \sum _ { h ^ { \prime } = h } ^ { H } r _ { h ^ { \prime } } ( s _ { h ^ { \prime } } , a _ { h ^ { \prime } } ) \mid s _ { h } = s ] .
+$$
+
+Similarly, we denote $Q _ { h } ^ { \pi } : \mathcal { S } \times \mathcal { A }  \mathbb { R }$ as the $Q$ -value function at step $h$ for policy $\pi$ , where
+
+$$
+Q _ { h } ^ { \pi } ( s , a ) : = \mathbb { E } _ { \pi } [ \sum _ { h ^ { \prime } = h } ^ { H } r _ { h ^ { \prime } } ( s _ { h ^ { \prime } } , a _ { h ^ { \prime } } ) \ | \ s _ { h } = s , a _ { h } = a ] .
+$$
+
+There exists an optimal policy $\pi ^ { \star }$ , which gives the optimal value function for all states [43], in the sense, $\begin{array} { r } { V _ { h } ^ { \pi ^ { \star } } ( s ) = \operatorname* { s u p } _ { \pi } \bar { V } _ { h } ^ { \pi } ( s ) } \end{array}$ for all $h \in [ H ]$ and $s \in S$ . For notational simplicity, we abbreviate $V ^ { \pi ^ { \star } }$ as $V ^ { \star }$ . We similarly define the optimal $Q$ -value function as $Q ^ { \star }$ . Recall that $Q ^ { \star }$ satisfies the Bellman optimality equation:
+
+$$
+Q _ { h } ^ { \star } ( s , a ) = ( { \cal T } _ { h } Q _ { h + 1 } ^ { \star } ) ( s , a ) : = r _ { h } ( s , a ) + \mathbb { E } _ { s ^ { \prime } \sim \mathbb { P } _ { h } ( \cdot \vert s , a ) } \operatorname* { m a x } _ { a ^ { \prime } \in \mathcal { A } } Q _ { h + 1 } ^ { \star } ( s ^ { \prime } , a ^ { \prime } ) .
+$$
+
+for all $( s , a , h ) \in \mathcal { S } \times \mathcal { A } \times [ H ]$ . We also call $\mathcal { T } _ { h }$ the Bellman operator at step $h$ .
+
+$\epsilon$ -optimality and regret We say a policy $\pi$ is $\epsilon$ -optimal if $V _ { 1 } ^ { \pi } ( s _ { 1 } ) \geq V _ { 1 } ^ { \star } ( s _ { 1 } ) - \epsilon$ . Suppose an agent interacts with the environment for $K$ episodes. Denote by $\pi ^ { k }$ the policy the agent follows in episode $k \in [ K ]$ . The (accumulative) regret is defined as
+
+$$
+\mathrm { R e g } ( K ) : = \sum _ { k = 1 } ^ { K } [ V _ { 1 } ^ { \star } ( s _ { 1 } ) - V _ { 1 } ^ { \pi ^ { k } } ( s _ { 1 } ) ] .
+$$
+
+The objective of reinforcement learning is to find an $\epsilon$ -optimal policy within a small number of interactions or to achieve sublinear regret.
+
+# 2.1 Function approximation
+
+In this paper, we consider reinforcement learning with value function approximation. Formally, the learner is given a function class $\mathcal { F } = \mathcal { F } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { F } _ { H }$ , where ${ \mathcal { F } } _ { h } \subseteq ( S \times { \mathcal { A } } \to [ 0 , 1 ] )$ offers a set of candidate functions to approximate $Q _ { h } ^ { \star }$ —the optimal $Q$ -value function at step $h$ . Since no reward is collected in the $( H + 1 ) ^ { \mathrm { t h } }$ steps, we always set $f _ { H + 1 } = 0$ .
+
+Reinforcement learning with function approximation in general is extremely challenging without further assumptions (see, e.g., hardness results in [16, 44]). Below, we present two assumptions about function approximation that are commonly adopted in the literature.
+
+Assumption 1 (Realizability). $Q _ { h } ^ { \star } \in \mathcal { F } _ { h }$ for all $h \in [ H ]$ .
+
+Realizability requires the function class is well-specified, i.e., function class $\mathcal { F }$ in fact contains the optimal $Q$ -value function $Q ^ { \star }$ with no approximation error.
+
+Assumption 2 (Completeness). $\mathcal T _ { h } \mathcal F _ { h + 1 } \subseteq \mathcal F _ { h }$ for all $h \in [ H ]$ .
+
+Note $\mathcal { T } _ { h } \mathcal { F } _ { h + 1 }$ is defined as $\{ T _ { h } f _ { h + 1 } : f _ { h + 1 } \in \mathcal { F } _ { h + 1 } \}$ . Completeness requires the function class $\mathcal { F }$ to be closed under the Bellman operator.
+
+When function class $\mathcal { F }$ has finite elements, we can use its cardinality $| \mathcal F |$ to measure the “size” of function class $\mathcal { F }$ . When addressing function classes with infinite elements, we need a notion similar to cardinality. We use the standard $\epsilon$ -covering number.
+
+Definition 3 ( $\epsilon$ -covering number). The $\epsilon$ -covering number of a set $\nu$ under metric $\rho ,$ , denoted as $\mathcal { N } ( \mathcal { V } , \epsilon , \rho )$ , is the minimum integer $n$ such that there exists a subset $\mathcal { V } _ { o } \subset \mathcal { V }$ with $| \nu _ { o } | = n$ , and for any $x \in \nu$ , there exists $y \in \mathcal { V } _ { o }$ such that $\rho ( x , y ) \le \epsilon$ .
+
+We refer readers to standard textbooks [see, e.g., 45] for further properties of covering number. In this paper, we will always apply the covering number on function class $\mathcal { F } = \mathcal { F } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { F } _ { H }$ , and use metric $\rho ( f , g ) = \operatorname* { m a x } _ { h } \| f _ { h } - g _ { h } \| _ { \infty }$ . For notational simplicity, we omit the metric dependence and denote the covering number as $\mathcal { N } _ { \mathcal { F } } ( \epsilon )$ .
+
+# 2.2 Eluder dimension
+
+One class of functions highly related to this paper is the function class of low Eluder dimension [18].
+
+Definition 4 ( $\epsilon$ -independence between points). Let $\mathcal { G }$ be a function class defined on $\mathcal { X }$ , and $z , x _ { 1 } , x _ { 2 }$ $\textstyle , \ldots , x _ { n } \in { \mathcal { X } }$ . We say $z$ is $\epsilon$ -independent of $\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \}$ with respect to $\mathcal { G }$ if there exist $g _ { 1 } , g _ { 2 } \in \mathcal { G }$ such that $\begin{array} { r } { \sqrt { \sum _ { i = 1 } ^ { n } ( g _ { 1 } ( x _ { i } ) - g _ { 2 } ( x _ { i } ) ) ^ { 2 } } \le \epsilon } \end{array}$ , but $g _ { 1 } ( z ) - g _ { 2 } ( z ) > \epsilon .$ .
+
+Intuitively, $z$ is independent of $\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \}$ means if that there exist two “certifying” functions $g _ { 1 }$ and $g _ { 2 }$ , so that their function values are similar at all points $\{ x _ { i } \} _ { i = 1 } ^ { n }$ , but the values are rather different at $z$ . This independence relation naturally induces the following complexity measure.
+
+Definition 5 (Eluder dimension). Let $\mathcal { G }$ be a function class defined on $\mathcal { X }$ . The Eluder dimension $\mathrm { d i m } _ { \mathrm { E } } ( \mathcal { G } , \epsilon )$ is the length of the longest sequence $\{ x _ { 1 } , \ldots , x _ { n } \} \subset { \mathcal { X } }$ such that there exists $\epsilon ^ { \prime } \geq \epsilon$ where $x _ { i }$ is $\epsilon ^ { \prime }$ -independent of $\{ x _ { 1 } , \dots , x _ { i - 1 } \}$ for all $i \in [ n ]$ .
+
+Recall that a vector space has dimension $d$ if and only if $d$ is the length of the longest sequence of elements $\{ x _ { 1 } , \ldots , x _ { d } \}$ such that $x _ { i }$ is linearly independent of $\{ x _ { 1 } , \dotsc , x _ { i - 1 } \}$ for all $i \in [ n ]$ . Eluder dimension generalizes the linear independence relation in standard vector space to capture both nonlinear independence and approximate independence, and thus is more general.
+
+# 3 Bellman Eluder Dimension
+
+In this section, we introduce our new complexity measure—Bellman Eluder (BE) dimension. As one of its most important properties, we will show that the family of problems with low BE dimension contains the two existing most general tractable problem classes in RL—problems with low Bellman rank, and problems with low Eluder dimension (see Figure 1).
+
+We start by developing a new distributional version of the original Eluder dimension proposed by Russo and Van Roy [18] (see Section 2.2 for more details).
+
+Definition 6 ( $\epsilon$ -independence between distributions). Let $\mathcal { G }$ be a function class defined on $\mathcal { X }$ , and $\nu , \mu _ { 1 } , \ldots , \mu _ { n }$ be probability measures over $\mathcal { X }$ . We say $\nu$ is $\epsilon$ -independent of $\{ \mu _ { 1 } , \mu _ { 2 } , \ldots , \mu _ { n } \}$ with respect to $\mathcal { G }$ if there exists $g \in { \mathcal { G } }$ such that $\begin{array} { r } { \sqrt { \sum _ { i = 1 } ^ { n } ( { \mathbb E } _ { \mu _ { i } } [ g ] ) ^ { 2 } } \le \epsilon } \end{array}$ , but $\begin{array} { r } { \left| \mathbb { E } _ { \nu } [ g ] \right| > \epsilon } \end{array}$ .
+
+Definition 7 (Distributional Eluder (DE) dimension). Let $\mathcal { G }$ be a function class defined on $\mathcal { X }$ , and $\Pi$ be a family of probability measures over $\mathcal { X }$ . The distributional Eluder dimension $\dim _ { \mathrm { D E } } ( { \mathcal { G } } , \Pi , \epsilon )$ is the length of the longest sequence $\{ \rho _ { 1 } , \ldots , \rho _ { n } \} \subset \Pi$ such that there exists $\epsilon ^ { \prime } \geq \epsilon$ where $\rho _ { i }$ is $\epsilon ^ { \prime }$ -independent of $\{ \rho _ { 1 } , . . . , \rho _ { i - 1 } \}$ for all $i \in [ n ]$ .
+
+Definition 6 and Definition 7 generalize Definition 4 and Definition 5 to their distributional versions, by inspecting the expected values of functions instead of the function values at points, and by restricting the candidate distributions to a certain family $\Pi$ . The main advantage of this generalization is exactly in the statistical setting, where estimating the expected values of functions with respect to a certain distribution family can be easier than estimating function values at each point (which is the case for RL in large state spaces).
+
+It is clear that the standard Eluder dimension is a special case of the distributional Eluder dimension, because if we choose $\Pi = \{ \delta _ { x } ( \cdot ) \mid x \in \mathcal { X } \}$ where $\delta _ { x } ( \cdot )$ is the dirac measure centered at $x$ , then $\dim _ { \operatorname { E } } ( { \mathcal { G } } , \epsilon ) = \dim _ { \operatorname { D E } } ( { \mathcal { G } } - { \mathcal { G } } , \Pi , \epsilon )$ where ${ \mathcal { G } } - { \mathcal { G } } = \{ g _ { 1 } - g _ { 2 } : g _ { 1 } , g _ { 2 } \in { \mathcal { G } } \}$ .
+
+Now we are ready to introduce the key notion in this paper—Bellman Eluder dimension.
+
+Definition 8 (Bellman Eluder (BE) dimension). Let $( I - \mathcal { T } _ { h } ) \mathcal { F } { : = } \{ f _ { h } - \mathcal { T } _ { h } f _ { h + 1 } : \mathrm { ~ } f \in \mathcal { F } \}$ be the set of Bellman residuals induced by $\mathcal { F }$ at step $h$ , and $\dot { \Pi } = \{ \Pi _ { h } \} _ { h = 1 } ^ { H }$ be a collection of $H$ probability measure families over $s \times { \mathcal { A } }$ . The $\epsilon$ -Bellman Eluder of $\mathcal { F }$ with respect to $\Pi$ is defined as
+
+$$
+\dim _ { \mathrm { B E } } ( { \mathcal F } , \Pi , \epsilon ) : = \operatorname* { m a x } _ { h \in [ H ] } \dim _ { \mathrm { D E } } \big ( ( I - { \mathcal T } _ { h } ) { \mathcal F } , \Pi _ { h } , \epsilon \big ) .
+$$
+
+Remark 9 (Q-type v.s. V-type). Definition 8 is based on the Bellman residuals functions that take a state-action pair as input, thus referred to as $Q$ -type $B E$ dimension. Alternatively, one can define $V -$ type BE dimension using a different set of Bellman residual functions that depend on states only (see Appendix $B$ ). We focus on $Q$ -type in the main paper, and present the results for $V .$ -type in Appendix B. Both variants are important, and they include different sets of examples (see Appendix B, C).
+
+In short, Bellman Eluder dimension is simply the distributional Eluder dimension on the function class of Bellman residuals, maximizing over all steps. In addition to function class $\mathcal { F }$ and error $\epsilon$ , Bellman Eluder dimension also depends on the choice of distribution family $\Pi$ . For the purpose of this paper, we focus on the following two specific choices.
+
+1. $\mathcal { D } _ { \mathcal { F } } : = \{ \mathcal { D } _ { \mathcal { F } , h } \} _ { h \in [ H ] }$ , where $\mathcal { D } _ { \mathcal { F } , h }$ denotes the collection of all probability measures over $\boldsymbol { S } \times \boldsymbol { A }$ at the $h ^ { \mathrm { t h } }$ step, which can be generated by executing the greedy policy $\pi _ { f }$ induced by any $f \in { \mathcal { F } }$ , i.e., $\bar { \pi } _ { f , h } ( \cdot ) = \mathrm { a r g m a } \bar { \mathrm { x } } _ { a \in \mathcal { A } } f _ { h } ( \cdot , \bar { a } )$ for all $h \in [ H ]$ . 2. $\mathcal { D } _ { \Delta } : = \{ \mathcal { D } _ { \Delta , h } \} _ { h \in [ H ] }$ , where ${ \mathcal { D } } _ { \Delta , h } = \{ \delta _ { ( s , a ) } ( \cdot ) | s \in S , a \in { \mathcal { A } } \}$ , i.e., the collections of probability measures that put measure 1 on a single state-action pair.
+
+We say a RL problem has low BE dimension if $\begin{array} { r } { \operatorname* { m i n } _ { \Pi \in \{ \mathcal { D } _ { \mathcal { F } } , \mathcal { D } _ { \Delta } \} } \mathrm { d i m } _ { \mathrm { B E } } ( \mathcal { F } , \Pi , \epsilon ) \mathrm { i s ~ s m a } } \end{array}$ ll.
+
+# 3.1 Relations with known tractable classes of RL problems
+
+Known tractable problem classes in RL include but not limited to tabular MDPs, linear MDPs [7], linear quadratic regulators [9], generalized linear MDPs [6], kernel MDPs (Appendix C), reactive POMDPs [16], reactive PSRs [46, 1]. There are two existing generic tractable problem classes that jointly contain all the examples mentioned above: the set of RL problems with low Bellman rank, and the set of RL problems with low Eluder dimension. However, for these two generic sets, one does not contain the other.
+
+In this section, we will show that our new class of RL problems with low BE dimension in fact contains both low Bellman rank problems and low Eluder dimension problems (see Figure 1). That is, our new problem class covers almost all existing tractable RL problems, and to our best knowledge, is the most generic tractable function class so far.
+
+Relation with low Bellman rank The seminal paper by Jiang et al. [1] proposes the complexity measure—Bellman rank, and shows that a majority of RL examples mentioned above have low Bellman rank. They also propose a hypothesis elimination based algorithm—OLIVE, that learns any low Bellman rank problem within polynomial samples. Formally,
+
+Definition 10 (Bellman rank). The Bellman rank is the minimum integer $d$ so that there exists $\phi _ { h }$ : $\mathcal { F }  \mathbb { R } ^ { d }$ and $\psi _ { h } : \mathcal { F }  \mathbb { R } ^ { d }$ for each $h \in [ H ]$ , such that for any $f , f ^ { \prime } \in \mathcal { F }$ , the average Bellman error.
+
+$$
+\mathcal { E } ( f , \pi _ { f ^ { \prime } } , h ) : = \mathbb { E } _ { \pi _ { f ^ { \prime } } } [ ( f _ { h } - \mathcal { T } _ { h } f _ { h + 1 } ) ( s _ { h } , a _ { h } ) ] = \langle \phi _ { h } ( f ) , \psi _ { h } ( f ^ { \prime } ) \rangle ,
+$$
+
+where $\| \phi _ { h } ( f ) \| _ { 2 } \cdot \| \psi _ { h } ( f ^ { \prime } ) \| _ { 2 } \leq \zeta$ , and $\zeta$ is the normalization parameter.
+
+We remark that similar to Bellman Eluder dimension, Bellman rank also has two variants—Q-type (Definition 10) and $\mathrm { V } .$ -type (see Appendix B). Recall that we use $\pi _ { f }$ to denote the greedy policy induced by value function $f$ . Intuitively, a problem with Bellman rank says its average Bellman error can be decomposed as the inner product of two $d$ -dimensional vectors, where one vector depends on the roll-in policy $\pi _ { f ^ { \prime } }$ , while the other vector depends on the value function $f$ . At a high level, it claims that the average Bellman error has a linear inner product structure.
+
+Proposition 11 (low Bellman rank $\subset$ low BE dimension). If an MDP with function class $\mathcal { F }$ has Bellman rank $d$ with normalization parameter $\zeta ,$ , then
+
+$$
+\dim _ { \mathrm { B E } } ( \mathcal { F } , \mathcal { D } _ { \mathcal { F } } , \epsilon ) \leq \mathcal { O } ( 1 + d \log ( 1 + \zeta / \epsilon ) ) .
+$$
+
+Proposition 11 claims that problems with low Bellman rank also have low BE dimension, with a small multiplicative factor that is only logarithmic in $\zeta$ and $\epsilon ^ { - 1 }$ .
+
+Relation with low Eluder dimension Wang et al. [17] study the setting where the function class $\mathcal { F }$ has low Eluder dimension, which includes generalized linear functions. They prove that, when the completeness assumption is satisfied,3 low Eluder dimension problems can be efficiently learned in polynomial samples.
+
+Proposition 12 (low Eluder dimension $\subset$ low BE dimension). Assume $\mathcal { F }$ satisfies completeness (Assumption 2). Then for all $\epsilon > 0$ ,
+
+$$
+\dim _ { \mathrm { B E } } \left( { \mathcal { F } } , { \mathcal { D } } _ { \Delta } , \epsilon \right) \leq \operatorname* { m a x } _ { h \in \left[ H \right] } \dim _ { \mathrm { E } } ( { \mathcal { F } } _ { h } , \epsilon ) .
+$$
+
+Proposition 12 asserts that problems with low Eluder dimension also have low BE dimension, which is a natural consequence of completeness and the fact that Eluder dimension is a special case of distributional Eluder dimension.
+
+Finally, we show that the set of low BE dimension problems is strictly larger than the union of low Eluder dimension problems and low Bellman rank problems.
+
+Proposition 13 (low BE dimension $\nless$ low Eluder dimension ∪ low Bellman rank). For any $m ~ \in ~ \mathbb { N } ^ { + }$ , there exists an MDP and a function class $\mathcal { F }$ so that for all $\epsilon ~ \in ~ ( 0 , 1 ]$ , we have $\dim _ { \mathrm { B E } } ( { \mathcal F } , { \mathcal D } _ { { \mathcal F } } , \epsilon ) = \dim _ { \mathrm { B E } } ( { \mathcal F } , { \mathcal D } _ { \Delta } , \epsilon ) \ \leq \ 5 ,$ , but ${ \mathrm { m i n } } \{ { \mathrm { m i n } } _ { h \in [ H ] }$ $\dim _ { \operatorname { E } } ( { \mathcal { F } } _ { h } , \epsilon )$ , Bellman $\mathrm { r a n k } \} \geq$ $m$ .
+
+In particular, the family of low BE dimension includes new examples such as kernel reactive POMDPs (Appendix C), which can not be addressed by the framework of either Bellman rank or Eluder dimension.
+
+1: Initialize: $\mathcal { D } _ { 1 } , . . . , \mathcal { D } _ { H }  \emptyset$ , $B ^ { 0 }  { \mathcal { F } }$ .   
