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+# Differential Privacy of Dirichlet Posterior Sampling
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We study the inherent privacy of releasing a single sample from a Dirichlet posterior   
+2 distribution. As a complement to the previous study that provides general theories   
+3 on the differential privacy of posterior sampling from exponential families, this   
+4 study focuses specifically on the Dirichlet posterior sampling and its privacy   
+5 guarantees. With the notion of truncated concentrated differential privacy (tCDP),   
+6 we are able to derive a simple privacy guarantee of the Dirichlet posterior sampling,   
+7 which effectively allows us to analyze its utility in various settings. Specifically,   
+8 we provide accuracy guarantees of the Dirichlet posterior sampling in Multinomial  
+9 Dirichlet sampling and private normalized histogram publishing.
+
+# 10 1 Introduction
+
+11 The Bayesian framework provides a way to perform statistical analysis by combining prior beliefs   
+12 with real-life evidence. At a high level, the belief and the evidence are assumed to be described   
+13 by probabilistic models. As we receive new data, our belief is updated accordingly via the Bayes’   
+14 theorem, resulting in the so-called posterior belief. The posterior tells us how much we are uncertain   
+15 about the model’s parameters.   
+16 The Dirichlet distribution is usually chosen as the prior when performing Bayesian analysis on discrete   
+17 variables, as it is a conjugate prior to the categorical and multinomial distributions. Specifically,   
+18 Dirichlet distributions are often used in discrete mixture models, where a Dirichlet prior is put on   
+19 the mixture weights [LW92; MMR05]. Such models have applications in NLP [PB98], biophysical   
+20 systems [Hin15], accident analysis [de 06], and genetics [BHW00; PM01; CWS03]. In all of these   
+21 studies, samplings from Dirichlet posteriors arise when performing Markov chain Monte Carlo   
+22 methods for approximate Bayesian inference.   
+23 Dirichlet posterior sampling also appears in other learning tasks. For example, in Bayesian active   
+24 learning, it arises in Gibbs sampling, which is used to approximate the posterior of the classifier over   
+25 the labeled sample [NLYCC13]. In Thompson sampling for multi-armed bandits, one repeatedly   
+26 draws a sample from the Dirichlet posterior of each arm, and picks the arm whose sample maximizes   
+27 the reward [ZHGSY20; AAFK20; NIK20]. And in Bayesian reinforcement learning, state-transition   
+28 probabilities are sampled from the Dirichlet posterior over past observed states [Str00; ORR13].   
+29 Dirichlet posterior sampling can also be used for data synthesis. Suppose that we have a histogram   
+30 $( x _ { 1 } , \ldots , x _ { d } )$ of actual data. An approximate discrete distribution of this histogram can be obtained by   
+31 drawing a sample $\mathbf { Y }$ from Dirichlet $( x _ { 1 } + \alpha _ { 1 } , \ldots , x _ { d } + \alpha _ { d } )$ , where $\alpha _ { 1 } , \ldots , \alpha _ { d }$ are prior parameters.   
+32 Then synthetic data is produced by repeatedly drawing from Multinomial(Y). There are many   
+33 studies on data synthesis that followed this approach [AV08; MKAGV08; RWZ14; PG14; SJGLY17].   
+34 In the above examples, the data that we integrate into these tasks might contain sensitive information.   
+35 Thus it is important to ask: how much of the information is protected from the Dirichlet samplings?   
+36 The goal of this study is to find an answer to this question.   
+37 The mathematical framework of differential privacy (DP) [DMNS06] allows us to quantify how much   
+38 the privacy of the Dirichlet posterior sampling is affected by the prior parameters $\alpha _ { 1 } , \ldots , \alpha _ { d }$ . In the   
+39 definition of DP, the privacy of a randomized algorithm is measured by how much its distribution   
+40 changes upon perturbing a single data point of the input. Nonetheless, this notion might be too   
+41 strict for the Dirichlet distribution, as a small perturbation of a near-zero parameter can cause a large   
+42 distribution shift. Thus, it might be more appropriate to rely on one of several relaxed notions of   
+43 DP, such as approximate differential privacy, Rényi differential privacy, or concentrated differential   
+44 privacy. It is natural to wonder if the Dirichlet posterior sampling satisfies any of these definitions.
+
+# 45 1.1 Overview of Our results
+
+46 This study focuses on the privacy and utility of Dirichlet posterior sampling. In summary, we provide   
+47 a closed-form privacy guarantee of the Dirichlet posterior sampling, which in turn allows us to   
+48 effectively analyze its utility in various settings.
+
+$\ S 3$ Privacy. We study the role of the prior parameters in the privacy of the Dirichlet posterior sampling. Theorem 1 is our main result, where we provide a guaranteed upper bound for truncated concentrated differential privacy (tCDP) of the Dirichlet posterior sampling. In addition, we convert the tCDP guarantee into an approximate differential privacy guarantee in Corollary 2.
+
+$\ S 4$ Utility. Using the tCDP guarantee, we investigate the utility of Dirichlet posterior sampling applied in two specific applications:
+
+• In Section 4.1, we consider one-time sampling from a Multinomial-Dirichlet distribution. But instead of directly sampling from this distribution, we sample from another distribution with larger prior parameters. The accuracy is then measured by the KL-divergence between the original and the private distributions.   
+• In Section 4.2, we use the Dirichlet posterior sampling for a private release of a normalized histogram. In this case, the accuracy is measured by the mean-squared error between the sample and the original normalized histogram.
+
+62 In both tasks, we compute the sample size that guarantees the desired level of accuracy. In the case   
+63 of private histogram publishing, we also compare the Dirichlet posterior sampling to the Gaussian   
+64 mechanism.
+
+# 65 1.2 Related work
+
+66 There are several studies on the differential privacy of posterior sampling. Wang, Fienberg, and   
+67 Smola [WFS15] showed that any posterior sampling with the log-likelihood bounded by $B$ is $4 B$ -   
+68 differentially private. However, the likelihoods that we study are not bounded away from zero; they   
+69 have the form $\Pi _ { i } p _ { i } ^ { x _ { i } }$ which becomes small when one of the $p _ { i }$ ’s is close to zero. Dimitrakakis, Nelson,   
+70 Zhang, Mitrokotsa, and Rubinstein [DNZMR17] showed that if the condition on the log-likelihood is   
+71 relaxed to the Lipschitz continuity with high probability, then one can obtain the approximate DP.   
+72 Nonetheless, with the Dirichlet density, it is difficult to compute the probability of events in which   
+73 the Lipschitz condition is satisfied.   
+74 In the case that the sufficient statistics $\mathbf { x }$ has finite $\ell ^ { 1 }$ -sensitivity, Foulds, Geumlek, Welling and   
+75 Chaudhuri [FGWC16] suggested adding Laplace noises to $\mathbf { x }$ . Suppose that y is the output; they   
+76 showed that sampling from $p ( \boldsymbol { \theta } | \mathbf { y } )$ is differentially private and as asymptotically efficient as sampling   
+77 from $p ( \boldsymbol { \theta } | \mathbf { x } )$ . However, for a small sample size, the posterior over the noisy statistics might be too   
+78 far away from the actual posterior. Bernstein and Sheldon [BS18] thus proposed to approximate the   
+79 joint distribution $p ( \boldsymbol { \theta } , \mathbf { x } , \mathbf { y } )$ using Gibbs sampling, which is then integrated over $\mathbf { x }$ to obtain a more   
+80 accurate posterior over $\mathbf { y }$ .   
+81 Geumlek, Song, and Chaudhuri [GSC17] were the first to study the posterior sampling with the   
+82 RDP. Even though they provided a general framework to find $( \lambda , \epsilon )$ -RDP guarantees for exponential   
+83 families, explicit forms of $\epsilon$ and the upper bound of $\lambda$ were not given. In contrast, our tCDP guarantees   
+84 of the Dirichlet posterior sampling imply an explicit expression for $\epsilon$ , and also an upper bound for $\lambda$ .   
+85 The privacy of data synthesis via sampling from Multinomial $( \mathbf { Y } )$ , where $\mathbf { Y }$ is a discrete distri  
+86 bution drawn from the Dirichlet posterior, was first studied by Machanavajjhala, Kifer, Abowd,   
+87 Gehrke, and Vilhuber [MKAGV08]. They showed that the data synthesis is $( \varepsilon , \delta )$ -probabilistic DP,   
+88 which implies $( \varepsilon , \delta )$ -approximate DP. However, as their privacy analysis includes the sampling from   
+89 Multinomial $( \mathbf { Y } )$ , their privacy guarantee depends on the number of synthetic samples. In contrast,   
+90 we show that the one-time sampling from the Dirichlet posterior is approximate DP, which by the   
+91 post-processing property allows us to sample from Multinomial $( \mathbf { Y } )$ as many times as we want while   
+92 retaining the same privacy guarantee.   
+93 The Dirichlet mechanism was first introduced by Gohari, Wu, Hawkins, Hale, and Topcu [GWHHT21].   
+94 Originally, the Dirichlet mechanism takes a discrete distribution $\mathbf { p } : = ( p _ { 1 } , \ldots , p _ { d } )$ and draws one   
+95 sample $\mathbf { Y } \sim \mathrm { D i r i c h l e t } ( r p _ { 1 } , \dots , r p _ { d } )$ . Note the absence of the prior parameters, which makes $\mathbf { Y }$ an   
+96 unbiased estimator of $\mathbf { p }$ . But this comes with a cost, as the worst case of privacy violation occurs   
+97 when almost all of the parameters are close to zero. The authors avoided this issue by restricting   
+98 the input space to a subset of the unit simplex, with some of the $p _ { i }$ ’s bounded below by a fixed   
+99 positive constant. This results in complicated expressions for the privacy guarantees as they involve   
+100 a minimization problem over the restricted domain. In this study, we take a different approach by   
+101 adding prior parameters to the Dirichlet mechanism. As a result, we obtain a biased algorithm that   
+102 requires no assumption on the input space and has simpler forms of privacy guarantees.
+
+# 1.3 Notations
+
+104 We let $\mathbb { R } _ { \geq 0 } ^ { d }$ be the set of $d$ -tuples of non-negative real numbers and $\mathbb { R } _ { > 0 } ^ { d }$ be the set of $d$ -tuples of   
+105 positive real numbers. We assume that all vectors are $d$ -dimensional where $d \geq 2$ . The notations for   
+106 all vectors are always in bold. Specifically, $\mathbf { x } : = ( x _ { 1 } , \ldots , x _ { d } ) \in \mathbb { R } _ { \geq 0 } ^ { d }$ consists of sample statistics of   
+107 the data and $\pmb { \alpha } : = ( \alpha _ { 1 } , \ldots , \alpha _ { d } ) \in \mathbb { R } _ { > 0 } ^ { d }$ consists of the prior parameters. The vector $\mathbf { p } : = ( p _ { 1 } , \ldots , p _ { d } )$   
+108 always satisfies $\textstyle \sum _ { i } p _ { i } = 1$ . The number of observations is always $N$ . We also denote $x _ { 0 } : = \textstyle \sum _ { i } x _ { i }$   
+109 and $\alpha _ { 0 } : = \textstyle \sum _ { i } \alpha _ { i }$ . For any vectors $\mathbf { x } , \mathbf { x } ^ { \prime }$ and scalar $r > 0$ , we write $\mathbf { x } + \mathbf { x } ^ { \prime } : = ( x _ { 1 } + x _ { 1 } ^ { \prime } , \ldots , x _ { d } + x _ { d } ^ { \prime } )$   
+110 and $r \mathbf { x } : = ( r x _ { 1 } , \ldots , r x _ { d } )$ . For any positive reals $x$ and $x ^ { \prime }$ , the notation $x \propto x ^ { \prime }$ means $x = C x ^ { \prime }$ for   
+111 some constant $C > 0$ , $x \approx x ^ { \prime }$ means $c x ^ { \prime } \leq x \leq C x ^ { \prime }$ for some $c , C > 0$ , and $x \lesssim x ^ { \prime }$ means $x \leq C x ^ { \prime }$   
+112 for some $C > 0$ . Lastly, $\| \mathbf { x } \| _ { \infty } : = \operatorname* { m a x } _ { i } | x _ { i } |$ is the $\ell ^ { \infty }$ norm of $\mathbf { x }$ .
+
+# 13 2 Background
+
+# 2.1 Privacy models
+
+Definition 2.1 (Pure and Approximate DP [DMNS06]). A randomized mechanism $M : \mathcal { X } ^ { n }  \mathcal { Y }$ is $( \varepsilon , \delta )$ -differentially private $( \varepsilon , \delta )$ -DP) if for any datasets $x , x ^ { \prime }$ differing on a single entry, and all events $E \subset \mathcal { V }$ ,
+
+$$
+\mathbb { P } [ M ( x ) \in E ] \leq e ^ { \varepsilon } \mathbb { P } [ M ( x ^ { \prime } ) \in E ] + \delta .
+$$
+
+118 If $M$ is $( \varepsilon , 0 )$ -DP, then we say that it is $\varepsilon$ -differential privacy $\dot { \varepsilon }$ -DP).
+
+19 The term pure differential privacy (pure DP) refers to $\epsilon$ -differential privacy, while approximate   
+20 differential privacy (approximate DP) refers to $( \varepsilon , \delta )$ -DP when $\delta > 0$ .
+
+121 In contrast to pure and approximate DP, the next definitions of differential privacy are defined in terms of the Rényi divergence between 122 $M ( x )$ and $M ( x ^ { \prime } )$ :
+
+123 Definition 2.2 (Rényi Divergence [Rén61]). Let $P$ and $Q$ be probability distributions. $\mathrm { F o r } \lambda \in ( 1 , \infty )$   
+124 the Rényi divergence of order $\lambda$ between $P$ and $Q$ is defined as
+
+$$
+\mathrm { D } _ { \lambda } ( P \| Q ) : = { \frac { 1 } { \lambda - 1 } } \log \int P ( y ) ^ { \lambda } Q ( y ) ^ { 1 - \lambda } d y = { \frac { 1 } { \lambda - 1 } } \log \biggl ( \operatorname { \mathbb { E } } _ { y \sim P } \biggl [ { \frac { P ( y ) ^ { \lambda - 1 } } { Q ( y ) ^ { \lambda - 1 } } } \biggr ] \biggr . . \biggr )
+$$
+
+125 Definition 2.3 (tCDP and zCDP [BDRS18; BS16]). A randomized mechanism $M : \mathcal { X } ^ { n }  \mathcal { Y }$ is   
+126 $\omega$ -truncated $\rho$ -concentrated differentially private $( ( \rho , \omega )$ -tCDP) if for any datasets $x , x ^ { \prime }$ differing on a   
+127 single entry and for all $\lambda \in ( 1 , \omega )$ ,
+
+$$
+\begin{array} { r } { \mathrm { D } _ { \lambda } ( M ( x ) \| M ( x ^ { \prime } ) ) \leq \lambda \rho . } \end{array}
+$$
+
+128 If $M$ is $( \rho , \infty )$ -tCDP, then we say that it is $\rho$ -zero-concentrated differential privacy ( $\rho$ -zCDP).
+
+129 Note that both tCDP and zCDP have the composition and post-processing properties. Intuitively, $\rho$ con  
+130 trols the expectation and standard deviation of the privacy loss random variable: Z = log P [M(x)=Y ]P [M(x0)=Y ] ,   
+131 where $Y$ has density $M ( x )$ , and $\omega$ controls the number of standard deviations for which $Z$ concen  
+132 trates like a Gaussian. A smaller $\rho$ and larger $\omega$ correspond to a stronger privacy guarantee. It turns   
+133 out that tCDP implies approximate DP:   
+134 Lemma 1 (From tCDP to Approximate DP [BDRS18]). Let $\delta > 0$ . If M is a $( \rho , \omega )$ -tCDP mechanism,   
+135 then it also satisfies $( \varepsilon , \delta )$ - $D P$ with
+
+$$
+\varepsilon = \left\{ \begin{array} { l l } { \rho + 2 \sqrt { \rho \log ( 1 / \delta ) } \quad } & { i f \log ( 1 / \delta ) \leq ( \omega - 1 ) ^ { 2 } \rho } \\ { \rho \omega + \frac { \log ( 1 / \delta ) } { \omega - 1 } } & { i f \log ( 1 / \delta ) > ( \omega - 1 ) ^ { 2 } \rho } \end{array} \right. .
+$$
+
+# 136 2.2 Dirichlet distribution
+
+137 For $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ , the Dirichlet distribution Dirichlet $( \alpha )$ is a continuous distribution of $d$ -dimensional   
+138 probability vectors i.e. vectors whose coordinate sum is equal to 1. The density function of $\mathbf { Y } \sim$   
+139 Dirichlet $( \alpha )$ is given by:
+
+$$
+p ( \mathbf { y } ) = \frac { 1 } { B ( \pmb { \alpha } ) } \prod _ { i = 1 } ^ { d } y _ { i } ^ { \alpha _ { i } - 1 } ,
+$$
+
+140 where $B ( \alpha )$ is the beta function, which can be written in terms of the gamma function:
+
+$$
+B ( \pmb { \alpha } ) = \frac { \prod _ { i } \Gamma ( \alpha _ { i } ) } { \Gamma ( \sum _ { i } \alpha _ { i } ) } .
+$$
+
+# 141 2.3 Dirichlet posterior sampling
+
+142 We consider the prior Dirichlet $( \alpha )$ and the likelihood of the form $\begin{array} { r } { p ( \mathbf { x } | \mathbf { y } ) \propto \prod _ { i = 1 } ^ { d } y _ { i } ^ { x _ { i } } } \end{array}$ where   
+143 $\mathbf { x } \in \mathbb { R } _ { \geq 0 } ^ { d }$ consists of sample statistics of the dataset. The Dirichlet posterior sampling is a one-time   
+144 sampling:
+
+$$
+\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { x } + \pmb { \alpha } ) .
+$$
+
+145 There is a modification of the sampling which introduces a concentration parameter $r > 0$ , and   
+146 instead we sample from Dirichlet $( r \mathbf { x } + \pmb { \alpha } )$ [GSC17; GWHHT21]. Smaller values of $r$ make the   
+147 sampling more private, and larger values of $r$ make $\mathbf { Y }$ a closer approximation of $\mathbf { x }$ . Even though the   
+148 case $r = 1$ is the main focus of this study, our main privacy results can be easily extended to other   
+149 values of $r$ as we will see at the end of Section 3.1.   
+150 Consider a special case where $\mathbf x = \mathbf p$ is an empirical distribution derived from the dataset, and we   
+151 want $\mathbf { Y }$ to be a private approximation of $\mathbf { p }$ ; the sampling $\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( r \mathbf { p } + \alpha )$ is called the   
+152 Dirichlet mechanism [GWHHT21]. It is interesting to note that the Dirichlet mechanism is a form of   
+153 the exponential mechanism [MT07]: let $r > 0$ be the privacy parameter, Dirichlet $( \alpha )$ be the prior,   
+154 and the negative KL-divergence be the score function of the exponential mechanism. Then the output   
+155 $\mathbf { Y }$ of this mechanism is distributed according to the following density function:
+
+$$
+\begin{array} { r l r } & { } & { \frac { \exp \left( - r \mathrm { D } _ { \mathrm { K L } } ( \mathbf { p } , \mathbf { y } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } } { \int \exp \left( - r \mathrm { D } _ { \mathrm { K L } } ( \mathbf { p } , \mathbf { y } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } d \mathbf { y } } \propto \exp \left( r \sum _ { i , p _ { i } \neq 0 } p _ { i } \log ( y _ { i } / p _ { i } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } } \\ & { } & { \propto \displaystyle \prod _ { i , p _ { i } \neq 0 } y _ { i } ^ { r p _ { i } } \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } = \prod _ { i } y _ { i } ^ { r p _ { i } + \alpha _ { i } - 1 } , } \end{array}
+$$
+
+156 which is exactly the density function of Dirichlet $( r \mathbf { p } + \alpha )$ .
+
+58 In most of this study, we take advantage of several nice properties of the log-gamma function and its derivatives. Specifically, $\begin{array} { r } { \psi ( x ) : = \frac { d } { d x } \log \Gamma ( x ) } \end{array}$ is concave and increasing, while its derivative $\psi ^ { \prime } ( x )$ is positive, convex, and decreasing. In addition, 0 $\psi ^ { \prime }$ can be approximated by the reciprocals:
+
+$$
+{ \frac { 1 } { x } } + { \frac { 1 } { 2 x ^ { 2 } } } < \psi ^ { \prime } ( x ) < { \frac { 1 } { x } } + { \frac { 1 } { x ^ { 2 } } } ,
+$$
+
+which implies that 161 $\textstyle \psi ^ { \prime } ( x ) \approx { \frac { 1 } { x ^ { 2 } } }$ as $x \to 0$ and $\begin{array} { r } { \psi ^ { \prime } ( x ) \approx \frac { 1 } { x } } \end{array}$ as $x \to \infty$
+
+# 3 Main privacy results
+
+# 3.1 Truncated concentrated differential privacy
+
+Theorem 1. Let $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ and $\alpha _ { m } : = \operatorname* { m i n } _ { i } \alpha _ { i }$ . Let $\gamma \in ( 0 , \alpha _ { m } )$ . Let $\Delta _ { 2 } , \Delta _ { \infty } > 0$ be constants that satisfy $\begin{array} { r } { \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \leq \Delta _ { 2 } ^ { 2 } } \end{array}$ and $\mathrm { m a x } _ { i } \left| x _ { i } - x _ { i } ^ { \prime } \right| \leq \Delta _ { \infty }$ whenever x, $\mathbf { \Delta } , \mathbf { x } ^ { \prime } \in \mathbb { R } _ { \geq 0 } ^ { 2 }$ are sample statistics of any two datasets differing on a single entry. The one-time sampling from Dirichlet $\left( \mathbf { x } + \alpha \right)$ is $( \rho , \omega )$ -tCDP, where $\begin{array} { r } { \omega = \frac { \gamma } { \Delta _ { \infty } } + 1 } \end{array}$ and
+
+$$
+\rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) .
+$$
+
+168 Note that $( \rho , \infty )$ -tCDP is not obtainable, as the ratio between two Dirichlet densities blows up as   
+169 $\omega  \infty$ . We present here a short proof that skips some calculations (see Appendix 1 for a full proof).
+
+proof. Consider any 170 $\begin{array} { r } { \lambda \in \left( 1 , \frac { \gamma } { \Delta _ { \infty } } + 1 \right) } \end{array}$ . Let $\mathbf { u } : = \mathbf { x } + \pmb { \alpha }$ and $\mathbf { u } ^ { \prime } : = \mathbf { x } ^ { \prime } + \alpha ^ { \prime }$ . Let $P ( \mathbf { y } )$ be the density of Dirichlet171 $\mathbf { \Pi } ( \mathbf { u } )$ and $P ^ { \prime } ( \mathbf { y } )$ be the density of Dirichlet $\mathbf { \Pi } ^ { ( \mathbf { u } ^ { \prime } ) }$ . A quick calculation shows that:
+
+$$
+\mathbb { E } _ { { \mathbf { y } } \sim P ( { \mathbf { y } } ) } \left[ \frac { P ( { \mathbf { y } } ) ^ { \lambda - 1 } } { P ^ { \prime } ( { \mathbf { y } } ) ^ { \lambda - 1 } } \right] = \frac { B ( { \mathbf { u } } ^ { \prime } ) ^ { \lambda - 1 } } { B ( { \mathbf { u } } ) ^ { \lambda - 1 } } \cdot \frac { B ( { \mathbf { u } } + ( \lambda - 1 ) ( { \mathbf { u } } - { \mathbf { u } } ^ { \prime } ) ) } { B ( { \mathbf { u } } ) } .
+$$
+
+172 We take the logarithm on both sides and apply the second-order Taylor expansion to the following   
+173 $G ( u _ { i } , u _ { i } ^ { \prime } )$ and $H ( u _ { i } , u _ { i } ^ { \prime } )$ terms that appear on the right-hand side. As a result, there exist $\xi$ between   
+174 $u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } )$ and $u _ { i }$ , and $\xi ^ { \prime }$ between $u _ { i }$ and $u _ { i } ^ { \prime }$ such that
+
+$$
+\begin{array} { r l r } {  { G ( u _ { i } , u _ { i } ^ { \prime } ) : = ( \lambda - 1 ) ( \log \Gamma ( u _ { i } ^ { \prime } ) - \log \Gamma ( u _ { i } ) ) } } \\ & { } & \\ & { } & { = - ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) \psi ( u _ { i } ) + \frac { 1 } { 2 } ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \xi ^ { \prime } ) } \\ & { } & { H ( u _ { i } , u _ { i } ^ { \prime } ) : = \log \Gamma ( u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } ) ) - \log \Gamma ( u _ { i } ) } \\ & { } & { = ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) \psi ( u _ { i } ) + \frac { 1 } { 2 } ( \lambda - 1 ) ^ { 2 } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \xi ) , } \end{array}
+$$
+
+Note that 175 $\psi ^ { \prime }$ is increasing. If $x _ { i } > x _ { i } ^ { \prime }$ , then $\xi$ and $\xi ^ { \prime }$ are bounded below by $u _ { i } ^ { \prime } \geq \alpha _ { m }$ . On the other hand, if 176 $x _ { i } \leq x _ { i } ^ { \prime }$ , then $\xi$ and $\xi ^ { \prime }$ are bounded below by $u _ { i } - ( \lambda - 1 ) | u _ { i } - u _ { i } ^ { \prime } |$ . The condition 177 $\begin{array} { r } { \lambda < \frac { \gamma } { \Delta _ { \infty } } + 1 } \end{array}$ guarantees that $u _ { i } - ( \lambda - 1 ) | u _ { i } - u _ { i } ^ { \prime } | > \alpha _ { m } - \gamma$ . All cases considered, we have
+
+$$
+\begin{array} { l } { G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) \leq \displaystyle \frac { 1 } { 2 } \big ( ( \lambda - 1 ) + ( \lambda - 1 ) ^ { 2 } \big ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \lambda ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) . } \end{array}
+$$
+
+Denoting 178 $u _ { 0 } : = \textstyle \sum _ { i } u _ { i }$ and $u _ { 0 } ^ { \prime } : = \textstyle \sum _ { i } u _ { i } ^ { \prime }$ , the same argument shows that $G ( u _ { 0 } , u _ { 0 } ^ { \prime } ) + H ( u _ { 0 } , u _ { 0 } ^ { \prime } ) > 0$ . 179 Therefore,
+
+$$
+\begin{array} { r l } & { D _ { \boldsymbol { \lambda } } ( P ( \mathbf { y } ) \| P ^ { \prime } ( \mathbf { y } ) ) = \displaystyle \frac { 1 } { \boldsymbol { \lambda } - 1 } \Biggl ( \sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) ) - G ( u _ { 0 } , u _ { 0 } ^ { \prime } ) - H ( u _ { 0 } , u _ { 0 } ^ { \prime } ) \Biggr ) } \\ & { \quad \quad < \displaystyle \frac { 1 } { \boldsymbol { \lambda } - 1 } \sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) ) } \\ & { \quad \quad \le \displaystyle \frac { 1 } { 2 } \boldsymbol { \lambda } \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) \le \frac { 1 } { 2 } \boldsymbol { \lambda } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) . } \end{array}
+$$
+
+![](images/9b723c7be81bab0fe812229760033b22531dcf9d6fe77c60ac3901a2063deaff.jpg)  
+Figure 1: Left: the actual values of $\begin{array} { r } { \rho = \frac { 1 } { 2 } \operatorname { D } _ { 2 } ( P \| P ^ { \prime } ) } \end{array}$ and the worst case $( \rho , 2 )$ -tCDP guarantees (2) at $\Delta _ { 2 } ^ { 2 } = \Delta _ { \infty } = 1$ . Here, $P$ and $P ^ { \prime }$ are Dirichlet posterior densities over $\mathbf { x } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 0 )$ , $\mathbf { x } ^ { \prime } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 1 )$ , and $\pmb { \alpha } = ( \alpha , \ldots , \alpha )$ . Right: comparison between $( \varepsilon , \delta )$ -DP guarantees of the Dirichlet posterior samplings (8) with different uniform priors: $\pmb { \alpha } = ( \alpha , \ldots , \alpha )$ .
+
+180 The guaranteed upper bound (2) is independent of the sample statistics. As a result, the bound applies   
+181 even in worst settings i.e., when $x _ { i } = 0$ and $x _ { i } ^ { \prime } = \Delta _ { \infty }$ , or vice versa, for some $i$ . As we can see in   
+182 Figure 1, the upper bound is a close approximation to the actual value of $\rho$ when $x _ { 6 } = 0$ and $x _ { 6 } ^ { \prime } = 1$ .   
+183 However, being a sample independent bound, the difference becomes substantial when all $x _ { i }$ ’s are   
+184 large. There is one way to get around this issue: if there is no privacy violation in assuming that   
+185 the sample statistics are always bounded below by some threshold $\tau$ , then we can incorporate the   
+186 threshold into the prior (thus $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ in (2) is replaced by $\psi ^ { \prime } ( \alpha _ { m } + \tau - \gamma ) )$ .   
+187 The parameter $\gamma$ allows us to adjust the moment bound $\omega$ as desired. Even though a higher $\omega$ usually   
+188 leads to a better privacy guarantee, there are two downsides to picking $\gamma$ close to $\alpha _ { m }$ in this case.   
+189 First, note that $\rho$ contains $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ ; as $\gamma \to \alpha _ { m }$ , the value of $\rho$ diverges to $\infty$ , leading to a weaker   
+190 privacy guarantee instead. Second, as the Taylor approximation (5) is accurate when $u _ { i }$ is close to   
+191 $u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } )$ , having a large value of $\lambda$ would push the guaranteed upper bound away from   
+192 the actual privacy loss. Thus it is recommended to pick $\gamma$ so that $\gamma / \Delta _ { \infty } \geq 1$ and $\alpha _ { m } - \gamma \gg 0$ .   
+193 Alternatively, we can choose the value of $\gamma$ that minimizes $\varepsilon$ when converting from tCDP to $( \varepsilon , \delta )$ -DP   
+194 using Lemma 1—this method will be explored in the next subsection.
+
+Theorem 1 can be easily applied to sampling from Dirichlet $( r \mathbf { x } + \alpha )$ . Replacing $\mathbf { x }$ with $r \mathbf { x }$ , we have $\Delta _ { 2 }$ replaced by $r \Delta _ { 2 }$ and $\Delta _ { \infty }$ replaced by $r \Delta _ { \infty }$ . Consequently, the sampling is $\begin{array} { r } { \left( \rho , \frac { \gamma } { r \Delta _ { \infty } } + 1 \right) } \end{array}$ -tCDP, where $\rho = { \textstyle { \frac { 1 } { 2 } } r ^ { 2 } \Delta _ { 2 } ^ { 2 } } \psi ^ { \prime } ( \alpha _ { m } - \gamma )$ . In Appendix 4, we analyze the scaling of $r$ in conjunction with $\alpha _ { m }$ at a fixed privacy budget $\rho$ .
+
+# 3.2 Approximate differential privacy
+
+200 We now convert the tCDP guarantee to an approximate DP guarantee. Let $\delta \in ( 0 , 1 )$ . Using Lemma 1,   
+201 the Dirichlet posterior sampling with Dirichlet $( \alpha )$ as the prior is $( \varepsilon , \delta )$ -DP with
+
+$$
+\varepsilon = \left\{ \begin{array} { l l } { \rho ( \gamma ) + 2 \sqrt { \rho ( \gamma ) \log ( 1 / \delta ) } } & { \mathrm { i f ~ } \log ( 1 / \delta ) \leq \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 } } \\ { \rho ( \gamma ) \Big ( \frac { \gamma } { \Delta _ { \infty } } + 1 \Big ) + \frac { \log ( 1 / \delta ) \Delta _ { \infty } } { \gamma } } & { \mathrm { i f ~ } \log ( 1 / \delta ) > \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 } } \end{array} , \right.
+$$
+
+where 202 $\begin{array} { r } { \rho ( \gamma ) = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) } \end{array}$
+
+203 We try to minimize $\epsilon$ by adjusting the value of $\gamma$ . First, we consider the case $\log ( 1 / \delta ) \leq \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 }$   
+204 Since $\rho ( \gamma )$ is a strictly increasing function of $\gamma$ , both $\rho ( \gamma ) + 2 \sqrt { \rho ( \gamma ) \log ( 1 / \delta ) }$ and $\gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 }$   
+205 are both strictly increasing function of $\gamma$ . Therefore, $\varepsilon$ is minimized at the minimum possible   
+206 value of $\gamma$ in this case, that is, at the unique $\gamma _ { M }$ that satisfies $\log ( 1 / \delta ) = \gamma _ { M } ^ { 2 } \rho ( \gamma _ { M } ) / \Delta _ { \infty } ^ { 2 } =$   
+207 $\begin{array} { r } { \frac { 1 } { 2 } \gamma _ { M } ^ { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma _ { M } ) / \Delta _ { \infty } ^ { 2 } } \end{array}$ .   
+208 Now we consider the second case, when $\gamma < \gamma _ { M }$ . As $\rho ( \gamma )$ is an increasing positive convex function   
+209 of $\gamma$ , the function
+
+$$
+f ( \gamma ) : = \frac 1 2 \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) \left( \frac { \gamma } { \Delta _ { \infty } } + 1 \right) + \frac { \log ( 1 / \delta ) \Delta _ { \infty } } { \gamma } ; \qquad \gamma \in ( 0 , \gamma _ { M } ] ,
+$$
+
+is also convex in $\gamma$ , and thus has a unique minimizer $\underline { { \gamma _ { m } } } \in ( 0 , \gamma _ { M } ]$ . Comparing to the first case, we have $f ( \gamma _ { m } ) \leq f ( \gamma _ { M } ) = \rho ( \gamma _ { M } ) + 2 \sqrt { \rho ( \gamma _ { M } ) \log ( 1 / \delta ) } .$ We then conclude that $\varepsilon = f ( \gamma _ { m } )$ .
+
+212 Theorem 2. Let $\alpha \in \mathbb { R } _ { > 0 } ^ { 2 }$ and denote $\alpha _ { m } = \operatorname* { m i n } _ { i } \alpha _ { i }$ . Let $\Delta _ { 2 } , \Delta _ { \infty } > 0$ be constants that satisfy   
+213 $\begin{array} { r } { \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \leq \Delta _ { 2 } ^ { 2 } } \end{array}$ and $\operatorname* { m a x } _ { i } \left| x _ { i } - x _ { i } ^ { \prime } \right| \leq \Delta _ { \infty }$ whenever x, $\mathbf { x } ^ { \prime } \in \mathbb { R } _ { \geq 0 } ^ { d }$ are sample statistics of any   
+214 two datasets differing on a single entry. For any $\delta \in ( 0 , 1 )$ , let $\gamma _ { M }$ be the solution to the equation   
+215 $\begin{array} { r } { \log ( 1 / \delta ) = \frac { 1 } { 2 } \gamma ^ { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) / \Delta _ { \infty } ^ { 2 } } \end{array}$ . The one-time sampling from Dirichlet $\left( \mathbf { x } + \alpha \right)$ is $( \varepsilon , \delta ) – D P$   
+216 where
+
+$$
+\varepsilon = \operatorname* { m i n } _ { \gamma \in ( 0 , \gamma _ { M } ] } f ( \gamma ) .
+$$
+
+17 Figure 1 shows how $\delta$ decays as a function of $\varepsilon$ at three different values of $\alpha _ { m }$
+
+# 4 Utility
+
+Using the results from the previous section, we analyze the Dirichlet posterior sampling’s utility in two specific tasks.
+
+# 4.1 Multinomial-Dirichlet sampling
+
+Suppose that we are observing $N$ trials, each of which has $d$ possible outcomes. For each $i \in$ $\{ 1 , \ldots , d \}$ , let $x _ { i }$ be the number of times the $i$ -th outcome was observed. Then we have the multinomial likelihood $\begin{array} { r } { p ( \mathbf { x } | \mathbf { y } ) \propto \prod _ { i } y _ { i } ^ { x _ { i } } } \end{array}$ . From this, we sample from the Dirichlet posterior:
+
+$$
+\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { x } + \pmb { \alpha } ) .
+$$
+
+225 Suppose that we want to sample from a true distribution $P _ { \mathbf { X } } \sim \mathrm { D i r i c h l e t } ( \mathbf { x } + { \pmb { \alpha } } )$ , but for privacy   
+226 reasons, we instead sample from $Q _ { \mathbf { x } } \sim \mathrm { D i r i c h l e t } ( \mathbf { x } + \pmb { \alpha } ^ { \prime } )$ where $\alpha _ { i } ^ { \prime } > \alpha _ { i }$ for all $i$ . The utility of the   
+227 privacy scheme is then measured by the KL-divergence between $P _ { \mathbf { x } }$ and $Q _ { \mathbf { x } }$ . Assuming that $\mathbf { x }$ is an   
+228 observation of Multinomial $\mathbf { \tau } ( \mathbf { p } )$ , the following Theorem tells us that, on average, the KL-divergence   
+229 is small when the sample size is large, and the $p _ { i }$ ’s are evenly distributed.
+
+Theorem 3. Let 30 $\mathbf { p } : = \left( p _ { 1 } , \ldots , p _ { d } \right)$ where $p _ { i } > 0$ for all $i$ and $\textstyle \sum _ { i } p _ { i } \ = \ 1$ . Define a random variable 31 $\mathbf { X } \sim$ Multinomial $\mathbf { \tau } ( \mathbf { p } )$ . Let $P _ { \mathbf { X } } \sim \mathrm { D i r i c h l e t } ( \mathbf { X } + { \boldsymbol { \alpha } } )$ and $Q _ { \mathbf { X } } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { X } + \mathbf { \alpha } \mathbf { \alpha } ^ { \prime } )$ where 32 $\alpha _ { i } ^ { \prime } \geq \alpha _ { i } \geq 1$ for all $i$ . The following estimate holds:
+
+$$
+\mathbb { E } _ { \mathbf { X } } [ \mathrm { D } _ { \mathrm { K L } } ( P _ { \mathbf { X } } \| Q _ { \mathbf { X } } ) ] \leq \frac { 1 } { N + 1 } \sum _ { i } ( \alpha _ { i } ^ { \prime } - \alpha _ { i } ) ^ { 2 } \cdot \frac { 1 } { p _ { i } } .
+$$
+
+233 The proof is given in Appendix 2. Let us consider a simple privacy scheme where we fix $s > 0$ and let 234 $\alpha _ { i } ^ { \prime } = \alpha _ { i } + s$ for all $i$ . Thus (10) becomes:
+
+$$
+\mathbb { E } _ { \mathbf { X } } [ \mathrm { D } _ { \mathrm { K L } } ( P _ { \mathbf { X } } \| Q _ { \mathbf { X } } ) ] \leq \frac { G ( \mathbf { p } ) s ^ { 2 } } { N + 1 } ,
+$$
+
+where 235 $G ( \mathbf { p } ) : = \textstyle \sum _ { i } 1 / p _ { i }$ . Now we take into account the privacy parameters. Let $\rho = \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma )$ and 236 $\rho ^ { \prime } = \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma )$ , where $\alpha _ { m } = \operatorname* { m i n } _ { i } \alpha _ { i }$ , $\alpha _ { m } ^ { \prime } = \operatorname* { m i n } _ { i } \alpha _ { i } ^ { \prime }$ , and $\gamma < \alpha _ { m }$ . Here, we approximate the values of 237 $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ and $\psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma )$ under two regimes:
+
+High-privacy regime: 238 $\alpha _ { m } ^ { \prime } \mathrm { ~ - ~ } \gamma \mathrm { ~ > ~ } 1$ . We have $\psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma ) \approx 1 / ( \alpha _ { m } ^ { \prime } - \gamma )$ , which implies 239 $\alpha _ { m } ^ { \prime } - \gamma \approx \Delta _ { 2 } ^ { 2 } / \rho ^ { \prime }$ . We also have $\alpha _ { m } - \gamma \approx \Delta _ { 2 } ^ { 2 } / \rho$ for $\alpha _ { m } - \gamma \geq 1$ and $\alpha _ { m } - \gamma > ( \alpha _ { m } - \gamma ) ^ { 2 } \approx \Delta _ { 2 } ^ { 2 } / \rho$ 240 for $\alpha - \gamma < 1$ . Thus we have the following bound for the right-hand side of (11):
+
+$$
+\frac { G ( { \bf p } ) s ^ { 2 } } { N + 1 } = \frac { G ( { \bf p } ) ( \alpha _ { m } ^ { \prime } - \alpha _ { m } ) ^ { 2 } } { N + 1 } \lesssim \frac { \Delta _ { 2 } ^ { 4 } G ( { \bf p } ) } { N + 1 } \biggl ( \frac { 1 } { \rho ^ { \prime } } - \frac { 1 } { \rho } \biggr ) ^ { 2 } < \frac { \Delta _ { 2 } ^ { 4 } G ( { \bf p } ) } { \rho ^ { \prime 2 } ( N + 1 ) } .
+$$
+
+Consequently, we have 241 $\mathrm { D } _ { \mathrm { K L } } ( P \| Q ) < \epsilon$ for $\begin{array} { r } { N = \Omega \left( \frac { \Delta _ { 2 } ^ { 4 } G ( \mathbf { p } ) } { \rho ^ { \prime 2 } \epsilon } \right) } \end{array}$
+
+242 Low-privacy regime: $1 > \alpha _ { m } ^ { \prime } - \gamma > 0$ . This is similar as above, except we have $\alpha _ { m } ^ { \prime } - \gamma \approx$   
+243 $\Delta _ { 2 } / \rho ^ { \prime 1 / 2 }$ and $\alpha _ { m } - \gamma \approx \Delta _ { 2 } / \rho ^ { 1 / 2 }$ . Similar computation as (12) shows that $\mathrm { D } _ { \mathrm { K L } } ( P \| Q ) < \epsilon$ when   
+244 $\begin{array} { r } { N = \Omega \left( \frac { \Delta _ { 2 } ^ { 2 } G ( \mathbf { p } ) } { \rho ^ { \prime } \epsilon } \right) } \end{array}$
+
+We observe that, in both regimes, the sample size scales faster with respect to $\epsilon$ with a higher value of $G ( \mathbf { p } )$ , which is associated with a higher number of outcomes $d$ , and more concentrated multinomial parameter $\mathbf { p }$ ; this agrees with the result of our simulation in Appendix 3. Moreover, for small $\rho ^ { \prime }$ the sample size scales as $1 / \rho ^ { \prime 2 }$ , while for large $\rho ^ { \prime }$ the sample size scales as $1 / \rho ^ { \prime }$ .
+
+# 4.2 Private normalized histograms
+
+250 Let $\mathbf { x } = ( x _ { 1 } , \ldots , x _ { d } )$ be a histogram of $N$ observations and $\mathbf { p } : = \mathbf { x } / N$ . We can privatize $\mathbf { p }$ by   
+251 sampling a probability vector: $\mathbf Y \sim \mathrm { D i r i c h l e t } ( { \mathbf x } + { \boldsymbol \alpha } )$ . Note that $\mathbf { Y }$ is a biased estimator of $\mathbf { p }$ .   
+252 Denoting $\alpha _ { 0 } : = \textstyle \sum _ { i } \alpha _ { i }$ , the bias of each component of $\mathbf { Y }$ is given by $\mathbb { E } [ \mathbf { Y } ] - p _ { i }$ . Hence,
+
+$$
+| \mathrm { B i a s } ( Y _ { i } ) | = \left| \frac { x _ { i } + \alpha _ { i } } { N + \alpha _ { 0 } } - p _ { i } \right| = \frac { | x _ { i } \alpha _ { 0 } - N \alpha _ { i } | } { N ( N + \alpha _ { 0 } ) } \leq \frac { N \alpha _ { 0 } } { N ( N + \alpha _ { 0 } ) } = \frac { \alpha _ { 0 } } { N + \alpha _ { 0 } } .
+$$
+
+Since 253 $Y _ { i } \sim \mathrm { B e t a } ( x _ { i } + \alpha _ { i } , N + \alpha _ { 0 } - x _ { i } - \alpha _ { i } )$ is $\frac { 1 } { 4 ( N + \alpha _ { 0 } + 1 ) }$ -sub-Gaussian [MA17], we have,
+
+$$
+\begin{array} { r l } & { \mathbb { P } [ | Y _ { i } - p _ { i } | > t + | \mathrm { B i a s } ( Y _ { i } ) | ] \le \mathbb { P } [ | Y _ { i } - \mathbb { E } [ Y _ { i } ] | + | \mathrm { B i a s } ( Y _ { i } ) | > t + | \mathrm { B i a s } ( Y _ { i } ) | ] } \\ & { \qquad = \mathbb { P } [ | Y _ { i } - \mathbb { E } [ Y _ { i } ] | > t ] } \\ & { \qquad \le 2 e ^ { - 2 t ^ { 2 } ( N + \alpha _ { 0 } + 1 ) } . } \end{array}
+$$
+
+With the union bound, we plug in 254 $\begin{array} { r } { t = \sqrt { \frac { \log ( 2 d / \beta ) } { 2 ( N + \alpha _ { 0 } + 1 ) } } } \end{array}$ , for any $\beta \in ( 0 , 1 )$ , to obtain the following 255 accuracy guarantee of the private normalized histogram:
+
+256 Theorem 4. Let $\mathbf Y \sim \mathrm { D i r i c h l e t } ( { \mathbf x } + { \boldsymbol \alpha } )$ , where $\mathbf { x } \in \mathbb { R } _ { \geq 0 } ^ { d }$ and $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ , and $\mathbf { p } : = \mathbf { x } / N$ . For any   
+257 $\beta \in ( 0 , 1 )$ , with probability at least $1 - \beta$ , the following inequality holds:
+
+$$
+\| \mathbf { Y } - \mathbf { p } \| _ { \infty } \leq \sqrt { \frac { \log ( 2 d / \beta ) } { 2 ( N + \alpha _ { 0 } + 1 ) } } + \frac { \alpha _ { 0 } } { N + \alpha _ { 0 } } .
+$$
+
+258 Given $\epsilon > 0$ , we use (13) to find a lower bound for $N$ that gives $\| \mathbf { Y } - \mathbf { p } \| _ { \infty } < \epsilon$ w.p. $1 - \beta$ when   
+259 $\mathbf { Y }$ is sampled with $\rho$ -tCDP. For simplicity, we consider a uniform prior: $\alpha _ { i } = \alpha > 0$ for all $i$ .   
+260 Thus, $\begin{array} { r } { \rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha - \gamma ) } \end{array}$ , where $\gamma$ might be chosen according to Corollary 2. We consider the two   
+261 following regimes:
+
+High-privacy regime: $\alpha - \gamma > 1$ . In this case, $\psi ^ { \prime } ( \alpha - \gamma ) \approx 1 / ( \alpha - \gamma )$ . From $\begin{array} { r } { \rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha - \gamma ) } \end{array}$ we have $\alpha \approx \Delta _ { 2 } ^ { 2 } / 2 \rho + \gamma$ . Replacing $\alpha _ { 0 }$ by $d \alpha$ in (13) yields the sample size:
+
+$$
+N = \Omega \biggl ( \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d } { \epsilon } \biggl ( \frac { \Delta _ { 2 } ^ { 2 } } { 2 \rho } + \gamma \biggr ) \biggr ) ,
+$$
+
+264 for the desired accuracy.
+
+265 Low-privacy regime: $\alpha - \gamma < 1$ . This is the same as above, except now we have $\psi ^ { \prime } ( \alpha - \gamma ) \approx$   
+266 $1 / ( \alpha - \gamma ) ^ { 2 }$ , which implies $\alpha \approx \Delta _ { 2 } / ( 2 \rho ) ^ { 1 / 2 } + \gamma$ . The sample size that guarantees the desired   
+267 accuracy is:
+
+$$
+N = \Omega \biggl ( \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d } { \epsilon } \biggl ( \frac { \Delta _ { 2 } } { \sqrt { 2 \rho } } + \gamma \biggr ) \biggr ) .
+$$
+
+268 Let us compare this result to the Gaussian mechanism, which adds a noise $\mathbf { Z } \sim N ( 0 , \sigma ^ { 2 } I _ { d } )$ to the   
+269 normalized histogram $\mathbf { p }$ directly. Thus the $\ell _ { 2 }$ -sensitivity in this case is $\Delta _ { 2 } / N$ . We have that the   
+270 Gaussian mechanism is $\rho$ -zCDP where $\begin{array} { r } { \rho = \frac { \Delta ^ { 2 } } { 2 N ^ { 2 } \sigma ^ { 2 } } } \end{array}$ [BS16]. Using the same argument as above, with   
+271 probability at least $1 - \beta$ , the following inequality holds for all $i$ :
+
+$$
+\| \mathbf { Z } \| _ { \infty } \leq \sqrt { \frac { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } } { N ^ { 2 } \rho } } .
+$$
+
+![](images/4d398e42217bd393cee6a912cf2c04e26395de3300d829c872c8f73d8e70fe67.jpg)  
+Figure 2: The $\ell ^ { \infty }$ -accuracy, as a function of $N$ , of Dirichlet posterior sampling $( \gamma = 1 )$ ) and Gaussian mechanisms for private normalized histograms $\Delta _ { 2 } ^ { 2 } = 2$ and $\Delta _ { \infty } = 1 $ ). For each $N , d$ and $\rho$ , we generated the inputs $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { 2 0 0 }$ , where $\mathbf { x } _ { k } \sim$ Multinomial $\left( \mathbf { q } _ { k } \right)$ and $\mathbf { q } _ { k } \sim \mathrm { D i r i c h l e t } ( 5 , \ldots , 5 )$ .
+
+Hence, the sample size of 272 $N = \Omega \Big ( \sqrt { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } / \rho \epsilon ^ { 2 } } \Big )$ guarantees the desired accuracy. Compar273 ing this to (14), if we assume $\epsilon < 1$ , the AM-GM inequality tells us that
+
+$$
+\frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d \Delta _ { 2 } ^ { 2 } } { \rho \epsilon } > \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { \Delta _ { 2 } ^ { 2 } } { \rho } \geq 2 \sqrt { \frac { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } } { \rho \epsilon ^ { 2 } } } .
+$$
+
+274 The inequality (17) implies that the Gaussian mechanism requires less sample than the Dirichlet   
+275 mechanism in order to guarantee the same level of accuracy. The Gaussian mechanism is also better   
+276 in the low-privacy regime as the $\rho$ in (15) satisfies ${ \sqrt { \rho } } < \rho$ and $\Delta _ { 2 } \approx \Delta _ { 2 } ^ { 2 }$ , leading to the same   
+277 inequality (17). Nonetheless, the decay in (16) is linear in $d$ , while that in (13) has $\alpha _ { 0 } = d \alpha$ in   
+278 the denominators. This observation suggests that, when $\mathbf { x }$ is a sparse histogram i.e. when $N \leq d$ ,   
+279 the $\ell ^ { \infty }$ -accuracy of the Dirichlet mechanism is smaller than that of the Gaussian mechanism. This   
+280 conclusion is supported by our simulation in Figure 2. We see that the $\ell ^ { \infty }$ -accuracy of the Dirichlet   
+281 mechanism is smaller than that of the Gaussian mechanism for small $N$ when $d = 1 0 0 0$ . The code   
+282 for all experiments in this study can be found in the supplemental material.