+2: for episode $k$ from 1 to $K$ do   
+3: Choose policy $\pi ^ { k } = \pi _ { f ^ { k } }$ , where $f ^ { k } = \operatorname { a r g m a x } _ { f \in { \mathcal { B } } ^ { k - 1 } } f { \big ( } s _ { 1 } , \pi _ { f } { \big ( } s _ { 1 } { \big ) } { \big ) }$ .   
+4: Collect a trajectory $\left( s _ { 1 } , a _ { 1 } , r _ { 1 } , \dots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 } \right)$ by following $\pi ^ { k }$ .   
+5: Augment $\mathcal { D } _ { h } = \mathcal { D } _ { h } \cup \left\{ \left( s _ { h } , a _ { h } , r _ { h } , s _ { h + 1 } \right) \right\}$ for all $h \in [ H ]$ .   
+6: Update   
+$\begin{array} { r l } & { \mathbf { \Phi } ^ { \mathrm { t e } } } \\ & { \quad \mathcal { B } ^ { k } = \left\{ f \in \mathcal { F } : \mathcal { L } _ { \mathcal { D } _ { h } } ( f _ { h } , f _ { h + 1 } ) \leq \displaystyle \operatorname* { i n f } _ { g \in \mathcal { G } _ { h } } \mathcal { L } _ { \mathcal { D } _ { h } } ( g , f _ { h + 1 } ) + \beta \mathrm { ~ f o r ~ a l l ~ } h \in [ H ] \right\} , } \\ & { \quad \quad \mathrm { w h e r e ~ } \mathcal { L } _ { \mathcal { D } _ { h } } ( \xi _ { h } , \zeta _ { h + 1 } ) = \displaystyle \sum _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } _ { h } } \left[ \xi _ { h } ( s , a ) - r - \operatorname* { m a x } _ { a ^ { \prime } \in \mathcal { A } } \zeta _ { h + 1 } ( s ^ { \prime } , a ^ { \prime } ) \right] ^ { 2 } . } \end{array}$
+
+7: Output $\pi ^ { \mathrm { { o u t } } }$ sampled uniformly at random from $\{ \pi ^ { k } \} _ { k = 1 } ^ { K }$
+
+# 4 Algorithm GOLF
+
+Section 3 defines a new class of RL problems with low BE dimension, and shows that the new class is rich, containing almost all the existing known tractable RL problems so far. In this section, we propose a new simple optimization-based algorithm—Global Optimism based on Local Fitting (GOLF). We prove that, low BE dimension problems are indeed tractable, i.e., GOLF can find nearoptimal policies for these problems within a polynomial number of samples.
+
+At a high level, GOLF can be viewed as an optimistic version of the classic algorithm—Fitted QIteration (FQI) [42]. GOLF generalizes the ELEANOR algorithm [8] from the special linear setting to the general setting with arbitrary function classes.
+
+The pseudocode of GOLF is given in Algorithm 1. GOLF initializes datasets $\{ \mathcal { D } _ { h } \} _ { h = 1 } ^ { H }$ to be empty sets, and confidence set $B ^ { 0 }$ to be $\mathcal { F }$ . Then, in each episode, GOLF performs two main steps:
+
+• Line 3 (Optimistic planning): compute the most optimistic value function $f ^ { k }$ from the confidence set $B ^ { k - 1 }$ constructed in the last episode , and choose $\pi ^ { k }$ to be its greedy policy. • Line 4-6 (Execute the policy and update the confidence set): execute policy $\pi ^ { k }$ for one episode, collect data, and update the confidence set using the new data.
+
+At the heart of GOLF is the way we construct the confidence set $B ^ { k }$ . For each $h \in [ H ]$ , GOLF maintains a local regression constraint using the collected transition data $\mathcal { D } _ { h }$ at this step
+
+$$
+\mathcal { L } _ { \mathcal { D } _ { h } } ( f _ { h } , f _ { h + 1 } ) \leq \operatorname* { i n f } _ { g \in \mathcal { G } _ { h } } \mathcal { L } _ { \mathcal { D } _ { h } } ( g , f _ { h + 1 } ) + \beta ,
+$$
+
+where $\beta$ is a confidence parameter, and $\mathcal { L } _ { \mathcal { D } _ { h } }$ is the squared loss defined in (2), which can be viewed as a proxy to the squared Bellman error at step $h$ . We remark that FQI algorithm [42] simply updates $\begin{array} { r } { \dot { f _ { h } } \gets \operatorname * { a r g m i n } _ { \phi \in \mathcal { F } _ { h } } \mathcal { L } _ { \mathcal { D } _ { h } } ( \phi , f _ { h + 1 } ) } \end{array}$ . Our constraint (3) can be viewed as a relaxed version of this update, which allows $f _ { h }$ to be not only the minimizer of the loss $\mathcal { L } _ { \mathcal { D } _ { h } } ( \cdot , f _ { h + 1 } )$ , but also any function whose loss is only slightly larger than the optimal loss over the auxiliary function class $\mathcal { G } _ { h }$ .
+
+We remark that in general, the optimization problem in Line 3 of GOLF can not be solved computationally efficiently.
+
+# 4.1 Theoretical guarantees
+
+In this subsection, we present the theoretical guarantees for GOLF, which hold under Assumption 1 (realizability) and the following generalized completeness assumption introduced in [47, 21]. Let $\mathcal { G } = \mathcal { G } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { G } _ { H }$ be an auxiliary function class provided to the learner where each ${ \mathcal { G } } _ { h } \subseteq ( S \times { \mathcal { A } } \to$ $[ 0 , 1 ] )$ . Generalized completeness requires the auxiliary function class $\mathcal { G }$ to be rich enough so that applying Bellman operator to any function in the primary function class $\mathcal { F }$ will end up in $\mathcal { G }$ .
+
+Assumption 14 (Generalized completeness). $\mathcal T _ { h } \mathcal F _ { h + 1 } \subseteq \mathcal G _ { h }$ for all $h \in [ H ]$ .
+
+If we choose $\mathcal { G } = \mathcal { F }$ , then Assumption 14 is equivalent to the standard completeness assumption (Assumption 2). Now, we are ready to present the main theorem for GOLF.
+
+Theorem 15 (Regret of GOLF). Under Assumption $^ { l }$ , 14, there exists an absolute constant c such that for any $\delta \in ( 0 , 1 ]$ , $K \in \mathbb N$ , if we choose parameter $\beta = c \log [ \mathcal { N _ { F \cup \mathcal { G } } } ( 1 / K ) \cdot K H / \delta ]$ in GOLF, then with probability at least $1 - \delta$ , for all $k \in [ K ]$ , we have
+
+$$
+\mathrm { R e g } ( k ) = \sum _ { t = 1 } ^ { k } \Big [ V _ { 1 } ^ { \star } \big ( s _ { 1 } \big ) - V _ { 1 } ^ { \pi ^ { t } } \big ( s _ { 1 } \big ) \Big ] \leq \mathcal { O } ( H \sqrt { d k \beta } ) ,
+$$
+
+where $\begin{array} { r } { d = \operatorname* { m i n } _ { \Pi \in \{ \mathcal { D } _ { \Delta } , \mathcal { D } _ { \mathcal { F } } \} } \mathrm { d i m } _ { \mathrm { B E } } \left( \mathcal { F } , \Pi , 1 / \sqrt { K } \right) } \end{array}$ is the BE dimension.
+
+Theorem 15 asserts that, under the realizability and completeness assumptions, the general class of RL problems with low BE dimension is indeed tractable: there exists an algorithm (GOLF) that can achieve $\sqrt { K }$ regret, whose multiplicative factor depends only polynomially on the horizon of MDP $H$ , the BE dimension $d$ , and the log covering number of the two function classes. Most importantly, the regret is independent of the number of the states, which is crucial for dealing with practical RL problems with function approximation, where the state spaces are typically exponentially large.
+
+We remark that when function class ${ \mathcal { F } } \cup { \mathcal { G } }$ has finite number of elements, its covering number is upper bounded by its cardinality $| { \mathcal { F } } \cup { \mathcal { G } } |$ . For a wide range of function classes in practice, the log $\epsilon ^ { \prime }$ -covering number has only logarithmic dependence on $\epsilon ^ { \prime }$ . Informally, we denote the log covering number as $\log \mathcal { N } _ { \mathcal { F } \cup \mathcal { G } }$ and omit its $\epsilon ^ { \prime }$ dependency for clean presentation. Theorem 15 claims that the regret scales as $\tilde { \mathcal { O } } ( H \sqrt { d K \log \mathcal { N _ { F \cup \mathcal { G } } } } )$ , where $\bar { \tilde { \mathcal { O } } } ( \cdot )$ omits absolute constants and logarithmic terms.4
+
+By the standard online-to-batch argument, we also derive the sample complexity of GOLF.
+
+Corollary 16 (Sample Complexity of GOLF). Under Assumption 1, 2, there exists an absolute constant c such that for any $\epsilon \in ( 0 , 1 ] ,$ , if we choose $\beta = c \log [ \mathcal { N _ { F \cup \mathcal { G } } } ( \epsilon ^ { 2 } / ( d H ^ { 2 } ) ) \cdot H K ]$ in GOLF, then the output policy $\pi ^ { o u t }$ is $\mathcal { O } ( \epsilon )$ -optimal with probability at least $1 / 2 , i f$
+
+$$
+K \geq \Omega \left( \frac { H ^ { 2 } d } { \epsilon ^ { 2 } } \cdot \log \left[ \mathcal { N } _ { \mathcal { F } \cup \mathcal { G } } \left( \frac { \epsilon ^ { 2 } } { H ^ { 2 } d } \right) \cdot \frac { H d } { \epsilon } \right] \right) ,
+$$
+
+where $\begin{array} { r } { d = \operatorname* { m i n } _ { \Pi \in \{ \mathcal { D } _ { \Delta } , \mathcal { D } _ { \mathcal { F } } \} } \mathrm { d i m } _ { \mathrm { B E } } \left( \mathcal { F } , \Pi , \epsilon / H \right) } \end{array}$ is the BE dimension.
+
+Corollary 16 claims that $\tilde { \mathcal { O } } ( H ^ { 2 } d \log ( \mathcal { N } _ { \mathcal { F } \cup \mathcal { G } } ) / \epsilon ^ { 2 } )$ samples are enough for GOLF to learn a nearoptimal policy of any low BE dimension problem. Our sample complexity scales linear in both the BE dimension $d$ , and the log covering number $\log ( \mathcal { N } _ { \mathcal { F } \cup \mathcal { G } } )$ .
+
+To showcase the sharpness of our results, we compare them to the previous results when restricted to the corresponding settings. (1) For linear function class with ambient dimension $d _ { \mathrm { l i n } }$ , we have√ BE dimension $d = \mathcal { \tilde { O } } ( d _ { \mathrm { l i n } } )$ and $\log ( \mathcal { N _ { F \cup \mathcal { G } } } ) = \tilde { \mathcal { O } } ( d _ { \mathrm { l i n } } )$ . Our regret bound becomes $\tilde { \mathcal { O } } ( H d _ { \mathrm { l i n } } \sqrt { K } )$ which matches the best known result [8] up to logarithmic factors; (2) For function class with low Eluder dimension [17], our results hold under weaker completeness assumptions. Our regret scales√ with $\sqrt { d _ { \mathrm { E } } }$ in terms of dependency on Eluder dimension $d _ { \mathrm { E } }$ , which improves the linear $d _ { \mathrm { E } }$ scaling in the regret of [17]; (3) Finally, for low Bellman rank problems, our sample complexity scales linearly with Bellman rank, which improves upon the quadratic dependence in [1]. We remark that all results mentioned above assume (approximate) realizability. All except [1] assume (approximate) completeness.
+
+# 4.2 Key ideas in proving Theorem 15
+
+In this subsection, we present a brief proof sketch for the regret bound of GOLF. We defer all the details to Appendix E. For simplicity, we only discuss the case of choosing $\mathcal { D } _ { \mathcal { F } }$ as the distribution family $\Pi$ in the definition of Bellman Eluder dimension (Definition 8). The proof for using $\mathcal { D } _ { \Delta }$ as the distribution family follows from similar arguments.
+
+Our proof strategy consists of three main steps.
+
+Step 1: Prove optimism. We firstly show that, with high probability, the optimal value function $Q ^ { \star }$ indeed lies in the confidence set $B ^ { k }$ for all $k \in [ K ]$ (Lemma 40 in Appendix E.1), which is a natural consequence of martingale concentration and the properties of the confidence set we designed. Because of $Q ^ { \star } \in B ^ { k }$ , the optimistic planning step (Line 3) in GOLF guarantees that $V _ { 1 } ^ { \star } ( \mathsf { \bar { s } } _ { 1 } ) \leq \operatorname* { m a x } _ { a } f _ { 1 } ^ { k } ( s _ { 1 } , \mathsf { \bar { a } } )$ for every episode $k$ . This optimism allows the following upper bound on regret
+
+$$
+\mathrm { R e g } ( K ) \leq \sum _ { k = 1 } ^ { K } \left( \operatorname* { m a x } _ { a } f _ { 1 } ^ { k } ( s _ { 1 } , a ) - V _ { 1 } ^ { \pi ^ { k } } ( s _ { 1 } ) \right) = \sum _ { h = 1 } ^ { H } \sum _ { k = 1 } ^ { K } \mathbb { E } _ { \pi ^ { k } } \left[ ( f _ { h } ^ { k } - \mathcal { T } f _ { h + 1 } ^ { k } ) ( s _ { h } , a _ { h } ) \right] ,
+$$
+
+where the right equality follows from the standard policy loss decomposition (see, e.g., Lemma 1 in [1]), and $\mathbb { E } _ { \pi }$ denotes the expectation taken over sequence $( s _ { 1 } , a _ { 1 } , \dotsc , s _ { H } , a _ { H } )$ when executing policy $\pi$ .
+
+Step 2: Utilize the sharpness of our confidence set. Recall that our construction of the confidence set in Line 6 of GOLF forces $f ^ { k }$ computed in episode $k$ to have a small loss $\mathcal { L } _ { \mathcal { D } _ { h } }$ , which is a proxy for empirical squared Bellman error under data $\mathcal { D } _ { h }$ . Since data $\mathcal { D } _ { h }$ in episode $k$ are collected by executing each $\pi ^ { i }$ for one episode for all $i < k$ , by standard martingale concentration arguments and the completeness assumption, we can show that with high probability (Lemma 39 in Appendix E.1)
+
+$$
+\sum _ { i = 1 } ^ { k - 1 } \mathbb { E } _ { \pi ^ { i } } \left[ ( f _ { h } ^ { k } - \mathcal { T } f _ { h + 1 } ^ { k } ) ( s _ { h } , a _ { h } ) \right] ^ { 2 } \leq \mathcal { O } ( \beta ) , \mathrm { ~ f o r ~ a l l ~ } ( k , h ) \in [ K ] \times [ H ] .
+$$
+
+Step 3: Establish relations between (4) and (5). So far, we want to upper-bound (4), while we know (5). We note that the RHS of (4) is very similar to the LHS of (5), except that the latter is the squared Bellman error, and the expectation is taken under previous policy $\pi ^ { i }$ for $i < k$ . To establish the connection between these two, it turns out that we need the Bellman Eluder dimension to be small. Concretely, we have the following lemma.
+
+Lemma 17. Given a function class $\Phi$ defined on $\mathcal { X }$ with $| \phi ( x ) | \le 1$ for all $( \phi , x ) \in \Phi \times \mathcal { X }$ , and $a$ family of probability measures $\Pi$ over $\mathcal { X }$ . Suppose sequence $\{ \phi _ { k } \} _ { k = 1 } ^ { K } \subset \dot { \Phi }$ and $\{ \mu _ { k } \} _ { k = 1 } ^ { K } \subset \Pi$ satisfy that for all $k \in [ K ] ,$ , $\begin{array} { r } { \sum _ { i = 1 } ^ { k - 1 } ( \mathbb { E } _ { \mu _ { i } } [ \phi _ { k } ] ) ^ { 2 } \ : \leq \ : \beta } \end{array}$ . Then for all $k \in [ K ]$ , $\begin{array} { r } { \sum _ { i = 1 } ^ { k } | \mathbb { E } _ { \mu _ { i } } [ \phi _ { i } ] | \ \leq } \end{array}$ $\mathcal { O } ( \sqrt { \mathrm { d i m } _ { \mathrm { D E } } ( \Phi , \Pi , 1 / k ) \beta k } )$ .
+
+Lemma 17 is a simplification of Lemma 41 in Appendix E, which is a modification of Lemma 2 in [18]. Intuitively, Lemma 17 can be viewed as an analogue of the pigeon-hole principle for DE dimension. Choose $\Phi$ to be the function class of Bellman residuals, and $\mu _ { k }$ to be the distribution under policy $\pi ^ { k }$ , we finish the proof.
+
+# 5 Conclusion
+
+In this paper, we propose a new complexity measure—Bellman Eluder (BE) dimension for reinforcement learning with function approximation. Our new complexity measure identifies a new rich class of RL problems that subsumes a majority of existing tractable problem classes in RL. We design a new optimization-based algorithm—GOLF, and provide a new analysis for algorithm OLIVE. Both algorithms show that the new rich class of RL problems we identified in fact can be learned within a polynomial number of samples. We hope our results shed light on the future research in finding the minimal structural assumptions that allow sample-efficient reinforcement learning.
+
+# Acknowledgement
+
+This work was partial supported by National Science Foundation Grant NSF-IIS-2107304.
+
+References   
+[1] 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.   
+[2] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013.   
+[3] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529 (7587):484–489, 2016.   
+[4] Jens Kober, J Andrew Bagnell, and Jan Peters. Reinforcement learning in robotics: A survey. The International Journal of Robotics Research, 32(11):1238–1274, 2013.   
+[5] Jiwei Li, Will Monroe, Alan Ritter, Michel Galley, Jianfeng Gao, and Dan Jurafsky. Deep reinforcement learning for dialogue generation. arXiv preprint arXiv:1606.01541, 2016.   
+[6] Yining Wang, Ruosong Wang, Simon S Du, and Akshay Krishnamurthy. Optimism in reinforcement learning with generalized linear function approximation. arXiv preprint arXiv:1912.04136, 2019.   
+[7] 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, 2020.   
+[8] Andrea Zanette, Alessandro Lazaric, Mykel Kochenderfer, and Emma Brunskill. Learning near optimal policies with low inherent bellman error. arXiv preprint arXiv:2003.00153, 2020.   
+[9] Brian DO Anderson and John B Moore. Optimal control: linear quadratic methods. Courier Corporation, 2007.   
+[10] Maryam Fazel, Rong Ge, Sham Kakade, and Mehran Mesbahi. Global convergence of policy gradient methods for the linear quadratic regulator. In International Conference on Machine Learning, pages 1467–1476. PMLR, 2018.   
+[11] Sarah Dean, Horia Mania, Nikolai Matni, Benjamin Recht, and Stephen Tu. On the sample complexity of the linear quadratic regulator. Foundations of Computational Mathematics, pages 1–47, 2019.   
+[12] Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 2013.   
+[13] Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):463–482, 2002.   
+[14] Nick Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine learning, 2(4):285–318, 1988.   
+[15] Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Online learning: Random averages, combinatorial parameters, and learnability. 2010.   
+[16] Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Pac reinforcement learning with rich observations. arXiv preprint arXiv:1602.02722, 2016.   
+[17] Ruosong Wang, Ruslan Salakhutdinov, and Lin F Yang. Provably efficient reinforcement learning with general value function approximation. arXiv preprint arXiv:2005.10804, 2020.   
+[18] Daniel Russo and Benjamin Van Roy. Eluder dimension and the sample complexity of optimistic exploration. In Advances in Neural Information Processing Systems, pages 2256–2264, 2013.   
+[19] Csaba Szepesvari and R ´ emi Munos. Finite time bounds for sampling based fitted value it- ´ eration. In Proceedings of the 22nd international conference on Machine learning, pages 880–887, 2005.   
+[20] Remi Munos and Csaba Szepesv ´ ari. Finite-time bounds for fitted value iteration. ´ Journal of Machine Learning Research, 9(May):815–857, 2008.   
+[21] Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. arXiv preprint arXiv:1905.00360, 2019.   
+[22] Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. arXiv preprint arXiv:2008.04990, 2020.   
+[23] Ronen I Brafman and Moshe Tennenholtz. R-max-a general polynomial time algorithm for near-optimal reinforcement learning. Journal of Machine Learning Research, 3(Oct):213–231, 2002.   
+[24] Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(4), 2010.   