+
+# 283 Potential negative societal impacts
+
+It is important to note that, when $\rho$ becomes unacceptably large (e.g., $\rho = 1 0 ^ { 4 }$ ), the sampling is far away from being private. Thus any organization that deploys the posterior sampling on sensitive data must not vacuously refer to this study and claim that its algorithm is private. It is the organization’s responsibility to fully publish the prior parameters, and educate its users/customers on differential privacy and how the privacy guarantees are calculated.
+
+It is desirable that differentially private algorithms are accurate for the task at hand, especially when the data is used for important decision-making. Thus, one needs to make sure that there is enough sample to achieve the desired level of accuracy. For a large differentially private system, privacy budgets need to be allocated to the parts that require accurate outputs.
+
+293 Lastly, one must be careful with the choice of prior parameters; if a uniform prior is used, smaller   
+294 groups will suffer a relatively larger statistical bias. As a result, private statistics of small populations   
+295 (such as ethnic or racial minorities) will be relatively less accurate. One way to get around this issue   
+296 is to (privately) impose larger prior parameters on larger populations.   
+297 References   
+298 [AAFK20] I. Aykin, B. Akgun, M. Feng, and M. Krunz. “MAMBA: A Multi-armed Bandit Framework for   
+299 Beam Tracking in Millimeter-wave Systems”. In: 39th IEEE Conference on Computer Com  
+300 munications, INFOCOM 2020, Toronto, ON, Canada, July 6-9, 2020. IEEE, 2020, pp. 1469–   
+301 1478.   
+302 [AV08] J. M. Abowd and L. Vilhuber. “How Protective Are Synthetic Data?” In: Privacy in Statistical   
+303 Databases, UNESCO Chair in Data Privacy International Conference, PSD 2008, Istan  
+304 bul, Turkey, September 24-26, 2008. Proceedings. Ed. by J. Domingo-Ferrer and Y. Saygin.   
+305 Vol. 5262. Lecture Notes in Computer Science. Springer, 2008, pp. 239–246.   
+306 [BDRS18] M. Bun, C. Dwork, G. N. Rothblum, and T. Steinke. “Composable and versatile privacy via   
+307 truncated CDP”. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of   
+308 Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018. Ed. by I. Diakonikolas,   
+309 D. Kempe, and M. Henzinger. ACM, 2018, pp. 74–86.   
+310 [BHW00] R. J. Boys, D. A. Henderson, and D. J. Wilkinson. “Detecting Homogeneous Segments in DNA   
+311 Sequences by Using Hidden Markov Models”. In: Journal of the Royal Statistical Society.   
+312 Series $C$ (Applied Statistics) 49.2 (2000), pp. 269–285. ISSN: 00359254, 14679876.   
+313 [BS16] M. Bun and T. Steinke. “Concentrated Differential Privacy: Simplifications, Extensions, and   
+314 Lower Bounds”. In: Theory of Cryptography. Ed. by M. Hirt and A. Smith. Berlin, Heidelberg:   
+315 Springer Berlin Heidelberg, 2016, pp. 635–658.   
+316 [BS18] G. Bernstein and D. R. Sheldon. “Differentially Private Bayesian Inference for Exponential   
+317 Families”. In: Advances in Neural Information Processing Systems 31: Annual Conference on   
+318 Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal,   
+319 Canada. Ed. by S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and   
+320 R. Garnett. 2018, pp. 2924–2934.   
+321 [CWS03] J. Corander, P. Waldmann, and M. J. Sillanpää. “Bayesian Analysis of Genetic Differentiation   
+322 Between Populations”. In: Genetics 163.1 (Jan. 2003), pp. 367–374. ISSN: 1943-2631.   
+323 [de 06] M. de Lapparent. “Empirical Bayesian analysis of accident severity for motorcyclists in large   
+324 French urban areas”. In: Accident Analysis & Prevention 38.2 (2006), pp. 260–268. ISSN:   
+325 0001-4575.   
+326 [DMNS06] C. Dwork, F. Mcsherry, K. Nissim, and A. Smith. “Calibrating noise to sensitivity in private   
+327 data analysis”. In: TCC. 2006.   
+328 [DNZMR17] C. Dimitrakakis, B. Nelson, Z. Zhang, A. Mitrokotsa, and B. I. P. Rubinstein. “Differential   
+329 Privacy for Bayesian Inference through Posterior Sampling”. In: J. Mach. Learn. Res. 18   
+330 (2017), 11:1–11:39.   
+331 [FGWC16] J. R. Foulds, J. Geumlek, M. Welling, and K. Chaudhuri. “On the Theory and Practice of   
+332 Privacy-Preserving Bayesian Data Analysis”. In: Proceedings of the Thirty-Second Conference   
+333 on Uncertainty in Artificial Intelligence, UAI 2016, June 25-29, 2016, New York City, NY, USA.   
+334 Ed. by A. T. Ihler and D. Janzing. AUAI Press, 2016.   
+335 [GSC17] J. Geumlek, S. Song, and K. Chaudhuri. “Renyi Differential Privacy Mechanisms for Posterior   
+336 Sampling”. In: Advances in Neural Information Processing Systems 30: Annual Conference   
+337 on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA.   
+338 Ed. by I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N. Vishwanathan,   
+339 and R. Garnett. 2017, pp. 5289–5298.   
+340 [GWHHT21] P. Gohari, B. Wu, C. Hawkins, M. T. Hale, and U. Topcu. “Differential Privacy on the Unit   
+341 Simplex via the Dirichlet Mechanism”. In: IEEE Trans. Inf. Forensics Secur. 16 (2021),   
+342 pp. 2326–2340.   
+343 [Hin15] K. Hines. “A Primer on Bayesian Inference for Biophysical Systems”. In: Biophysical Journal   
+344 108.9 (2015), pp. 2103–2113. ISSN: 0006-3495.   
+345 [LW92] M. Lavine and M. West. “A Bayesian method for classification and discrimination”. In:   
+346 Canadian Journal of Statistics 20.4 (1992), pp. 451–461.   
+347 [MA17] O. Marchal and J. Arbel. “On the sub-Gaussianity of the Beta and Dirichlet distributions”. In:   
+348 Electronic Communications in Probability 22.none (2017), pp. 1 –14.   
+349 [MKAGV08] A. Machanavajjhala, D. Kifer, J. M. Abowd, J. Gehrke, and L. Vilhuber. “Privacy: Theory   
+350 meets Practice on the Map”. In: Proceedings of the 24th International Conference on Data   
+351 Engineering, ICDE 2008, April 7-12, 2008, Cancún, Mexico. Ed. by G. Alonso, J. A. Blakeley,   
+352 and A. L. P. Chen. IEEE Computer Society, 2008, pp. 277–286.   
+353 [MMR05] J.-M. Marin, K. Mengersen, and C. P. Robert. “Bayesian Modelling and Inference on Mixtures   
+354 of Distributions”. In: Bayesian Thinking. Ed. by D. Dey and C. Rao. Vol. 25. Handbook of   
+355 Statistics. Elsevier, 2005, pp. 459–507.   
+356 [MT07] F. McSherry and K. Talwar. “Mechanism Design via Differential Privacy”. In: 48th Annual   
+357 IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007,   
+358 Providence, RI, USA, Proceedings. IEEE Computer Society, 2007, pp. 94–103.   
+359 [NIK20] I. Nasim, A. S. Ibrahim, and S. Kim. “Learning-Based Beamforming for Multi-User Vehicular   
+360 Communications: A Combinatorial Multi-Armed Bandit Approach”. In: IEEE Access 8 (2020),   
+361 pp. 219891–219902.   
+362 [NLYCC13] V. C. Nguyen, W. S. Lee, N. Ye, K. M. A. Chai, and H. L. Chieu. “Active Learning for   
+363 Probabilistic Hypotheses Using the Maximum Gibbs Error Criterion”. In: Advances in Neural   
+364 Information Processing Systems 26: 27th Annual Conference on Neural Information Processing   
+365 Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada,   
+366 United States. Ed. by C. J. C. Burges, L. Bottou, Z. Ghahramani, and K. Q. Weinberger. 2013,   
+367 pp. 1457–1465.   
+368 [ORR13] I. Osband, D. Russo, and B. V. Roy. “(More) Efficient Reinforcement Learning via Poste  
+369 rior Sampling”. In: Advances in Neural Information Processing Systems 26: 27th Annual   
+370 Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held   
+371 December 5-8, 2013, Lake Tahoe, Nevada, United States. Ed. by C. J. C. Burges, L. Bottou,   
+372 Z. Ghahramani, and K. Q. Weinberger. 2013, pp. 3003–3011.   
+373 [PB98] T. Pedersen and R. F. Bruce. “Knowledge Lean Word-Sense Disambiguation”. In: Proceedings   
+374 of the Fifteenth National Conference on Artificial Intelligence and Tenth Innovative Appli  
+375 cations of Artificial Intelligence Conference, AAAI 98, IAAI 98, July 26-30, 1998, Madison,   
+376 Wisconsin, USA. Ed. by J. Mostow and C. Rich. AAAI Press / The MIT Press, 1998, pp. 800–   
+377 805.   
+378 [PG14] Y. Park and J. Ghosh. “PeGS: Perturbed Gibbs Samplers that Generate Privacy-Compliant   
+379 Synthetic Data”. In: Trans. Data Priv. 7.3 (2014), pp. 253–282.   
+380 [PM01] J. Pella and M. Masuda. “Bayesian methods for analysis of stock mixtures from genetic   
+381 characters”. English. In: Fishery Bulletin 99 (Jan. 2001). 1, p. 151. ISSN: 00900656.   
+382 [Rén61] A. Rényi. “On measures of entropy and information”. In: Proceedings of the Fourth Berkeley   
+383 Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory   
+384 of Statistics. The Regents of the University of California. 1961.   
+385 [RWZ14] J. P. Reiter, Q. Wang, and B. Zhang. “Bayesian Estimation of Disclosure Risks for Multiply   
+386 Imputed, Synthetic Data”. In: J. Priv. Confidentiality 6.1 (2014).   
+387 [SJGLY17] M. J. Schneider, S. Jagpal, S. Gupta, S. Li, and Y. Yu. “Protecting customer privacy when   
+388 marketing with second-party data”. In: International Journal of Research in Marketing 34.3   
+389 (2017), pp. 593–603. ISSN: 0167-8116.   
+390 [Str00] M. J. A. Strens. “A Bayesian Framework for Reinforcement Learning”. In: Proceedings of the   
+391 Seventeenth International Conference on Machine Learning (ICML 2000), Stanford University,   
+392 Stanford, CA, USA, June 29 - July 2, 2000. Ed. by P. Langley. Morgan Kaufmann, 2000,   
+393 pp. 943–950.   
+394 [WFS15] Y. Wang, S. E. Fienberg, and A. J. Smola. “Privacy for Free: Posterior Sampling and Stochastic   
+395 Gradient Monte Carlo”. In: Proceedings of the 32nd International Conference on Machine   
+396 Learning, ICML 2015, Lille, France, 6-11 July 2015. Ed. by F. R. Bach and D. M. Blei. Vol. 37.   
+397 JMLR Workshop and Conference Proceedings. JMLR.org, 2015, pp. 2493–2502.   
+398 [ZHGSY20] J. Zhu, X. Huang, X. Gao, Z. Shao, and Y. Yang. “Multi-Interface Channel Allocation in Fog   
+399 Computing Systems using Thompson Sampling”. In: 2020 IEEE International Conference on   
+400 Communications, ICC 2020, Dublin, Ireland, June 7-11, 2020. IEEE, 2020, pp. 1–6.
+
+# 401 Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We gave a simple guaranteed upper bound of tCDP (2) for the Dirichlet posterior sampling and illustrated how it can be used to derive accuracy guarantees in Section 4.   
+(b) Did you describe the limitations of your work? [Yes] We discussed a limitation of the guaranteed upper bound of tCDP in the paragraph following Theorem 1. We also described a situation under which the Gaussian mechanism is preferable to the Dirichlet posterior sampling at the end of Section 4.2.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See the section on potential negative societal impacts at the end of the paper.
+
+(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? [No] The proofs of all theorems are given in the main paper, except that of Theorem 3 which is given in Appendix 2.
+
+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] The code and the instructions for our simulations are included in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We specified the details of our simulations in the figures’ captions.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We reported the error bars in Figure 2   
+(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)? [N/A] Our experiments are not computationally intensive.
+
+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? [N/A]   
+(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? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/Pk972F7J8Um/Pk972F7J8Um_content_list.json

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+        "type": "text",
+        "text": "Differential Privacy of Dirichlet Posterior Sampling ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 We study the inherent privacy of releasing a single sample from a Dirichlet posterior   \n2 distribution. As a complement to the previous study that provides general theories   \n3 on the differential privacy of posterior sampling from exponential families, this   \n4 study focuses specifically on the Dirichlet posterior sampling and its privacy   \n5 guarantees. With the notion of truncated concentrated differential privacy (tCDP),   \n6 we are able to derive a simple privacy guarantee of the Dirichlet posterior sampling,   \n7 which effectively allows us to analyze its utility in various settings. Specifically,   \n8 we provide accuracy guarantees of the Dirichlet posterior sampling in Multinomial  \n9 Dirichlet sampling and private normalized histogram publishing. ",
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+    {
+        "type": "text",
+        "text": "10 1 Introduction ",
+        "text_level": 1,
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+        ],
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+    },
+    {
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+        "text": "11 The Bayesian framework provides a way to perform statistical analysis by combining prior beliefs   \n12 with real-life evidence. At a high level, the belief and the evidence are assumed to be described   \n13 by probabilistic models. As we receive new data, our belief is updated accordingly via the Bayes’   \n14 theorem, resulting in the so-called posterior belief. The posterior tells us how much we are uncertain   \n15 about the model’s parameters.   \n16 The Dirichlet distribution is usually chosen as the prior when performing Bayesian analysis on discrete   \n17 variables, as it is a conjugate prior to the categorical and multinomial distributions. Specifically,   \n18 Dirichlet distributions are often used in discrete mixture models, where a Dirichlet prior is put on   \n19 the mixture weights [LW92; MMR05]. Such models have applications in NLP [PB98], biophysical   \n20 systems [Hin15], accident analysis [de 06], and genetics [BHW00; PM01; CWS03]. In all of these   \n21 studies, samplings from Dirichlet posteriors arise when performing Markov chain Monte Carlo   \n22 methods for approximate Bayesian inference.   \n23 Dirichlet posterior sampling also appears in other learning tasks. For example, in Bayesian active   \n24 learning, it arises in Gibbs sampling, which is used to approximate the posterior of the classifier over   \n25 the labeled sample [NLYCC13]. In Thompson sampling for multi-armed bandits, one repeatedly   \n26 draws a sample from the Dirichlet posterior of each arm, and picks the arm whose sample maximizes   \n27 the reward [ZHGSY20; AAFK20; NIK20]. And in Bayesian reinforcement learning, state-transition   \n28 probabilities are sampled from the Dirichlet posterior over past observed states [Str00; ORR13].   \n29 Dirichlet posterior sampling can also be used for data synthesis. Suppose that we have a histogram   \n30 $( x _ { 1 } , \\ldots , x _ { d } )$ of actual data. An approximate discrete distribution of this histogram can be obtained by   \n31 drawing a sample $\\mathbf { Y }$ from Dirichlet $( x _ { 1 } + \\alpha _ { 1 } , \\ldots , x _ { d } + \\alpha _ { d } )$ , where $\\alpha _ { 1 } , \\ldots , \\alpha _ { d }$ are prior parameters.   \n32 Then synthetic data is produced by repeatedly drawing from Multinomial(Y). There are many   \n33 studies on data synthesis that followed this approach [AV08; MKAGV08; RWZ14; PG14; SJGLY17].   \n34 In the above examples, the data that we integrate into these tasks might contain sensitive information.   \n35 Thus it is important to ask: how much of the information is protected from the Dirichlet samplings?   \n36 The goal of this study is to find an answer to this question.   \n37 The mathematical framework of differential privacy (DP) [DMNS06] allows us to quantify how much   \n38 the privacy of the Dirichlet posterior sampling is affected by the prior parameters $\\alpha _ { 1 } , \\ldots , \\alpha _ { d }$ . In the   \n39 definition of DP, the privacy of a randomized algorithm is measured by how much its distribution   \n40 changes upon perturbing a single data point of the input. Nonetheless, this notion might be too   \n41 strict for the Dirichlet distribution, as a small perturbation of a near-zero parameter can cause a large   \n42 distribution shift. Thus, it might be more appropriate to rely on one of several relaxed notions of   \n43 DP, such as approximate differential privacy, Rényi differential privacy, or concentrated differential   \n44 privacy. It is natural to wonder if the Dirichlet posterior sampling satisfies any of these definitions. ",
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+        "type": "text",
+        "text": "45 1.1 Overview of Our results ",
+        "text_level": 1,
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+        "text": "46 This study focuses on the privacy and utility of Dirichlet posterior sampling. In summary, we provide   \n47 a closed-form privacy guarantee of the Dirichlet posterior sampling, which in turn allows us to   \n48 effectively analyze its utility in various settings. ",
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+        "text": "$\\ S 3$ Privacy. We study the role of the prior parameters in the privacy of the Dirichlet posterior sampling. Theorem 1 is our main result, where we provide a guaranteed upper bound for truncated concentrated differential privacy (tCDP) of the Dirichlet posterior sampling. In addition, we convert the tCDP guarantee into an approximate differential privacy guarantee in Corollary 2. ",
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+        "text": "$\\ S 4$ Utility. Using the tCDP guarantee, we investigate the utility of Dirichlet posterior sampling applied in two specific applications: ",
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+        "text": "• In Section 4.1, we consider one-time sampling from a Multinomial-Dirichlet distribution. But instead of directly sampling from this distribution, we sample from another distribution with larger prior parameters. The accuracy is then measured by the KL-divergence between the original and the private distributions.   \n• In Section 4.2, we use the Dirichlet posterior sampling for a private release of a normalized histogram. In this case, the accuracy is measured by the mean-squared error between the sample and the original normalized histogram. ",
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+        "text": "62 In both tasks, we compute the sample size that guarantees the desired level of accuracy. In the case   \n63 of private histogram publishing, we also compare the Dirichlet posterior sampling to the Gaussian   \n64 mechanism. ",
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+        "text": "65 1.2 Related work ",
+        "text_level": 1,
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+        "text": "66 There are several studies on the differential privacy of posterior sampling. Wang, Fienberg, and   \n67 Smola [WFS15] showed that any posterior sampling with the log-likelihood bounded by $B$ is $4 B$ -   \n68 differentially private. However, the likelihoods that we study are not bounded away from zero; they   \n69 have the form $\\Pi _ { i } p _ { i } ^ { x _ { i } }$ which becomes small when one of the $p _ { i }$ ’s is close to zero. Dimitrakakis, Nelson,   \n70 Zhang, Mitrokotsa, and Rubinstein [DNZMR17] showed that if the condition on the log-likelihood is   \n71 relaxed to the Lipschitz continuity with high probability, then one can obtain the approximate DP.   \n72 Nonetheless, with the Dirichlet density, it is difficult to compute the probability of events in which   \n73 the Lipschitz condition is satisfied.   \n74 In the case that the sufficient statistics $\\mathbf { x }$ has finite $\\ell ^ { 1 }$ -sensitivity, Foulds, Geumlek, Welling and   \n75 Chaudhuri [FGWC16] suggested adding Laplace noises to $\\mathbf { x }$ . Suppose that y is the output; they   \n76 showed that sampling from $p ( \\boldsymbol { \\theta } | \\mathbf { y } )$ is differentially private and as asymptotically efficient as sampling   \n77 from $p ( \\boldsymbol { \\theta } | \\mathbf { x } )$ . However, for a small sample size, the posterior over the noisy statistics might be too   \n78 far away from the actual posterior. Bernstein and Sheldon [BS18] thus proposed to approximate the   \n79 joint distribution $p ( \\boldsymbol { \\theta } , \\mathbf { x } , \\mathbf { y } )$ using Gibbs sampling, which is then integrated over $\\mathbf { x }$ to obtain a more   \n80 accurate posterior over $\\mathbf { y }$ .   \n81 Geumlek, Song, and Chaudhuri [GSC17] were the first to study the posterior sampling with the   \n82 RDP. Even though they provided a general framework to find $( \\lambda , \\epsilon )$ -RDP guarantees for exponential   \n83 families, explicit forms of $\\epsilon$ and the upper bound of $\\lambda$ were not given. In contrast, our tCDP guarantees   \n84 of the Dirichlet posterior sampling imply an explicit expression for $\\epsilon$ , and also an upper bound for $\\lambda$ .   \n85 The privacy of data synthesis via sampling from Multinomial $( \\mathbf { Y } )$ , where $\\mathbf { Y }$ is a discrete distri  \n86 bution drawn from the Dirichlet posterior, was first studied by Machanavajjhala, Kifer, Abowd,   \n87 Gehrke, and Vilhuber [MKAGV08]. They showed that the data synthesis is $( \\varepsilon , \\delta )$ -probabilistic DP,   \n88 which implies $( \\varepsilon , \\delta )$ -approximate DP. However, as their privacy analysis includes the sampling from   \n89 Multinomial $( \\mathbf { Y } )$ , their privacy guarantee depends on the number of synthetic samples. In contrast,   \n90 we show that the one-time sampling from the Dirichlet posterior is approximate DP, which by the   \n91 post-processing property allows us to sample from Multinomial $( \\mathbf { Y } )$ as many times as we want while   \n92 retaining the same privacy guarantee.   \n93 The Dirichlet mechanism was first introduced by Gohari, Wu, Hawkins, Hale, and Topcu [GWHHT21].   \n94 Originally, the Dirichlet mechanism takes a discrete distribution $\\mathbf { p } : = ( p _ { 1 } , \\ldots , p _ { d } )$ and draws one   \n95 sample $\\mathbf { Y } \\sim \\mathrm { D i r i c h l e t } ( r p _ { 1 } , \\dots , r p _ { d } )$ . Note the absence of the prior parameters, which makes $\\mathbf { Y }$ an   \n96 unbiased estimator of $\\mathbf { p }$ . But this comes with a cost, as the worst case of privacy violation occurs   \n97 when almost all of the parameters are close to zero. The authors avoided this issue by restricting   \n98 the input space to a subset of the unit simplex, with some of the $p _ { i }$ ’s bounded below by a fixed   \n99 positive constant. This results in complicated expressions for the privacy guarantees as they involve   \n100 a minimization problem over the restricted domain. In this study, we take a different approach by   \n101 adding prior parameters to the Dirichlet mechanism. As a result, we obtain a biased algorithm that   \n102 requires no assumption on the input space and has simpler forms of privacy guarantees. ",
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+        "text": "1.3 Notations ",
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+        "text": "104 We let $\\mathbb { R } _ { \\geq 0 } ^ { d }$ be the set of $d$ -tuples of non-negative real numbers and $\\mathbb { R } _ { > 0 } ^ { d }$ be the set of $d$ -tuples of   \n105 positive real numbers. We assume that all vectors are $d$ -dimensional where $d \\geq 2$ . The notations for   \n106 all vectors are always in bold. Specifically, $\\mathbf { x } : = ( x _ { 1 } , \\ldots , x _ { d } ) \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ consists of sample statistics of   \n107 the data and $\\pmb { \\alpha } : = ( \\alpha _ { 1 } , \\ldots , \\alpha _ { d } ) \\in \\mathbb { R } _ { > 0 } ^ { d }$ consists of the prior parameters. The vector $\\mathbf { p } : = ( p _ { 1 } , \\ldots , p _ { d } )$   \n108 always satisfies $\\textstyle \\sum _ { i } p _ { i } = 1$ . The number of observations is always $N$ . We also denote $x _ { 0 } : = \\textstyle \\sum _ { i } x _ { i }$   \n109 and $\\alpha _ { 0 } : = \\textstyle \\sum _ { i } \\alpha _ { i }$ . For any vectors $\\mathbf { x } , \\mathbf { x } ^ { \\prime }$ and scalar $r > 0$ , we write $\\mathbf { x } + \\mathbf { x } ^ { \\prime } : = ( x _ { 1 } + x _ { 1 } ^ { \\prime } , \\ldots , x _ { d } + x _ { d } ^ { \\prime } )$   \n110 and $r \\mathbf { x } : = ( r x _ { 1 } , \\ldots , r x _ { d } )$ . For any positive reals $x$ and $x ^ { \\prime }$ , the notation $x \\propto x ^ { \\prime }$ means $x = C x ^ { \\prime }$ for   \n111 some constant $C > 0$ , $x \\approx x ^ { \\prime }$ means $c x ^ { \\prime } \\leq x \\leq C x ^ { \\prime }$ for some $c , C > 0$ , and $x \\lesssim x ^ { \\prime }$ means $x \\leq C x ^ { \\prime }$   \n112 for some $C > 0$ . Lastly, $\\| \\mathbf { x } \\| _ { \\infty } : = \\operatorname* { m a x } _ { i } | x _ { i } |$ is the $\\ell ^ { \\infty }$ norm of $\\mathbf { x }$ . ",
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+    {
+        "type": "text",
+        "text": "13 2 Background ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "2.1 Privacy models ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 (Pure and Approximate DP [DMNS06]). A randomized mechanism $M : \\mathcal { X } ^ { n }  \\mathcal { Y }$ is $( \\varepsilon , \\delta )$ -differentially private $( \\varepsilon , \\delta )$ -DP) if for any datasets $x , x ^ { \\prime }$ differing on a single entry, and all events $E \\subset \\mathcal { V }$ , ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a03fcdf947f012874d645b12dc14503ac64d8f6d03d5b3a81a3e0270a6008422.jpg",
+        "text": "$$\n\\mathbb { P } [ M ( x ) \\in E ] \\leq e ^ { \\varepsilon } \\mathbb { P } [ M ( x ^ { \\prime } ) \\in E ] + \\delta .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "118 If $M$ is $( \\varepsilon , 0 )$ -DP, then we say that it is $\\varepsilon$ -differential privacy $\\dot { \\varepsilon }$ -DP). ",
+        "bbox": [
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+            753,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "19 The term pure differential privacy (pure DP) refers to $\\epsilon$ -differential privacy, while approximate   \n20 differential privacy (approximate DP) refers to $( \\varepsilon , \\delta )$ -DP when $\\delta > 0$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "121 In contrast to pure and approximate DP, the next definitions of differential privacy are defined in terms of the Rényi divergence between 122 $M ( x )$ and $M ( x ^ { \\prime } )$ : ",
+        "bbox": [
+            150,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "123 Definition 2.2 (Rényi Divergence [Rén61]). Let $P$ and $Q$ be probability distributions. $\\mathrm { F o r } \\lambda \\in ( 1 , \\infty )$   \n124 the Rényi divergence of order $\\lambda$ between $P$ and $Q$ is defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/93eccc79cfb463d3011599e5ccc70bc2df5473e9e983b54fac6d7edb496536f4.jpg",
+        "text": "$$\n\\mathrm { D } _ { \\lambda } ( P \\| Q ) : = { \\frac { 1 } { \\lambda - 1 } } \\log \\int P ( y ) ^ { \\lambda } Q ( y ) ^ { 1 - \\lambda } d y = { \\frac { 1 } { \\lambda - 1 } } \\log \\biggl ( \\operatorname { \\mathbb { E } } _ { y \\sim P } \\biggl [ { \\frac { P ( y ) ^ { \\lambda - 1 } } { Q ( y ) ^ { \\lambda - 1 } } } \\biggr ] \\biggr . . \\biggr )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "125 Definition 2.3 (tCDP and zCDP [BDRS18; BS16]). A randomized mechanism $M : \\mathcal { X } ^ { n }  \\mathcal { Y }$ is   \n126 $\\omega$ -truncated $\\rho$ -concentrated differentially private $( ( \\rho , \\omega )$ -tCDP) if for any datasets $x , x ^ { \\prime }$ differing on a   \n127 single entry and for all $\\lambda \\in ( 1 , \\omega )$ , ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ada4dfd5ac3d4ef86be7aac385827fcfd1809248cc9c468a693ec0b99e679e3a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { D } _ { \\lambda } ( M ( x ) \\| M ( x ^ { \\prime } ) ) \\leq \\lambda \\rho . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "128 If $M$ is $( \\rho , \\infty )$ -tCDP, then we say that it is $\\rho$ -zero-concentrated differential privacy ( $\\rho$ -zCDP). ",
+        "bbox": [
+            151,
+            162,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "129 Note that both tCDP and zCDP have the composition and post-processing properties. Intuitively, $\\rho$ con  \n130 trols the expectation and standard deviation of the privacy loss random variable: Z = log P [M(x)=Y ]P [M(x0)=Y ] ,   \n131 where $Y$ has density $M ( x )$ , and $\\omega$ controls the number of standard deviations for which $Z$ concen  \n132 trates like a Gaussian. A smaller $\\rho$ and larger $\\omega$ correspond to a stronger privacy guarantee. It turns   \n133 out that tCDP implies approximate DP:   \n134 Lemma 1 (From tCDP to Approximate DP [BDRS18]). Let $\\delta > 0$ . If M is a $( \\rho , \\omega )$ -tCDP mechanism,   \n135 then it also satisfies $( \\varepsilon , \\delta )$ - $D P$ with ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5682ac841ef523f83696996e82de41e02241bf842345449d1713bf0934043f2d.jpg",
+        "text": "$$\n\\varepsilon = \\left\\{ \\begin{array} { l l } { \\rho + 2 \\sqrt { \\rho \\log ( 1 / \\delta ) } \\quad } & { i f \\log ( 1 / \\delta ) \\leq ( \\omega - 1 ) ^ { 2 } \\rho } \\\\ { \\rho \\omega + \\frac { \\log ( 1 / \\delta ) } { \\omega - 1 } } & { i f \\log ( 1 / \\delta ) > ( \\omega - 1 ) ^ { 2 } \\rho } \\end{array} \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "136 2.2 Dirichlet distribution ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            359,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "137 For $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ , the Dirichlet distribution Dirichlet $( \\alpha )$ is a continuous distribution of $d$ -dimensional   \n138 probability vectors i.e. vectors whose coordinate sum is equal to 1. The density function of $\\mathbf { Y } \\sim$   \n139 Dirichlet $( \\alpha )$ is given by: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb865a389ca943847c9b7ca4d3ac4e3e6cafefda1521cf6aeef1ab38819aa540.jpg",
+        "text": "$$\np ( \\mathbf { y } ) = \\frac { 1 } { B ( \\pmb { \\alpha } ) } \\prod _ { i = 1 } ^ { d } y _ { i } ^ { \\alpha _ { i } - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "140 where $B ( \\alpha )$ is the beta function, which can be written in terms of the gamma function: ",
+        "bbox": [
+            137,
+            474,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4de47ec9a7935d45799b02e33e5087d394647393a8890ed4359090d66b6b2981.jpg",
+        "text": "$$\nB ( \\pmb { \\alpha } ) = \\frac { \\prod _ { i } \\Gamma ( \\alpha _ { i } ) } { \\Gamma ( \\sum _ { i } \\alpha _ { i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "141 2.3 Dirichlet posterior sampling ",
+        "text_level": 1,
+        "bbox": [
+            145,
+            541,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "142 We consider the prior Dirichlet $( \\alpha )$ and the likelihood of the form $\\begin{array} { r } { p ( \\mathbf { x } | \\mathbf { y } ) \\propto \\prod _ { i = 1 } ^ { d } y _ { i } ^ { x _ { i } } } \\end{array}$ where   \n143 $\\mathbf { x } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ consists of sample statistics of the dataset. The Dirichlet posterior sampling is a one-time   \n144 sampling: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81c53d2e40427e02c3b08c62fc781df0d80965bd295a8ecfc387de285a339621.jpg",
+        "text": "$$\n\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { x } + \\pmb { \\alpha } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "145 There is a modification of the sampling which introduces a concentration parameter $r > 0$ , and   \n146 instead we sample from Dirichlet $( r \\mathbf { x } + \\pmb { \\alpha } )$ [GSC17; GWHHT21]. Smaller values of $r$ make the   \n147 sampling more private, and larger values of $r$ make $\\mathbf { Y }$ a closer approximation of $\\mathbf { x }$ . Even though the   \n148 case $r = 1$ is the main focus of this study, our main privacy results can be easily extended to other   \n149 values of $r$ as we will see at the end of Section 3.1.   \n150 Consider a special case where $\\mathbf x = \\mathbf p$ is an empirical distribution derived from the dataset, and we   \n151 want $\\mathbf { Y }$ to be a private approximation of $\\mathbf { p }$ ; the sampling $\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( r \\mathbf { p } + \\alpha )$ is called the   \n152 Dirichlet mechanism [GWHHT21]. It is interesting to note that the Dirichlet mechanism is a form of   \n153 the exponential mechanism [MT07]: let $r > 0$ be the privacy parameter, Dirichlet $( \\alpha )$ be the prior,   \n154 and the negative KL-divergence be the score function of the exponential mechanism. Then the output   \n155 $\\mathbf { Y }$ of this mechanism is distributed according to the following density function: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            713,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/32fa8491332f190dd97efb2a6ec88182fc644c266ee63558b2bdeccc8187550c.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\frac { \\exp \\left( - r \\mathrm { D } _ { \\mathrm { K L } } ( \\mathbf { p } , \\mathbf { y } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } } { \\int \\exp \\left( - r \\mathrm { D } _ { \\mathrm { K L } } ( \\mathbf { p } , \\mathbf { y } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } d \\mathbf { y } } \\propto \\exp \\left( r \\sum _ { i , p _ { i } \\neq 0 } p _ { i } \\log ( y _ { i } / p _ { i } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } } \\\\ & { } & { \\propto \\displaystyle \\prod _ { i , p _ { i } \\neq 0 } y _ { i } ^ { r p _ { i } } \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } = \\prod _ { i } y _ { i } ^ { r p _ { i } + \\alpha _ { i } - 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "156 which is exactly the density function of Dirichlet $( r \\mathbf { p } + \\alpha )$ . ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "58 In most of this study, we take advantage of several nice properties of the log-gamma function and its derivatives. Specifically, $\\begin{array} { r } { \\psi ( x ) : = \\frac { d } { d x } \\log \\Gamma ( x ) } \\end{array}$ is concave and increasing, while its derivative $\\psi ^ { \\prime } ( x )$ is positive, convex, and decreasing. In addition, 0 $\\psi ^ { \\prime }$ can be approximated by the reciprocals: ",
+        "bbox": [
+            156,
+            117,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43598d4f5aab518fd3cbc576f85848eaad039f6a414b2615b48ed5e3a2340362.jpg",
+        "text": "$$\n{ \\frac { 1 } { x } } + { \\frac { 1 } { 2 x ^ { 2 } } } < \\psi ^ { \\prime } ( x ) < { \\frac { 1 } { x } } + { \\frac { 1 } { x ^ { 2 } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
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+            194
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which implies that 161 $\\textstyle \\psi ^ { \\prime } ( x ) \\approx { \\frac { 1 } { x ^ { 2 } } }$ as $x \\to 0$ and $\\begin{array} { r } { \\psi ^ { \\prime } ( x ) \\approx \\frac { 1 } { x } } \\end{array}$ as $x \\to \\infty$ ",
+        "bbox": [
+            148,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Main privacy results ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            229,
+            379,
+            247
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Truncated concentrated differential privacy ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            258,
+            517,
+            275
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ and $\\alpha _ { m } : = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ . Let $\\gamma \\in ( 0 , \\alpha _ { m } )$ . Let $\\Delta _ { 2 } , \\Delta _ { \\infty } > 0$ be constants that satisfy $\\begin{array} { r } { \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\leq \\Delta _ { 2 } ^ { 2 } } \\end{array}$ and $\\mathrm { m a x } _ { i } \\left| x _ { i } - x _ { i } ^ { \\prime } \\right| \\leq \\Delta _ { \\infty }$ whenever x, $\\mathbf { \\Delta } , \\mathbf { x } ^ { \\prime } \\in \\mathbb { R } _ { \\geq 0 } ^ { 2 }$ are sample statistics of any two datasets differing on a single entry. The one-time sampling from Dirichlet $\\left( \\mathbf { x } + \\alpha \\right)$ is $( \\rho , \\omega )$ -tCDP, where $\\begin{array} { r } { \\omega = \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 } \\end{array}$ and ",
+        "bbox": [
+            171,
+            285,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c071f26e5c9b105ed606cd08d89805f7dc5069443a3294706e91b05f41c9b8e.jpg",
+        "text": "$$\n\\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "168 Note that $( \\rho , \\infty )$ -tCDP is not obtainable, as the ratio between two Dirichlet densities blows up as   \n169 $\\omega  \\infty$ . We present here a short proof that skips some calculations (see Appendix 1 for a full proof). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "proof. Consider any 170 $\\begin{array} { r } { \\lambda \\in \\left( 1 , \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\right) } \\end{array}$ . Let $\\mathbf { u } : = \\mathbf { x } + \\pmb { \\alpha }$ and $\\mathbf { u } ^ { \\prime } : = \\mathbf { x } ^ { \\prime } + \\alpha ^ { \\prime }$ . Let $P ( \\mathbf { y } )$ be the density of Dirichlet171 $\\mathbf { \\Pi } ( \\mathbf { u } )$ and $P ^ { \\prime } ( \\mathbf { y } )$ be the density of Dirichlet $\\mathbf { \\Pi } ^ { ( \\mathbf { u } ^ { \\prime } ) }$ . A quick calculation shows that: ",
+        "bbox": [
+            142,
+            433,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d51b1105be5d627508b18a044d72525a2b62b72c1b7e105ea02b3abafae54e6.jpg",
+        "text": "$$\n\\mathbb { E } _ { { \\mathbf { y } } \\sim P ( { \\mathbf { y } } ) } \\left[ \\frac { P ( { \\mathbf { y } } ) ^ { \\lambda - 1 } } { P ^ { \\prime } ( { \\mathbf { y } } ) ^ { \\lambda - 1 } } \\right] = \\frac { B ( { \\mathbf { u } } ^ { \\prime } ) ^ { \\lambda - 1 } } { B ( { \\mathbf { u } } ) ^ { \\lambda - 1 } } \\cdot \\frac { B ( { \\mathbf { u } } + ( \\lambda - 1 ) ( { \\mathbf { u } } - { \\mathbf { u } } ^ { \\prime } ) ) } { B ( { \\mathbf { u } } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "172 We take the logarithm on both sides and apply the second-order Taylor expansion to the following   \n173 $G ( u _ { i } , u _ { i } ^ { \\prime } )$ and $H ( u _ { i } , u _ { i } ^ { \\prime } )$ terms that appear on the right-hand side. As a result, there exist $\\xi$ between   \n174 $u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } )$ and $u _ { i }$ , and $\\xi ^ { \\prime }$ between $u _ { i }$ and $u _ { i } ^ { \\prime }$ such that ",
+        "bbox": [
+            143,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eda561c84699fa451ed7b193c850b548da587691345bf8f7becc94c8820d1cfd.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { G ( u _ { i } , u _ { i } ^ { \\prime } ) : = ( \\lambda - 1 ) ( \\log \\Gamma ( u _ { i } ^ { \\prime } ) - \\log \\Gamma ( u _ { i } ) ) } } \\\\ & { } & \\\\ & { } & { = - ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) \\psi ( u _ { i } ) + \\frac { 1 } { 2 } ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\xi ^ { \\prime } ) } \\\\ & { } & { H ( u _ { i } , u _ { i } ^ { \\prime } ) : = \\log \\Gamma ( u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } ) ) - \\log \\Gamma ( u _ { i } ) } \\\\ & { } & { = ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) \\psi ( u _ { i } ) + \\frac { 1 } { 2 } ( \\lambda - 1 ) ^ { 2 } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\xi ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            556,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that 175 $\\psi ^ { \\prime }$ is increasing. If $x _ { i } > x _ { i } ^ { \\prime }$ , then $\\xi$ and $\\xi ^ { \\prime }$ are bounded below by $u _ { i } ^ { \\prime } \\geq \\alpha _ { m }$ . On the other hand, if 176 $x _ { i } \\leq x _ { i } ^ { \\prime }$ , then $\\xi$ and $\\xi ^ { \\prime }$ are bounded below by $u _ { i } - ( \\lambda - 1 ) | u _ { i } - u _ { i } ^ { \\prime } |$ . The condition 177 $\\begin{array} { r } { \\lambda < \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 } \\end{array}$ guarantees that $u _ { i } - ( \\lambda - 1 ) | u _ { i } - u _ { i } ^ { \\prime } | > \\alpha _ { m } - \\gamma$ . All cases considered, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/924a0a8d6141a741238a6777371a02cd2f841049f9963563d324459104d00d79.jpg",
+        "text": "$$\n\\begin{array} { l } { G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) \\leq \\displaystyle \\frac { 1 } { 2 } \\big ( ( \\lambda - 1 ) + ( \\lambda - 1 ) ^ { 2 } \\big ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\lambda ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Denoting 178 $u _ { 0 } : = \\textstyle \\sum _ { i } u _ { i }$ and $u _ { 0 } ^ { \\prime } : = \\textstyle \\sum _ { i } u _ { i } ^ { \\prime }$ , the same argument shows that $G ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) + H ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) > 0$ . 179 Therefore, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdd25e6e1558eba61028d51c44db09f492a18b03b4ee8237fc3befaaf821011a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { D _ { \\boldsymbol { \\lambda } } ( P ( \\mathbf { y } ) \\| P ^ { \\prime } ( \\mathbf { y } ) ) = \\displaystyle \\frac { 1 } { \\boldsymbol { \\lambda } - 1 } \\Biggl ( \\sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) ) - G ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) - H ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) \\Biggr ) } \\\\ & { \\quad \\quad < \\displaystyle \\frac { 1 } { \\boldsymbol { \\lambda } - 1 } \\sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) ) } \\\\ & { \\quad \\quad \\le \\displaystyle \\frac { 1 } { 2 } \\boldsymbol { \\lambda } \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) \\le \\frac { 1 } { 2 } \\boldsymbol { \\lambda } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b723c7be81bab0fe812229760033b22531dcf9d6fe77c60ac3901a2063deaff.jpg",
+        "image_caption": [
+            "Figure 1: Left: the actual values of $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\operatorname { D } _ { 2 } ( P \\| P ^ { \\prime } ) } \\end{array}$ and the worst case $( \\rho , 2 )$ -tCDP guarantees (2) at $\\Delta _ { 2 } ^ { 2 } = \\Delta _ { \\infty } = 1$ . Here, $P$ and $P ^ { \\prime }$ are Dirichlet posterior densities over $\\mathbf { x } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 0 )$ , $\\mathbf { x } ^ { \\prime } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 1 )$ , and $\\pmb { \\alpha } = ( \\alpha , \\ldots , \\alpha )$ . Right: comparison between $( \\varepsilon , \\delta )$ -DP guarantees of the Dirichlet posterior samplings (8) with different uniform priors: $\\pmb { \\alpha } = ( \\alpha , \\ldots , \\alpha )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "180 The guaranteed upper bound (2) is independent of the sample statistics. As a result, the bound applies   \n181 even in worst settings i.e., when $x _ { i } = 0$ and $x _ { i } ^ { \\prime } = \\Delta _ { \\infty }$ , or vice versa, for some $i$ . As we can see in   \n182 Figure 1, the upper bound is a close approximation to the actual value of $\\rho$ when $x _ { 6 } = 0$ and $x _ { 6 } ^ { \\prime } = 1$ .   \n183 However, being a sample independent bound, the difference becomes substantial when all $x _ { i }$ ’s are   \n184 large. There is one way to get around this issue: if there is no privacy violation in assuming that   \n185 the sample statistics are always bounded below by some threshold $\\tau$ , then we can incorporate the   \n186 threshold into the prior (thus $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ in (2) is replaced by $\\psi ^ { \\prime } ( \\alpha _ { m } + \\tau - \\gamma ) )$ .   \n187 The parameter $\\gamma$ allows us to adjust the moment bound $\\omega$ as desired. Even though a higher $\\omega$ usually   \n188 leads to a better privacy guarantee, there are two downsides to picking $\\gamma$ close to $\\alpha _ { m }$ in this case.   \n189 First, note that $\\rho$ contains $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ ; as $\\gamma \\to \\alpha _ { m }$ , the value of $\\rho$ diverges to $\\infty$ , leading to a weaker   \n190 privacy guarantee instead. Second, as the Taylor approximation (5) is accurate when $u _ { i }$ is close to   \n191 $u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } )$ , having a large value of $\\lambda$ would push the guaranteed upper bound away from   \n192 the actual privacy loss. Thus it is recommended to pick $\\gamma$ so that $\\gamma / \\Delta _ { \\infty } \\geq 1$ and $\\alpha _ { m } - \\gamma \\gg 0$ .   \n193 Alternatively, we can choose the value of $\\gamma$ that minimizes $\\varepsilon$ when converting from tCDP to $( \\varepsilon , \\delta )$ -DP   \n194 using Lemma 1—this method will be explored in the next subsection. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 can be easily applied to sampling from Dirichlet $( r \\mathbf { x } + \\alpha )$ . Replacing $\\mathbf { x }$ with $r \\mathbf { x }$ , we have $\\Delta _ { 2 }$ replaced by $r \\Delta _ { 2 }$ and $\\Delta _ { \\infty }$ replaced by $r \\Delta _ { \\infty }$ . Consequently, the sampling is $\\begin{array} { r } { \\left( \\rho , \\frac { \\gamma } { r \\Delta _ { \\infty } } + 1 \\right) } \\end{array}$ -tCDP, where $\\rho = { \\textstyle { \\frac { 1 } { 2 } } r ^ { 2 } \\Delta _ { 2 } ^ { 2 } } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ . In Appendix 4, we analyze the scaling of $r$ in conjunction with $\\alpha _ { m }$ at a fixed privacy budget $\\rho$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Approximate differential privacy ",
+        "text_level": 1,
+        "bbox": [
+            169,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "200 We now convert the tCDP guarantee to an approximate DP guarantee. Let $\\delta \\in ( 0 , 1 )$ . Using Lemma 1,   \n201 the Dirichlet posterior sampling with Dirichlet $( \\alpha )$ as the prior is $( \\varepsilon , \\delta )$ -DP with ",
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+    {
+        "type": "equation",
+        "img_path": "images/6298ff4207b3ce4534b84df6d385b7bcce1b93402f660e9b6ae174c7d896ac30.jpg",
+        "text": "$$\n\\varepsilon = \\left\\{ \\begin{array} { l l } { \\rho ( \\gamma ) + 2 \\sqrt { \\rho ( \\gamma ) \\log ( 1 / \\delta ) } } & { \\mathrm { i f ~ } \\log ( 1 / \\delta ) \\leq \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\\\ { \\rho ( \\gamma ) \\Big ( \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\Big ) + \\frac { \\log ( 1 / \\delta ) \\Delta _ { \\infty } } { \\gamma } } & { \\mathrm { i f ~ } \\log ( 1 / \\delta ) > \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array} , \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where 202 $\\begin{array} { r } { \\rho ( \\gamma ) = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) } \\end{array}$ ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "203 We try to minimize $\\epsilon$ by adjusting the value of $\\gamma$ . First, we consider the case $\\log ( 1 / \\delta ) \\leq \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 }$   \n204 Since $\\rho ( \\gamma )$ is a strictly increasing function of $\\gamma$ , both $\\rho ( \\gamma ) + 2 \\sqrt { \\rho ( \\gamma ) \\log ( 1 / \\delta ) }$ and $\\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 }$   \n205 are both strictly increasing function of $\\gamma$ . Therefore, $\\varepsilon$ is minimized at the minimum possible   \n206 value of $\\gamma$ in this case, that is, at the unique $\\gamma _ { M }$ that satisfies $\\log ( 1 / \\delta ) = \\gamma _ { M } ^ { 2 } \\rho ( \\gamma _ { M } ) / \\Delta _ { \\infty } ^ { 2 } =$   \n207 $\\begin{array} { r } { \\frac { 1 } { 2 } \\gamma _ { M } ^ { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma _ { M } ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array}$ .   \n208 Now we consider the second case, when $\\gamma < \\gamma _ { M }$ . As $\\rho ( \\gamma )$ is an increasing positive convex function   \n209 of $\\gamma$ , the function ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcd8f2f31dac6a5156ad364cfe9c4a90ce60b51aeb6769e198459e174b0e14bb.jpg",
+        "text": "$$\nf ( \\gamma ) : = \\frac 1 2 \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) \\left( \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\right) + \\frac { \\log ( 1 / \\delta ) \\Delta _ { \\infty } } { \\gamma } ; \\qquad \\gamma \\in ( 0 , \\gamma _ { M } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            121,
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+            156
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "is also convex in $\\gamma$ , and thus has a unique minimizer $\\underline { { \\gamma _ { m } } } \\in ( 0 , \\gamma _ { M } ]$ . Comparing to the first case, we have $f ( \\gamma _ { m } ) \\leq f ( \\gamma _ { M } ) = \\rho ( \\gamma _ { M } ) + 2 \\sqrt { \\rho ( \\gamma _ { M } ) \\log ( 1 / \\delta ) } .$ We then conclude that $\\varepsilon = f ( \\gamma _ { m } )$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "212 Theorem 2. Let $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { 2 }$ and denote $\\alpha _ { m } = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ . Let $\\Delta _ { 2 } , \\Delta _ { \\infty } > 0$ be constants that satisfy   \n213 $\\begin{array} { r } { \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\leq \\Delta _ { 2 } ^ { 2 } } \\end{array}$ and $\\operatorname* { m a x } _ { i } \\left| x _ { i } - x _ { i } ^ { \\prime } \\right| \\leq \\Delta _ { \\infty }$ whenever x, $\\mathbf { x } ^ { \\prime } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ are sample statistics of any   \n214 two datasets differing on a single entry. For any $\\delta \\in ( 0 , 1 )$ , let $\\gamma _ { M }$ be the solution to the equation   \n215 $\\begin{array} { r } { \\log ( 1 / \\delta ) = \\frac { 1 } { 2 } \\gamma ^ { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array}$ . The one-time sampling from Dirichlet $\\left( \\mathbf { x } + \\alpha \\right)$ is $( \\varepsilon , \\delta ) – D P$   \n216 where ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5347fdf65d8f5bd7b8451bda92c6cba6c425f16d7d5ff29dc3ed4b319a7a3d60.jpg",
+        "text": "$$\n\\varepsilon = \\operatorname* { m i n } _ { \\gamma \\in ( 0 , \\gamma _ { M } ] } f ( \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "17 Figure 1 shows how $\\delta$ decays as a function of $\\varepsilon$ at three different values of $\\alpha _ { m }$ ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "4 Utility ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            328,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using the results from the previous section, we analyze the Dirichlet posterior sampling’s utility in two specific tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Multinomial-Dirichlet sampling ",
+        "text_level": 1,
+        "bbox": [
+            166,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Suppose that we are observing $N$ trials, each of which has $d$ possible outcomes. For each $i \\in$ $\\{ 1 , \\ldots , d \\}$ , let $x _ { i }$ be the number of times the $i$ -th outcome was observed. Then we have the multinomial likelihood $\\begin{array} { r } { p ( \\mathbf { x } | \\mathbf { y } ) \\propto \\prod _ { i } y _ { i } ^ { x _ { i } } } \\end{array}$ . From this, we sample from the Dirichlet posterior: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e9407c845bb1087a07c2694a9563c1955bd334025bf84648bde6447d9bc518b.jpg",
+        "text": "$$\n\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { x } + \\pmb { \\alpha } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "225 Suppose that we want to sample from a true distribution $P _ { \\mathbf { X } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { x } + { \\pmb { \\alpha } } )$ , but for privacy   \n226 reasons, we instead sample from $Q _ { \\mathbf { x } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { x } + \\pmb { \\alpha } ^ { \\prime } )$ where $\\alpha _ { i } ^ { \\prime } > \\alpha _ { i }$ for all $i$ . The utility of the   \n227 privacy scheme is then measured by the KL-divergence between $P _ { \\mathbf { x } }$ and $Q _ { \\mathbf { x } }$ . Assuming that $\\mathbf { x }$ is an   \n228 observation of Multinomial $\\mathbf { \\tau } ( \\mathbf { p } )$ , the following Theorem tells us that, on average, the KL-divergence   \n229 is small when the sample size is large, and the $p _ { i }$ ’s are evenly distributed. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Let 30 $\\mathbf { p } : = \\left( p _ { 1 } , \\ldots , p _ { d } \\right)$ where $p _ { i } > 0$ for all $i$ and $\\textstyle \\sum _ { i } p _ { i } \\ = \\ 1$ . Define a random variable 31 $\\mathbf { X } \\sim$ Multinomial $\\mathbf { \\tau } ( \\mathbf { p } )$ . Let $P _ { \\mathbf { X } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { X } + { \\boldsymbol { \\alpha } } )$ and $Q _ { \\mathbf { X } } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { X } + \\mathbf { \\alpha } \\mathbf { \\alpha } ^ { \\prime } )$ where 32 $\\alpha _ { i } ^ { \\prime } \\geq \\alpha _ { i } \\geq 1$ for all $i$ . The following estimate holds: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcc77176e1fb57707beeb51e095d0ecb850dd9043b7f7c0e2b40ea53a17d8b2f.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { X } } [ \\mathrm { D } _ { \\mathrm { K L } } ( P _ { \\mathbf { X } } \\| Q _ { \\mathbf { X } } ) ] \\leq \\frac { 1 } { N + 1 } \\sum _ { i } ( \\alpha _ { i } ^ { \\prime } - \\alpha _ { i } ) ^ { 2 } \\cdot \\frac { 1 } { p _ { i } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "233 The proof is given in Appendix 2. Let us consider a simple privacy scheme where we fix $s > 0$ and let 234 $\\alpha _ { i } ^ { \\prime } = \\alpha _ { i } + s$ for all $i$ . Thus (10) becomes: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c990b3fed25d5be5f403f085524958192030cc19922bc62f624f795a50d9cc4e.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { X } } [ \\mathrm { D } _ { \\mathrm { K L } } ( P _ { \\mathbf { X } } \\| Q _ { \\mathbf { X } } ) ] \\leq \\frac { G ( \\mathbf { p } ) s ^ { 2 } } { N + 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 235 $G ( \\mathbf { p } ) : = \\textstyle \\sum _ { i } 1 / p _ { i }$ . Now we take into account the privacy parameters. Let $\\rho = \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ and 236 $\\rho ^ { \\prime } = \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ , where $\\alpha _ { m } = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ , $\\alpha _ { m } ^ { \\prime } = \\operatorname* { m i n } _ { i } \\alpha _ { i } ^ { \\prime }$ , and $\\gamma < \\alpha _ { m }$ . Here, we approximate the values of 237 $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ and $\\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ under two regimes: ",