+[25] Christoph Dann and Emma Brunskill. Sample complexity of episodic fixed-horizon reinforcement learning. In Advances in Neural Information Processing Systems, pages 2818–2826, 2015.   
+[26] Shipra Agrawal and Randy Jia. Optimistic posterior sampling for reinforcement learning: worst-case regret bounds. In Advances in Neural Information Processing Systems, pages 1184– 1194, 2017.   
+[27] Mohammad Gheshlaghi Azar, Ian Osband, and Remi Munos. Minimax regret bounds for ´ reinforcement learning. arXiv preprint arXiv:1703.05449, 2017.   
+[28] Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. arXiv preprint arXiv:1901.00210, 2019.   
+[29] Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. Is q-learning provably efficient? In Advances in Neural Information Processing Systems, pages 4863–4873, 2018.   
+[30] Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learning via reference-advantage decomposition. arXiv preprint arXiv:2004.10019, 2020.   
+[31] Omar Darwiche Domingues, Pierre Menard, Emilie Kaufmann, and Michal Valko. Episodic ´ reinforcement learning in finite mdps: Minimax lower bounds revisited. In Algorithmic Learning Theory, pages 578–598. PMLR, 2021.   
+[32] Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. arXiv preprint arXiv:1912.05830, 2019.   
+[33] Andrea Zanette, Alessandro Lazaric, Mykel J Kochenderfer, and Emma Brunskill. Provably efficient reward-agnostic navigation with linear value iteration. Advances in Neural Information Processing Systems, 33, 2020.   
+[34] Alekh Agarwal, Sham Kakade, Akshay Krishnamurthy, and Wen Sun. Flambe: Structural complexity and representation learning of low rank mdps. Advances in Neural Information Processing Systems, 33, 2020.   
+[35] Gergely Neu and Ciara Pike-Burke. A unifying view of optimism in episodic reinforcement learning. Advances in Neural Information Processing Systems, 33, 2020.   
+[36] Wen Sun, Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Modelbased rl in contextual decision processes: Pac bounds and exponential improvements over model-free approaches. In Conference on Learning Theory, pages 2898–2933, 2019.   
+[37] Ian Osband and Benjamin Van Roy. Model-based reinforcement learning and the eluder dimension. In Advances in Neural Information Processing Systems, pages 1466–1474, 2014.   
+[38] Kefan Dong, Jian Peng, Yining Wang, and Yuan Zhou. Root-n-regret for learning in markov decision processes with function approximation and low bellman rank. In Conference on Learning Theory, pages 1554–1557. PMLR, 2020.   
+[39] Zhuoran Yang, Chi Jin, Zhaoran Wang, Mengdi Wang, and Michael I Jordan. Bridging exploration and general function approximation in reinforcement learning: Provably efficient kernel and neural value iterations. arXiv preprint arXiv:2011.04622, 2020.   
+[40] Dylan J Foster, Alexander Rakhlin, David Simchi-Levi, and Yunzong Xu. Instance-dependent complexity of contextual bandits and reinforcement learning: A disagreement-based perspective. arXiv preprint arXiv:2010.03104, 2020.   
+[41] Simon S Du, Sham M Kakade, Jason D Lee, Shachar Lovett, Gaurav Mahajan, Wen Sun, and Ruosong Wang. Bilinear classes: A structural framework for provable generalization in rl. arXiv preprint arXiv:2103.10897, 2021.   
+[42] Csaba Szepesvari. Algorithms for reinforcement learning. ´ Synthesis lectures on artificial intelligence and machine learning, 4(1):1–103, 2010.   
+[43] Martin L Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014.   
+[44] Gellert Weisz, Philip Amortila, and Csaba Szepesvari. Exponential lower bounds for ´ planning in mdps with linearly-realizable optimal action-value functions. arXiv preprint arXiv:2010.01374, 2020.   
+[45] Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint, volume 48. Cambridge University Press, 2019.   
+[46] Satinder Singh, Michael James, and Matthew Rudary. Predictive state representations: A new theory for modeling dynamical systems. arXiv preprint arXiv:1207.4167, 2012.   
+[47] Andras Antos, Csaba Szepesv ´ ari, and R ´ emi Munos. Learning near-optimal policies with ´ bellman-residual minimization based fitted policy iteration and a single sample path. Machine Learning, 71(1):89–129, 2008.   
+[48] Christoph Dann, Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. On oracle-efficient pac rl with rich observations. In Advances in neural information processing systems, pages 1422–1432, 2018.   
+[49] Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning, pages 1638–1646, 2014.

parse/train/b8Kl8mcK6tb/b8Kl8mcK6tb_content_list.json

@@ -0,0 +1,1588 @@
+[
+    {
+        "type": "text",
+        "text": "Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms ",
+        "text_level": 1,
+        "bbox": [
+            181,
+            122,
+            815,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chi Jin Princeton University chij@princeton.edu ",
+        "bbox": [
+            256,
+            226,
+            434,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Qinghua Liu Princeton University qinghual@princeton.edu ",
+        "bbox": [
+            522,
+            226,
+            741,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sobhan Miryoosefi Princeton University miryoosefi@cs.princeton.edu ",
+        "bbox": [
+            364,
+            289,
+            632,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            367,
+            535,
+            383
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Finding the minimal structural assumptions that empower sample-efficient learning is one of the most important research directions in Reinforcement Learning (RL). This paper advances our understanding of this fundamental question by introducing a new complexity measure—Bellman Eluder (BE) dimension. We show that the family of RL problems of low BE dimension is remarkably rich, which subsumes a vast majority of existing tractable RL problems including but not limited to tabular MDPs, linear MDPs, reactive POMDPs, low Bellman rank problems as well as low Eluder dimension problems. This paper further designs a new optimization-based algorithm—GOLF, and reanalyzes a hypothesis eliminationbased algorithm—OLIVE [proposed in 1]. We prove that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret and sample complexity results match or improve the best existing results for several well-known subclasses of low BE dimension problems. ",
+        "bbox": [
+            233,
+            398,
+            764,
+            606
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            632,
+            310,
+            650
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modern Reinforcement Learning (RL) commonly engages practical problems with an enormous number of states, where function approximation must be deployed to approximate the true value function using functions from a prespecified function class. Function approximation, especially based on deep neural networks, lies at the heart of the recent practical successes of RL in domains such as Atari [2], Go [3], robotics [4], and dialogue systems [5]. ",
+        "bbox": [
+            174,
+            665,
+            823,
+            734
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite its empirical success, RL with function approximation raises a new series of theoretical challenges when comparing to the classic tabular RL: (1) generalization, to generalize knowledge from the visited states to the unvisited states due to the enormous state space. (2) limited expressiveness, to handle the complicated issues where true value functions or intermediate steps computed in the algorithm can be functions outside the prespecified function class. (3) exploration, to address the tradeoff between exploration and exploitation when above challenges are present. ",
+        "bbox": [
+            174,
+            741,
+            825,
+            824
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Consequently, most existing theoretical results on efficient RL with function approximation rely on relatively strong structural assumptions. For instance, many require that the MDP admits a linear approximation [6–8], or that the model is precisely Linear Quadratic Regulator (LQR) [9–11]. Most of these structural assumptions rarely hold in practical applications. This naturally leads to one of the most fundamental questions in RL. ",
+        "bbox": [
+            174,
+            830,
+            823,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/77c926de0e00917b5f40e684448ed4d632520e6ebd7f236362d3c5c22bd1676f.jpg",
+        "image_caption": [
+            "Figure 1: A schematic summarizing relations among families of RL problems1 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            92,
+            687,
+            255
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "What are the minimal structural assumptions that empower sample-efficient RL? ",
+        "text_level": 1,
+        "bbox": [
+            212,
+            308,
+            781,
+            321
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We advance our understanding of this grand question via the following two steps: (1) identify a rich class of RL problems (with weak structural assumptions) that cover many practical applications of interests; (2) design sample-efficient algorithms that provably learn any RL problem in this class. ",
+        "bbox": [
+            174,
+            333,
+            825,
+            376
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The attempts to find weak or minimal structural assumptions that allow statistical learning can be traced in supervised learning where VC dimension [12] or Rademacher complexity [13] is proposed, or in online learning where Littlestone dimension [14] or sequential Rademacher complexity [15] is developed. ",
+        "bbox": [
+            174,
+            381,
+            825,
+            436
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the area of reinforcement learning, there are two intriguing lines of recent works that have made significant progress in this direction. To begin with, Jiang et al. [1] introduces a generic complexity notion—Bellman rank, which can be proved small for many RL problems including linear MDPs [7], reactive POMDPs [16], etc. [1] further propose an hypothesis elimination-based algorithm— OLIVE for sample-efficient learning of problems with low Bellman rank. On the other hand, recent work by Wang et al. [17] considers general function approximation with low Eluder dimension [18], and designs a UCB-style algorithm with regret guarantee. Noticeably, generalized linear MDPs [6] and kernel MDPs (see Appendix C) are subclasses of low Eluder dimension problems, but not low Bellman rank. ",
+        "bbox": [
+            174,
+            443,
+            825,
+            568
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we make the following three contributions. ",
+        "bbox": [
+            174,
+            574,
+            544,
+            588
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We introduce a new complexity measure for RL—Bellman Eluder (BE) dimension. We prove that the family of RL problems of low BE dimension is remarkably rich, which subsumes both low Bellman rank problems and low Eluder dimension problems—two arguably most generic tractable function classes so far in the literature (see Figure 1). The family of low BE dimension further includes new problems such as kernel reactive POMDPs (see Appendix C) which were not known to be sample-efficiently learnable. We design a new optimization-based algorithm—GOLF, which provably learns nearoptimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret or sample complexity guarantees match [8] which is minimax optimal when specified to the linear setting. Our rates further improve upon [1, 17] in low Bellman rank and low Eluder dimension settings, respectively. We reanalyze the hypothesis elimination based algorithm—OLIVE proposed in [1]. We show it can also learn RL problems with low BE dimension sample-efficiently, under slightly weaker assumptions but with worse sample complexity comparing to GOLF. ",
+        "bbox": [
+            215,
+            599,
+            825,
+            818
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 Related works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            834,
+            313,
+            849
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This section reviews prior theoretical works on RL, under Markov Decision Process (MDP) models. ",
+        "bbox": [
+            178,
+            859,
+            820,
+            875
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We remark that there has been a long line of research on function approximation in the batch RL setting [see, e.g., 19–22]. In this setting, agents are provided with exploratory data or simulator, so that they do not need to explicitly address the challenge of exploration. In this paper, we do not make such assumption, and attack the exploration problem directly. In the following we focus exclusively on the RL results in the general setting where exploration is required. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Tabular RL. Tabular RL concerns MDPs with a small number of states and actions, which has been thoroughly studied in recent years [see, e.g., 23–30]. In the episodic setting with non-stationary dynamics, the best regret bound $\\tilde { \\mathcal { O } } ( \\sqrt { H ^ { 2 } | \\boldsymbol { S } | | \\boldsymbol { A } | T } )$ is achieved by both model-based [27] and modelfree [30] algorithms. Moreover, the bound is proved to be minimax-optimal [29, 31]. This minimax bound suggests that when the state-action space is enormous, RL is information-theoretically hard without further structural assumptions. ",
+        "bbox": [
+            174,
+            176,
+            825,
+            262
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "RL with linear function approximation. A recent line of work studies RL with linear function approximation [see, e.g., 7, 6, 32, 8, 33–36] These papers assume certain completeness conditions, as well as the optimal value function can be well approximated by linear functions. Under one√ formulation of linear approximation, the minimax regret bound $\\tilde { \\mathcal { O } } ( \\dot { d } \\sqrt { T } )$ is achieved by algorithm ELEANOR [8], where $d$ is the ambient dimension of the feature space. ",
+        "bbox": [
+            174,
+            276,
+            825,
+            348
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "RL with general function approximation. Beyond the linear setting, there is a flurry line of research studying RL with general function approximation [see, e.g., 37, 1, 36, 38, 17, 39, 40]. Among them, [1] and [17] are the closest to our work. ",
+        "bbox": [
+            174,
+            361,
+            825,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Jiang et al. [1] propose a complexity measure named Bellman rank and design an algorithm OLIVE with PAC guarantees for problems with low Bellman rank. We note that low Bellman rank is a special case of low BE dimension. When specialized to the low Bellman rank setting, our result for OLIVE exactly matches the guarantee in [1]. Our result for GOLF requires an additional completeness assumption, but provides sharper sample complexity guarantee. ",
+        "bbox": [
+            174,
+            409,
+            825,
+            478
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Wang et al. [17] propose a UCB-type algorithm with a regret guarantee under the assumption that the function class has a low eluder dimension. Again, we will show that low Eluder dimension is a special case of low BE dimension. Comparing to [17], our algorithm GOLF works under a weaker completeness assumption, with a better regret guarantee. ",
+        "bbox": [
+            174,
+            484,
+            825,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Finally, we remark that the algorithms proposed in [1, 33, 41] and this paper are all computationally inefficient in general. We notice several existing works [e.g., 7, 17] can be computationally efficient given suitable regression oracles but they require stronger representation conditions and also achieve worse regret guarantees. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            602
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Relation to bilinear classes Concurrent to this work, Du et al. [41] propose a new general tractable class of RL problems—bilinear class with low effective dimension (also known as low critical information gain in Du et al. [41]). We comment on the similarities and differences between two works as follows. ",
+        "bbox": [
+            174,
+            614,
+            825,
+            670
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In terms of algorithms, both Algorithm 2 in this paper and the algorithm proposed in Du et al. [41] are based on OLIVE originally proposed in Jiang et al. [1]. The two algorithms share similar guarantees in terms of assumptions and complexity results. More importantly, our work further develops a new type of algorithm for general function approximation—GOLF, a natural and clean algorithm which can be viewed as an optimistic version of classical algorithm—Fitted Q-Iteration [42]. GOLF gives much sharper sample complexity guarantees compared to [41] for various settings, and is minimaxoptimal when applied to the linear setting [8]. ",
+        "bbox": [
+            173,
+            676,
+            825,
+            773
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In terms of richness of new classes identified, it depends on (a) what structure of MDP the complexity measures are applied to, and (b) what complexity measures are used. For (a), BE dimension applies to the Bellman error, while the bilinear class allows general surrogate losses of the Bellman error. For (b), this paper uses Eluder dimension while Du et al. [41] uses effective dimension. It can be shown that low effective dimension always implies low Eluder dimension (see Appendix C.2). In short, Du et al. [41] is more general in (a), while our work is more general in (b). As a result, neither work fully captures the other. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            877
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In particular, our BE framework covers a majority of the examples identified in Du et al. [41] including low occupancy complexity, linear $Q ^ { \\star } / V ^ { \\star }$ , $Q ^ { \\star }$ state aggregation, feature selection/FLAMBE. ",
+        "bbox": [
+            174,
+            882,
+            820,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Nevertheless, our work can not address examples with model-based function approximation (e.g., low witness rank [36]) while [41] can. On the other hand, Du et al. [41] can not address the class of RL problems with low Eluder dimension [17] while our work can. Moreover, for several classes of RL problems that both works cover, our complexity measure is sharper. For example, in the setting of function approximation with generalized linear functions, the BE dimension is ${ \\tilde { O } } ( d )$ where $d$ is the ambient dimension of the feature vectors, while the effective dimension under the generalized bilinear framework of Du et al. [41] is at least $\\tilde { \\Omega } ( d ^ { 2 } )$ . ",
+        "bbox": [
+            173,
+            90,
+            825,
+            193
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1.2 Paper organization ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            210,
+            346,
+            226
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We present preliminaries in Section 2, the definition of Bellman-Eluder dimension as well as its relations to existing complexity notions in Section 3. We present the results for algorithm GOLF in Section 4, and conclude in Section 5. Due to space limit, we postpone the results for algorithm OLIVE to Appendix A. Further discussions on Q-type versus V-type variants of BE dimension, as well as the practical examples will be provided in Appendix B and C. All the proofs are postponed to the appendix. ",
+        "bbox": [
+            173,
+            237,
+            825,
+            321
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
+            318,
+            361
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider episodic Markov Decision Process (MDP), denoted by $\\mathcal { M } = ( \\mathcal { S } , \\mathcal { A } , H , \\mathbb { P } , r )$ , where $s$   \nis the state space, $\\mathcal { A }$ is the action space, $H$ is the number of steps in each episode, $\\mathbb { P } = \\left\\{ \\mathbb { P } _ { h } \\right\\} _ { h \\in [ H ] }$   \nis the collection of transition measures with $\\mathbb { P } _ { h } ( s ^ { \\prime } \\mid s , a )$ equal to the probability of transiting to   \n$s ^ { \\prime }$ after taking action $a$ at state $s$ at the $h ^ { \\mathrm { t h } }$ step, and $r ~ { = } ~ \\bar { \\{ } r _ { h } \\} _ { h \\in \\{ H \\} }$ is the collection of reward   \nfunthe for ns with step. 2 possible $r _ { h } ( s , a )$ eoue stic reward received after takinme reward is non-negative, and . $a$ $s$ $h ^ { \\mathrm { t h } }$ $\\begin{array} { r } { \\sum _ { h = 1 } ^ { H } r _ { h } ( s _ { h } , a _ { h } ) \\le 1 } \\end{array}$ $( s _ { 1 } , a _ { 1 } , \\dotsc , s _ { H } , a _ { H } )$ ",
+        "bbox": [
+            173,
+            376,
+            825,
+            481
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In each episode, the agent starts at a fixed initial state $s _ { 1 }$ . Then, at each step $h \\in [ H ]$ , the agent observes its current state $s _ { h }$ , takes action $a _ { h }$ , receives reward $r _ { h } ( s _ { h } , a _ { h } )$ , and causes the environment to transit to $s _ { h + 1 } \\sim \\mathbb { P } _ { h } ( \\cdot \\mid s _ { h } , a _ { h } )$ . Without loss of generality, we assume there is a terminating state $s _ { \\mathrm { e n d } }$ which the environment will always transit to at step $H + 1$ , and the episode terminates when $s _ { \\mathrm { e n d } }$ is reached. ",
+        "bbox": [
+            173,
+            484,
+            825,
+            555
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Policy and value functions A (deterministic) policy $\\mathcal { A } \\} _ { h = 1 } ^ { H }$ . We denote ed sum of th $V _ { h } ^ { \\pi } : { \\cal S }  \\mathbb { R }$ as the value function at step ewards received under policy is a collection of $h$ for policy , starting f $\\pi$ $H$ , sm functions $V _ { h } ^ { \\pi } ( s )$ $\\{ \\pi _ { h } : { \\mathcal { S } } \\to$ gives thehe end of $\\pi$ $s _ { h } = s$ the episode. In symbol, ",