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+    {
+        "type": "text",
+        "text": "High-privacy regime: 238 $\\alpha _ { m } ^ { \\prime } \\mathrm { ~ - ~ } \\gamma \\mathrm { ~ > ~ } 1$ . We have $\\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma ) \\approx 1 / ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ , which implies 239 $\\alpha _ { m } ^ { \\prime } - \\gamma \\approx \\Delta _ { 2 } ^ { 2 } / \\rho ^ { \\prime }$ . We also have $\\alpha _ { m } - \\gamma \\approx \\Delta _ { 2 } ^ { 2 } / \\rho$ for $\\alpha _ { m } - \\gamma \\geq 1$ and $\\alpha _ { m } - \\gamma > ( \\alpha _ { m } - \\gamma ) ^ { 2 } \\approx \\Delta _ { 2 } ^ { 2 } / \\rho$ 240 for $\\alpha - \\gamma < 1$ . Thus we have the following bound for the right-hand side of (11): ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43f3c38a4e99d254010fca3b14c8d659b5a9d81394f89e4fcf805698b459a86b.jpg",
+        "text": "$$\n\\frac { G ( { \\bf p } ) s ^ { 2 } } { N + 1 } = \\frac { G ( { \\bf p } ) ( \\alpha _ { m } ^ { \\prime } - \\alpha _ { m } ) ^ { 2 } } { N + 1 } \\lesssim \\frac { \\Delta _ { 2 } ^ { 4 } G ( { \\bf p } ) } { N + 1 } \\biggl ( \\frac { 1 } { \\rho ^ { \\prime } } - \\frac { 1 } { \\rho } \\biggr ) ^ { 2 } < \\frac { \\Delta _ { 2 } ^ { 4 } G ( { \\bf p } ) } { \\rho ^ { \\prime 2 } ( N + 1 ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Consequently, we have 241 $\\mathrm { D } _ { \\mathrm { K L } } ( P \\| Q ) < \\epsilon$ for $\\begin{array} { r } { N = \\Omega \\left( \\frac { \\Delta _ { 2 } ^ { 4 } G ( \\mathbf { p } ) } { \\rho ^ { \\prime 2 } \\epsilon } \\right) } \\end{array}$ ",
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "242 Low-privacy regime: $1 > \\alpha _ { m } ^ { \\prime } - \\gamma > 0$ . This is similar as above, except we have $\\alpha _ { m } ^ { \\prime } - \\gamma \\approx$   \n243 $\\Delta _ { 2 } / \\rho ^ { \\prime 1 / 2 }$ and $\\alpha _ { m } - \\gamma \\approx \\Delta _ { 2 } / \\rho ^ { 1 / 2 }$ . Similar computation as (12) shows that $\\mathrm { D } _ { \\mathrm { K L } } ( P \\| Q ) < \\epsilon$ when   \n244 $\\begin{array} { r } { N = \\Omega \\left( \\frac { \\Delta _ { 2 } ^ { 2 } G ( \\mathbf { p } ) } { \\rho ^ { \\prime } \\epsilon } \\right) } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We observe that, in both regimes, the sample size scales faster with respect to $\\epsilon$ with a higher value of $G ( \\mathbf { p } )$ , which is associated with a higher number of outcomes $d$ , and more concentrated multinomial parameter $\\mathbf { p }$ ; this agrees with the result of our simulation in Appendix 3. Moreover, for small $\\rho ^ { \\prime }$ the sample size scales as $1 / \\rho ^ { \\prime 2 }$ , while for large $\\rho ^ { \\prime }$ the sample size scales as $1 / \\rho ^ { \\prime }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Private normalized histograms ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "250 Let $\\mathbf { x } = ( x _ { 1 } , \\ldots , x _ { d } )$ be a histogram of $N$ observations and $\\mathbf { p } : = \\mathbf { x } / N$ . We can privatize $\\mathbf { p }$ by   \n251 sampling a probability vector: $\\mathbf Y \\sim \\mathrm { D i r i c h l e t } ( { \\mathbf x } + { \\boldsymbol \\alpha } )$ . Note that $\\mathbf { Y }$ is a biased estimator of $\\mathbf { p }$ .   \n252 Denoting $\\alpha _ { 0 } : = \\textstyle \\sum _ { i } \\alpha _ { i }$ , the bias of each component of $\\mathbf { Y }$ is given by $\\mathbb { E } [ \\mathbf { Y } ] - p _ { i }$ . Hence, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/acfa7af0b43f43c6c250eb27ee7cd6953e76746f9898d49a1224abbc50b2e3f6.jpg",
+        "text": "$$\n| \\mathrm { B i a s } ( Y _ { i } ) | = \\left| \\frac { x _ { i } + \\alpha _ { i } } { N + \\alpha _ { 0 } } - p _ { i } \\right| = \\frac { | x _ { i } \\alpha _ { 0 } - N \\alpha _ { i } | } { N ( N + \\alpha _ { 0 } ) } \\leq \\frac { N \\alpha _ { 0 } } { N ( N + \\alpha _ { 0 } ) } = \\frac { \\alpha _ { 0 } } { N + \\alpha _ { 0 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            300,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since 253 $Y _ { i } \\sim \\mathrm { B e t a } ( x _ { i } + \\alpha _ { i } , N + \\alpha _ { 0 } - x _ { i } - \\alpha _ { i } )$ is $\\frac { 1 } { 4 ( N + \\alpha _ { 0 } + 1 ) }$ -sub-Gaussian [MA17], we have, ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/abf09d4f3a689aaa55b0df6994813c7534a229d387d704d8a3da2faeda62cfbe.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } [ | Y _ { i } - p _ { i } | > t + | \\mathrm { B i a s } ( Y _ { i } ) | ] \\le \\mathbb { P } [ | Y _ { i } - \\mathbb { E } [ Y _ { i } ] | + | \\mathrm { B i a s } ( Y _ { i } ) | > t + | \\mathrm { B i a s } ( Y _ { i } ) | ] } \\\\ & { \\qquad = \\mathbb { P } [ | Y _ { i } - \\mathbb { E } [ Y _ { i } ] | > t ] } \\\\ & { \\qquad \\le 2 e ^ { - 2 t ^ { 2 } ( N + \\alpha _ { 0 } + 1 ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "With the union bound, we plug in 254 $\\begin{array} { r } { t = \\sqrt { \\frac { \\log ( 2 d / \\beta ) } { 2 ( N + \\alpha _ { 0 } + 1 ) } } } \\end{array}$ , for any $\\beta \\in ( 0 , 1 )$ , to obtain the following 255 accuracy guarantee of the private normalized histogram: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "256 Theorem 4. Let $\\mathbf Y \\sim \\mathrm { D i r i c h l e t } ( { \\mathbf x } + { \\boldsymbol \\alpha } )$ , where $\\mathbf { x } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ and $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ , and $\\mathbf { p } : = \\mathbf { x } / N$ . For any   \n257 $\\beta \\in ( 0 , 1 )$ , with probability at least $1 - \\beta$ , the following inequality holds: ",
+        "bbox": [
+            145,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0d54e562ae47567af0e5d3fafa75d73322673a6540b4c7b9479beacf8ae913f.jpg",
+        "text": "$$\n\\| \\mathbf { Y } - \\mathbf { p } \\| _ { \\infty } \\leq \\sqrt { \\frac { \\log ( 2 d / \\beta ) } { 2 ( N + \\alpha _ { 0 } + 1 ) } } + \\frac { \\alpha _ { 0 } } { N + \\alpha _ { 0 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "258 Given $\\epsilon > 0$ , we use (13) to find a lower bound for $N$ that gives $\\| \\mathbf { Y } - \\mathbf { p } \\| _ { \\infty } < \\epsilon$ w.p. $1 - \\beta$ when   \n259 $\\mathbf { Y }$ is sampled with $\\rho$ -tCDP. For simplicity, we consider a uniform prior: $\\alpha _ { i } = \\alpha > 0$ for all $i$ .   \n260 Thus, $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha - \\gamma ) } \\end{array}$ , where $\\gamma$ might be chosen according to Corollary 2. We consider the two   \n261 following regimes: ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "High-privacy regime: $\\alpha - \\gamma > 1$ . In this case, $\\psi ^ { \\prime } ( \\alpha - \\gamma ) \\approx 1 / ( \\alpha - \\gamma )$ . From $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha - \\gamma ) } \\end{array}$ we have $\\alpha \\approx \\Delta _ { 2 } ^ { 2 } / 2 \\rho + \\gamma$ . Replacing $\\alpha _ { 0 }$ by $d \\alpha$ in (13) yields the sample size: ",
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b2e0f2a9c4ecb55ff40d739b01c28e86d4c732b58831a487349c9317886cd59.jpg",
+        "text": "$$\nN = \\Omega \\biggl ( \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d } { \\epsilon } \\biggl ( \\frac { \\Delta _ { 2 } ^ { 2 } } { 2 \\rho } + \\gamma \\biggr ) \\biggr ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "264 for the desired accuracy. ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "265 Low-privacy regime: $\\alpha - \\gamma < 1$ . This is the same as above, except now we have $\\psi ^ { \\prime } ( \\alpha - \\gamma ) \\approx$   \n266 $1 / ( \\alpha - \\gamma ) ^ { 2 }$ , which implies $\\alpha \\approx \\Delta _ { 2 } / ( 2 \\rho ) ^ { 1 / 2 } + \\gamma$ . The sample size that guarantees the desired   \n267 accuracy is: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7142b5d9c6b72b545f886db6868236a551d39d6dbd61e35e32d88d5fc2393253.jpg",
+        "text": "$$\nN = \\Omega \\biggl ( \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d } { \\epsilon } \\biggl ( \\frac { \\Delta _ { 2 } } { \\sqrt { 2 \\rho } } + \\gamma \\biggr ) \\biggr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "268 Let us compare this result to the Gaussian mechanism, which adds a noise $\\mathbf { Z } \\sim N ( 0 , \\sigma ^ { 2 } I _ { d } )$ to the   \n269 normalized histogram $\\mathbf { p }$ directly. Thus the $\\ell _ { 2 }$ -sensitivity in this case is $\\Delta _ { 2 } / N$ . We have that the   \n270 Gaussian mechanism is $\\rho$ -zCDP where $\\begin{array} { r } { \\rho = \\frac { \\Delta ^ { 2 } } { 2 N ^ { 2 } \\sigma ^ { 2 } } } \\end{array}$ [BS16]. Using the same argument as above, with   \n271 probability at least $1 - \\beta$ , the following inequality holds for all $i$ : ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02c4415ff1cb154a29b826fadec2a5e08c020ede9ac6400f3cb8e56b453a6080.jpg",
+        "text": "$$\n\\| \\mathbf { Z } \\| _ { \\infty } \\leq \\sqrt { \\frac { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } } { N ^ { 2 } \\rho } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/4d398e42217bd393cee6a912cf2c04e26395de3300d829c872c8f73d8e70fe67.jpg",
+        "image_caption": [
+            "Figure 2: The $\\ell ^ { \\infty }$ -accuracy, as a function of $N$ , of Dirichlet posterior sampling $( \\gamma = 1 )$ ) and Gaussian mechanisms for private normalized histograms $\\Delta _ { 2 } ^ { 2 } = 2$ and $\\Delta _ { \\infty } = 1 $ ). For each $N , d$ and $\\rho$ , we generated the inputs $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { 2 0 0 }$ , where $\\mathbf { x } _ { k } \\sim$ Multinomial $\\left( \\mathbf { q } _ { k } \\right)$ and $\\mathbf { q } _ { k } \\sim \\mathrm { D i r i c h l e t } ( 5 , \\ldots , 5 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hence, the sample size of 272 $N = \\Omega \\Big ( \\sqrt { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } / \\rho \\epsilon ^ { 2 } } \\Big )$ guarantees the desired accuracy. Compar273 ing this to (14), if we assume $\\epsilon < 1$ , the AM-GM inequality tells us that ",
+        "bbox": [
+            138,
+            426,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71774da6e8c49643e0365d0aff1d38983a994504e1f2b72a5cccd376a2a41f0c.jpg",
+        "text": "$$\n\\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d \\Delta _ { 2 } ^ { 2 } } { \\rho \\epsilon } > \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { \\Delta _ { 2 } ^ { 2 } } { \\rho } \\geq 2 \\sqrt { \\frac { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } } { \\rho \\epsilon ^ { 2 } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            468,
+            705,
+            511
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "274 The inequality (17) implies that the Gaussian mechanism requires less sample than the Dirichlet   \n275 mechanism in order to guarantee the same level of accuracy. The Gaussian mechanism is also better   \n276 in the low-privacy regime as the $\\rho$ in (15) satisfies ${ \\sqrt { \\rho } } < \\rho$ and $\\Delta _ { 2 } \\approx \\Delta _ { 2 } ^ { 2 }$ , leading to the same   \n277 inequality (17). Nonetheless, the decay in (16) is linear in $d$ , while that in (13) has $\\alpha _ { 0 } = d \\alpha$ in   \n278 the denominators. This observation suggests that, when $\\mathbf { x }$ is a sparse histogram i.e. when $N \\leq d$ ,   \n279 the $\\ell ^ { \\infty }$ -accuracy of the Dirichlet mechanism is smaller than that of the Gaussian mechanism. This   \n280 conclusion is supported by our simulation in Figure 2. We see that the $\\ell ^ { \\infty }$ -accuracy of the Dirichlet   \n281 mechanism is smaller than that of the Gaussian mechanism for small $N$ when $d = 1 0 0 0$ . The code   \n282 for all experiments in this study can be found in the supplemental material. ",
+        "bbox": [
+            138,
+            515,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "283 Potential negative societal impacts ",
+        "text_level": 1,
+        "bbox": [
+            140,
+            670,
+            464,
+            688
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is important to note that, when $\\rho$ becomes unacceptably large (e.g., $\\rho = 1 0 ^ { 4 }$ ), the sampling is far away from being private. Thus any organization that deploys the posterior sampling on sensitive data must not vacuously refer to this study and claim that its algorithm is private. It is the organization’s responsibility to fully publish the prior parameters, and educate its users/customers on differential privacy and how the privacy guarantees are calculated. ",
+        "bbox": [
+            166,
+            700,
+            825,
+            777
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is desirable that differentially private algorithms are accurate for the task at hand, especially when the data is used for important decision-making. Thus, one needs to make sure that there is enough sample to achieve the desired level of accuracy. For a large differentially private system, privacy budgets need to be allocated to the parts that require accurate outputs. ",
+        "bbox": [
+            171,
+            784,
+            823,
+            844
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "293 Lastly, one must be careful with the choice of prior parameters; if a uniform prior is used, smaller   \n294 groups will suffer a relatively larger statistical bias. As a result, private statistics of small populations   \n295 (such as ethnic or racial minorities) will be relatively less accurate. One way to get around this issue   \n296 is to (privately) impose larger prior parameters on larger populations.   \n297 References   \n298 [AAFK20] I. Aykin, B. Akgun, M. Feng, and M. Krunz. “MAMBA: A Multi-armed Bandit Framework for   \n299 Beam Tracking in Millimeter-wave Systems”. In: 39th IEEE Conference on Computer Com  \n300 munications, INFOCOM 2020, Toronto, ON, Canada, July 6-9, 2020. IEEE, 2020, pp. 1469–   \n301 1478.   \n302 [AV08] J. M. Abowd and L. Vilhuber. “How Protective Are Synthetic Data?” In: Privacy in Statistical   \n303 Databases, UNESCO Chair in Data Privacy International Conference, PSD 2008, Istan  \n304 bul, Turkey, September 24-26, 2008. Proceedings. Ed. by J. Domingo-Ferrer and Y. Saygin.   \n305 Vol. 5262. Lecture Notes in Computer Science. Springer, 2008, pp. 239–246.   \n306 [BDRS18] M. Bun, C. Dwork, G. N. Rothblum, and T. Steinke. “Composable and versatile privacy via   \n307 truncated CDP”. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of   \n308 Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018. Ed. by I. Diakonikolas,   \n309 D. Kempe, and M. Henzinger. ACM, 2018, pp. 74–86.   \n310 [BHW00] R. J. Boys, D. A. Henderson, and D. J. Wilkinson. “Detecting Homogeneous Segments in DNA   \n311 Sequences by Using Hidden Markov Models”. In: Journal of the Royal Statistical Society.   \n312 Series $C$ (Applied Statistics) 49.2 (2000), pp. 269–285. ISSN: 00359254, 14679876.   \n313 [BS16] M. Bun and T. Steinke. “Concentrated Differential Privacy: Simplifications, Extensions, and   \n314 Lower Bounds”. In: Theory of Cryptography. Ed. by M. Hirt and A. Smith. Berlin, Heidelberg:   \n315 Springer Berlin Heidelberg, 2016, pp. 635–658.   \n316 [BS18] G. Bernstein and D. R. Sheldon. “Differentially Private Bayesian Inference for Exponential   \n317 Families”. In: Advances in Neural Information Processing Systems 31: Annual Conference on   \n318 Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal,   \n319 Canada. Ed. by S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and   \n320 R. Garnett. 2018, pp. 2924–2934.   \n321 [CWS03] J. Corander, P. Waldmann, and M. J. Sillanpää. “Bayesian Analysis of Genetic Differentiation   \n322 Between Populations”. In: Genetics 163.1 (Jan. 2003), pp. 367–374. ISSN: 1943-2631.   \n323 [de 06] M. de Lapparent. “Empirical Bayesian analysis of accident severity for motorcyclists in large   \n324 French urban areas”. In: Accident Analysis & Prevention 38.2 (2006), pp. 260–268. ISSN:   \n325 0001-4575.   \n326 [DMNS06] C. Dwork, F. Mcsherry, K. Nissim, and A. Smith. “Calibrating noise to sensitivity in private   \n327 data analysis”. In: TCC. 2006.   \n328 [DNZMR17] C. Dimitrakakis, B. Nelson, Z. Zhang, A. Mitrokotsa, and B. I. P. Rubinstein. “Differential   \n329 Privacy for Bayesian Inference through Posterior Sampling”. In: J. Mach. Learn. Res. 18   \n330 (2017), 11:1–11:39.   \n331 [FGWC16] J. R. Foulds, J. Geumlek, M. Welling, and K. Chaudhuri. “On the Theory and Practice of   \n332 Privacy-Preserving Bayesian Data Analysis”. In: Proceedings of the Thirty-Second Conference   \n333 on Uncertainty in Artificial Intelligence, UAI 2016, June 25-29, 2016, New York City, NY, USA.   \n334 Ed. by A. T. Ihler and D. Janzing. AUAI Press, 2016.   \n335 [GSC17] J. Geumlek, S. Song, and K. Chaudhuri. “Renyi Differential Privacy Mechanisms for Posterior   \n336 Sampling”. In: Advances in Neural Information Processing Systems 30: Annual Conference   \n337 on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA.   \n338 Ed. by I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N. Vishwanathan,   \n339 and R. Garnett. 2017, pp. 5289–5298.   \n340 [GWHHT21] P. Gohari, B. Wu, C. Hawkins, M. T. Hale, and U. Topcu. “Differential Privacy on the Unit   \n341 Simplex via the Dirichlet Mechanism”. In: IEEE Trans. Inf. Forensics Secur. 16 (2021),   \n342 pp. 2326–2340.   \n343 [Hin15] K. Hines. “A Primer on Bayesian Inference for Biophysical Systems”. In: Biophysical Journal   \n344 108.9 (2015), pp. 2103–2113. ISSN: 0006-3495.   \n345 [LW92] M. Lavine and M. West. “A Bayesian method for classification and discrimination”. In:   \n346 Canadian Journal of Statistics 20.4 (1992), pp. 451–461.   \n347 [MA17] O. Marchal and J. Arbel. “On the sub-Gaussianity of the Beta and Dirichlet distributions”. In:   \n348 Electronic Communications in Probability 22.none (2017), pp. 1 –14.   \n349 [MKAGV08] A. Machanavajjhala, D. Kifer, J. M. Abowd, J. Gehrke, and L. Vilhuber. “Privacy: Theory   \n350 meets Practice on the Map”. In: Proceedings of the 24th International Conference on Data   \n351 Engineering, ICDE 2008, April 7-12, 2008, Cancún, Mexico. Ed. by G. Alonso, J. A. Blakeley,   \n352 and A. L. P. Chen. IEEE Computer Society, 2008, pp. 277–286.   \n353 [MMR05] J.-M. Marin, K. Mengersen, and C. P. Robert. “Bayesian Modelling and Inference on Mixtures   \n354 of Distributions”. In: Bayesian Thinking. Ed. by D. Dey and C. Rao. Vol. 25. Handbook of   \n355 Statistics. Elsevier, 2005, pp. 459–507.   \n356 [MT07] F. McSherry and K. Talwar. “Mechanism Design via Differential Privacy”. In: 48th Annual   \n357 IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007,   \n358 Providence, RI, USA, Proceedings. IEEE Computer Society, 2007, pp. 94–103.   \n359 [NIK20] I. Nasim, A. S. Ibrahim, and S. Kim. “Learning-Based Beamforming for Multi-User Vehicular   \n360 Communications: A Combinatorial Multi-Armed Bandit Approach”. In: IEEE Access 8 (2020),   \n361 pp. 219891–219902.   \n362 [NLYCC13] V. C. Nguyen, W. S. Lee, N. Ye, K. M. A. Chai, and H. L. Chieu. “Active Learning for   \n363 Probabilistic Hypotheses Using the Maximum Gibbs Error Criterion”. In: Advances in Neural   \n364 Information Processing Systems 26: 27th Annual Conference on Neural Information Processing   \n365 Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada,   \n366 United States. Ed. by C. J. C. Burges, L. Bottou, Z. Ghahramani, and K. Q. Weinberger. 2013,   \n367 pp. 1457–1465.   \n368 [ORR13] I. Osband, D. Russo, and B. V. Roy. “(More) Efficient Reinforcement Learning via Poste  \n369 rior Sampling”. In: Advances in Neural Information Processing Systems 26: 27th Annual   \n370 Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held   \n371 December 5-8, 2013, Lake Tahoe, Nevada, United States. Ed. by C. J. C. Burges, L. Bottou,   \n372 Z. Ghahramani, and K. Q. Weinberger. 2013, pp. 3003–3011.   \n373 [PB98] T. Pedersen and R. F. Bruce. “Knowledge Lean Word-Sense Disambiguation”. In: Proceedings   \n374 of the Fifteenth National Conference on Artificial Intelligence and Tenth Innovative Appli  \n375 cations of Artificial Intelligence Conference, AAAI 98, IAAI 98, July 26-30, 1998, Madison,   \n376 Wisconsin, USA. Ed. by J. Mostow and C. Rich. AAAI Press / The MIT Press, 1998, pp. 800–   \n377 805.   \n378 [PG14] Y. Park and J. Ghosh. “PeGS: Perturbed Gibbs Samplers that Generate Privacy-Compliant   \n379 Synthetic Data”. In: Trans. Data Priv. 7.3 (2014), pp. 253–282.   \n380 [PM01] J. Pella and M. Masuda. “Bayesian methods for analysis of stock mixtures from genetic   \n381 characters”. English. In: Fishery Bulletin 99 (Jan. 2001). 1, p. 151. ISSN: 00900656.   \n382 [Rén61] A. Rényi. “On measures of entropy and information”. In: Proceedings of the Fourth Berkeley   \n383 Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory   \n384 of Statistics. The Regents of the University of California. 1961.   \n385 [RWZ14] J. P. Reiter, Q. Wang, and B. Zhang. “Bayesian Estimation of Disclosure Risks for Multiply   \n386 Imputed, Synthetic Data”. In: J. Priv. Confidentiality 6.1 (2014).   \n387 [SJGLY17] M. J. Schneider, S. Jagpal, S. Gupta, S. Li, and Y. Yu. “Protecting customer privacy when   \n388 marketing with second-party data”. In: International Journal of Research in Marketing 34.3   \n389 (2017), pp. 593–603. ISSN: 0167-8116.   \n390 [Str00] M. J. A. Strens. “A Bayesian Framework for Reinforcement Learning”. In: Proceedings of the   \n391 Seventeenth International Conference on Machine Learning (ICML 2000), Stanford University,   \n392 Stanford, CA, USA, June 29 - July 2, 2000. Ed. by P. Langley. Morgan Kaufmann, 2000,   \n393 pp. 943–950.   \n394 [WFS15] Y. Wang, S. E. Fienberg, and A. J. Smola. “Privacy for Free: Posterior Sampling and Stochastic   \n395 Gradient Monte Carlo”. In: Proceedings of the 32nd International Conference on Machine   \n396 Learning, ICML 2015, Lille, France, 6-11 July 2015. Ed. by F. R. Bach and D. M. Blei. Vol. 37.   \n397 JMLR Workshop and Conference Proceedings. JMLR.org, 2015, pp. 2493–2502.   \n398 [ZHGSY20] J. Zhu, X. Huang, X. Gao, Z. Shao, and Y. Yang. “Multi-Interface Channel Allocation in Fog   \n399 Computing Systems using Thompson Sampling”. In: 2020 IEEE International Conference on   \n400 Communications, ICC 2020, Dublin, Ireland, June 7-11, 2020. IEEE, 2020, pp. 1–6. ",
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parse/train/RzYrn625bu8/RzYrn625bu8.md

@@ -0,0 +1,312 @@
+# VATT: Transformers for Multimodal Self-Supervised Learning from Raw Video, Audio and Text
+
+Hassan Akbari∗ Columbia University ha2436@columbia.edu
+
+Liangzhe Yuan Google lzyuan@google.com
+
+Rui Qian∗ Cornell University rq49@cornell.edu
+
+Wei-Hong Chuang Google whchuang@google.com
+
+Shih-Fu Chang Columbia University sc250@columbia.edu
+
+Yin Cui Google yincui@google.com
+
+Boqing Gong   
+Google   
+bgong@google.com
+
+# Abstract
+
+We present a framework for learning multimodal representations from unlabeled data using convolution-free Transformer architectures. Specifically, our VideoAudio-Text Transformer (VATT) takes raw signals as inputs and extracts multimodal representations that are rich enough to benefit a variety of downstream tasks. We train VATT end-to-end from scratch using multimodal contrastive losses and evaluate its performance by the downstream tasks of video action recognition, audio event classification, image classification, and text-to-video retrieval. Furthermore, we study a modality-agnostic, single-backbone Transformer by sharing weights among the three modalities. We show that the convolution-free VATT outperforms state-of-the-art ConvNet-based architectures in the downstream tasks. Especially, VATT’s vision Transformer achieves the top-1 accuracy of $8 2 . 1 \%$ on Kinetics-400, $8 3 . 6 \%$ on Kinetics-600, $7 2 . 7 \%$ on Kinetics-700, and $4 1 . 1 \%$ on Moments in Time, new records while avoiding supervised pre-training. Transferring to image classification leads to $7 8 . 7 \%$ top-1 accuracy on ImageNet compared to ${ \bar { 6 4 . 7 \% } }$ by training the same Transformer from scratch, showing the generalizability of our model despite the domain gap between videos and images. VATT’s audio Transformer also sets a new record on waveform-based audio event recognition by achieving the mAP of $3 9 . 4 \%$ on AudioSet without any supervised pre-training. VATT’s source code is publicly available.2
+
+# 1 Introduction
+
+Convolutional neural networks (CNNs) [53, 51] have triumphed over various computer vision tasks. The inductive bias induced by convolutions, namely translation invariance and locality, are proven effective for the visual data. In the meantime, however, we witness in the natural language processing (NLP) community a paradigm shift from the models with strong inductive biases, such as recurrent neural networks [43, 7] and CNNs [104, 32], to more general architectures constructed upon selfattention. Particularly, Transformers [88] have become the de facto model architecture for NLP tasks [23, 70, 71, 10]. Pre-training a Transformer on large text corpora followed by fine-tuning gives rise to state-of-the-art results for different downstream tasks.