+        "bbox": [
+            173,
+            571,
+            825,
+            628
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/364bcf5462a870d49b75f878fff975f46f32720dd2857db460f03511e0daaff0.jpg",
+        "text": "$$\nV _ { h } ^ { \\pi } ( s ) : = \\mathbb { E } _ { \\pi } [ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } ( s _ { h ^ { \\prime } } , a _ { h ^ { \\prime } } ) \\mid s _ { h } = s ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            631,
+            638,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we denote $Q _ { h } ^ { \\pi } : \\mathcal { S } \\times \\mathcal { A }  \\mathbb { R }$ as the $Q$ -value function at step $h$ for policy $\\pi$ , where ",
+        "bbox": [
+            171,
+            679,
+            776,
+            695
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbddcbe9d0b7fcd08bbaa1e1b876fdef13667184442a770bfc987720f38df955.jpg",
+        "text": "$$\nQ _ { h } ^ { \\pi } ( s , a ) : = \\mathbb { E } _ { \\pi } [ \\sum _ { h ^ { \\prime } = h } ^ { H } r _ { h ^ { \\prime } } ( s _ { h ^ { \\prime } } , a _ { h ^ { \\prime } } ) \\ | \\ s _ { h } = s , a _ { h } = a ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            704,
+            673,
+            750
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There exists an optimal policy $\\pi ^ { \\star }$ , which gives the optimal value function for all states [43], in the sense, $\\begin{array} { r } { V _ { h } ^ { \\pi ^ { \\star } } ( s ) = \\operatorname* { s u p } _ { \\pi } \\bar { V } _ { h } ^ { \\pi } ( s ) } \\end{array}$ for all $h \\in [ H ]$ and $s \\in S$ . For notational simplicity, we abbreviate $V ^ { \\pi ^ { \\star } }$ as $V ^ { \\star }$ . We similarly define the optimal $Q$ -value function as $Q ^ { \\star }$ . Recall that $Q ^ { \\star }$ satisfies the Bellman optimality equation: ",
+        "bbox": [
+            173,
+            763,
+            825,
+            824
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/374d1edb02431e9851836f077c9e0977fe6f34a5b54c1b45b5bb14ed7c95e196.jpg",
+        "text": "$$\nQ _ { h } ^ { \\star } ( s , a ) = ( { \\cal T } _ { h } Q _ { h + 1 } ^ { \\star } ) ( s , a ) : = r _ { h } ( s , a ) + \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathbb { P } _ { h } ( \\cdot \\vert s , a ) } \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\mathcal { A } } Q _ { h + 1 } ^ { \\star } ( s ^ { \\prime } , a ^ { \\prime } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            833,
+            745,
+            856
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for all $( s , a , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ]$ . We also call $\\mathcal { T } _ { h }$ the Bellman operator at step $h$ . ",
+        "bbox": [
+            174,
+            866,
+            697,
+            883
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\epsilon$ -optimality and regret We say a policy $\\pi$ is $\\epsilon$ -optimal if $V _ { 1 } ^ { \\pi } ( s _ { 1 } ) \\geq V _ { 1 } ^ { \\star } ( s _ { 1 } ) - \\epsilon$ . Suppose an agent interacts with the environment for $K$ episodes. Denote by $\\pi ^ { k }$ the policy the agent follows in episode $k \\in [ K ]$ . The (accumulative) regret is defined as ",
+        "bbox": [
+            174,
+            90,
+            825,
+            135
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e4edc5fe579df180c282f73bd762326244c035e38856be99c3ad862c3c456005.jpg",
+        "text": "$$\n\\mathrm { R e g } ( K ) : = \\sum _ { k = 1 } ^ { K } [ V _ { 1 } ^ { \\star } ( s _ { 1 } ) - V _ { 1 } ^ { \\pi ^ { k } } ( s _ { 1 } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            140,
+            622,
+            184
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The objective of reinforcement learning is to find an $\\epsilon$ -optimal policy within a small number of interactions or to achieve sublinear regret. ",
+        "bbox": [
+            173,
+            189,
+            825,
+            218
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.1 Function approximation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            233,
+            383,
+            250
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we consider reinforcement learning with value function approximation. Formally, the learner is given a function class $\\mathcal { F } = \\mathcal { F } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { F } _ { H }$ , where ${ \\mathcal { F } } _ { h } \\subseteq ( S \\times { \\mathcal { A } } \\to [ 0 , 1 ] )$ offers a set of candidate functions to approximate $Q _ { h } ^ { \\star }$ —the optimal $Q$ -value function at step $h$ . Since no reward is collected in the $( H + 1 ) ^ { \\mathrm { t h } }$ steps, we always set $f _ { H + 1 } = 0$ . ",
+        "bbox": [
+            174,
+            258,
+            825,
+            318
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning with function approximation in general is extremely challenging without further assumptions (see, e.g., hardness results in [16, 44]). Below, we present two assumptions about function approximation that are commonly adopted in the literature. ",
+        "bbox": [
+            174,
+            323,
+            825,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 (Realizability). $Q _ { h } ^ { \\star } \\in \\mathcal { F } _ { h }$ for all $h \\in [ H ]$ . ",
+        "bbox": [
+            174,
+            368,
+            549,
+            385
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Realizability requires the function class is well-specified, i.e., function class $\\mathcal { F }$ in fact contains the optimal $Q$ -value function $Q ^ { \\star }$ with no approximation error. ",
+        "bbox": [
+            174,
+            393,
+            823,
+            422
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 (Completeness). $\\mathcal T _ { h } \\mathcal F _ { h + 1 } \\subseteq \\mathcal F _ { h }$ for all $h \\in [ H ]$ . ",
+        "bbox": [
+            173,
+            426,
+            591,
+            443
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note $\\mathcal { T } _ { h } \\mathcal { F } _ { h + 1 }$ is defined as $\\{ T _ { h } f _ { h + 1 } : f _ { h + 1 } \\in \\mathcal { F } _ { h + 1 } \\}$ . Completeness requires the function class $\\mathcal { F }$ to be closed under the Bellman operator. ",
+        "bbox": [
+            174,
+            450,
+            823,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When function class $\\mathcal { F }$ has finite elements, we can use its cardinality $| \\mathcal F |$ to measure the “size” of function class $\\mathcal { F }$ . When addressing function classes with infinite elements, we need a notion similar to cardinality. We use the standard $\\epsilon$ -covering number. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            529
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 3 ( $\\epsilon$ -covering number). The $\\epsilon$ -covering number of a set $\\nu$ under metric $\\rho ,$ , denoted as $\\mathcal { N } ( \\mathcal { V } , \\epsilon , \\rho )$ , is the minimum integer $n$ such that there exists a subset $\\mathcal { V } _ { o } \\subset \\mathcal { V }$ with $| \\nu _ { o } | = n$ , and for any $x \\in \\nu$ , there exists $y \\in \\mathcal { V } _ { o }$ such that $\\rho ( x , y ) \\le \\epsilon$ . ",
+        "bbox": [
+            173,
+            532,
+            825,
+            575
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We refer readers to standard textbooks [see, e.g., 45] for further properties of covering number. In this paper, we will always apply the covering number on function class $\\mathcal { F } = \\mathcal { F } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { F } _ { H }$ , and use metric $\\rho ( f , g ) = \\operatorname* { m a x } _ { h } \\| f _ { h } - g _ { h } \\| _ { \\infty }$ . For notational simplicity, we omit the metric dependence and denote the covering number as $\\mathcal { N } _ { \\mathcal { F } } ( \\epsilon )$ . ",
+        "bbox": [
+            174,
+            585,
+            825,
+            642
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2 Eluder dimension ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            656,
+            338,
+            671
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One class of functions highly related to this paper is the function class of low Eluder dimension [18]. ",
+        "bbox": [
+            173,
+            683,
+            821,
+            698
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 4 ( $\\epsilon$ -independence between points). Let $\\mathcal { G }$ be a function class defined on $\\mathcal { X }$ , and $z , x _ { 1 } , x _ { 2 }$ $\\textstyle , \\ldots , x _ { n } \\in { \\mathcal { X } }$ . We say $z$ is $\\epsilon$ -independent of $\\{ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } \\}$ with respect to $\\mathcal { G }$ if there exist $g _ { 1 } , g _ { 2 } \\in \\mathcal { G }$ such that $\\begin{array} { r } { \\sqrt { \\sum _ { i = 1 } ^ { n } ( g _ { 1 } ( x _ { i } ) - g _ { 2 } ( x _ { i } ) ) ^ { 2 } } \\le \\epsilon } \\end{array}$ , but $g _ { 1 } ( z ) - g _ { 2 } ( z ) > \\epsilon .$ . ",
+        "bbox": [
+            173,
+            700,
+            825,
+            748
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, $z$ is independent of $\\{ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } \\}$ means if that there exist two “certifying” functions $g _ { 1 }$ and $g _ { 2 }$ , so that their function values are similar at all points $\\{ x _ { i } \\} _ { i = 1 } ^ { n }$ , but the values are rather different at $z$ . This independence relation naturally induces the following complexity measure. ",
+        "bbox": [
+            173,
+            756,
+            825,
+            799
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 5 (Eluder dimension). Let $\\mathcal { G }$ be a function class defined on $\\mathcal { X }$ . The Eluder dimension $\\mathrm { d i m } _ { \\mathrm { E } } ( \\mathcal { G } , \\epsilon )$ is the length of the longest sequence $\\{ x _ { 1 } , \\ldots , x _ { n } \\} \\subset { \\mathcal { X } }$ such that there exists $\\epsilon ^ { \\prime } \\geq \\epsilon$ where $x _ { i }$ is $\\epsilon ^ { \\prime }$ -independent of $\\{ x _ { 1 } , \\dots , x _ { i - 1 } \\}$ for all $i \\in [ n ]$ . ",
+        "bbox": [
+            173,
+            803,
+            823,
+            847
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recall that a vector space has dimension $d$ if and only if $d$ is the length of the longest sequence of elements $\\{ x _ { 1 } , \\ldots , x _ { d } \\}$ such that $x _ { i }$ is linearly independent of $\\{ x _ { 1 } , \\dotsc , x _ { i - 1 } \\}$ for all $i \\in [ n ]$ . Eluder dimension generalizes the linear independence relation in standard vector space to capture both nonlinear independence and approximate independence, and thus is more general. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Bellman Eluder Dimension ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            431,
+            107
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our new complexity measure—Bellman Eluder (BE) dimension. As one of its most important properties, we will show that the family of problems with low BE dimension contains the two existing most general tractable problem classes in RL—problems with low Bellman rank, and problems with low Eluder dimension (see Figure 1). ",
+        "bbox": [
+            173,
+            119,
+            825,
+            176
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We start by developing a new distributional version of the original Eluder dimension proposed by Russo and Van Roy [18] (see Section 2.2 for more details). ",
+        "bbox": [
+            173,
+            181,
+            821,
+            210
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 6 ( $\\epsilon$ -independence between distributions). Let $\\mathcal { G }$ be a function class defined on $\\mathcal { X }$ , and $\\nu , \\mu _ { 1 } , \\ldots , \\mu _ { n }$ be probability measures over $\\mathcal { X }$ . We say $\\nu$ is $\\epsilon$ -independent of $\\{ \\mu _ { 1 } , \\mu _ { 2 } , \\ldots , \\mu _ { n } \\}$ with respect to $\\mathcal { G }$ if there exists $g \\in { \\mathcal { G } }$ such that $\\begin{array} { r } { \\sqrt { \\sum _ { i = 1 } ^ { n } ( { \\mathbb E } _ { \\mu _ { i } } [ g ] ) ^ { 2 } } \\le \\epsilon } \\end{array}$ , but $\\begin{array} { r } { \\left| \\mathbb { E } _ { \\nu } [ g ] \\right| > \\epsilon } \\end{array}$ . ",
+        "bbox": [
+            173,
+            214,
+            823,
+            261
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 7 (Distributional Eluder (DE) dimension). Let $\\mathcal { G }$ be a function class defined on $\\mathcal { X }$ , and $\\Pi$ be a family of probability measures over $\\mathcal { X }$ . The distributional Eluder dimension $\\dim _ { \\mathrm { D E } } ( { \\mathcal { G } } , \\Pi , \\epsilon )$ is the length of the longest sequence $\\{ \\rho _ { 1 } , \\ldots , \\rho _ { n } \\} \\subset \\Pi$ such that there exists $\\epsilon ^ { \\prime } \\geq \\epsilon$ where $\\rho _ { i }$ is $\\epsilon ^ { \\prime }$ -independent of $\\{ \\rho _ { 1 } , . . . , \\rho _ { i - 1 } \\}$ for all $i \\in [ n ]$ . ",
+        "bbox": [
+            174,
+            262,
+            825,
+            320
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 6 and Definition 7 generalize Definition 4 and Definition 5 to their distributional versions, by inspecting the expected values of functions instead of the function values at points, and by restricting the candidate distributions to a certain family $\\Pi$ . The main advantage of this generalization is exactly in the statistical setting, where estimating the expected values of functions with respect to a certain distribution family can be easier than estimating function values at each point (which is the case for RL in large state spaces). ",
+        "bbox": [
+            173,
+            329,
+            825,
+            414
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "It is clear that the standard Eluder dimension is a special case of the distributional Eluder dimension, because if we choose $\\Pi = \\{ \\delta _ { x } ( \\cdot ) \\mid x \\in \\mathcal { X } \\}$ where $\\delta _ { x } ( \\cdot )$ is the dirac measure centered at $x$ , then $\\dim _ { \\operatorname { E } } ( { \\mathcal { G } } , \\epsilon ) = \\dim _ { \\operatorname { D E } } ( { \\mathcal { G } } - { \\mathcal { G } } , \\Pi , \\epsilon )$ where ${ \\mathcal { G } } - { \\mathcal { G } } = \\{ g _ { 1 } - g _ { 2 } : g _ { 1 } , g _ { 2 } \\in { \\mathcal { G } } \\}$ . ",
+        "bbox": [
+            173,
+            419,
+            825,
+            463
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now we are ready to introduce the key notion in this paper—Bellman Eluder dimension. ",
+        "bbox": [
+            173,
+            467,
+            753,
+            482
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 8 (Bellman Eluder (BE) dimension). Let $( I - \\mathcal { T } _ { h } ) \\mathcal { F } { : = } \\{ f _ { h } - \\mathcal { T } _ { h } f _ { h + 1 } : \\mathrm { ~ } f \\in \\mathcal { F } \\}$ be the set of Bellman residuals induced by $\\mathcal { F }$ at step $h$ , and $\\dot { \\Pi } = \\{ \\Pi _ { h } \\} _ { h = 1 } ^ { H }$ be a collection of $H$ probability measure families over $s \\times { \\mathcal { A } }$ . The $\\epsilon$ -Bellman Eluder of $\\mathcal { F }$ with respect to $\\Pi$ is defined as ",
+        "bbox": [
+            173,
+            484,
+            826,
+            527
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7684c55c79e7aae92818aee655b029b28d502f9bc2ba8596b64bbde2307b01f.jpg",
+        "text": "$$\n\\dim _ { \\mathrm { B E } } ( { \\mathcal F } , \\Pi , \\epsilon ) : = \\operatorname* { m a x } _ { h \\in [ H ] } \\dim _ { \\mathrm { D E } } \\big ( ( I - { \\mathcal T } _ { h } ) { \\mathcal F } , \\Pi _ { h } , \\epsilon \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            532,
+            674,
+            559
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 9 (Q-type v.s. V-type). Definition 8 is based on the Bellman residuals functions that take a state-action pair as input, thus referred to as $Q$ -type $B E$ dimension. Alternatively, one can define $V -$ type BE dimension using a different set of Bellman residual functions that depend on states only (see Appendix $B$ ). We focus on $Q$ -type in the main paper, and present the results for $V .$ -type in Appendix B. Both variants are important, and they include different sets of examples (see Appendix B, C). ",
+        "bbox": [
+            173,
+            564,
+            825,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In short, Bellman Eluder dimension is simply the distributional Eluder dimension on the function class of Bellman residuals, maximizing over all steps. In addition to function class $\\mathcal { F }$ and error $\\epsilon$ , Bellman Eluder dimension also depends on the choice of distribution family $\\Pi$ . For the purpose of this paper, we focus on the following two specific choices. ",
+        "bbox": [
+            174,
+            643,
+            825,
+            702
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. $\\mathcal { D } _ { \\mathcal { F } } : = \\{ \\mathcal { D } _ { \\mathcal { F } , h } \\} _ { h \\in [ H ] }$ , where $\\mathcal { D } _ { \\mathcal { F } , h }$ denotes the collection of all probability measures over $\\boldsymbol { S } \\times \\boldsymbol { A }$ at the $h ^ { \\mathrm { t h } }$ step, which can be generated by executing the greedy policy $\\pi _ { f }$ induced by any $f \\in { \\mathcal { F } }$ , i.e., $\\bar { \\pi } _ { f , h } ( \\cdot ) = \\mathrm { a r g m a } \\bar { \\mathrm { x } } _ { a \\in \\mathcal { A } } f _ { h } ( \\cdot , \\bar { a } )$ for all $h \\in [ H ]$ . 2. $\\mathcal { D } _ { \\Delta } : = \\{ \\mathcal { D } _ { \\Delta , h } \\} _ { h \\in [ H ] }$ , where ${ \\mathcal { D } } _ { \\Delta , h } = \\{ \\delta _ { ( s , a ) } ( \\cdot ) | s \\in S , a \\in { \\mathcal { A } } \\}$ , i.e., the collections of probability measures that put measure 1 on a single state-action pair. ",
+        "bbox": [
+            210,
+            710,
+            825,
+            790
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We say a RL problem has low BE dimension if $\\begin{array} { r } { \\operatorname* { m i n } _ { \\Pi \\in \\{ \\mathcal { D } _ { \\mathcal { F } } , \\mathcal { D } _ { \\Delta } \\} } \\mathrm { d i m } _ { \\mathrm { B E } } ( \\mathcal { F } , \\Pi , \\epsilon ) \\mathrm { i s ~ s m a } } \\end{array}$ ll. ",
+        "bbox": [
+            169,
+            799,
+            756,
+            816
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 Relations with known tractable classes of RL problems ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            830,
+            594,
+            845
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Known tractable problem classes in RL include but not limited to tabular MDPs, linear MDPs [7], linear quadratic regulators [9], generalized linear MDPs [6], kernel MDPs (Appendix C), reactive POMDPs [16], reactive PSRs [46, 1]. There are two existing generic tractable problem classes that jointly contain all the examples mentioned above: the set of RL problems with low Bellman rank, and the set of RL problems with low Eluder dimension. However, for these two generic sets, one does not contain the other. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will show that our new class of RL problems with low BE dimension in fact contains both low Bellman rank problems and low Eluder dimension problems (see Figure 1). That is, our new problem class covers almost all existing tractable RL problems, and to our best knowledge, is the most generic tractable function class so far. ",
+        "bbox": [
+            173,
+            125,
+            825,
+            181
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Relation with low Bellman rank The seminal paper by Jiang et al. [1] proposes the complexity measure—Bellman rank, and shows that a majority of RL examples mentioned above have low Bellman rank. They also propose a hypothesis elimination based algorithm—OLIVE, that learns any low Bellman rank problem within polynomial samples. Formally, ",
+        "bbox": [
+            173,
+            196,
+            825,
+            253
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Definition 10 (Bellman rank). The Bellman rank is the minimum integer $d$ so that there exists $\\phi _ { h }$ : $\\mathcal { F }  \\mathbb { R } ^ { d }$ and $\\psi _ { h } : \\mathcal { F }  \\mathbb { R } ^ { d }$ for each $h \\in [ H ]$ , such that for any $f , f ^ { \\prime } \\in \\mathcal { F }$ , the average Bellman error. ",
+        "bbox": [
+            173,
+            257,
+            825,
+            299
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0b2c3eedd07d09f7622957c613d0dec65e5d9aba0a1f0f7ec292e77a19e01cf.jpg",
+        "text": "$$\n\\mathcal { E } ( f , \\pi _ { f ^ { \\prime } } , h ) : = \\mathbb { E } _ { \\pi _ { f ^ { \\prime } } } [ ( f _ { h } - \\mathcal { T } _ { h } f _ { h + 1 } ) ( s _ { h } , a _ { h } ) ] = \\langle \\phi _ { h } ( f ) , \\psi _ { h } ( f ^ { \\prime } ) \\rangle ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            299,
+            714,
+            319
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\| \\phi _ { h } ( f ) \\| _ { 2 } \\cdot \\| \\psi _ { h } ( f ^ { \\prime } ) \\| _ { 2 } \\leq \\zeta$ , and $\\zeta$ is the normalization parameter. ",
+        "bbox": [
+            176,
+            321,
+            648,
+            338
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We remark that similar to Bellman Eluder dimension, Bellman rank also has two variants—Q-type (Definition 10) and $\\mathrm { V } .$ -type (see Appendix B). Recall that we use $\\pi _ { f }$ to denote the greedy policy induced by value function $f$ . Intuitively, a problem with Bellman rank says its average Bellman error can be decomposed as the inner product of two $d$ -dimensional vectors, where one vector depends on the roll-in policy $\\pi _ { f ^ { \\prime } }$ , while the other vector depends on the value function $f$ . At a high level, it claims that the average Bellman error has a linear inner product structure. ",
+        "bbox": [
+            173,
+            348,