+
+![](images/0682c4b5ff5d083c190cbad0b248ec16501c3f63dda533f65af4f09ebbaca9f4.jpg)  
+Figure 1: Overview of the VATT architecture and the self-supervised, multimodal learning strategy. VATT linearly projects each modality into a feature vector and feeds it into a Transformer encoder. We define a semantically hierarchical common space to account for the granularity of different modalities and employ the Noise Contrastive Estimation (NCE) to train the model.
+
+In view of the success of the attention mechanism in NLP, there has been a rich line of works exploring its potential in computer vision. Early work studied hybrid models consisting of both convolutions and attention modules [89, 94, 36, 105]. Recent studies showed that convolution-free, specially designed all-attention models can match CNNs’ performance on image recognition tasks [106, 44, 73]. Most recently, [25] achieved impressive performance on several image recognition tasks, including ImageNet [22], using a pre-trained Transformer with minimal architecture changes. Their work delivered a compelling message that “large scale (supervised) training trumps inductive bias (for image classification).” This conclusion was further extended to video recognition tasks by [9, 5].
+
+However, the large-scale supervised training of Transformers is essentially troubling for two main reasons. First, it rules out the much larger other part of “big visual data,” i.e, the vast amount of unlabeled, unstructured visual data. As a result, the supervised training strategy could produce biased systems that require even more labeled data to correct their biases. Second, this strategy fundamentally limits the application scope of Transformers in computer vision because it is costly and extremely time-consuming to collect enough labeled images or videos for training the millions of parameters, choosing hyper-parameters, and validating their expected generalization.
+
+Hence, this work poses another pressing question about the Transformers that take raw signals as input. How to empower them with large-scale, unlabeled visual data? To answer this question, we draw insights from NLP. BERT [23] and GPT [70, 71, 10] use masked language modeling as their pre-training tasks. Natural languages are organic supervision for Transformers. They sequentially place words, phrases, and sentences into context, granting them semantics and syntax. For visual data, the most organic supervision is arguably the multimodal videos. They are abundantly available in the digital world, and their temporal, cross-modality regulation, and therefore supervision, requires no human annotation. The extreme scale of multimodal videos is potentially capable to teach Transformers necessary priors, as opposed to predefined inductive biases, to model the visual world.
+
+To this end, we study self-supervised, multimodal pre-training of three Transformers [88], which take as input the raw RGB frames of internet videos, audio waveforms, and text transcripts of the speech audio, respectively. We call the video, audio, text Transformers VATT. Figure 1 illustrates the architecture. VATT borrows the exact architecture from BERT [23] and ViT [25] except the layer of tokenization and linear projection reserved for each modality separately. This design shares the same spirit as ViT that we make the minimal changes to the architecture so that the learned model can transfer its weights to various frameworks and tasks. Furthermore, the self-supervised, multimodal learning strategy resonates the spirit of BERT and GPT that the pre-training requires minimal human curated labels.
+
+We evaluate the pre-trained Transformers on a variety of downstream tasks: image classification, video action recognition, audio event classification, and zero-shot text-to-video retrieval. Fine-tuning the vision-modality Transformer on ImageNet [22] obtains the top-1 accuracy of $7 8 . 7 \%$ , which is comparable to $7 9 . { \dot { 9 } } \%$ achieved by ViT. This result is especially appealing considering the domain gap between videos and images, and that ViT is pre-trained using a large-scale, human-curated image dataset. Furthermore, we set new records on Kinetics-400 [14], Kinetics-600 [15], Moments in Time [61], and AudioSet [33] without supervised pre-training.
+
+Our VATT results, along with others reported for NLP tasks [23, 10], image recognition [25], semantic segmentation [108], point cloud classification [107], and action recoginition [9], demonstrate that Transformer is a versatile general-purpose architecture for different types of data.
+
+To move one step forward, we challenge the Transformers in VATT by a seemingly too strong constraint: sharing weights among the video, audio, and text modalities. The idea is to test whether there exists a single, general-purpose model for all the modalities — of course, they still have their own layers of tokenization and linear projection. Preliminary results are encouraging. This modality-agnostic Transformer is on par with three modality-specific ones of slightly smaller sizes.
+
+Finally, another contribution of this work is DropToken, a simple and yet effective technique to reduce the training complexity with a minor reduction of the end Transformers’ performance. DropToken randomly drops a portion of the video and audio tokens from each input sequence during training, allowing for high-resolution inputs and leveraging their abundance. This is significant for Transformers because their computational complexity is quadratic with respect to the number of input tokens.
+
+# 2 Related work
+
+# 2.1 Transformers in Vision
+
+Transformer was originally built for NLP tasks [88] and the design of multi-head attention shows its effectiveness on modeling long-term correlation of words. A few attempts have been made to use Transformer for vision tasks like image super-resolution [99], object detection [11] and multimodal video understanding [84, 19, 57]. However these methods still rely on the feature extracted by CNNs. Recently, [25] proposes a set of convolution-free vision Transformers which directly work on raw images and obtain competitive performance with CNNs. [86] improves the training data efficiency of [25] by using stronger data augmentations and knowledge distillation. Since then, the pure Transformer design has been adopted to various vision tasks including semantic segmentation [108], point cloud classification [107], action recoginition [9, 78, 5]. To the best of our knowledge, our VATT is the first Transformer model on raw multimodal inputs of video, audio and text.
+
+# 2.2 Self-Supervised Learning
+
+Single vision modality. Early work of self-supervised visual representation learning usually learns from unlabeled images via manually specified pretext tasks, like auto-encoding [64, 102, 103], patch location prediction [24], solving jigsaw puzzles [63], and image rotation prediction [35]. [95] propose a novel instance discrimination objective. The recent trend of contrastive learning [40, 17, 100, 37, 41, 85] integrates data augmentations and instance discrimination by maintaining relative consistency between representations of an image and its augmented view. Clustering can also provide an effective addition [12]. Recently, [18] conduct contrastive learning using ViT [25] and achieve impressive results. As for the video domain, it is natural to exploit the temporal signals as the pretext task. Examples include predicting the future frame [82], motion and appearance statistics [90], speed [8, 91] and encodings [56, 38, 39], sorting frames or video clips [54, 97, 45, 31]. Recently, [68] apply contrastive learning to videos with a temporal sampling strategy and temporally consistent spatial augmentation.
+
+Multimodal video. Video is a natural source of multimodal data. Multimodal self-supervised learning can be achieved by predicting whether a video has correspondence with an audio stream [3, 4, 62, 50], cross-modality clustering [2], and evolving losses [67]. Recently, [1] use contrastive loss to learn from video, audio and text; [74] learn to predict a broad view that spans a longer temporal context from a narrow view. VATT serves as a first work combining the strength of convolution-free Transformer and multimodal contrastive learning.
+
+# 3 Approach
+
+In this section, we introduce our convolution-free VATT architecture and elaborate on the selfsupervised multimodal objectives for training VATT from scratch.
+
+Figure 1 is an overview of the architecture. We feed each modality to a tokenization layer, where the raw input is projected to an embedding vector followed by a Transformer. There are two major settings: 1) The backbone Transformers are separate and have specific weights for each modality, and 2) The Transformers share weights, namely, there is a single backbone Transformer applied to any of the modalities. In either setting, the backbone extracts modality-specific representations, which are then mapped to common spaces to be compared with each other by contrastive losses. We describe each module in the following.
+
+# 3.1 Tokenization and Positional Encoding
+
+VATT operates on raw signals. The vision-modality input consists of 3-channel RGB pixels of video frames, the audio input is in the form of air density amplitudes (waveforms), and the text input is a sequence of words. We first define a modality-specific tokenization layer that takes as input the raw signals and returns a sequence of vectors to be fed to the Transformers. Besides, each modality has its own positional encoding, which injects the order of tokens into Transformers [88]. We partition an entire video clip of size $T \times H \times W$ to a sequence of $\lceil T / t \rceil \cdot \lceil H / h \rceil \cdot \lceil W / w \rceil$ patches, where each patch contains $t \times h \times w \times 3$ voxels. We apply a linear projection on the entire voxels in each patch to get a $d$ -dimensional vector representation. This projection is performed by a learnable weight $W _ { v p } \in \mathbb { R } ^ { t \cdot h \cdot w \cdot 3 \times d }$ . This can be seen as a 3D extension of the patching mechanism proposed in [25]. To encode the position of these patches, we define a dimension-specific sequence of learnable embeddings as follows:
+
+$$
+\begin{array} { r l } { e _ { i , j , k } = e _ { \mathrm { T e m p o r a l } _ { i } } + e _ { \mathrm { H o r i z o n t a l } _ { j } } + e _ { \mathrm { V e r t i c a l } _ { k } } , } & { } \\ { E _ { \mathrm { T e m p o r a l } } \in \mathbb { R } ^ { \lceil T / t \rceil \times d } , } & { E _ { \mathrm { H o r i z o n t a l } } \in \mathbb { R } ^ { \lceil H / h \rceil \times d } , \quad E _ { \mathrm { V e r t i c a l } } \in \mathbb { R } ^ { \lceil W / w \rceil \times d } } \end{array}
+$$
+
+where $e _ { i }$ is the $i$ -th row of $\pmb { { \cal E } }$ . This scheme allows us to use $\lceil T / t \rceil + \lceil H / h \rceil + \lceil W / w \rceil$ positional embeddings to encode all the $\lceil T / t \rceil \cdot \lceil H / h \rceil \cdot \lceil W / w \rceil$ patches in a video clip. The raw audio waveform is a 1D input with length $T ^ { \prime }$ , and we partition it to $\lceil T ^ { \prime } / t ^ { \prime } \rceil$ segments each containing $t ^ { \prime }$ waveform amplitudes. Similar to video, we apply a linear projection with a learnable weight $W _ { a p } \in \mathbb { R } ^ { t ^ { \prime } \times d }$ to all elements in a patch to get a $d$ -dimensional vector representation. We use $\lceil T ^ { \prime } / t ^ { \prime } \rceil$ learnable embeddings to encode the position of each waveform segment. For text, we first construct a vocabulary of size $v$ out of all words in our training dataset. For an input text sequence, we then map each word to a $v$ -dimensional one-hot vector followed by a linear projection with a learnable weight $W _ { t p } \in \mathbb { R } ^ { v \times d }$ . This is equivalent to an embedding dictionary lookup, which has been widely used in natural language understanding [60].
+
+# 3.1.1 DropToken
+
+We introduce DropToken, a simple and yet effective strategy to reduce the computational complexity during training. Once we get the token sequence for the video or audio modality, we randomly sample a portion of the tokens and then feed the sampled sequence, not the complete set of tokens, to the Transformer. This is crucial for reducing the computational cost because a Transformer’s computation complexity is quadratic, $O ( N ^ { 2 } )$ , where $N$ is number of tokens in the input sequence. Any effort on reducing the input length would reduce the number of FLOPs quadratically. This has an immediate impact on the wall clock time for training these models and makes it possible to host large models in limited hardware. We argue that instead of reducing the resolution or dimension of the raw inputs, it is better to keep a high-fidelity input and randomly sample the tokens via DropToken. DropToken is appealing especially with the raw video and audio inputs, which may contain high redundancies.
+
+# 3.2 The Transformer Architecture
+
+For simplicity, we adopt the most established Transformer architecture [23], which has been widely used in NLP. Similar to ViT [25], we do not tweak the architecture so that our weights can be easily transferred to any standard Transformer implementation. We will briefly elaborate on the pipeline (also illustrated in Figure 1 middle panel) and refer the reader to [25, 23] for more details of the standard Transformer architecture. The sequence of input tokens to the Transformer follows the below formulation:
+
+$$
+z _ { \mathrm { i n } } = [ x _ { \mathrm { A G G } } ; ~ x _ { 0 } W _ { P } ; ~ x _ { 1 } W _ { P } ; \ldots ; ~ x _ { N } W _ { P } ] + e _ { \mathrm { P O S } }
+$$
+
+where ${ \mathbf { \mathcal { x } } } _ { n }$ is the input patches sequence and $\pmb { x } _ { \mathrm { A G G } }$ is the learnable embedding of a special aggregation token whose corresponding output in the Transformer $( z _ { \mathrm { o u t } } ^ { 0 } )$ is used as the aggregated representation for the entire input sequence. This will be later used for classification and common space mapping. We use a standard self-attention [88] as the Multi-Head-Attention (MHA) module, and GeLU [42] as the activation in the MLP layer. We also use Layer Normalization [6] before the MHA and MLP modules. In our text model, we remove the position encoding $e _ { \mathrm { P O S } }$ and add a learnable relative bias to each attention score of the first layer in the MHA module. This simple change makes our text model’s weights directly transferable to the state-of-the-art text model T5 [72].
+
+# 3.3 Common Space Projection
+
+We use common space projection and contrastive learning in that common space to train our networks. More specifically, given a video-audio-text triplet, we define a semantically hierarchical common space mapping that enables us to directly compare video-audio pairs as well as video-text pairs by the cosine similarity. As argued in [1], such comparison is more feasible if we assume there are different levels of semantic granularity for these modalities. To achieve this, we define multi-level projections as follows:
+
+$$
+\begin{array} { r l } { z _ { v , v a } = g _ { v \to v a } ( z _ { \mathrm { o u t } } ^ { \mathrm { v i d e o } } ) , \ } & { z _ { a , v a } = g _ { a \to v a } ( z _ { \mathrm { o u t } } ^ { \mathrm { a u d i o } } ) } \\ { z _ { t , v t } = g _ { t \to v t } ( z _ { \mathrm { o u t } } ^ { \mathrm { t e x t } } ) , \ } & { z _ { v , v t } = g _ { v \to v t } ( z _ { v , v a } ) } \end{array}
+$$
+
+where $g _ { v  v a }$ and $g _ { a  v a }$ are the projection heads to respectively map the video and audio Transformers’ outputs to the video-audio common space $\boldsymbol { S _ { v a } }$ . Moreover, $g _ { t  v t }$ and $g _ { v  v t }$ project the text Transformer’s outputs and the video embedding in the $\boldsymbol { S } _ { v a }$ space to video-text common space, $\boldsymbol { S _ { v t } }$ . This multi-level common space projection is depicted in Figure 1 (the rightmost panel). The main intuition behind this hierarchy is that different modalities have different levels of semantic granularity, so we should impose this as an inductive bias in the common space projection. Similar to [1], we use a linear projection for $g _ { a  v a } ( . )$ , $g _ { t  v t } ( . )$ , and $g _ { v  v t } ( . )$ , and a two-layer projection with ReLU in between for $g _ { v  v a } ( . )$ . To ease the training, a batch normalization is used after each linear layer.
+
+# 3.4 Multimodal Contrastive Learning
+
+Inspired by [1, 3, 59], we use Noise Contrastive Estimation (NCE) to align video-audio pairs and Multiple Instance Learning NCE (MIL-NCE) to align video-text pairs. The pairs are composed from different temporal locations in the video-audio-text stream. Positive pairs from two modalities are constructed by sampling their corresponding streams from the same location in the video, and negative pairs are constructed by sampling from any non-matching locations in the video [1]. Concretely, given the common space specified in Section 3, the loss objectives can be written as follows:
+
+$$
+\mathrm { N C E } ( z _ { v , v a } , z _ { a , v a } ) = - \log \left( \frac { \exp ( z _ { v , v a } ^ { \top } z _ { a , v a } / \tau ) } { \exp ( z _ { v , v a } ^ { \top } z _ { a , v a } / \tau ) + \sum _ { z ^ { \prime } \in \mathcal { N } } \exp ( z _ { v , v a } ^ { \prime \top } z _ { a , v a } ^ { \prime } / \tau ) } \right) ,
+$$
+
+$$
+\begin{array} { r } { \mathcal { A } \mathrm { I L } \mathrm { - N C E } ( z _ { v , v t } , \{ z _ { t , v t } \} ) = - \log \left( \frac { \sum _ { z _ { k , v t } \in \mathcal { P } } \exp ( z _ { v , v t } ^ { \top } z _ { t , v t } / \tau ) } { \sum _ { z _ { t , v t } \in \mathcal { P } } \exp ( z _ { v , v t } ^ { \top } z _ { t , v t } / \tau ) + \sum _ { z ^ { \prime } \in \mathcal { N } } \exp ( z _ { v , v t } ^ { \prime \top } z _ { t , v t } ^ { \prime } / \tau ) } \right) , } \end{array}
+$$
+
+where $\mathcal { N }$ contains all non-matching pairs in a batch. In Equation 5, $\mathcal { P }$ contains five text clips that are nearest neighbors to the video clip in time. $\tau$ is a temperature to adjust the softness of the objectives in distinguishing the positive pairs from the negative pairs.
+
+The overall per-sample objective for training the entire VATT model end-to-end is as follows:
+
+$$
+\mathcal { L } = \mathrm { N C E } ( z _ { v , v a } , z _ { a , v a } ) + \lambda \mathrm { M I L } \mathrm { - N C E } ( z _ { v , v t } , \{ z _ { t , v t } \} ) ,
+$$
+
+where $\lambda$ balances the two losses. The model is optimized based on the back-propagation of the average loss calculated over a batch of samples.
+
+# 4 Experiments
+
+In this section, we first briefly describe the experimental setup for the pre-training and downstream evaluation, and then present the results and analytic interpretation of VATT in different tasks. We refer the reader to the Appendix for a more detailed description of all experimental settings.
+
+# 4.1 Experimental Setup
+
+Pre-train: we use a combination of AudioSet [33] and HowTo100M [58] datasets to pre-train VATT— we use only a subset of the HowTo100M dataset in compliance with Youtube’s policies. Following [1], we use video-audio-text triplets from HowTo100M clips while only using video-audio pairs from AudioSet. We sample 32 frames at 10 fps with a spatial size of $2 2 4 \times 2 2 4$ following a random crop, horizontal flip and color augmentation (details in A.2.1). Accordingly, we sample audio waveforms in sync at $4 8 \mathrm { k H z }$ . Both video and audio are normalized between [-1,1]. We use patch sizes of $4 \times 1 6 \times 1 6$ and 128 for video and raw waveform tokenization, respectively (ablation in A.5). We use one-hot vectors to encode text sequences (capped to 16 tokens) with the vocabulary size of $2 ^ { 1 6 }$ . In all pre-training experiments, we use DropToken with drop rate $50 \%$ . We train our models using the Adam optimizer [46] with a quarter-period cosine scheduled learning rate from 1e-4 to 5e-5 and 10k warmup steps. Optimization is performed on totally $5 0 0 \mathrm { k }$ steps with batch size 2048 (512 in exploration experiments). Following the previously established practice [1] for the projection to the common spaces $\boldsymbol { S _ { v a } }$ and $\boldsymbol { S _ { v t } }$ , we use $d _ { v a } = 5 1 2$ and $d _ { v t } = 2 5 6$ . We also use the temperature of $\tau = 0 . 0 7$ and the weight of $\lambda = 1$ in the loss in Equation 6. We use 4 network sizes in our experiments (details in A.2.2). We use the Medium model (155M parameters) for our modality-agnostic variant (VATT-MA), and 3 variants for the modality-specific video-audio-text backbones: Base-Base-Small (BBS; 197M), Medium-Base-Small (MBS; 264M), and Large-Base-Small (LBS; 415M). Pre-training an MBS VATT with batch size 2048 on 256 TPUs (v3) takes less than 3 days. Pre-training with batch size 512 takes less than 1 day.