+            825,
+            433
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 11 (low Bellman rank $\\subset$ low BE dimension). If an MDP with function class $\\mathcal { F }$ has Bellman rank $d$ with normalization parameter $\\zeta ,$ , then ",
+        "bbox": [
+            174,
+            436,
+            823,
+            465
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05a1198e489301233e7d23d30cd0c0711778a6ac5e6f8e47708954de77d3288b.jpg",
+        "text": "$$\n\\dim _ { \\mathrm { B E } } ( \\mathcal { F } , \\mathcal { D } _ { \\mathcal { F } } , \\epsilon ) \\leq \\mathcal { O } ( 1 + d \\log ( 1 + \\zeta / \\epsilon ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            473,
+            647,
+            491
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 11 claims that problems with low Bellman rank also have low BE dimension, with a small multiplicative factor that is only logarithmic in $\\zeta$ and $\\epsilon ^ { - 1 }$ . ",
+        "bbox": [
+            171,
+            503,
+            823,
+            534
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Relation with low Eluder dimension Wang et al. [17] study the setting where the function class $\\mathcal { F }$ has low Eluder dimension, which includes generalized linear functions. They prove that, when the completeness assumption is satisfied,3 low Eluder dimension problems can be efficiently learned in polynomial samples. ",
+        "bbox": [
+            173,
+            547,
+            825,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 12 (low Eluder dimension $\\subset$ low BE dimension). Assume $\\mathcal { F }$ satisfies completeness (Assumption 2). Then for all $\\epsilon > 0$ , ",
+        "bbox": [
+            171,
+            608,
+            821,
+            637
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ff0a11d0cc3927d32b30d4abf17768d4c2c49ab0c24c9a306f26067f4c90580.jpg",
+        "text": "$$\n\\dim _ { \\mathrm { B E } } \\left( { \\mathcal { F } } , { \\mathcal { D } } _ { \\Delta } , \\epsilon \\right) \\leq \\operatorname* { m a x } _ { h \\in \\left[ H \\right] } \\dim _ { \\mathrm { E } } ( { \\mathcal { F } } _ { h } , \\epsilon ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            643,
+            635,
+            670
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 12 asserts that problems with low Eluder dimension also have low BE dimension, which is a natural consequence of completeness and the fact that Eluder dimension is a special case of distributional Eluder dimension. ",
+        "bbox": [
+            173,
+            684,
+            825,
+            727
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we show that the set of low BE dimension problems is strictly larger than the union of low Eluder dimension problems and low Bellman rank problems. ",
+        "bbox": [
+            174,
+            733,
+            820,
+            762
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 13 (low BE dimension $\\nless$ low Eluder dimension ∪ low Bellman rank). For any $m ~ \\in ~ \\mathbb { N } ^ { + }$ , there exists an MDP and a function class $\\mathcal { F }$ so that for all $\\epsilon ~ \\in ~ ( 0 , 1 ]$ , we have $\\dim _ { \\mathrm { B E } } ( { \\mathcal F } , { \\mathcal D } _ { { \\mathcal F } } , \\epsilon ) = \\dim _ { \\mathrm { B E } } ( { \\mathcal F } , { \\mathcal D } _ { \\Delta } , \\epsilon ) \\ \\leq \\ 5 ,$ , but ${ \\mathrm { m i n } } \\{ { \\mathrm { m i n } } _ { h \\in [ H ] }$ $\\dim _ { \\operatorname { E } } ( { \\mathcal { F } } _ { h } , \\epsilon )$ , Bellman $\\mathrm { r a n k } \\} \\geq$ $m$ . ",
+        "bbox": [
+            173,
+            765,
+            825,
+            821
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In particular, the family of low BE dimension includes new examples such as kernel reactive POMDPs (Appendix C), which can not be addressed by the framework of either Bellman rank or Eluder dimension. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            876
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize: $\\mathcal { D } _ { 1 } , . . . , \\mathcal { D } _ { H }  \\emptyset$ , $B ^ { 0 }  { \\mathcal { F } }$ .   \n2: for episode $k$ from 1 to $K$ do   \n3: Choose policy $\\pi ^ { k } = \\pi _ { f ^ { k } }$ , where $f ^ { k } = \\operatorname { a r g m a x } _ { f \\in { \\mathcal { B } } ^ { k - 1 } } f { \\big ( } s _ { 1 } , \\pi _ { f } { \\big ( } s _ { 1 } { \\big ) } { \\big ) }$ .   \n4: Collect a trajectory $\\left( s _ { 1 } , a _ { 1 } , r _ { 1 } , \\dots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 } \\right)$ by following $\\pi ^ { k }$ .   \n5: Augment $\\mathcal { D } _ { h } = \\mathcal { D } _ { h } \\cup \\left\\{ \\left( s _ { h } , a _ { h } , r _ { h } , s _ { h + 1 } \\right) \\right\\}$ for all $h \\in [ H ]$ .   \n6: Update   \n$\\begin{array} { r l } & { \\mathbf { \\Phi } ^ { \\mathrm { t e } } } \\\\ & { \\quad \\mathcal { B } ^ { k } = \\left\\{ f \\in \\mathcal { F } : \\mathcal { L } _ { \\mathcal { D } _ { h } } ( f _ { h } , f _ { h + 1 } ) \\leq \\displaystyle \\operatorname* { i n f } _ { g \\in \\mathcal { G } _ { h } } \\mathcal { L } _ { \\mathcal { D } _ { h } } ( g , f _ { h + 1 } ) + \\beta \\mathrm { ~ f o r ~ a l l ~ } h \\in [ H ] \\right\\} , } \\\\ & { \\quad \\quad \\mathrm { w h e r e ~ } \\mathcal { L } _ { \\mathcal { D } _ { h } } ( \\xi _ { h } , \\zeta _ { h + 1 } ) = \\displaystyle \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } _ { h } } \\left[ \\xi _ { h } ( s , a ) - r - \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\mathcal { A } } \\zeta _ { h + 1 } ( s ^ { \\prime } , a ^ { \\prime } ) \\right] ^ { 2 } . } \\end{array}$ ",
+        "bbox": [
+            181,
+            111,
+            779,
+            257
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7: Output $\\pi ^ { \\mathrm { { o u t } } }$ sampled uniformly at random from $\\{ \\pi ^ { k } \\} _ { k = 1 } ^ { K }$ ",
+        "bbox": [
+            179,
+            258,
+            583,
+            275
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 Algorithm GOLF ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            303,
+            349,
+            320
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Section 3 defines a new class of RL problems with low BE dimension, and shows that the new class is rich, containing almost all the existing known tractable RL problems so far. In this section, we propose a new simple optimization-based algorithm—Global Optimism based on Local Fitting (GOLF). We prove that, low BE dimension problems are indeed tractable, i.e., GOLF can find nearoptimal policies for these problems within a polynomial number of samples. ",
+        "bbox": [
+            173,
+            333,
+            825,
+            405
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "At a high level, GOLF can be viewed as an optimistic version of the classic algorithm—Fitted QIteration (FQI) [42]. GOLF generalizes the ELEANOR algorithm [8] from the special linear setting to the general setting with arbitrary function classes. ",
+        "bbox": [
+            174,
+            410,
+            825,
+            452
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The pseudocode of GOLF is given in Algorithm 1. GOLF initializes datasets $\\{ \\mathcal { D } _ { h } \\} _ { h = 1 } ^ { H }$ to be empty sets, and confidence set $B ^ { 0 }$ to be $\\mathcal { F }$ . Then, in each episode, GOLF performs two main steps: ",
+        "bbox": [
+            173,
+            458,
+            821,
+            488
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Line 3 (Optimistic planning): compute the most optimistic value function $f ^ { k }$ from the confidence set $B ^ { k - 1 }$ constructed in the last episode , and choose $\\pi ^ { k }$ to be its greedy policy. • Line 4-6 (Execute the policy and update the confidence set): execute policy $\\pi ^ { k }$ for one episode, collect data, and update the confidence set using the new data. ",
+        "bbox": [
+            215,
+            498,
+            825,
+            560
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "At the heart of GOLF is the way we construct the confidence set $B ^ { k }$ . For each $h \\in [ H ]$ , GOLF maintains a local regression constraint using the collected transition data $\\mathcal { D } _ { h }$ at this step ",
+        "bbox": [
+            176,
+            570,
+            825,
+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/89e3162bbe4ebd2dc51b46a277be3e64f9698f45469ddab5ffcdb46dd2b89975.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathcal { D } _ { h } } ( f _ { h } , f _ { h + 1 } ) \\leq \\operatorname* { i n f } _ { g \\in \\mathcal { G } _ { h } } \\mathcal { L } _ { \\mathcal { D } _ { h } } ( g , f _ { h + 1 } ) + \\beta ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            606,
+            640,
+            631
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta$ is a confidence parameter, and $\\mathcal { L } _ { \\mathcal { D } _ { h } }$ is the squared loss defined in (2), which can be viewed as a proxy to the squared Bellman error at step $h$ . We remark that FQI algorithm [42] simply updates $\\begin{array} { r } { \\dot { f _ { h } } \\gets \\operatorname * { a r g m i n } _ { \\phi \\in \\mathcal { F } _ { h } } \\mathcal { L } _ { \\mathcal { D } _ { h } } ( \\phi , f _ { h + 1 } ) } \\end{array}$ . Our constraint (3) can be viewed as a relaxed version of this update, which allows $f _ { h }$ to be not only the minimizer of the loss $\\mathcal { L } _ { \\mathcal { D } _ { h } } ( \\cdot , f _ { h + 1 } )$ , but also any function whose loss is only slightly larger than the optimal loss over the auxiliary function class $\\mathcal { G } _ { h }$ . ",
+        "bbox": [
+            173,
+            636,
+            825,
+            708
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We remark that in general, the optimization problem in Line 3 of GOLF can not be solved computationally efficiently. ",
+        "bbox": [
+            173,
+            713,
+            823,
+            742
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 Theoretical guarantees ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            758,
+            372,
+            773
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we present the theoretical guarantees for GOLF, which hold under Assumption 1 (realizability) and the following generalized completeness assumption introduced in [47, 21]. Let $\\mathcal { G } = \\mathcal { G } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { G } _ { H }$ be an auxiliary function class provided to the learner where each ${ \\mathcal { G } } _ { h } \\subseteq ( S \\times { \\mathcal { A } } \\to$ $[ 0 , 1 ] )$ . Generalized completeness requires the auxiliary function class $\\mathcal { G }$ to be rich enough so that applying Bellman operator to any function in the primary function class $\\mathcal { F }$ will end up in $\\mathcal { G }$ . ",
+        "bbox": [
+            173,
+            784,
+            825,
+            853
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Assumption 14 (Generalized completeness). $\\mathcal T _ { h } \\mathcal F _ { h + 1 } \\subseteq \\mathcal G _ { h }$ for all $h \\in [ H ]$ . ",
+        "bbox": [
+            174,
+            857,
+            678,
+            873
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "If we choose $\\mathcal { G } = \\mathcal { F }$ , then Assumption 14 is equivalent to the standard completeness assumption (Assumption 2). Now, we are ready to present the main theorem for GOLF. ",
+        "bbox": [
+            173,
+            882,
+            821,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 15 (Regret of GOLF). Under Assumption $^ { l }$ , 14, there exists an absolute constant c such that for any $\\delta \\in ( 0 , 1 ]$ , $K \\in \\mathbb N$ , if we choose parameter $\\beta = c \\log [ \\mathcal { N _ { F \\cup \\mathcal { G } } } ( 1 / K ) \\cdot K H / \\delta ]$ in GOLF, then with probability at least $1 - \\delta$ , for all $k \\in [ K ]$ , we have ",
+        "bbox": [
+            173,
+            90,
+            825,
+            135
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e31567478f5dd79e645aed2d066a7489289e0c68fd08faf4586a9cb8e7a42526.jpg",
+        "text": "$$\n\\mathrm { R e g } ( k ) = \\sum _ { t = 1 } ^ { k } \\Big [ V _ { 1 } ^ { \\star } \\big ( s _ { 1 } \\big ) - V _ { 1 } ^ { \\pi ^ { t } } \\big ( s _ { 1 } \\big ) \\Big ] \\leq \\mathcal { O } ( H \\sqrt { d k \\beta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            140,
+            673,
+            183
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { d = \\operatorname* { m i n } _ { \\Pi \\in \\{ \\mathcal { D } _ { \\Delta } , \\mathcal { D } _ { \\mathcal { F } } \\} } \\mathrm { d i m } _ { \\mathrm { B E } } \\left( \\mathcal { F } , \\Pi , 1 / \\sqrt { K } \\right) } \\end{array}$ is the BE dimension. ",
+        "bbox": [
+            174,
+            189,
+            642,
+            208
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 15 asserts that, under the realizability and completeness assumptions, the general class of RL problems with low BE dimension is indeed tractable: there exists an algorithm (GOLF) that can achieve $\\sqrt { K }$ regret, whose multiplicative factor depends only polynomially on the horizon of MDP $H$ , the BE dimension $d$ , and the log covering number of the two function classes. Most importantly, the regret is independent of the number of the states, which is crucial for dealing with practical RL problems with function approximation, where the state spaces are typically exponentially large. ",
+        "bbox": [
+            173,
+            215,
+            826,
+            301
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We remark that when function class ${ \\mathcal { F } } \\cup { \\mathcal { G } }$ has finite number of elements, its covering number is upper bounded by its cardinality $| { \\mathcal { F } } \\cup { \\mathcal { G } } |$ . For a wide range of function classes in practice, the log $\\epsilon ^ { \\prime }$ -covering number has only logarithmic dependence on $\\epsilon ^ { \\prime }$ . Informally, we denote the log covering number as $\\log \\mathcal { N } _ { \\mathcal { F } \\cup \\mathcal { G } }$ and omit its $\\epsilon ^ { \\prime }$ dependency for clean presentation. Theorem 15 claims that the regret scales as $\\tilde { \\mathcal { O } } ( H \\sqrt { d K \\log \\mathcal { N _ { F \\cup \\mathcal { G } } } } )$ , where $\\bar { \\tilde { \\mathcal { O } } } ( \\cdot )$ omits absolute constants and logarithmic terms.4 ",
+        "bbox": [
+            173,
+            306,
+            825,
+            381
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "By the standard online-to-batch argument, we also derive the sample complexity of GOLF. ",
+        "bbox": [
+            171,
+            385,
+            766,
+            400
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Corollary 16 (Sample Complexity of GOLF). Under Assumption 1, 2, there exists an absolute constant c such that for any $\\epsilon \\in ( 0 , 1 ] ,$ , if we choose $\\beta = c \\log [ \\mathcal { N _ { F \\cup \\mathcal { G } } } ( \\epsilon ^ { 2 } / ( d H ^ { 2 } ) ) \\cdot H K ]$ in GOLF, then the output policy $\\pi ^ { o u t }$ is $\\mathcal { O } ( \\epsilon )$ -optimal with probability at least $1 / 2 , i f$ ",
+        "bbox": [
+            176,
+            404,
+            823,
+            446
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7443575baee63ee3c220a986051406e313ac03fa624834ad6fefdf70f66a589c.jpg",
+        "text": "$$\nK \\geq \\Omega \\left( \\frac { H ^ { 2 } d } { \\epsilon ^ { 2 } } \\cdot \\log \\left[ \\mathcal { N } _ { \\mathcal { F } \\cup \\mathcal { G } } \\left( \\frac { \\epsilon ^ { 2 } } { H ^ { 2 } d } \\right) \\cdot \\frac { H d } { \\epsilon } \\right] \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            452,
+            655,
+            487
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { d = \\operatorname* { m i n } _ { \\Pi \\in \\{ \\mathcal { D } _ { \\Delta } , \\mathcal { D } _ { \\mathcal { F } } \\} } \\mathrm { d i m } _ { \\mathrm { B E } } \\left( \\mathcal { F } , \\Pi , \\epsilon / H \\right) } \\end{array}$ is the BE dimension. ",
+        "bbox": [
+            174,
+            492,
+            625,
+            511
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Corollary 16 claims that $\\tilde { \\mathcal { O } } ( H ^ { 2 } d \\log ( \\mathcal { N } _ { \\mathcal { F } \\cup \\mathcal { G } } ) / \\epsilon ^ { 2 } )$ samples are enough for GOLF to learn a nearoptimal policy of any low BE dimension problem. Our sample complexity scales linear in both the BE dimension $d$ , and the log covering number $\\log ( \\mathcal { N } _ { \\mathcal { F } \\cup \\mathcal { G } } )$ . ",
+        "bbox": [
+            174,
+            521,
+            825,
+            565
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To showcase the sharpness of our results, we compare them to the previous results when restricted to the corresponding settings. (1) For linear function class with ambient dimension $d _ { \\mathrm { l i n } }$ , we have√ BE dimension $d = \\mathcal { \\tilde { O } } ( d _ { \\mathrm { l i n } } )$ and $\\log ( \\mathcal { N _ { F \\cup \\mathcal { G } } } ) = \\tilde { \\mathcal { O } } ( d _ { \\mathrm { l i n } } )$ . Our regret bound becomes $\\tilde { \\mathcal { O } } ( H d _ { \\mathrm { l i n } } \\sqrt { K } )$ which matches the best known result [8] up to logarithmic factors; (2) For function class with low Eluder dimension [17], our results hold under weaker completeness assumptions. Our regret scales√ with $\\sqrt { d _ { \\mathrm { E } } }$ in terms of dependency on Eluder dimension $d _ { \\mathrm { E } }$ , which improves the linear $d _ { \\mathrm { E } }$ scaling in the regret of [17]; (3) Finally, for low Bellman rank problems, our sample complexity scales linearly with Bellman rank, which improves upon the quadratic dependence in [1]. We remark that all results mentioned above assume (approximate) realizability. All except [1] assume (approximate) completeness. ",
+        "bbox": [
+            173,
+            570,
+            825,
+            712
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.2 Key ideas in proving Theorem 15 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            727,
+            442,
+            742
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we present a brief proof sketch for the regret bound of GOLF. We defer all the details to Appendix E. For simplicity, we only discuss the case of choosing $\\mathcal { D } _ { \\mathcal { F } }$ as the distribution family $\\Pi$ in the definition of Bellman Eluder dimension (Definition 8). The proof for using $\\mathcal { D } _ { \\Delta }$ as the distribution family follows from similar arguments. ",
+        "bbox": [
+            173,
+            752,
+            825,
+            809
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our proof strategy consists of three main steps. ",
+        "bbox": [
+            174,
+            815,
+            482,
+            829
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Step 1: Prove optimism. We firstly show that, with high probability, the optimal value function $Q ^ { \\star }$ indeed lies in the confidence set $B ^ { k }$ for all $k \\in [ K ]$ (Lemma 40 in Appendix E.1), which is a natural consequence of martingale concentration and the properties of the confidence set we designed. Because of $Q ^ { \\star } \\in B ^ { k }$ , the optimistic planning step (Line 3) in GOLF guarantees that $V _ { 1 } ^ { \\star } ( \\mathsf { \\bar { s } } _ { 1 } ) \\leq \\operatorname* { m a x } _ { a } f _ { 1 } ^ { k } ( s _ { 1 } , \\mathsf { \\bar { a } } )$ for every episode $k$ . This optimism allows the following upper bound on regret ",
+        "bbox": [
+            173,
+            843,
+            825,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb189fdc09bb764e7ba53689d725ea8b3734e362b81e50ceee1a8c8229146624.jpg",
+        "text": "$$\n\\mathrm { R e g } ( K ) \\leq \\sum _ { k = 1 } ^ { K } \\left( \\operatorname* { m a x } _ { a } f _ { 1 } ^ { k } ( s _ { 1 } , a ) - V _ { 1 } ^ { \\pi ^ { k } } ( s _ { 1 } ) \\right) = \\sum _ { h = 1 } ^ { H } \\sum _ { k = 1 } ^ { K } \\mathbb { E } _ { \\pi ^ { k } } \\left[ ( f _ { h } ^ { k } - \\mathcal { T } f _ { h + 1 } ^ { k } ) ( s _ { h } , a _ { h } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            154,
+            784,
+            198
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where the right equality follows from the standard policy loss decomposition (see, e.g., Lemma 1 in [1]), and $\\mathbb { E } _ { \\pi }$ denotes the expectation taken over sequence $( s _ { 1 } , a _ { 1 } , \\dotsc , s _ { H } , a _ { H } )$ when executing policy $\\pi$ . ",
+        "bbox": [
+            173,
+            203,
+            826,
+            246
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Utilize the sharpness of our confidence set. Recall that our construction of the confidence set in Line 6 of GOLF forces $f ^ { k }$ computed in episode $k$ to have a small loss $\\mathcal { L } _ { \\mathcal { D } _ { h } }$ , which is a proxy for empirical squared Bellman error under data $\\mathcal { D } _ { h }$ . Since data $\\mathcal { D } _ { h }$ in episode $k$ are collected by executing each $\\pi ^ { i }$ for one episode for all $i < k$ , by standard martingale concentration arguments and the completeness assumption, we can show that with high probability (Lemma 39 in Appendix E.1) ",
+        "bbox": [
+            173,
+            258,
+            825,
+            340
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0fa7cd064dedc9d689f336698477714ab20071e31d29bf1111cb94a90e3a185d.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { k - 1 } \\mathbb { E } _ { \\pi ^ { i } } \\left[ ( f _ { h } ^ { k } - \\mathcal { T } f _ { h + 1 } ^ { k } ) ( s _ { h } , a _ { h } ) \\right] ^ { 2 } \\leq \\mathcal { O } ( \\beta ) , \\mathrm { ~ f o r ~ a l l ~ } ( k , h ) \\in [ K ] \\times [ H ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            334,
+            735,
+            377
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Step 3: Establish relations between (4) and (5). So far, we want to upper-bound (4), while we know (5). We note that the RHS of (4) is very similar to the LHS of (5), except that the latter is the squared Bellman error, and the expectation is taken under previous policy $\\pi ^ { i }$ for $i < k$ . To establish the connection between these two, it turns out that we need the Bellman Eluder dimension to be small. Concretely, we have the following lemma. ",