+
+Downstream: we evaluate the pre-trained VATT models on 4 major downstream tasks using a total of 10 datasets. We use UCF101 [81], HMDB51 [52], Kinetics-400 [14], Kinetics-600 [15], and Moments in Time [61] for video action recognition. We use ESC50 [66] and AudioSet [33] for audio event classification, and we evaluate the quality of our video-text common space representations by zero-shot text-to-video retrieval on YouCook2 [109] and MSR-VTT [98]. Finally, we evaluate the transferability of the vision backbone by fine-tuning it on ImageNet classification [22]. Since HMDB51, UCF101, and ESC50 are very small datasets compared to the size of our networks, we only use them to train a linear classifier on top of the frozen pre-trained backbones. In our exploration experiments, we report linear classification accuracy and zero-shot video retrieval metrics. We refer to the Appendix for a detailed description of the datasets and the experimental setup.
+
+# 4.2 Results
+
+# 4.2.1 Fine-tuning for video action recognition
+
+We fine-tune VATT’s vision Transformer on Kinetics-400, Kinetics-600, and Moments in Time, three of the arguably most established large-scale datasets for video action recognition. We use the final checkpoints of four pre-train settings for these experiments: three modality-specific variations (LBS, MBS, BBS), and one modality-agnostic (Medium). Table 1 shows the results compared with the state-of-the-art video models. On all three datasets, we achieve higher accuracy than previous works including TimeSFormer [9], a recent effort in fine-tuning the ViT checkpoints obtained by supervised pre-training. In contrast, our pre-training does not rely on any labels curated by humans. To the best of our knowledge, VATT provides the first vision Transformer backbone that is pre-trained from scratch using self-supervision on multimodal videos and achieves state-of-the-art results on video action recognition. It is also worth mentioning that fine-tuning VATT on the most recent Kinetics-700 dataset results in a top-1 accuracy of $7 2 . 7 \%$ , which outperforms the state-of-the-art top-1 accuracy of $7 2 . 4 \%$ in [47].
+
+To further quantify how much the multimodal self-supervised pre-training helps in achieving these numbers, we train a variant from scratch without any pre-training and observe the top-1 and top-5 accuracies of $2 6 . 4 \%$ and $5 1 . 8 \%$ on Kinetics-400, respectively. The low accuracies verify the efficacy of our pre-training strategy for VATT. Finally, we find that VATT-MA-Medium, the modality-agnostic backbone shared by the video, audio, and text modalities, is on par with the modality-specific VATTBase when fine-tuned for the video action recognition. This result is encouraging as it indicates the potential of unifying three data modalities by a single Transformer backbone.
+
+<table><tr><td></td><td colspan="2">Kinetics-400</td><td colspan="2">Kinetics-600</td><td colspan="2">Moments in Time</td><td rowspan="2">TFLOPS</td></tr><tr><td>METHOD</td><td>ToP-1</td><td>TOP-5</td><td>TOP-1</td><td>TOP-5</td><td>TOP-1</td><td>TOP-5</td></tr><tr><td>I3D[13]</td><td>71.1</td><td>89.3</td><td>71.9</td><td>90.1</td><td>29.5</td><td>56.1</td><td></td></tr><tr><td>R(2+1)D [26]</td><td>72.0</td><td>90.0</td><td>-</td><td>-</td><td>1</td><td>1</td><td>17.5</td></tr><tr><td>bLVNet [27]</td><td>73.5</td><td>91.2</td><td>-</td><td>/</td><td>31.4</td><td>59.3</td><td>0.84</td></tr><tr><td>S3D-G [96]</td><td>74.7</td><td>93.4</td><td>=</td><td></td><td>-</td><td>-</td><td>-</td></tr><tr><td>Oct-I3D+NL [20]</td><td>75.7</td><td>1</td><td>76.0</td><td></td><td>=</td><td>1</td><td>0.84</td></tr><tr><td>D3D[83]</td><td>75.9</td><td>=</td><td>77.9</td><td></td><td></td><td></td><td>=</td></tr><tr><td>I3D+NL [93]</td><td>77.7</td><td>93.3</td><td>-</td><td></td><td></td><td></td><td>10.8</td></tr><tr><td>ip-CSN-152 [87]</td><td>77.8</td><td>92.8</td><td>1</td><td></td><td></td><td>=</td><td>3.3</td></tr><tr><td>AttentionNAS [92]</td><td>-</td><td>1</td><td>79.8</td><td>94.4</td><td>32.5</td><td>60.3</td><td>1.0</td></tr><tr><td>AssembleNet-101 [77]</td><td>-</td><td>=</td><td>-</td><td>-</td><td>34.3</td><td>62.7</td><td>1</td></tr><tr><td>MoViNet-A5 [47]</td><td>78.2</td><td>1</td><td>82.7</td><td>1</td><td>39.1</td><td>-</td><td>0.29</td></tr><tr><td>LGD-3D-101 [69]</td><td>79.4</td><td>94.4</td><td>81.5</td><td>95.6</td><td>-</td><td>=</td><td>-</td></tr><tr><td>SlowFast-R101-NL [30]</td><td>79.8</td><td>93.9</td><td>81.8</td><td>95.1</td><td></td><td></td><td>7.0</td></tr><tr><td>X3D-XL [29]</td><td>79.1</td><td>93.9</td><td>81.9</td><td>95.5</td><td>=</td><td></td><td>1.5</td></tr><tr><td>X3D-XXL [29]</td><td>80.4</td><td>94.6</td><td>1</td><td>-</td><td></td><td></td><td>5.8</td></tr><tr><td>TimeSFormer-L [9]</td><td>80.7</td><td>94.7</td><td>82.2</td><td>95.6</td><td>-</td><td>=</td><td>7.14</td></tr><tr><td>VATT-Base</td><td>79.6</td><td>94.9</td><td>80.5</td><td>95.5</td><td>38.7</td><td>67.5</td><td>9.09</td></tr><tr><td>VATT-Medium</td><td>81.1</td><td>95.6</td><td>82.4</td><td>96.1</td><td>39.5</td><td>68.2</td><td>15.02</td></tr><tr><td>VATT-Large</td><td>82.1</td><td>95.5</td><td>83.6</td><td>96.6</td><td>41.1</td><td>67.7</td><td>29.80</td></tr><tr><td>VATT-MA-Medium</td><td>79.9</td><td>94.9</td><td>80.8</td><td>95.5</td><td>37.8</td><td>65.9</td><td>15.02</td></tr></table>
+
+Table 1: Video action recognition accuracy on Kinetics-400, Kinetics-600, and Moments in Time.
+
+# 4.2.2 Fine-tuning for audio event classification
+
+We fine-tune VATT’s audio Transformer on AudioSet, which benchmarks the task of multi-label audio event classification. We use the final checkpoints of two pre-train settings: one modality-specific (BBS), and one modality-agnostic (Medium). Table 2 shows the results compared to state-of-the-art models. Following common practice [34, 48], we report mean Average Precision (mAP), Area Under Curve (AUC), and d-prime (based on AUC) [34]. Our audio Transformer consistently outperforms the existing CNN-based models in all metrics. More interestingly, fine-tuning the modality-agnostic backbone (VATT-MA-Medium) is on par with fine-tuning the modality-specific one (VATT-Base). To the best of our knowledge, VATT is the first Transformer that outperforms CNN-based models in audio event recognition. VATT operates on raw waveforms and does not utilize any handcrafted features.
+
+# 4.2.3 Fine-tuning for image classification
+
+In this section, we show that our pipeline is capable of transferring the learned knowledge into another domain by performing the image classification task, even though the models are pre-trained in the multimodal video domain. We fine-tune the vision Transformer in VATT-BBS on ImageNet without any modification to the backbone architecture. Instead, to satisfy the voxel-to-patch layer we replicate the input image 4 times and feed it to the network. The network sees the input as a single-frame video clip and performs spatial self-attention. Table 3 shows the results for fine-tuning the vision Transformer end-to-end on ImageNet. We can see that our pre-training leads to a significant boost in the accuracy compared to training from scratch. We also observe that even though the self-supervised pre-training happens in the video domain, we still achieve competitive results to the supervised pre-training using large-scale image data [25].
+
+# 4.2.4 Zero-shot text-to-video retrieval
+
+We feed video-text pairs to VATT-MBS, and extract representations in the $\boldsymbol { S _ { v t } }$ space. We then calculate the similarity between each video-text pair from YouCook2 and MSR-VTT. Given a text query, we rank the videos based on their similarities to the text. We then measure the recall for the correct video in the top-10 videos. We also measure the median of the rank of the correct video. Table 4 compares our video retrieval results to two baselines. In our experiments we observe that the zero-shot retrieval results are heavily affected by the batch size and number of epochs, confirming the observation made in [1]. That said, our model still delivers comparable results to MMV [1] while being pre-trained with a half number of epochs and a half batch size of theirs. We also experiment with a larger batch size 8192 and longer pre-training for 6 epochs, arriving at exactly the same results as MIL-NCE [59] on YouCook2 and the $\mathbf { R } @ 1 0$ of 29.2 and MedR of 42 on MSR-VTT. We also notice that, probably due to the noisy nature of text transcripts, a sophisticated language model like ours is underrated. As shown in [1], using a simple linear projection would still perform reasonably well. It is worth exploring other, higher-quality text sources in future work.
+
+Table 2: Finetuning results for AudioSet event classification.   
+
+<table><tr><td>METHOD</td><td>mAP AUC d-prime</td></tr><tr><td>DaiNet [21]</td><td>29.5 95.8 2.437</td></tr><tr><td>LeeNet11 [55]</td><td>26.6 95.3 2.371</td></tr><tr><td>LeeNet24 [55]</td><td>33.6 96.3 2.525</td></tr><tr><td>Res1dNet31 [49]</td><td>36.5 95.8 2.444</td></tr><tr><td>Res1dNet51 [49]</td><td>35.5 94.8 2.295</td></tr><tr><td>Wavegram-CNN [49]</td><td>38.9 96.8 2.612</td></tr><tr><td>VATT-Base</td><td>39.4 97.1 2.895</td></tr><tr><td>VATT-MA-Medium</td><td>39.3 97.0 2.884</td></tr></table>
+
+Table 3: Finetuning results for ImageNet classification.   
+
+<table><tr><td>METHOD</td><td>PRE-TRAINING DATA</td><td>TOP-1</td><td>TOP-5</td></tr><tr><td>iGPT-L [16]</td><td>ImageNet</td><td>72.6</td><td>1</td></tr><tr><td>ViT-Base [25]</td><td>JFT</td><td>79.9</td><td>1</td></tr><tr><td>VATT-Base</td><td>-</td><td>64.7</td><td>83.9</td></tr><tr><td>VATT-Base</td><td>HowTo100M</td><td>78.7</td><td>93.9</td></tr></table>
+
+Table 4: Zero-shot text-to-video retrieval.   
+
+<table><tr><td></td><td colspan="4">YouCook2 MSR-VTT</td></tr><tr><td>METHOD</td><td>BATCH EPOCH R@10 MedR R@ 10 MedR 27</td><td></td><td></td><td></td></tr><tr><td>MIL-NCE [59]</td><td>8192</td><td>51.2</td><td>10 32.4</td><td>30</td></tr><tr><td>MMV[1]</td><td>4096 8</td><td>45.4</td><td>13 31.1</td><td>38</td></tr><tr><td>VATT-MBS</td><td>2048 4</td><td>45.5</td><td>13 29.7</td><td>49</td></tr><tr><td>VATT-MA-Medium</td><td>2048 4</td><td>40.6</td><td>17</td><td>23.6 67</td></tr></table>
+
+# 4.2.5 Feature visualization
+
+We take our modality-specific and modality-agnostic VATT fine-tuned on Kinetics-400 and visualize their output feature representations using t-SNE. For comparison, we also include the feature visualization of the vision Transformer trained from scratch on Kinetics-400. From Figure 2, we observe that the fine-tuned VATT yields a much better separation than the model trained from scratch. Furthermore, it is worth noting that there is no clear difference between the modality-agnostic features and the modality-specific ones.
+
+We further investigate the VATT backbones without any fine-tuning. We randomly choose 1k video clips from the YouCook2 dataset and store the representations from two points of a pre-trained VATT model. One is after the tokenization layer (input space of the Transformer), and the other is after the common space projection (output space), where the loss is computed. Figure 3-top visualizes the representations, comparing modality-specific VATT to modality-agnostic VATT. Interestingly, we observe that the representations are slightly more mixed together in the modality-agnostic setting compared to the modality-specific ones, implying that the modality-agnostic backbone sees different modalities as different symbols describing the same concept. This is analogous to a unified language model in NLP that supports multiple languages.
+
+To see how well VATT distinguishes positive video-text pairs from randomly sampled pairs, we calculate pair-wise similarities for all possible pairs and perform a Kernel Density Estimation (KDE) to visualize the distributions of the similarities of the positive pairs vs. negative pairs. We perform this procedure for both input and output spaces of the modality-specific and modality-agnostic backbones. Figure 3-bottom shows the KDE curves of these similarities. We can see that VATT in both settings separates the positive and negative pairs in its output space. This verifies VATT’s efficacy in learning a semantic common space for different modalities, even if we share the backbone across modalities.
+
+# 4.2.6 Model Activations
+
+We measure the average activation of the modality-agnostic VATT when a full multimodal input is fed to the model. More specifically, we sample $1 0 0 \mathrm { k }$ short video clips from the test split of HowTo100M along with their corresponding audio and text and feed them to the model separately. For each modality, we calculate the average activation of each node at the output of the MLP module, before the residual addition (Figure 1-Transformer Encoder). Figure 4 shows the average activations across all nodes in a Medium-size model. We observe that earlier nodes in the model are activated with the text inputs, while the middle-to-later nodes are activated with video and audio modalities. However, the nodes in the last layers of the network are activated with all modalities almost equally. This might suggest that the model allocates different nodes to certain modalities while reaching the same level of semantic perception for all modalities in the later layers. Such observation encourages further studies on the possibility of utilizing Mixture-of-Experts [79, 28, 76] to increase the model’s capacity for simultaneous multimodal perception. We leave this direction of research for future work.
+
+![](images/264dcf3b3a980aec5d2a5f2a0fee6ce340c013a3c55bff2abf50990cf4d4afe8.jpg)  
+Figure 2: t-SNE visualization of the feature representations extracted by the vision Transformer in different training settings. For better visualization, we show 100 random classes from Kinetics-400.
+
+![](images/c6f8e58fd69ca0d895846cdad125dcf844cc929bef9f7744a0c560ca0c9eb205.jpg)  
+Figure 3: t-SNE visualization and distribution of pair-wise similarities of the input space vs. output space for modality-specific and modality-agnostic backbones when different modalities are fed.
+
+# 4.2.7 Effect of DropToken
+
+We introduced a new method to reduce the redundancy in high-resolution data. To study the effect of the proposed DropToken method on downstream applications and the pre-training computation, we perform pre-training by randomly dropping $7 5 \%$ , $5 0 \%$ , $2 5 \%$ , and $0 \%$ (no drop) of the tokens from the video and audio inputs. Table 5 shows the accuracy of linear classification on HMDB51, UCF101, ESC50 and $\mathrm { R @ 1 0 }$ on YouCook2 and MSR-VTT vs. the drop rate along with GFLOPs during a forward call. We choose $5 0 \%$ sampling rate for our large-scale pre-training as it offers a good trade-off between accuracy and computational costs. We then take the final checkpoint of the pre-trained VATT with $5 0 \%$ DropToken rate and perform fine-tuning on Kinetics-400 at different DropToken rates and at different spatial and temporal resolutions to see how high-resolution inputs coupled with DropToken compare to low-resolution inputs with no tokens dropped during fine-tuning. Table 6 shows the top-1 accuracy on Kinetics-400. We argue against using low-resolution inputs, which is the most common approach to reduce the computational cost during training. Instead, we suggest using high-resolution inputs with DropToken, whose accuracy and training cost are comparable to or better than low-resolution counterparts.
+
+![](images/78326d8ea06d3a0ed997fe60a987577bc3f5e4296c66b02b900d7bc3adee2312.jpg)  
+Late Activation for Video & Audio   
+Figure 4: The average node activation across the Modality-Agnostic-Medium VATT while feeding a multimodal video-audio-text triplet to the model.
+
+<table><tr><td></td><td>DropToken Drop Rate 75%</td></tr><tr><td>Multimodal GFLOPs 188.1</td><td>50% 25% 0% 375.4 574.2 784.8</td></tr><tr><td>HMDB51 62.5</td><td>65.6 66.4</td></tr><tr><td>UCF101</td><td>64.8 85.5 87.2</td></tr><tr><td>84.0 ESC50 78.9</td><td>84.1 84.6</td></tr><tr><td>YouCookII 17.9</td><td>20.7 24.2</td></tr><tr><td>14.1</td><td>14.6 15.1</td></tr></table>
+
+Table 5: Top-1 accuracy of linear classification and $\mathrm { R @ 1 0 }$ of video retrieval vs. drop rate vs. inference GFLOPs in the VATT-MBS.
+
+<table><tr><td rowspan="2">Resolution/ FLOPs</td><td colspan="4">DropToken Drop Rate</td></tr><tr><td>75%</td><td>50%</td><td>25%</td><td>0%</td></tr><tr><td>32 × 224 × 224 Inference (GFLOPs)</td><td></td><td></td><td>=</td><td>79.9 548.1</td></tr><tr><td>64×224 ×224 Inference (GFLOPs)</td><td>= -</td><td>=</td><td>= 1</td><td>80.8 1222.1</td></tr><tr><td>32 ×320×320 Inference (GFLOPs)</td><td>79.3 279.8</td><td>80.2 572.5</td><td>80.7 898.9</td><td>81.1 1252.3</td></tr></table>
+
+Table 6: Top-1 accuracy of video action recognition on Kinetics400 using high-resolution inputs coupled with DropToken vs. low-resolution inputs.
+
+# 5 Conclusion and Discussion
+
+In this paper, we present a self-supervised multimodal representation learning framework based on Transformers. Our study suggests that Transformers are effective for learning semantic video/audio/text representations — even if one model is shared across modalities — and multimodal self-supervised pre-training is promising for reducing their dependency on large-scale labeled data. We show that DropToken can significantly reduce the pre-training complexity with video and audio modalities and have minor impact on the models’ generalization. We report new records of results on video action recognition and audio event classification and competitive performance on image classification and video retrieval. Having these results, we still see some limitations in our work. Firstly, not all videos have organic audio or speech, while our approach depends on meaningful multimodal correspondences. Besides, the text modality currently consists of speech transcripts, which are noisy and sometimes sparse. Potential negative Societal Impacts are mainly concerned with applications. The models could be biased if one applies our approach to the multimodal videos that are not representative enough. Finally, our method is still demanding in computation, though we managed to avoid the need for human labels. Future work can improve upon these limitations.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank Min-Hsuan Tsai, Jean-Baptise Alayrac, Andrew Audibert, Yeqing Li, Vidush Mukund, and the TensorFlow team for their help with codes, infrastructure, and insightful discussions.
+
+# References
+
+[1] Jean-Baptiste Alayrac, Adrià Recasens, Rosalia Schneider, Relja Arandjelovic, Jason Rama- ´ puram, Jeffrey De Fauw, Lucas Smaira, Sander Dieleman, and Andrew Zisserman. Selfsupervised multimodal versatile networks. In NeurIPS, 2020. 3, 5, 6, 8, 17, 18, 19, 20 [2] Humam Alwassel, Dhruv Mahajan, Bruno Korbar, Lorenzo Torresani, Bernard Ghanem, and Du Tran. Self-supervised learning by cross-modal audio-video clustering. arXiv preprint arXiv:1911.12667, 2019. 3, 20 [3] Relja Arandjelovic and Andrew Zisserman. Look, listen and learn. In CVPR, 2017. 3, 5 [4] Relja Arandjelovic and Andrew Zisserman. Objects that sound. In ECCV, 2018. 3 [5] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia ´ Schmid. Vivit: A video vision transformer. arXiv preprint arXiv:2103.15691, 2021. 2, 3 [6] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. 5 [7] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR, 2015. 1 [8] Sagie Benaim, Ariel Ephrat, Oran Lang, Inbar Mosseri, William T Freeman, Michael Rubinstein, Michal Irani, and Tali Dekel. Speednet: Learning the speediness in videos. In CVPR,   
+2020. 3 [9] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is space-time attention all you need for video understanding? arXiv preprint arXiv:2102.05095, 2021. 2, 3, 6, 7 [10] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. arXiv preprint arXiv:2005.14165, 2020. 2, 3 [11] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In ECCV, 2020. 3 [12] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. In NeurIPS, 2020.   