+        "bbox": [
+            173,
+            390,
+            825,
+            460
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lemma 17. Given a function class $\\Phi$ defined on $\\mathcal { X }$ with $| \\phi ( x ) | \\le 1$ for all $( \\phi , x ) \\in \\Phi \\times \\mathcal { X }$ , and $a$ family of probability measures $\\Pi$ over $\\mathcal { X }$ . Suppose sequence $\\{ \\phi _ { k } \\} _ { k = 1 } ^ { K } \\subset \\dot { \\Phi }$ and $\\{ \\mu _ { k } \\} _ { k = 1 } ^ { K } \\subset \\Pi$ satisfy that for all $k \\in [ K ] ,$ , $\\begin{array} { r } { \\sum _ { i = 1 } ^ { k - 1 } ( \\mathbb { E } _ { \\mu _ { i } } [ \\phi _ { k } ] ) ^ { 2 } \\ : \\leq \\ : \\beta } \\end{array}$ . Then for all $k \\in [ K ]$ , $\\begin{array} { r } { \\sum _ { i = 1 } ^ { k } | \\mathbb { E } _ { \\mu _ { i } } [ \\phi _ { i } ] | \\ \\leq } \\end{array}$ $\\mathcal { O } ( \\sqrt { \\mathrm { d i m } _ { \\mathrm { D E } } ( \\Phi , \\Pi , 1 / k ) \\beta k } )$ . ",
+        "bbox": [
+            173,
+            463,
+            825,
+            527
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lemma 17 is a simplification of Lemma 41 in Appendix E, which is a modification of Lemma 2 in [18]. Intuitively, Lemma 17 can be viewed as an analogue of the pigeon-hole principle for DE dimension. Choose $\\Phi$ to be the function class of Bellman residuals, and $\\mu _ { k }$ to be the distribution under policy $\\pi ^ { k }$ , we finish the proof. ",
+        "bbox": [
+            173,
+            536,
+            825,
+            593
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            612,
+            299,
+            630
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a new complexity measure—Bellman Eluder (BE) dimension for reinforcement learning with function approximation. Our new complexity measure identifies a new rich class of RL problems that subsumes a majority of existing tractable problem classes in RL. We design a new optimization-based algorithm—GOLF, and provide a new analysis for algorithm OLIVE. Both algorithms show that the new rich class of RL problems we identified in fact can be learned within a polynomial number of samples. We hope our results shed light on the future research in finding the minimal structural assumptions that allow sample-efficient reinforcement learning. ",
+        "bbox": [
+            173,
+            643,
+            825,
+            741
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            760,
+            330,
+            777
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was partial supported by National Science Foundation Grant NSF-IIS-2107304. ",
+        "bbox": [
+            171,
+            791,
+            764,
+            806
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] 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.   \n[2] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013.   \n[3] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529 (7587):484–489, 2016.   \n[4] Jens Kober, J Andrew Bagnell, and Jan Peters. Reinforcement learning in robotics: A survey. The International Journal of Robotics Research, 32(11):1238–1274, 2013.   \n[5] Jiwei Li, Will Monroe, Alan Ritter, Michel Galley, Jianfeng Gao, and Dan Jurafsky. Deep reinforcement learning for dialogue generation. arXiv preprint arXiv:1606.01541, 2016.   \n[6] Yining Wang, Ruosong Wang, Simon S Du, and Akshay Krishnamurthy. Optimism in reinforcement learning with generalized linear function approximation. arXiv preprint arXiv:1912.04136, 2019.   \n[7] 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, 2020.   \n[8] Andrea Zanette, Alessandro Lazaric, Mykel Kochenderfer, and Emma Brunskill. Learning near optimal policies with low inherent bellman error. arXiv preprint arXiv:2003.00153, 2020.   \n[9] Brian DO Anderson and John B Moore. Optimal control: linear quadratic methods. Courier Corporation, 2007.   \n[10] Maryam Fazel, Rong Ge, Sham Kakade, and Mehran Mesbahi. Global convergence of policy gradient methods for the linear quadratic regulator. In International Conference on Machine Learning, pages 1467–1476. PMLR, 2018.   \n[11] Sarah Dean, Horia Mania, Nikolai Matni, Benjamin Recht, and Stephen Tu. On the sample complexity of the linear quadratic regulator. Foundations of Computational Mathematics, pages 1–47, 2019.   \n[12] Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 2013.   \n[13] Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):463–482, 2002.   \n[14] Nick Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine learning, 2(4):285–318, 1988.   \n[15] Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Online learning: Random averages, combinatorial parameters, and learnability. 2010.   \n[16] Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Pac reinforcement learning with rich observations. arXiv preprint arXiv:1602.02722, 2016.   \n[17] Ruosong Wang, Ruslan Salakhutdinov, and Lin F Yang. Provably efficient reinforcement learning with general value function approximation. arXiv preprint arXiv:2005.10804, 2020.   \n[18] Daniel Russo and Benjamin Van Roy. Eluder dimension and the sample complexity of optimistic exploration. In Advances in Neural Information Processing Systems, pages 2256–2264, 2013.   \n[19] Csaba Szepesvari and R ´ emi Munos. Finite time bounds for sampling based fitted value it- ´ eration. In Proceedings of the 22nd international conference on Machine learning, pages 880–887, 2005.   \n[20] Remi Munos and Csaba Szepesv ´ ari. Finite-time bounds for fitted value iteration. ´ Journal of Machine Learning Research, 9(May):815–857, 2008.   \n[21] Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. arXiv preprint arXiv:1905.00360, 2019.   \n[22] Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. arXiv preprint arXiv:2008.04990, 2020.   \n[23] Ronen I Brafman and Moshe Tennenholtz. R-max-a general polynomial time algorithm for near-optimal reinforcement learning. Journal of Machine Learning Research, 3(Oct):213–231, 2002.   \n[24] Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(4), 2010.   \n[25] Christoph Dann and Emma Brunskill. Sample complexity of episodic fixed-horizon reinforcement learning. In Advances in Neural Information Processing Systems, pages 2818–2826, 2015.   \n[26] Shipra Agrawal and Randy Jia. Optimistic posterior sampling for reinforcement learning: worst-case regret bounds. In Advances in Neural Information Processing Systems, pages 1184– 1194, 2017.   \n[27] Mohammad Gheshlaghi Azar, Ian Osband, and Remi Munos. Minimax regret bounds for ´ reinforcement learning. arXiv preprint arXiv:1703.05449, 2017.   \n[28] Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. arXiv preprint arXiv:1901.00210, 2019.   \n[29] Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. Is q-learning provably efficient? In Advances in Neural Information Processing Systems, pages 4863–4873, 2018.   \n[30] Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learning via reference-advantage decomposition. arXiv preprint arXiv:2004.10019, 2020.   \n[31] Omar Darwiche Domingues, Pierre Menard, Emilie Kaufmann, and Michal Valko. Episodic ´ reinforcement learning in finite mdps: Minimax lower bounds revisited. In Algorithmic Learning Theory, pages 578–598. PMLR, 2021.   \n[32] Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. arXiv preprint arXiv:1912.05830, 2019.   \n[33] Andrea Zanette, Alessandro Lazaric, Mykel J Kochenderfer, and Emma Brunskill. Provably efficient reward-agnostic navigation with linear value iteration. Advances in Neural Information Processing Systems, 33, 2020.   \n[34] Alekh Agarwal, Sham Kakade, Akshay Krishnamurthy, and Wen Sun. Flambe: Structural complexity and representation learning of low rank mdps. Advances in Neural Information Processing Systems, 33, 2020.   \n[35] Gergely Neu and Ciara Pike-Burke. A unifying view of optimism in episodic reinforcement learning. Advances in Neural Information Processing Systems, 33, 2020.   \n[36] Wen Sun, Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Modelbased rl in contextual decision processes: Pac bounds and exponential improvements over model-free approaches. In Conference on Learning Theory, pages 2898–2933, 2019.   \n[37] Ian Osband and Benjamin Van Roy. Model-based reinforcement learning and the eluder dimension. In Advances in Neural Information Processing Systems, pages 1466–1474, 2014.   \n[38] Kefan Dong, Jian Peng, Yining Wang, and Yuan Zhou. Root-n-regret for learning in markov decision processes with function approximation and low bellman rank. In Conference on Learning Theory, pages 1554–1557. PMLR, 2020.   \n[39] Zhuoran Yang, Chi Jin, Zhaoran Wang, Mengdi Wang, and Michael I Jordan. Bridging exploration and general function approximation in reinforcement learning: Provably efficient kernel and neural value iterations. arXiv preprint arXiv:2011.04622, 2020.   \n[40] Dylan J Foster, Alexander Rakhlin, David Simchi-Levi, and Yunzong Xu. Instance-dependent complexity of contextual bandits and reinforcement learning: A disagreement-based perspective. arXiv preprint arXiv:2010.03104, 2020.   \n[41] Simon S Du, Sham M Kakade, Jason D Lee, Shachar Lovett, Gaurav Mahajan, Wen Sun, and Ruosong Wang. Bilinear classes: A structural framework for provable generalization in rl. arXiv preprint arXiv:2103.10897, 2021.   \n[42] Csaba Szepesvari. Algorithms for reinforcement learning. ´ Synthesis lectures on artificial intelligence and machine learning, 4(1):1–103, 2010.   \n[43] Martin L Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014.   \n[44] Gellert Weisz, Philip Amortila, and Csaba Szepesvari. Exponential lower bounds for ´ planning in mdps with linearly-realizable optimal action-value functions. arXiv preprint arXiv:2010.01374, 2020.   \n[45] Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint, volume 48. Cambridge University Press, 2019.   \n[46] Satinder Singh, Michael James, and Matthew Rudary. Predictive state representations: A new theory for modeling dynamical systems. arXiv preprint arXiv:1207.4167, 2012.   \n[47] Andras Antos, Csaba Szepesv ´ ari, and R ´ emi Munos. Learning near-optimal policies with ´ bellman-residual minimization based fitted policy iteration and a single sample path. Machine Learning, 71(1):89–129, 2008.   \n[48] Christoph Dann, Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. On oracle-efficient pac rl with rich observations. In Advances in neural information processing systems, pages 1422–1432, 2018.   \n[49] Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In International Conference on Machine Learning, pages 1638–1646, 2014. ",
+        "bbox": [
+            173,
+            87,
+            826,
+            915
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            79,
+            826,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            88,
+            826,
+            647
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rygf-kSYwH/rygf-kSYwH.md

@@ -0,0 +1,464 @@
+# Behaviour Suite for Reinforcement Learning
+
+Ian Osband∗, Yotam Doron, Matteo Hessel, John Aslanides Eren Sezener, Andre Saraiva, Katrina McKinney, Tor Lattimore, Csaba Szepesvari Satinder Singh, Benjamin Van Roy, Richard Sutton, David Silver, Hado Van Hasselt
+
+DeepMind
+
+# Abstract
+
+This paper introduces the Behaviour Suite for Reinforcement Learning, or bsuite for short. bsuite is a collection of carefully-designed experiments that investigate core capabilities of reinforcement learning (RL) agents with two objectives. First, to collect clear, informative and scalable problems that capture key issues in the design of general and efficient learning algorithms. Second, to study agent behaviour through their performance on these shared benchmarks. To complement this effort, we open source github.com/deepmind/bsuite, which automates evaluation and analysis of any agent on bsuite. This library facilitates reproducible and accessible research on the core issues in RL, and ultimately the design of superior learning algorithms. Our code is Python, and easy to use within existing projects. We include examples with OpenAI Baselines, Dopamine as well as new reference implementations. Going forward, we hope to incorporate more excellent experiments from the research community, and commit to a periodic review of bsuite from a committee of prominent researchers.
+
+# 1 Introduction
+
+The reinforcement learning (RL) problem describes an agent interacting with an environment with the goal of maximizing cumulative reward through time (Sutton $\&$ Barto, 2017). Unlike other branches of control, the dynamics of the environment are not fully known to the agent, but can be learned through experience. Unlike other branches of statistics and machine learning, an RL agent must consider the effects of its actions upon future experience. An efficient RL agent must address three challenges simultaneously:
+
+1. Generalization: be able to learn efficiently from data it collects.   
+2. Exploration: prioritize the right experience to learn from.   
+3. Long-term consequences: consider effects beyond a single timestep.
+
+The great promise of reinforcement learning are agents that can learn to solve a wide range of important problems. According to some definitions, an agent that can learn to perform at or above human level across a wide variety of tasks is an artificial general intelligence (AGI) (Minsky, 1961; Legg et al., 2007).
+
+Interest in artificial intelligence has undergone a resurgence in recent years. Part of this interest is driven by the constant stream of innovation and success on high profile challenges previously deemed impossible for computer systems. Improvements in image recognition are a clear example of these accomplishments, progressing from individual digit recognition (LeCun et al., 1998), to mastering ImageNet in only a few years (Deng et al., 2009; Krizhevsky et al., 2012). The advances in RL systems have been similarly impressive: from checkers (Samuel, 1959), to Backgammon (Tesauro, 1995), to Atari games (Mnih et al., 2015a), to competing with professional players at DOTA (Pachocki et al., 2019) or StarCraft (Vinyals et al., 2019) and beating world champions at Go (Silver et al., 2016). Outside of playing games, decision systems are increasingly guided by AI systems (Evans & Gao, 2016).
+
+As we look towards the next great challenges for RL and AI, we need to understand our systems better (Henderson et al., 2017). This includes the scalability of our RL algorithms, the environments where we expect them to perform well, and the key issues outstanding in the design of a general intelligence system. We have the existence proof that a single self-learning RL agent can master the game of Go purely from self-play (Silver et al., 2018). We do not have a clear picture of whether such a learning algorithm will perform well at driving a car, or managing a power plant. If we want to take the next leaps forward, we need to continue to enhance our understanding.
+
+# 1.1 Practical theory often lags practical algorithms
+
+The practical success of RL algorithms has built upon a base of theory including gradient descent (Bottou, 2010), temporal difference learning (Sutton, 1988) and other foundational algorithms. Good theory provides insight into our algorithms beyond the particular, and a route towards general improvements beyond ad-hoc tinkering. As the psychologist Kurt Lewin said, ‘there is nothing as practical as good theory’ (Lewin, 1943). If we hope to use RL to tackle important problems, then we must continue to solidify these foundations. This need is particularly clear for RL with nonlinear function approximation, or ‘deep RL’. At the same time, theory often lags practice, particularly in difficult problems. We should not avoid practical progress that can be made before we reach a full theoretical understanding. The successful development of algorithms and theory typically moves in tandem, with each side enriched by the insights of the other.
+
+The evolution of neural network research, or deep learning, provides a poignant illustration of how theory and practice can develop together (LeCun et al., 2015). Many of the key ideas for deep learning have been around, and with successful demonstrations, for many years before the modern deep learning explosion (Rosenblatt, 1958; Ivakhnenko, 1968; Fukushima, 1979). However, most of these techniques remained outside the scope of developed learning theory, partly due to their complex and non-convex loss functions. Much of the field turned away from these techniques in a ‘neural network winter’, focusing instead of function approximation under convex loss (Cortes & Vapnik, 1995). These convex methods were almost completely dominant until the emergence of benchmark problems, mostly for image recognition, where deep learning methods were able to clearly and objectively demonstrate their superiority (LeCun et al., 1998; Krizhevsky et al., 2012). It is only now, several years after these high profile successes, that learning theory has begun to turn its attention back to deep learning (Kawaguchi, 2016; Bartlett et al., 2017; Belkin et al., 2018). The current theory of deep RL is still in its infancy. In the absence of a comprehensive theory, the community needs principled benchmarks that help to develop an understanding of the strengths and weakenesses of our algorithms.
+
+# 1.2 An ‘MNIST’ for reinforcement learning
+
+In this paper we introduce the Behaviour Suite for Reinforcement Learning (or bsuite for short): a collection of experiments designed to highlight key aspects of agent scalability. Our aim is that these experiments can help provide a bridge between theory and practice, with benefits to both sides. These experiments embody fundamental issues, such as ‘exploration’ or ‘memory’ in a way that can be easily tested and iterated. For the development of theory, they force us to instantiate measurable and falsifiable hypotheses that we might later formalize into provable guarantees. While a full theory of RL may remain out of reach, the development of clear experiments that instantiate outstanding challenges for the field is a powerful driver for progress. We provide a description of the current suite of experiments and the key issues they identify in Section 2.
+
+Our work on bsuite is part of a research process, rather than a final offering. We do not claim to capture all, or even most, of the important issues in RL. Instead, we hope to provide a simple library that collects the best available experiments, and makes them easily accessible to the community. As part of an ongoing commitment, we are forming a bsuite committee that will periodically review the experiments included in the official bsuite release. We provide more details on what makes an ‘excellent’ experiment in Section 2, and on how to engage in their construction for future iterations in Section 5.
+
+The Behaviour Suite for Reinforcement Learning is a not a replacement for ‘grand challenge’ undertakings in artificial intelligence, or a leaderboard to climb. Instead it is a collection of diagnostic experiments designed to provide insight into key aspects of agent behaviour. Just as the MNIST dataset offers a clean, sanitised, test of image recognition as a stepping stone to advanced computer vision; so too bsuite aims to instantiate targeted experiments for the development of key RL capabilities.
+
+The successful use of illustrative benchmark problems is not unique to machine learning, and our work is similar in spirit to the Mixed Integer Programming Library (MIPLIB) (miplib2017). In mixed integer programming, and unlike linear programming, the majority of algorithmic advances have (so far) eluded theoretical analysis. In this field, MIPLIB serves to instantiate key properties of problems (or types of problems), and evaluation on MIPLIB is a typical component of any new algorithm. We hope that bsuite can grow to perform a similar role in RL research, at least for those parts that continue to elude a unified theory of artificial intelligence. We provide guidelines for how researchers can use bsuite effectively in Section 3.
+
+# 1.3 Open source code, reproducible research
+
+As part of this project we open source github.com/deepmind/bsuite, which instantiates all experiments in code and automates the evaluation and analysis of any RL agent on bsuite. This library serves to facilitate reproducible and accessible research on the core issues in reinforcement learning. It includes:
+
+• Canonical implementations of all experiments, as described in Section 2.   
+• Reference implementations of several reinforcement learning algorithms.   
+Example usage of bsuite with alternative codebases, including ‘OpenAI Gym’.   
+• Launch scripts for Google cloud that automate large scale compute at low cost.1 • A ready-made bsuite Jupyter notebook with analyses for all experiments.   
+• Automated LATEX appendix, suitable for inclusion in conference submission.
+
+We provide more details on code and usage in Section 4.
+
+We hope the Behaviour Suite for Reinforcement Learning, and its open source code, will provide significant value to the RL research community, and help to make key conceptual issues concrete and precise. bsuite can highlight bottlenecks in general algorithms that are not amenable to hacks, and reveal properties and scalings of algorithms outside the scope of current analytical techniques. We believe this offers an avenue towards great leaps on key issues, separate to the challenges of large-scale engineering (Nair et al., 2015). Further, bsuite facilitates clear, targeted and unified experiments across different code frameworks, something that can help to remedy issues of reproducibility in RL research (Tanner & White, 2009; Henderson et al., 2017).
+
+# 1.4 Related work
+
+The Behaviour Suite for Reinforcement Learning fits into a long history of RL benchmarks. From the beginning, research into general learning algorithms has been grounded by the performance on specific environments (Sutton & Barto, 2017). At first, these environments were typically motivated by small MDPs that instantiate the general learning problem. ‘CartPole’ (Barto et al., 1983) and ‘MountainCar’ (Moore, 1990) are examples of classic benchmarks that has provided a testing ground for RL algorithm development. Similarly, when studying specific capabilities of learning algorithms, it has often been helpful to design diagnostic environments with that capability in mind. Examples of this include ‘RiverSwim’ for exploration (Strehl & Littman, 2008) or ‘Taxi’ for temporal abstraction (Dietterich, 2000). Performance in these environments provide a targeted signal for particular aspects of algorithm development.
+
+As the capabilities or RL algorithms have advanced, so has the complexity of the benchmark problems. The Arcade Learning Environment (ALE) has been instrumental in driving progress in deep RL through surfacing dozens of Atari 2600 games as learning environments (Bellemare et al., 2013). Similar projects have been crucial to progress in continuous control (Duan et al., 2016; Tassa et al., 2018), model-based RL (Wang et al., 2019) and even rich 3D games (Beattie et al., 2016). Performing well in these complex environments requires the integration of many core agent capabilities. We might think of these benchmarks as natural successors to ‘CartPole’ or ‘MountainCar’.
+
+The Behaviour Suite for Reinforcement Learning offers a complementary approach to existing benchmarks in RL, with several novel components:
+
+1. bsuite experiments enforce a specific methodology for agent evaluation beyond just the environment definition. This is crucial for scientific comparisons and something that has become a major problem for many benchmark suites (Machado et al., 2017) (Section 2). 2. bsuite aims to isolate core capabilities with targeted ‘unit tests’, rather than integrate general learning ability. Other benchmarks evolve by increasing complexity, bsuite aims to remove all confounds from the core agent capabilities of interest (Section 3). 3. bsuite experiments are designed with an emphasis on scalability rather than final performance. Previous ‘unit tests’ (such as ‘Taxi’ or ‘RiverSwim’) are of fixed size, bsuite experiments are specifically designed to vary the complexity smoothly (Section 2). 4. github.com/deepmind/bsuite has an extraordinary emphasis on the ease of use, and compatibility with RL agents not specifically designed for bsuite. Evaluating an agent on bsuite is practical even for agents designed for a different benchmark (Section 4).
+
+# 2 Experiments
+
+This section outlines the experiments included in the Behaviour Suite for Reinforcement Learning 2019 release. In the context of bsuite, an experiment consists of three parts:
+
+1. Environments: a fixed set of environments determined by some parameters.   
+2. Interaction: a fixed regime of agent/environment interaction (e.g. 100 episodes).   
+3. Analysis: a fixed procedure that maps agent behaviour to results and plots.
+
+One crucial part of each bsuite analysis defines a ‘score’ that maps agent performance on the task to $[ 0 , 1 ]$ . This score allows for agent comparison ‘at a glance’, the Jupyter notebook includes further detailed analysis for each experiment. All experiments in bsuite only measure behavioural aspects of RL agents. This means that they only measure properties that can be observed in the environment, and are not internal to the agent. It is this choice that allows bsuite to easily generate and compare results across different algorithms and codebases. Researchers may still find it useful to investigate internal aspects of their agents on bsuite environments, but it is not part of the standard analysis.
+
+Every current and future bsuite experiment should target some key issue in RL. We aim for simple behavioural experiments, where agents that implement some concept well score better than those that don’t. For an experiment to be included in bsuite it should embody five key qualities:
+
+• Targeted: performance in this task corresponds to a key issue in RL.   
+• Simple: strips away confounding/confusing factors in research.   
+• Challenging: pushes agents beyond the normal range.   
+• Scalable: provides insight on scalability, not performance on one environment.   
+• Fast: iteration from launch to results in under 30min on standard CPU.
+
+Where our current experiments fall short, we see this as an opportunity to improve the Behaviour Suite for Reinforcement Learning in future iterations. We can do this both through replacing experiments with improved variants, and through broadening the scope of issues that we consider.
+
+We maintain the full description of each of our experiments through the code and accompanying documentation at github.com/deepmind/bsuite. In the following subsections, we pick two bsuite experiments to review in detail: ‘memory length’ and ‘deep sea’, and review these examples in detail. By presenting these experiments as examples, we can emphasize what we think makes bsuite a valuable tool for investigating core RL issues. We do provide a high level summary of all other current experiments in Appendix A.
+
+To accompany our experiment descriptions, we present results and analysis comparing three baseline algorithms on bsuite: DQN (Mnih et al., 2015a), A2C (Mnih et al., 2016) and Bootstrapped DQN (Osband et al., 2016). As part of our open source effort, we include full code for these agents and more at bsuite/baselines. All plots and analysis are generated through the automated bsuite Jupyter notebook, and give a flavour for the sort of agent comparisons that are made easy by bsuite.
+
+# 2.1 Example experiment: memory length
+
+Almost everyone agrees that a competent learning system requires memory, and almost everyone finds the concept of memory intuitive. Nevertheless, it can be difficult to provide a rigorous definition for memory. Even in human minds, there is evidence for distinct types of ‘memory’ handled by distinct regions of the brain (Milner et al., 1998). The assessment of memory only becomes more difficult to analyse in the context of general learning algorithms, which may differ greatly from human models of cognition. Which types of memory should we analyse? How can we inspect belief models for arbitrary learning systems? Our approach in bsuite is to sidestep these debates through simple behavioural experiments.
+
+We refer to this experiment as memory length; it is designed to test the number of sequential steps an agent can remember a single bit. The underlying environment is based on a stylized T-maze (O’Keefe & Dostrovsky, 1971), parameterized by a length $N \in \mathbb { N }$ . Each episode lasts $N$ steps with observation $o _ { t } = ( c _ { t } , t / N )$ for $t = 1 , . . , N$ and action space $\mathcal { A } = \{ - 1 , + 1 \}$ . The context $c _ { 1 } \sim \mathrm { U n i f } ( \mathcal { A } )$ and $c _ { t } = 0$ for all $t \geq 2$ . The reward $r _ { t } = 0$ for all $t < N$ , but $r _ { N } = \mathrm { S i g n } ( a _ { N } = c _ { 1 } )$ . For the bsuite experiment we run the agent on sizes $N = 1 , . . , 1 0 0$ exponentially spaced and look at the average regret compared to optimal after 10k episodes. The summary ‘score’ is the percentage of runs for which the average regret is less than $7 5 \%$ of that achieved by a uniformly random policy.
+
+![](images/5fba235f7cc9fe8a44f3a05fb6242801bf0737a0878ecf8655fe36dd7b275bb1.jpg)  
+Figure 1: Illustration of the ‘memory length’ environment
+
+Memory length is a good bsuite experiment because it is targeted, simple, challenging, scalable and fast. By construction, an agent that performs well on this task has mastered some use of memory over multiple timesteps. Our summary ‘score’ provides a quick and dirty way to compare agent performance at a high level. Our sweep over different lengths $N$ provides empirical evidence about the scaling properties of the algorithm beyond a simple pass/fail. Figure 2a gives a quick snapshot of the performance of baseline algorithms. Unsurprisingly, actor-critic with a recurrent neural network greatly outperforms the feedforward DQN and Bootstrapped DQN. Figure 2b gives us a more detailed analysis of the same underlying data. Both DQN and Bootstrapped DQN are unable to learn anything for length $> 1$ , they lack functioning memory. A2C performs well for all $N \ \leq \ 3 0$ and essentially random for all $N > 3 0$ , with quite a sharp cutoff. While it is not surprising that the recurrent agent outperforms feedforward architectures on a memory task, Figure 2b gives an excellent insight into the scaling properties of this architecture. In this case, we have a clear explanation for the observed performance: the RNN agent was trained via backprop-through-time with length 30. bsuite recovers an empirical evaluation of the scaling properties we would expect from theory.
+
+# 2.2 Example experiment: deep sea
+
+Reinforcement learning calls for a sophisticated form of exploration called deep exploration (Osband et al., 2017). Just as an agent seeking to ‘exploit’ must consider the long term consequences of its actions towards cumulative rewards, an agent seeking to ‘explore’ must consider how its actions can position it to learn more effectively in future timesteps. The literature on efficient exploration broadly states that only agents that perform deep exploration can expect polynomial sample complexity in learning (Kearns & Singh, 2002). This literature has focused, for the most part, on uncovering possible strategies for deep exploration through studying the tabular setting analytically (Jaksch et al., 2010; Azar et al., 2017). Our approach in bsuite is to complement this understanding through a series of behavioural experiments that highlight the need for efficient exploration.
+
+![](images/b14ce0f5f2981daa185a93293264a95c8755b30c32be1507672aedfc51ffbb96.jpg)  
+Figure 2: Selected output from bsuite evaluation on ‘memory length’.
+
+The deep sea problem is implemented as an $N \times N$ grid with a one-hot encoding for state. The agent begins each episode in the top left corner of the grid and descends one row per timestep. Each episode terminates after $N$ steps, when the agent reaches the bottom row. In each state there is a random but fixed mapping between actions $\mathcal { A } = \{ 0 , 1 \}$ and the transitions ‘left’ and ‘right’. At each timestep there is a small cost $r = - 0 . 0 1 / N$ of moving right, and $r = 0$ for moving left. However, should the agent transition right at every timestep of the episode it will be rewarded with an additional reward of $+ 1$ . This presents a particularly challenging exploration problem for two reasons. First, following the ‘gradient’ of small intermediate rewards leads the agent away from the optimal policy. Second, a policy that explores with actions uniformly at random has probability $2 ^ { - N }$ of reaching the rewarding state in any episode. For the bsuite experiment we run the agent on sizes $N = 1 0 , 1 2 , . . , 5 0$ and look at the average regret compared to optimal after 10k episodes. The summary ‘score’ computes the percentage of runs for which the average regret drops below 0.9 faster than the $2 ^ { N }$ episodes expected by dithering.
+
+![](images/50b2723be4f1b41e9b292fd87a0e75bec57e3986b2f1efb1d914002f9dd94bde.jpg)  
+Figure 3: Deep-sea exploration: a simple example where deep exploration is critical.
+
+Deep Sea is a good bsuite experiment because it is targeted, simple, challenging, scalable and fast. By construction, an agent that performs well on this task has mastered some key properties of deep exploration. Our summary score provides a ‘quick and dirty’ way to compare agent performance at a high level. Our sweep over different sizes $N$ can help to provide empirical evidence of the scaling properties of an algorithm beyond a simple pass/fail. Figure 3 presents example output comparing A2C, DQN and Bootstrapped DQN on this task. Figure 4a gives a quick snapshot of performance. As expected, only Bootstrapped DQN, which was developed for efficient exploration, scores well. Figure 4b gives a more detailed analysis of the same underlying data. When we compare the scaling of learning with problem size $N$ it is clear that only Bootstrapped DQN scales gracefully to large problem sizes. Although our experiment was only run to size 50, the regular progression of learning times suggest we might expect this algorithm to scale towards $N > 5 0$ .
+
+![](images/2ac0d82a1954f38d9d2ef8e9b96275bedc51bb126fea0c4f4076f61b08c2790c.jpg)  
+Figure 4: Selected output from bsuite evaluation on ‘deep sea’.
+
+# 3 How to use bsuite
+
+This section describes some of the ways you can use bsuite in your research and development of RL algorithms. Our aim is to present a high-level description of some research and engineering use cases, rather than a tutorial for the code installation and use. We provide examples of specific investigations using bsuite in Appendixes C, D and E. Section 4 provides an outline of our code and implementation. Full details and tutorials are available at github.com/deepmind/bsuite.
+
+A bsuite experiment is defined by a set of environments and number of episodes of interaction. Since loading the environment via bsuite handles the logging automatically, any agent interacting with that environment will generate the data required for required for analysis through the Jupyter notebook we provide (P´erez & Granger, 2007). Generating plots and analysis via the notebook only requires users to provide the path to the logged data. The ‘radar plot’ (Figure 5) at the start of the notebook provides a snapshot of agent behaviour, based on summary scores. The notebook also contains a complete description of every experiment, summary scoring and in-depth analysis of each experiment. You can interact with the full report at bit.ly/bsuite-agents.
+
+![](images/9b6810dff7ffef5a1c445303c8577c724df91665cf1bec6e6540082eb4987422.jpg)  
+Figure 5: We aggregate experiment performance with a snapshot of 7 core capabilities.
+
+If you are developing an algorithm to make progress on fundamental issues in RL, running on bsuite provides a simple way to replicate benchmark experiments in the field. Although many of these problems are ‘small’, in the sense that their solution does not necessarily require large neural architecture, they are designed to highlight key challenges in RL. Further, although these experiments do offer a summary ‘score’, the plots and analysis are designed to provide much more information than just a leaderboard ranking. By using this common code and analysis, it is easy to benchmark your agents and provide reproducible and verifiable research.
+
+If you are using RL as a tool to crack a ‘grand challenge’ in AI, such as beating a world champion at Go, then taking on bsuite gridworlds might seem like small fry. We argue that one of the most valuable uses of bsuite is as a diagnostic ‘unit-test’ for large-scale algorithm development. Imagine you believe that ‘better exploration’ is key to improving your performance on some challenge, but when you try your ‘improved’ agent, the performance does not improve. Does this mean your agent does not do good exploration? Or maybe that exploration is not the bottleneck in this problem? Worse still, these experiments might take days and thousands of dollars of compute to run, and even then the information you get might not be targeted to the key RL issues. Running on bsuite, you can test key capabilities of your agent and diagnose potential improvements much faster, and more cheaply. For example, you might see that your algorithm completely fails at credit assignment beyond $n = 2 0$ steps. If this is the case, maybe this lack of credit-assignment over long horizons is the bottleneck and not necessarily exploration. This can allow for much faster, and much better informed agent development - just like a good suite of tests for software development.
+
+Another benefit of bsuite is to disseminate your results more easily and engage with the research community. For example, if you write a conference paper targeting some improvement to hierarchical reinforcement learning, you will likely provide some justification for your results in terms of theorems or experiments targeted to this setting.2 However, it is typically a large amount of work to evaluate your algorithm according to alternative metrics, such as exploration. This means that some fields may evolve without realising the connections and distinctions between related concepts. If you run on bsuite, you can automatically generate a one-page Appendix, with a link to a notebook report hosted online. This can help provide a scientific evaluation of your algorithmic changes, and help to share your results in an easily-digestible format, compatible with ICML, ICLR and NeurIPS formatting. We provide examples of these experiment reports in Appendices B, C, D and E.
+
+# 4 Code structure
+
+To avoid discrepancies between this paper and the source code, we suggest that you take practical tutorials directly from github.com/deepmind/bsuite. A good starting point is bit.ly/bsuite-tutorial: a Jupyter notebook where you can play the code right from your browser, without installing anything. The purpose of this section is to provide a high-level overview of the code that we open source. In particular, we want to stress is that bsuite is designed to be a library for RL research, not a framework. We provide implementations for all the environments, analysis, run loop and even baseline agents. However, it is not necessary that you make use of them all in order to make use of bsuite.
+
+The recommended method is to implement your RL agent as a class that implements a policy method for action selection, and an update method for learning from transitions and rewards. Then, simply pass your agent to our run loop, which enumerates all the necessary bsuite experiments and logs all the data automatically. If you do this, then all the experiments and analysis will be handled automatically and generate your results via the included Jupyter notebook. We provide examples of running these scripts locally, and via Google cloud through our tutorials.
+
+If you have an existing codebase, you can still use bsuite without migrating to our run loop or agent structure. Simply replace your environment with environment $=$ bsuite.load and record(bsuite id) and add the flag bsuite id to your code. You can then complete a full bsuite evaluation by iterating over the bsuite ids defined in sweep.SWEEP. Since the environments handle the logging themselves, your don’t need any additional logging for the standard analysis. Although full bsuite includes many separate evaluations, no single bsuite environment takes more than 30 minutes to run and the sweep is naturally parallel. As such, we recommend launching in parallel using multiple processes or multiple machines. Our examples include a simple approach using Python’s multiprocessing module with Google cloud compute. We also provide examples of running bsuite from OpenAI baselines (Dhariwal et al., 2017) and Dopamine (Castro et al., 2018).
+
+Designing a single RL agent compatible with diverse environments can cause problems, particularly for specialized neural networks. bsuite alleviates this problem by specifying an observation spec that surfaces the necessary information for adaptive network creation. By default, bsuite environments implement the dm env standards (Muldal et al., 2017), but we also include a wrapper for use through Openai gym (Brockman et al., 2016). However, if your agent is hardcoded for a format, bsuite offers the option to output each environment with the observation spec of your choosing via linear interpolation. This means that, if you are developing a network suitable for Atari and particular observation spec, you can choose to swap in bsuite without any changes to your agent.
+
+# 5 Future iterations
+
+This paper introduces the Behaviour Suite for Reinforcement Learning, and marks the start of its ongoing development. With our opensource effort, we chose a specific collection of experiments as the bsuite2019 release, but expect this collection to evolve in future iterations. We are reaching out to researchers and practitioners to help collate the most informative, targeted, scalable and clear experiments possible for reinforcement learning. To do this, submissions should implement a sweep that determines the selection of environments to include and logs the necessary data, together with an analysis that parses this data.
+
+In order to review and collate these submissions we will be forming a bsuite committee. The committee will meet annually during the NeurIPS conference to decide which experiments will be included in the bsuite release. We are reaching out to a select group of researchers, and hope to build a strong core formed across industry and academia. If you would like to submit an experiment to bsuite or propose a committee member, you can do this via github pull request, or via email to bsuite.committee@gmail.com.
+
+We believe that bsuite can be a valuable tool for the RL community, and particularly for research in deep RL. So far, the great success of deep RL has been to leverage large amounts of computation to improve performance. With bsuite, we hope to leverage largescale computation for improved understanding. By collecting clear, informative and scalable experiments; and providing accessible tools for reproducible evaluation we hope to facilitate progress in reinforcement learning research.
+
+# References
+
+Mart´ın Abadi et al. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/. Software available from tensorflow.org.
+
+Mohammad Gheshlaghi Azar, Ian Osband, and R´emi Munos. Minimax regret bounds for reinforcement learning. In Proc. of ICML, 2017.
+
+Peter L Bartlett, Dylan J Foster, and Matus J Telgarsky. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems 30, pp. 6241–6250, 2017.
+
+Andrew G Barto, Richard S Sutton, and Charles W Anderson. Neuronlike adaptive elements that can solve difficult learning control problems. IEEE transactions on systems, man, and cybernetics, (5):834–846, 1983.
+
+Charles Beattie, Joel Z Leibo, Denis Teplyashin, Tom Ward, Marcus Wainwright, Heinrich K¨uttler, Andrew Lefrancq, Simon Green, V´ıctor Vald´es, Amir Sadik, et al. Deepmind lab. arXiv preprint arXiv:1612.03801, 2016.
+
+Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machine learning and the bias-variance trade-off. arXiv preprint arXiv:1812.11118, 2018.
+
+Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The Arcade Learning Environment: An Evaluation Platform for General Agents. Journal of Artificial Intelligence Research, 47:253–279, 2013.
+
+L´eon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of COMPSTAT’2010, pp. 177–186. Springer, 2010.
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. CoRR, abs/1606.01540, 2016. URL http://arxiv.org/abs/ 1606.01540.
+
+Pablo Samuel Castro, Subhodeep Moitra, Carles Gelada, Saurabh Kumar, and Marc G. Bellemare. Dopamine: A Research Framework for Deep Reinforcement Learning. 2018. URL http://arxiv. org/abs/1812.06110.
+
+Corinna Cortes and Vladimir Vapnik. Support-vector networks. Machine learning, 20(3):273–297, 1995.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009.
+
+Prafulla Dhariwal, Christopher Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert, Alec Radford, John Schulman, Szymon Sidor, Yuhuai Wu, and Peter Zhokhov. Openai baselines. https: //github.com/openai/baselines, 2017.
+
+Thomas G Dietterich. Hierarchical reinforcement learning with the maxq value function decomposition. Journal of artificial intelligence research, 13:227–303, 2000.
+
+Yan Duan, Xi Chen, Rein Houthooft, John Schulman, and Pieter Abbeel. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning, pp. 1329–1338, 2016.
+
+Richard Evans and Jim Gao. Deepmind AI reduces google data centre cooling bill by 40 https: //deepmind.com/blog/deepmind-ai-reduces-google-data-centre-cooling-bill-40/, 2016.
+
+Kunihiko Fukushima. Neural network model for a mechanism of pattern recognition unaffected by shift in position-neocognitron. IEICE Technical Report, A, 62(10):658–665, 1979.
+
+John C Gittins. Bandit processes and dynamic allocation indices. Journal of the Royal Statistical Society: Series B (Methodological), 41(2):148–164, 1979.
+
+Peter Henderson, Riashat Islam, Philip Bachman, Joelle Pineau, Doina Precup, and David Meger. Deep reinforcement learning that matters. CoRR, abs/1709.06560, 2017. URL http://arxiv. org/abs/1709.06560.
+
+Alexey Grigorevich Ivakhnenko. The group method of data of handling; a rival of the method of stochastic approximation. Soviet Automatic Control, 13:43–55, 1968.
+
+Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(Apr):1563–1600, 2010.
+
+Kenji Kawaguchi. Deep learning without poor local minima. In Advances in neural information processing systems, pp. 586–594, 2016.
+
+M. Kearns and S. Singh. Near-optimal reinforcement learning in polynomial time. Machine Learning, 49, 2002.
+
+Jeannette Kiefer and Jacob Wolfowitz. Stochastic estimation of the maximum of a regression function. 1952.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. URL http://arxiv.org/abs/1412.6980.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems 25, pp. 1097–1105, 2012.
+
+Yann LeCun, L´eon Bottou, Yoshua Bengio, Patrick Haffner, et al. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436, 2015.
+
+Shane Legg, Marcus Hutter, et al. A collection of definitions of intelligence. Frontiers in Artificial Intelligence and applications, 157:17, 2007.
+
+Kurt Lewin. Psychology and the process of group living. The Journal of Social Psychology, 17(1): 113–131, 1943.
+
+Xiuyuan Lu and Benjamin Van Roy. Ensemble sampling. In Advances in Neural Information Processing Systems, pp. 3260–3268, 2017.
+
+Marlos C Machado, Marc G Bellemare, Erik Talvitie, Joel Veness, Matthew Hausknecht, and Michael Bowling. Revisiting the Arcade Learning Environment: Evaluation protocols and open problems for general agents. arXiv preprint arXiv:1709.06009, 2017.
+
+Brenda Milner, Larry R Squire, and Eric R Kandel. Cognitive neuroscience and the study of memory. Neuron, 20(3):445–468, 1998.
+
+Marvin Minsky. Steps towards artificial intelligence. Proceedings of the IRE, 1961.
+
+miplib2017. MIPLIB 2017, 2018. http://miplib.zib.de.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level Control through Deep Reinforcement Learning. Nature, 518(7540):529–533, 2015a.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015b.
+
+Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proc. of ICML, 2016.
+
+Andrew William Moore. Efficient memory-based learning for robot control. 1990.
+
+Alistair Muldal, Yotam Doron, and John Aslanides. dm env. https://github.com/deepmind/dm_ env, 2017.
+
+Arun Nair, Praveen Srinivasan, Sam Blackwell, Cagdas Alcicek, Rory Fearon, et al. Massively Parallel Methods for Deep Reinforcement Learning. In ICML Workshop on Deep Learning, 2015.
+
+John O’Keefe and Jonathan Dostrovsky. The hippocampus as a spatial map: preliminary evidence from unit activity in the freely-moving rat. Brain research, 1971.
+
+Ian Osband, Charles Blundell, Alexander Pritzel, and Benjamin Van Roy. Deep exploration via bootstrapped DQN. In Advances In Neural Information Processing Systems 29, pp. 4026–4034, 2016.
+
+Ian Osband, Daniel Russo, Zheng Wen, and Benjamin Van Roy. Deep exploration via randomized value functions. arXiv preprint arXiv:1703.07608, 2017.
+
+Ian Osband, John Aslanides, and Albin Cassirer. Randomized prior functions for deep reinforcement learning. In Advances in Neural Information Processing Systems 31, pp. 8617–8629. Curran Associates, Inc., 2018. URL http://papers.nips.cc/paper/ 8080-randomized-prior-functions-for-deep-reinforcement-learning.pdf.
+
+Ian Osband, Yotam Doron, Matteo Hessel, John Aslanides, , Eren Sezener, Andre Saraiva, Katrina McKinney, Tor Lattimore, Csaba Szepesvari, Satinder Singh, Benjamin Van Roy, Richard Sutton, David Silver, and Hado Van Hasselt. Behaviour suite for reinforcement learning. 2019.
+
+Jakub Pachocki, David Farhi, Szymon Sidor, Greg Brockman, Filip Wolski, Henrique PondÃľ, Jie Tang, Jonathan Raiman, Michael Petrov, Christy Dennison, Brooke Chan, Susan Zhang, RafaÅĆ JÃşzefowicz, and PrzemysÅĆaw DÄŹbiak. Openai five. https://openai.com/five, 2019.
+
+Fernando P´erez and Brian E. Granger. IPython: a system for interactive scientific computing. Computing in Science and Engineering, 9(3):21–29, May 2007. ISSN 1521-9615. doi: 10.1109/ MCSE.2007.53. URL https://ipython.org.
+
+Frank Rosenblatt. The perceptron: a probabilistic model for information storage and organization in the brain. Psychological review, 65(6):386, 1958.
+
+Daniel Russo, Benjamin Van Roy, Abbas Kazerouni, and Ian Osband. A tutorial on Thompson sampling. arXiv preprint arXiv:1707.02038, 2017.
+
+AL Samuel. Some studies oin machine learning using the game of checkers. IBM Journal of Researchand Development, 3:211–229, 1959.
+
+David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of Go with deep neural networks and tree search. Nature, 529(7587):484–489, 2016.
+
+David Silver, Thomas Hubert, Julian Schrittwieser, Ioannis Antonoglou, Matthew Lai, Arthur Guez, Marc Lanctot, Laurent Sifre, Dharshan Kumaran, Thore Graepel, Timothy Lillicrap, Karen Simonyan, and Demis Hassabis. A general reinforcement learning algorithm that masters chess, shogi, and go through self-play. Science, 362(6419):1140–1144, 2018. ISSN 0036-8075. doi: 10.1126/science.aar6404. URL https://science.sciencemag.org/content/362/6419/1140.
+
+Alexander L Strehl and Michael L Littman. An analysis of model-based interval estimation for markov decision processes. Journal of Computer and System Sciences, 74(8):1309–1331, 2008.
+
+Richard Sutton and Andrew Barto. Reinforcement Learning: An Introduction. MIT Press, 2017.
+
+R.S. Sutton. Learning to predict by the methods of temporal differences. Machine learning, 3, 1988.
+
+Brian Tanner and Adam White. Rl-glue: Language-independent software for reinforcement-learning experiments. Journal of Machine Learning Research, 10(Sep):2133–2136, 2009.
+
+Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdolmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind control suite. arXiv preprint arXiv:1801.00690, 2018.
+
+Gerald Tesauro. Temporal difference learning and TD-gammon. Communications of the ACM, 38 (3):58–68, 1995.
+
+T. Tieleman and G. Hinton. Lecture 6.5—RmsProp: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 2012.
+
+Hado van Hasselt, Arthur Guez, and David Silver. Deep Reinforcement Learning with Double Q-Learning. In Proceedings of the AAAI Conference on Artificial Intelligence, 2016.
+
+Oriol Vinyals, Igor Babuschkin, Junyoung Chung, Michael Mathieu, Jaderberg, et al. AlphaStar: Mastering the Real-Time Strategy Game StarCraft II. https://deepmind.com/blog/ alphastar-mastering-real-time-strategy-game-starcraft-ii/, 2019.
+
+Tingwu Wang, Xuchan Bao, Ignasi Clavera, Jerrick Hoang, Yeming Wen, Eric Langlois, Shunshi Zhang, Guodong Zhang, Pieter Abbeel, and Jimmy Ba. Benchmarking model-based reinforcement learning. CoRR, abs/1907.02057, 2019. URL http://arxiv.org/abs/1907.02057.
+
+# A Experiment summary
+
+This appendix outlines the experiments that make up the bsuite 2019 release. In the interests of brevity, we provide only an outline of each experiment here. Full documentation for the environments, interaction and analysis are kept with code at github.com/deepmind/bsuite.
+
+# A.1 Basic learning
+
+We begin with a collection of very simple decision problems, and standard analysis that confirms an agent’s competence at learning a rewarding policy within them. We call these experiments ‘basic’, since they are not particularly targeted at specific core issues in RL, but instead test a general base level of competence we expect all general agents to attain.
+
+# A.1.1 Simple bandit
+
+![](images/ef97416bdbde16221bd45e95e4a9613584874df1212509fb6f68b91a237227e9.jpg)
+
+# A.1.2 MNIST
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Finite-armed bandit with deterministic re- wards [0,0.1,..1] (Gittins, 1979). 20 seeds.</td></tr><tr><td> interaction</td><td>10k episodes, record regret vs optimal.</td></tr><tr><td> score</td><td> regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>basic</td></tr></table>
+
+![](images/a92674e5270923d4c96ecb0a25d54d68a0c0484cb26334715d3e48c4fb453fd9.jpg)
+
+A.1.3 Catch
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Contextual bandit classification of MNIST with ±1 rewards (LeCun et al.,1998). 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td> score</td><td> regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>basic, generalization</td></tr></table>
+
+![](images/9abeccfe27e47e808cc80794a4014c47247c21e78e5883527285787b4229e587.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>A 10x5 Tetris-grid with single block falling per column. The agent can move left/right in the bottom row to‘catch’ the block. 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td> regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>basic, credit assignment</td></tr></table>
+
+A.1.4 Cartpole
+
+![](images/6a7a3a4c5c088db232e4cb408366bc089144e0fccf86a7d95a55edec7bafb552.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td> environments</td><td>Agent can move a cart left/right on a plane to keep a balanced pole upright (Barto et al., 1983), 20 seeds.</td></tr><tr><td> interaction</td><td>10k episodes, record average regret.</td></tr><tr><td> score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>basic, credit assignment, generalization</td></tr></table>
+
+# A.1.5 Mountain car
+
+![](images/d579b04d0310ca37943a037dd6c3fa19fd7d8d0d8d39d769261bf68b828f0cc2.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Agent drives an underpowered car up a hil (Moore,1990), 20 seeds.</td></tr><tr><td> interaction</td><td>10k episodes, record average regret.</td></tr><tr><td> score</td><td> regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>basic, credit assignment, generalization</td></tr></table>
+
+# A.2 Stochasticity
+
+To investigate the robustness of RL agents to noisy rewards, we repeat the experiments from Section A.1 under differing levels of Gaussian noise. This time we allocate the 20 different seeds across 5 levels of Gaussian noise $N ( 0 , \sigma ^ { 2 } )$ for $\sigma = [ 0 . 1 , 0 . 3 , 1 , 3 , 1 0 ]$ with 4 seeds each.
+
+# A.3 Problem scale
+
+To investigate the robustness of RL agents to problem scale, we repeat the experiments from Section A.1 under differing reward scales. This time we allocate the 20 different seeds across 5 levels of reward scaling, where we multiply the observed rewards by $\lambda =$ $[ 0 . 0 1 , 0 . 1 , 1 , 1 0 , 1 0 0 ]$ with 4 seeds each.
+
+# A.4 Exploration
+
+As an agent interacts with its environment, it observes the outcomes that result from previous states and actions, and learns about the system dynamics. This leads to a fundamental tradeoff: by exploring poorly-understood states and actions the agent can learn to improve future performance, but it may attain better short-run performance by exploiting its existing knowledge. Exploration is the challenge of prioritizing useful information for learning, and the experiments in this section are designed to necessitate efficient exploration for good performance.
+
+# A.4.1 Deep sea
+
+![](images/6fb014a631c5ce402f98627f84e2ac7cde7fab432fe04e4ea8251574ea2cf2b0.jpg)
+
+# A.4.2 Stochastic deep sea
+
+![](images/32928f4151ef5170229369be92d0e2e4e87ec7b1643370c2e9ce76c64b6f6e3f.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Deep sea chain environments with stochastic transitions, N(O,1) reward noise, size N=[5..50].</td></tr><tr><td> interaction</td><td>10k episodes, record average regret.</td></tr><tr><td> score</td><td>% of runs with ave regret &lt; 90% random</td></tr><tr><td> issues</td><td>exploration, stochasticity</td></tr></table>
+
+# A.4.3 Cartpole swingup
+
+![](images/0c3e4c2cca89bc6eb5df8b70e66253f0e23a362b8f20f77f7b0538f0f6f1942d.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td> environments</td><td>Cartpole ‘swing up’ problem with sparse re- ward (Barto et al.,1983), heigh limit x=[0, 0.5, .., 0.95].</td></tr><tr><td> interaction</td><td>1k episodes, record average regret.</td></tr><tr><td> score</td><td>% of runs with average return &gt; 0</td></tr><tr><td> issues</td><td>exploration, generalization</td></tr></table>
+
+# A.5 Credit assignment
+
+Reinforcement learning extends contextual bandit decision problem to allow long term consequences in decision problems. This means that actions in one timestep can effect dynamics in future timesteps. One of the challenges of this setting is that of credit assignment, and the experiments in this section are designed to highlight these issues.
+
+# A.5.1 Umbrella length
+
+![](images/5be62617046129f9a126e1dfe1faef3398971f9e49d8786f855b74a29118d7e2.jpg)
+
+# A.5.2 Umbrella features
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Stylized‘umbrella problem&#x27;,where only the first decision matters and long chain of con- founding variables. Vary length 1..10o loga- rithmically.</td></tr><tr><td> interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>credit assignment, stochasticity</td></tr></table>
+
+![](images/72b3f1164a425f8526bef64ae751dee450926e756eae837f5a4dbba01d650ff7.jpg)  
+A.5.3 Discounting chain
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Stylized ‘umbrella problem&#x27;,where only the first decision matters and long chain of con- founding variables. Vary features 1..10o loga- rithmically.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret .</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>credit assignment, stochasticity</td></tr></table>
+
+<table><tr><td rowspan="7">oo+ Reward 1 at =- Reward 1.l at t=3 。O Reward 1 at t-10 。o Reward 1 at t=30 oo</td><td>component</td><td>description</td></tr><tr><td> environments</td><td>Experiment designed to highlight issues of dis- counting horizon.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score Reward l at t=100</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>credit assignment</td></tr></table>
+
+# A.6 Memory
+
+Memory is the challenge that an agent should be able to curate an effective state representation from a series of observations. In this section we review a series of experiments in which agents with memory can perform much better than those that only have access to the immediate observation.
+
+# A.6.1 Memory length
+
+![](images/b55cba6830e354ea429514e45d8dae150702387b89be3201fe5eddd8aad71c35.jpg)  
+A.6.2 Memory bits
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>T-maze with a single binary context, grow length 1..100 logarithmically.</td></tr><tr><td> interaction</td><td>1k episodes, record average regret.</td></tr><tr><td> score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>credit assignment</td></tr></table>
+
+![](images/97f25769d712b5bee4ecc4ad5179f7d692088854ad3fe3b1a254f79c67b19e04.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>T-maze with length 2, vary number of bits to remember 1..100 logarithmically.</td></tr><tr><td> interaction</td><td>1k episodes, record average regret.</td></tr><tr><td> score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td> issues</td><td>credit assignment</td></tr></table>
+
+# B bsuite report as conference appendix
+
+If you run an agent on bsuite, and you want to share these results as part of a conference submission, we make it easy to share a single-page ‘bsuite report’ as part of your appendix. We provide a simple LATEXfile that you can copy/paste into your paper, and is compatible out-the-box with ICLR, ICML and NeurIPS style files. This single page summary displays the summary scores for experiment evaluations for one or more agents, with plots generated automatically from the included ipython notebook. In each report, two sections are left for the authors to fill in: one describing the variants of the agents examined and another to give some brief commentary on the results. We suggest that authors promote more in-depth analysis to their main papers, or simply link to a hosted version of the full bsuite analysis online. You can find more details on our automated reports at github.com/deepmind/bsuite.
+
+The sections that follow are example bsuite reports, that give some example of how these report appendixes might be used. We believe that these simple reports can be a good complement to conference submissions in RL research, that ‘sanity check’ the elementary properties of algorithmic implementations. An added bonus of bsuite is that it is easy to set up a like for like experiment between agents from different ‘frameworks’ in a way that would be extremely laborious for an individual researcher. If you are writing a conference paper on a new RL algorithm, we believe that it makes sense for you to include a bsuite report in the appendix by default.
+
+# C bsuite report: benchmarking baseline agents
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# C.1 Agent definition
+
+In this experiment all implementations are taken from bsuite/baselines with default configurations. We provide a brief summary of the agents run on bsuite2019:
+
+• random: selects action uniformly at random each timestep.   
+• dqn: Deep Q-networks (Mnih et al., 2015b).   
+• boot dqn: bootstrapped DQN with prior networks (Osband et al., 2016; 2018).   
+• actor critic rnn: an actor critic with recurrent neural network (Mnih et al., 2016).
+
+# C.2 Summary scores
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory bit.ly/bsuite-agents.
+
+![](images/42cdd217b4392223ef039dd74cff482c673aa7bfc194372eeb4befaf857a3478.jpg)  
+Figure 6: A snapshot of agent behaviour.
+
+![](images/c96f970f02435d58ffeca4e171da11a6fada35ed3986226fb1f5da4e76ccf688.jpg)  
+Figure 7: Score for each bsuite experiment.
+
+# C.3 Results commentary
+
+• random performs uniformly poorly, confirming the scores are working as intended.
+
+• dqn performs well on basic tasks, and quite well on credit assignment, generalization, noise and scale. DQN performs extremely poorly across memory and exploration tasks. The feedforward MLP has no mechanism for memory, and $\epsilon { = } 5 \%$ -greedy action selection is inefficient exploration.
+
+• boot dqn is mostly identically to DQN, except for exploration where it greatly outperforms. This result matches our understanding of Bootstrapped DQN as a variant of DQN designed to estimate uncertainty and use this to guide deep exploration.
+
+• actor critic rnn typically performs worse than either DQN or Bootstrapped DQN on all tasks apart from memory. This agent is the only one able to perform better than random due to its recurrent network architecture.
+
+# D bsuite report: optimization algorithm in DQN
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# D.1 Agent definition
+
+All agents correspond to different instantiations of the DQN agent (Mnih et al., 2015b), as implemented in bsuite/baselines but with differnet optimizers from Tensorflow (Abadi et al., 2015). In each case we tune a learning rate to optimize performance on ‘basic’ tasks from {1e-1, 1e-2, 1e-3}, keeping all other parameters constant at default value.
+
+• sgd: vanilla stochastic gradient descent with learning rate 1e-2 (Kiefer & Wolfowitz, 1952).   
+• rmsprop: RMSProp with learning rate 1e-3 (Tieleman & Hinton, 2012).   
+• adam: Adam with learning rate 1e-3 (Kingma & Ba, 2015).
+
+# D.2 Summary scores
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory: bit.ly/bsuite-optim.
+
+![](images/3970f5b55d39aec9dd37b475a8a48c52557bc12b476d7443288e3855238f2303.jpg)  
+Figure 8: A snapshot of agent behaviour.
+
+![](images/29de21ce937e30d387d018c4a6adbd7d4049d57c4a4eeb7d887202c84d55c374.jpg)  
+Figure 9: Score for each bsuite experiment.
+
+# D.3 Results commentary
+
+Both RMSProp and Adam perform better than SGD in every category. In most categories, Adam slightly outperforms RMSprop, although this difference is much more minor. SGD performs particularly badly on environments that require generalization and/or scale. This is not particularly surprising, since we expect the non-adaptive SGD may be more sensitive to learning rate optimization or annealing.
+
+In Figure 11 we can see that the differences are particularly pronounced on the cartpole domains. We hypothesize that this task requires more efficient neural network optimization, and the nonadaptive SGD is prone to numerical issues.
+
+# E bsuite report: ensemble size in Bootstrapped DQN
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# E.1 Agent definition
+
+In this experiment, all agents correspond to different instantiations of a Bootstrapped DQN with prior networks (Osband et al., 2016; 2018). We take the default implementation from bsuite/baselines. We investigate the effect of the number of models used in the ensemble, sweeping over {1, 3, 10, 30}.
+
+# E.2 Summary scores
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory: bit.ly/bsuite-ensemble.
+
+![](images/2b54632e488669188b1d36b0e5ff530e9a5a96cdef42321992da3f84c21c1dca.jpg)  
+Figure 10: A snapshot of agent behaviour. Figure 11: Score for each bsuite experiment.
+
+# E.3 Results commentary
+
+Generally, increasing the size of the ensemble improves bsuite performance across the board. However, we do see signficantly decreasing returns to ensemble size, so that ensemble 30 does not perform much better than size 10. These results are not predicted by the theoretical scaling of proven bounds (Lu & Van Roy, 2017), but are consistent with previous empirical findings (Osband et al., 2017; Russo et al., 2017). The gains are most extreme in the exploration tasks, where ensemble sizes less than 10 are not able to solve large ‘deep sea’ tasks, but larger ensembles solve them reliably.
+
+Even for large ensemble sizes, our implementation does not completely solve every cartpole swingup instance. Further examination learning curves suggests this may be due to some instability issues, which might be helped by using Double DQN to combat value overestimation (van Hasselt et al., 2016).