+3 [13] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In CVPR, 2017. 7 [14] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In CVPR, 2017. 3, 6, 17 [15] Joao Carreira, Eric Noland, Andras Banki-Horvath, Chloe Hillier, and Andrew Zisserman. A short note about kinetics-600. arXiv preprint arXiv:1808.01340, 2018. 3, 6, 17 [16] Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Heewoo Jun, David Luan, and Ilya Sutskever. Generative pretraining from pixels. In ICML, 2020. 8 [17] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In ICML, 2020. 3 [18] Xinlei Chen, Saining Xie, and Kaiming He. An empirical study of training self-supervised visual transformers. arXiv preprint arXiv:2104.02057, 2021. 3 [19] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed El Kholy, Faisal Ahmed, Zhe Gan, Yu Cheng, and Jingjing Liu. Uniter: Universal image-text representation learning. In ECCV, 2020. 3 [20] Yunpeng Chen, Haoqi Fan, Bing Xu, Zhicheng Yan, Yannis Kalantidis, Marcus Rohrbach, Shuicheng Yan, and Jiashi Feng. Drop an octave: Reducing spatial redundancy in convolutional neural networks with octave convolution. In ICCV, 2019. 7   
+[21] Wei Dai, Chia Dai, Shuhui Qu, Juncheng Li, and Samarjit Das. Very deep convolutional neural networks for raw waveforms. In ICASSP, 2017. 8   
+[22] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009. 2, 3, 6, 17   
+[23] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL, 2019. 2, 3, 4   
+[24] Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In ICCV, 2015. 3   
+[25] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR, 2021. 2, 3, 4, 7, 8   
+[26] Heng Wang Du Tran, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. 2018 ieee. In CVPR, 2017. 7   
+[27] Quanfu Fan, Chun-Fu (Ricarhd) Chen, Hilde Kuehne, Marco Pistoia, and David Cox. More Is Less: Learning Efficient Video Representations by Temporal Aggregation Modules. In NeurIPS. 2019. 7   
+[28] William Fedus, Barret Zoph, and Noam Shazeer. Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity. arXiv preprint arXiv:2101.03961, 2021. 9   
+[29] Christoph Feichtenhofer. X3d: Expanding architectures for efficient video recognition. In CVPR, 2020. 7   
+[30] Christoph Feichtenhofer, Haoqi Fan, Jitendra Malik, and Kaiming He. Slowfast networks for video recognition. In ICCV, 2019. 7, 18   
+[31] Basura Fernando, Hakan Bilen, Efstratios Gavves, and Stephen Gould. Self-supervised video representation learning with odd-one-out networks. In CVPR, 2017. 3   
+[32] Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. Convolutional sequence to sequence learning. In ICML, 2017. 1   
+[33] Jort F. Gemmeke, Daniel P. W. Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R. Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In ICASSP, 2017. 3, 6, 17   
+[34] Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In ICASSP, 2017. 7   
+[35] Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. In ICLR, 2018. 3   
+[36] Rohit Girdhar and Deva Ramanan. Attentional pooling for action recognition. In NeurIPS, 2017. 2   
+[37] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. In NeurIPS, 2020. 3   
+[38] Tengda Han, Weidi Xie, and Andrew Zisserman. Video representation learning by dense predictive coding. In ICCV Workshops, 2019. 3   
+[39] Tengda Han, Weidi Xie, and Andrew Zisserman. Memory-augmented dense predictive coding for video representation learning. In ECCV, 2020. 3 [40] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, 2020. 3 [41] Olivier J Hénaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019. 3 [42] Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415, 2016. 5 [43] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation,   
+1997. 1 [44] Han Hu, Zheng Zhang, Zhenda Xie, and Stephen Lin. Local relation networks for image recognition. In ICCV, 2019. 2 [45] Dahun Kim, Donghyeon Cho, and In So Kweon. Self-supervised video representation learning with space-time cubic puzzles. In AAAI, 2019. 3 [46] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. 6, 18 [47] Dan Kondratyuk, Liangzhe Yuan, Yandong Li, Li Zhang, Mingxing Tan, Matthew Brown, and Boqing Gong. Movinets: Mobile video networks for efficient video recognition. In CVPR,   
+2021. 6, 7 [48] Qiuqiang Kong, Changsong Yu, Yong Xu, Turab Iqbal, Wenwu Wang, and Mark D Plumbley. Weakly labelled audioset tagging with attention neural networks. TASLP, 2019. 7 [49] Qiuqiang Kong, Yin Cao, Turab Iqbal, Yuxuan Wang, Wenwu Wang, and Mark D Plumbley. Panns: Large-scale pretrained audio neural networks for audio pattern recognition. TASLP,   
+2020. 8, 19 [50] Bruno Korbar, Du Tran, and Lorenzo Torresani. Cooperative learning of audio and video models from self-supervised synchronization. NeurIPS, 2018. 3, 20 [51] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NeurIPS, 2012. 1 [52] H. Kuehne, H. Jhuang, E. Garrote, T. Poggio, and T. Serre. HMDB: a large video database for human motion recognition. In ICCV, 2011. 6, 17 [53] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 1998. 1 [54] Hsin-Ying Lee, Jia-Bin Huang, Maneesh Singh, and Ming-Hsuan Yang. Unsupervised representation learning by sorting sequences. In ICCV, 2017. 3 [55] Jongpil Lee, Jiyoung Park, Keunhyoung Luke Kim, and Juhan Nam. Sample-level deep convolutional neural networks for music auto-tagging using raw waveforms. arXiv preprint arXiv:1703.01789, 2017. 8 [56] William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning. arXiv preprint arXiv:1605.08104, 2016. 3 [57] Huaishao Luo, Lei Ji, Botian Shi, Haoyang Huang, Nan Duan, Tianrui Li, Xilin Chen, and Ming Zhou. Univilm: A unified video and language pre-training model for multimodal understanding and generation. arXiv preprint arXiv:2002.06353, 2020. 3 [58] Antoine Miech, Dimitri Zhukov, Jean-Baptiste Alayrac, Makarand Tapaswi, Ivan Laptev, and Josef Sivic. Howto100m: Learning a text-video embedding by watching hundred million narrated video clips. In ICCV, 2019. 6, 17
+
+[59] Antoine Miech, Jean-Baptiste Alayrac, Lucas Smaira, Ivan Laptev, Josef Sivic, and Andrew Zisserman. End-to-end learning of visual representations from uncurated instructional videos. In CVPR, 2020. 5, 8, 17, 20
+
+[60] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013. 4, 18
+
+[61] Mathew Monfort, Alex Andonian, Bolei Zhou, Kandan Ramakrishnan, Sarah Adel Bargal, Tom Yan, Lisa Brown, Quanfu Fan, Dan Gutfreund, Carl Vondrick, et al. Moments in time dataset: one million videos for event understanding. TPAMI, 2019. 3, 6, 17
+
+[62] Pedro Morgado, Nuno Vasconcelos, and Ishan Misra. Audio-visual instance discrimination with cross-modal agreement. arXiv preprint arXiv:2004.12943, 2020. 3
+
+[63] Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In ECCV, 2016. 3
+
+[64] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In CVPR, 2016. 3
+
+[65] Mandela Patrick, Yuki M Asano, Polina Kuznetsova, Ruth Fong, João F Henriques, Geoffrey Zweig, and Andrea Vedaldi. Multi-modal self-supervision from generalized data transformations. arXiv preprint arXiv:2003.04298, 2020. 20
+
+[66] Karol J. Piczak. ESC: Dataset for Environmental Sound Classification. In ACM MM, 2015. 6, 17
+
+[67] AJ Piergiovanni, Anelia Angelova, and Michael S Ryoo. Evolving losses for unsupervised video representation learning. In CVPR, 2020. 3, 20
+
+[68] Rui Qian, Tianjian Meng, Boqing Gong, Ming-Hsuan Yang, Huisheng Wang, Serge Belongie, and Yin Cui. Spatiotemporal contrastive video representation learning. In CVPR, 2021. 3 [69] Zhaofan Qiu, Ting Yao, Chong-Wah Ngo, Xinmei Tian, and Tao Mei. Learning spatio-temporal representation with local and global diffusion. In CVPR, 2019. 7
+
+[70] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018. 2
+
+[71] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI blog, 2019. 2
+
+[72] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. JMLR, 2020. 5
+
+[73] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan Bello, Anselm Levskaya, and Jonathon Shlens. Stand-alone self-attention in vision models. arXiv preprint arXiv:1906.05909, 2019. 2
+
+[74] Adrià Recasens, Pauline Luc, Jean-Baptiste Alayrac, Luyu Wang, Florian Strub, Corentin Tallec, Mateusz Malinowski, Viorica Patraucean, Florent Altché, Michal Valko, et al. Broaden your views for self-supervised video learning. arXiv preprint arXiv:2103.16559, 2021. 3
+
+[75] Steffen Rendle. Factorization machines. In ICDM, 2010. 19 [76] Carlos Riquelme, Joan Puigcerver, Basil Mustafa, Maxim Neumann, Rodolphe Jenatton, André Susano Pinto, Daniel Keysers, and Neil Houlsby. Scaling vision with sparse mixture of experts. arXiv preprint arXiv:2106.05974, 2021. 9
+
+[77] Michael S Ryoo, AJ Piergiovanni, Mingxing Tan, and Anelia Angelova. Assemblenet: Searching for multi-stream neural connectivity in video architectures. arXiv preprint arXiv:1905.13209, 2019. 7
+
+[78] Gilad Sharir, Asaf Noy, and Lihi Zelnik-Manor. An image is worth 16x16 words, what is a video worth? arXiv preprint arXiv:2103.13915, 2021. 3
+
+[79] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. arXiv preprint arXiv:1701.06538, 2017. 9
+
+[80] Laura Smith, Nikita Dhawan, Marvin Zhang, Pieter Abbeel, and Sergey Levine. Avid: Learning multi-stage tasks via pixel-level translation of human videos. arXiv preprint arXiv:1912.04443, 2019. 20
+
+[81] Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. Ucf101: A dataset of 101 human actions classes from videos in the wild. arXiv preprint arXiv:1212.0402, 2012. 6, 17
+
+[82] Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In ICML, 2015. 3
+
+[83] Jonathan Stroud, David Ross, Chen Sun, Jia Deng, and Rahul Sukthankar. D3d: Distilled 3d networks for video action recognition. In WACV, 2020. 7
+
+[84] Chen Sun, Fabien Baradel, Kevin Murphy, and Cordelia Schmid. Learning video representations using contrastive bidirectional transformer. arXiv preprint arXiv:1906.05743, 2019. 3
+
+[85] Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. In ECCV, 2020. 3
+
+[86] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image transformers & distillation through attention. arXiv preprint arXiv:2012.12877, 2020. 3
+
+[87] Du Tran, Heng Wang, Lorenzo Torresani, and Matt Feiszli. Video classification with channelseparated convolutional networks. In ICCV, 2019. 7
+
+[88] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017. 1, 2, 3, 4, 5
+
+[89] Fei Wang, Mengqing Jiang, Chen Qian, Shuo Yang, Cheng Li, Honggang Zhang, Xiaogang Wang, and Xiaoou Tang. Residual attention network for image classification. In CVPR, 2017. 2
+
+[90] Jiangliu Wang, Jianbo Jiao, Linchao Bao, Shengfeng He, Yunhui Liu, and Wei Liu. Selfsupervised spatio-temporal representation learning for videos by predicting motion and appearance statistics. In CVPR, 2019. 3
+
+[91] Jiangliu Wang, Jianbo Jiao, and Yun-Hui Liu. Self-supervised video representation learning by pace prediction. In ECCV, 2020. 3
+
+[92] Xiaofang Wang, Xuehan Xiong, Maxim Neumann, AJ Piergiovanni, Michael S Ryoo, Anelia Angelova, Kris M Kitani, and Wei Hua. Attentionnas: Spatiotemporal attention cell search for video classification. In ECCV, 2020. 7
+
+[93] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In CVPR, 2018. 7
+
+[94] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon. Cbam: Convolutional block attention module. In ECCV, 2018. 2
+
+[95] Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In CVPR, 2018. 3
+
+[96] Saining Xie, Chen Sun, Jonathan Huang, Zhuowen Tu, and Kevin Murphy. Rethinking spatiotemporal feature learning: Speed-accuracy trade-offs in video classification. In ECCV, 2018. 7
+
+[97] Dejing Xu, Jun Xiao, Zhou Zhao, Jian Shao, Di Xie, and Yueting Zhuang. Self-supervised spatiotemporal learning via video clip order prediction. In CVPR, 2019. 3 [98] Jun Xu, Tao Mei, Ting Yao, and Yong Rui. Msr-vtt: A large video description dataset for bridging video and language. In CVPR, 2016. 6, 17 [99] Fuzhi Yang, Huan Yang, Jianlong Fu, Hongtao Lu, and Baining Guo. Learning texture transformer network for image super-resolution. In CVPR, 2020. 3 [100] Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang. Unsupervised embedding learning via invariant and spreading instance feature. In CVPR, 2019. 3 [101] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017. 19 [102] Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In ECCV,   
+2016. 3 [103] Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning by cross-channel prediction. In CVPR, 2017. 3 [104] Xiang Zhang, Junbo Zhao, and Yann Lecun. Character-level convolutional networks for text classification. NeurIPS, 2015. 1 [105] Y Zhang, K Li, K Li, B Zhong, and Y Fu. Residual non-local attention networks for image restoration. In ICLR, 2019. 2 [106] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun. Exploring self-attention for image recognition. In CVPR, 2020. 2 [107] Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip Torr, and Vladlen Koltun. Point transformer. arXiv preprint arXiv:2012.09164, 2020. 3 [108] Sixiao Zheng, Jiachen Lu, Hengshuang Zhao, Xiatian Zhu, Zekun Luo, Yabiao Wang, Yanwei Fu, Jianfeng Feng, Tao Xiang, Philip HS Torr, et al. Rethinking semantic segmentation from a sequence-to-sequence perspective with transformers. arXiv preprint arXiv:2012.15840, 2020.   
+3
+
+[109] Luowei Zhou, Chenliang Xu, and Jason J Corso. Towards automatic learning of procedures from web instructional videos. In AAAI, 2018. 6, 17

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+        "text": "VATT: Transformers for Multimodal Self-Supervised Learning from Raw Video, Audio and Text ",
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+        "text": "Hassan Akbari∗ Columbia University ha2436@columbia.edu ",
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+        "text": "Abstract ",
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+        "text": "We present a framework for learning multimodal representations from unlabeled data using convolution-free Transformer architectures. Specifically, our VideoAudio-Text Transformer (VATT) takes raw signals as inputs and extracts multimodal representations that are rich enough to benefit a variety of downstream tasks. We train VATT end-to-end from scratch using multimodal contrastive losses and evaluate its performance by the downstream tasks of video action recognition, audio event classification, image classification, and text-to-video retrieval. Furthermore, we study a modality-agnostic, single-backbone Transformer by sharing weights among the three modalities. We show that the convolution-free VATT outperforms state-of-the-art ConvNet-based architectures in the downstream tasks. Especially, VATT’s vision Transformer achieves the top-1 accuracy of $8 2 . 1 \\%$ on Kinetics-400, $8 3 . 6 \\%$ on Kinetics-600, $7 2 . 7 \\%$ on Kinetics-700, and $4 1 . 1 \\%$ on Moments in Time, new records while avoiding supervised pre-training. Transferring to image classification leads to $7 8 . 7 \\%$ top-1 accuracy on ImageNet compared to ${ \\bar { 6 4 . 7 \\% } }$ by training the same Transformer from scratch, showing the generalizability of our model despite the domain gap between videos and images. VATT’s audio Transformer also sets a new record on waveform-based audio event recognition by achieving the mAP of $3 9 . 4 \\%$ on AudioSet without any supervised pre-training. VATT’s source code is publicly available.2 ",
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+        "text": "1 Introduction ",
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+        "text": "Convolutional neural networks (CNNs) [53, 51] have triumphed over various computer vision tasks. The inductive bias induced by convolutions, namely translation invariance and locality, are proven effective for the visual data. In the meantime, however, we witness in the natural language processing (NLP) community a paradigm shift from the models with strong inductive biases, such as recurrent neural networks [43, 7] and CNNs [104, 32], to more general architectures constructed upon selfattention. Particularly, Transformers [88] have become the de facto model architecture for NLP tasks [23, 70, 71, 10]. Pre-training a Transformer on large text corpora followed by fine-tuning gives rise to state-of-the-art results for different downstream tasks. ",
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+            "Figure 1: Overview of the VATT architecture and the self-supervised, multimodal learning strategy. VATT linearly projects each modality into a feature vector and feeds it into a Transformer encoder. We define a semantically hierarchical common space to account for the granularity of different modalities and employ the Noise Contrastive Estimation (NCE) to train the model. "
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+        "text": "In view of the success of the attention mechanism in NLP, there has been a rich line of works exploring its potential in computer vision. Early work studied hybrid models consisting of both convolutions and attention modules [89, 94, 36, 105]. Recent studies showed that convolution-free, specially designed all-attention models can match CNNs’ performance on image recognition tasks [106, 44, 73]. Most recently, [25] achieved impressive performance on several image recognition tasks, including ImageNet [22], using a pre-trained Transformer with minimal architecture changes. Their work delivered a compelling message that “large scale (supervised) training trumps inductive bias (for image classification).” This conclusion was further extended to video recognition tasks by [9, 5]. ",
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+        "text": "However, the large-scale supervised training of Transformers is essentially troubling for two main reasons. First, it rules out the much larger other part of “big visual data,” i.e, the vast amount of unlabeled, unstructured visual data. As a result, the supervised training strategy could produce biased systems that require even more labeled data to correct their biases. Second, this strategy fundamentally limits the application scope of Transformers in computer vision because it is costly and extremely time-consuming to collect enough labeled images or videos for training the millions of parameters, choosing hyper-parameters, and validating their expected generalization. ",
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+    {
+        "type": "text",
+        "text": "Hence, this work poses another pressing question about the Transformers that take raw signals as input. How to empower them with large-scale, unlabeled visual data? To answer this question, we draw insights from NLP. BERT [23] and GPT [70, 71, 10] use masked language modeling as their pre-training tasks. Natural languages are organic supervision for Transformers. They sequentially place words, phrases, and sentences into context, granting them semantics and syntax. For visual data, the most organic supervision is arguably the multimodal videos. They are abundantly available in the digital world, and their temporal, cross-modality regulation, and therefore supervision, requires no human annotation. The extreme scale of multimodal videos is potentially capable to teach Transformers necessary priors, as opposed to predefined inductive biases, to model the visual world. ",
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+        "type": "text",
+        "text": "To this end, we study self-supervised, multimodal pre-training of three Transformers [88], which take as input the raw RGB frames of internet videos, audio waveforms, and text transcripts of the speech audio, respectively. We call the video, audio, text Transformers VATT. Figure 1 illustrates the architecture. VATT borrows the exact architecture from BERT [23] and ViT [25] except the layer of tokenization and linear projection reserved for each modality separately. This design shares the same spirit as ViT that we make the minimal changes to the architecture so that the learned model can transfer its weights to various frameworks and tasks. Furthermore, the self-supervised, multimodal learning strategy resonates the spirit of BERT and GPT that the pre-training requires minimal human curated labels. ",
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+        "type": "text",
+        "text": "We evaluate the pre-trained Transformers on a variety of downstream tasks: image classification, video action recognition, audio event classification, and zero-shot text-to-video retrieval. Fine-tuning the vision-modality Transformer on ImageNet [22] obtains the top-1 accuracy of $7 8 . 7 \\%$ , which is comparable to $7 9 . { \\dot { 9 } } \\%$ achieved by ViT. This result is especially appealing considering the domain gap between videos and images, and that ViT is pre-trained using a large-scale, human-curated image dataset. Furthermore, we set new records on Kinetics-400 [14], Kinetics-600 [15], Moments in Time [61], and AudioSet [33] without supervised pre-training. ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Our VATT results, along with others reported for NLP tasks [23, 10], image recognition [25], semantic segmentation [108], point cloud classification [107], and action recoginition [9], demonstrate that Transformer is a versatile general-purpose architecture for different types of data. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "To move one step forward, we challenge the Transformers in VATT by a seemingly too strong constraint: sharing weights among the video, audio, and text modalities. The idea is to test whether there exists a single, general-purpose model for all the modalities — of course, they still have their own layers of tokenization and linear projection. Preliminary results are encouraging. This modality-agnostic Transformer is on par with three modality-specific ones of slightly smaller sizes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, another contribution of this work is DropToken, a simple and yet effective technique to reduce the training complexity with a minor reduction of the end Transformers’ performance. DropToken randomly drops a portion of the video and audio tokens from each input sequence during training, allowing for high-resolution inputs and leveraging their abundance. This is significant for Transformers because their computational complexity is quadratic with respect to the number of input tokens. ",
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 Transformers in Vision ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Transformer was originally built for NLP tasks [88] and the design of multi-head attention shows its effectiveness on modeling long-term correlation of words. A few attempts have been made to use Transformer for vision tasks like image super-resolution [99], object detection [11] and multimodal video understanding [84, 19, 57]. However these methods still rely on the feature extracted by CNNs. Recently, [25] proposes a set of convolution-free vision Transformers which directly work on raw images and obtain competitive performance with CNNs. [86] improves the training data efficiency of [25] by using stronger data augmentations and knowledge distillation. Since then, the pure Transformer design has been adopted to various vision tasks including semantic segmentation [108], point cloud classification [107], action recoginition [9, 78, 5]. To the best of our knowledge, our VATT is the first Transformer model on raw multimodal inputs of video, audio and text. ",
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+    {
+        "type": "text",
+        "text": "2.2 Self-Supervised Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Single vision modality. Early work of self-supervised visual representation learning usually learns from unlabeled images via manually specified pretext tasks, like auto-encoding [64, 102, 103], patch location prediction [24], solving jigsaw puzzles [63], and image rotation prediction [35]. [95] propose a novel instance discrimination objective. The recent trend of contrastive learning [40, 17, 100, 37, 41, 85] integrates data augmentations and instance discrimination by maintaining relative consistency between representations of an image and its augmented view. Clustering can also provide an effective addition [12]. Recently, [18] conduct contrastive learning using ViT [25] and achieve impressive results. As for the video domain, it is natural to exploit the temporal signals as the pretext task. Examples include predicting the future frame [82], motion and appearance statistics [90], speed [8, 91] and encodings [56, 38, 39], sorting frames or video clips [54, 97, 45, 31]. Recently, [68] apply contrastive learning to videos with a temporal sampling strategy and temporally consistent spatial augmentation. ",
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+    {
+        "type": "text",
+        "text": "Multimodal video. Video is a natural source of multimodal data. Multimodal self-supervised learning can be achieved by predicting whether a video has correspondence with an audio stream [3, 4, 62, 50], cross-modality clustering [2], and evolving losses [67]. Recently, [1] use contrastive loss to learn from video, audio and text; [74] learn to predict a broad view that spans a longer temporal context from a narrow view. VATT serves as a first work combining the strength of convolution-free Transformer and multimodal contrastive learning. ",
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+    {
+        "type": "text",
+        "text": "3 Approach ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we introduce our convolution-free VATT architecture and elaborate on the selfsupervised multimodal objectives for training VATT from scratch. ",
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+        "type": "text",
+        "text": "Figure 1 is an overview of the architecture. We feed each modality to a tokenization layer, where the raw input is projected to an embedding vector followed by a Transformer. There are two major settings: 1) The backbone Transformers are separate and have specific weights for each modality, and 2) The Transformers share weights, namely, there is a single backbone Transformer applied to any of the modalities. In either setting, the backbone extracts modality-specific representations, which are then mapped to common spaces to be compared with each other by contrastive losses. We describe each module in the following. ",
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+    {
+        "type": "text",
+        "text": "3.1 Tokenization and Positional Encoding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "VATT operates on raw signals. The vision-modality input consists of 3-channel RGB pixels of video frames, the audio input is in the form of air density amplitudes (waveforms), and the text input is a sequence of words. We first define a modality-specific tokenization layer that takes as input the raw signals and returns a sequence of vectors to be fed to the Transformers. Besides, each modality has its own positional encoding, which injects the order of tokens into Transformers [88]. We partition an entire video clip of size $T \\times H \\times W$ to a sequence of $\\lceil T / t \\rceil \\cdot \\lceil H / h \\rceil \\cdot \\lceil W / w \\rceil$ patches, where each patch contains $t \\times h \\times w \\times 3$ voxels. We apply a linear projection on the entire voxels in each patch to get a $d$ -dimensional vector representation. This projection is performed by a learnable weight $W _ { v p } \\in \\mathbb { R } ^ { t \\cdot h \\cdot w \\cdot 3 \\times d }$ . This can be seen as a 3D extension of the patching mechanism proposed in [25]. To encode the position of these patches, we define a dimension-specific sequence of learnable embeddings as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/6adce449c3de86090036ff5f81e4ec11c880952d7de9e2bcf5c4c81ceebdbcc8.jpg",
+        "text": "$$\n\\begin{array} { r l } { e _ { i , j , k } = e _ { \\mathrm { T e m p o r a l } _ { i } } + e _ { \\mathrm { H o r i z o n t a l } _ { j } } + e _ { \\mathrm { V e r t i c a l } _ { k } } , } & { } \\\\ { E _ { \\mathrm { T e m p o r a l } } \\in \\mathbb { R } ^ { \\lceil T / t \\rceil \\times d } , } & { E _ { \\mathrm { H o r i z o n t a l } } \\in \\mathbb { R } ^ { \\lceil H / h \\rceil \\times d } , \\quad E _ { \\mathrm { V e r t i c a l } } \\in \\mathbb { R } ^ { \\lceil W / w \\rceil \\times d } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $e _ { i }$ is the $i$ -th row of $\\pmb { { \\cal E } }$ . This scheme allows us to use $\\lceil T / t \\rceil + \\lceil H / h \\rceil + \\lceil W / w \\rceil$ positional embeddings to encode all the $\\lceil T / t \\rceil \\cdot \\lceil H / h \\rceil \\cdot \\lceil W / w \\rceil$ patches in a video clip. The raw audio waveform is a 1D input with length $T ^ { \\prime }$ , and we partition it to $\\lceil T ^ { \\prime } / t ^ { \\prime } \\rceil$ segments each containing $t ^ { \\prime }$ waveform amplitudes. Similar to video, we apply a linear projection with a learnable weight $W _ { a p } \\in \\mathbb { R } ^ { t ^ { \\prime } \\times d }$ to all elements in a patch to get a $d$ -dimensional vector representation. We use $\\lceil T ^ { \\prime } / t ^ { \\prime } \\rceil$ learnable embeddings to encode the position of each waveform segment. For text, we first construct a vocabulary of size $v$ out of all words in our training dataset. For an input text sequence, we then map each word to a $v$ -dimensional one-hot vector followed by a linear projection with a learnable weight $W _ { t p } \\in \\mathbb { R } ^ { v \\times d }$ . This is equivalent to an embedding dictionary lookup, which has been widely used in natural language understanding [60]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1.1 DropToken ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We introduce DropToken, a simple and yet effective strategy to reduce the computational complexity during training. Once we get the token sequence for the video or audio modality, we randomly sample a portion of the tokens and then feed the sampled sequence, not the complete set of tokens, to the Transformer. This is crucial for reducing the computational cost because a Transformer’s computation complexity is quadratic, $O ( N ^ { 2 } )$ , where $N$ is number of tokens in the input sequence. Any effort on reducing the input length would reduce the number of FLOPs quadratically. This has an immediate impact on the wall clock time for training these models and makes it possible to host large models in limited hardware. We argue that instead of reducing the resolution or dimension of the raw inputs, it is better to keep a high-fidelity input and randomly sample the tokens via DropToken. DropToken is appealing especially with the raw video and audio inputs, which may contain high redundancies. ",
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+    {
+        "type": "text",
+        "text": "3.2 The Transformer Architecture ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For simplicity, we adopt the most established Transformer architecture [23], which has been widely used in NLP. Similar to ViT [25], we do not tweak the architecture so that our weights can be easily transferred to any standard Transformer implementation. We will briefly elaborate on the pipeline (also illustrated in Figure 1 middle panel) and refer the reader to [25, 23] for more details of the standard Transformer architecture. The sequence of input tokens to the Transformer follows the below formulation: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/6f96dd30b74d4fbcbd295dbd7078b11d58db1de9c972aefcc9c77e196fc50f18.jpg",
+        "text": "$$\nz _ { \\mathrm { i n } } = [ x _ { \\mathrm { A G G } } ; ~ x _ { 0 } W _ { P } ; ~ x _ { 1 } W _ { P } ; \\ldots ; ~ x _ { N } W _ { P } ] + e _ { \\mathrm { P O S } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where ${ \\mathbf { \\mathcal { x } } } _ { n }$ is the input patches sequence and $\\pmb { x } _ { \\mathrm { A G G } }$ is the learnable embedding of a special aggregation token whose corresponding output in the Transformer $( z _ { \\mathrm { o u t } } ^ { 0 } )$ is used as the aggregated representation for the entire input sequence. This will be later used for classification and common space mapping. We use a standard self-attention [88] as the Multi-Head-Attention (MHA) module, and GeLU [42] as the activation in the MLP layer. We also use Layer Normalization [6] before the MHA and MLP modules. In our text model, we remove the position encoding $e _ { \\mathrm { P O S } }$ and add a learnable relative bias to each attention score of the first layer in the MHA module. This simple change makes our text model’s weights directly transferable to the state-of-the-art text model T5 [72]. ",
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+    {
+        "type": "text",
+        "text": "3.3 Common Space Projection ",
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+        "type": "text",
+        "text": "We use common space projection and contrastive learning in that common space to train our networks. More specifically, given a video-audio-text triplet, we define a semantically hierarchical common space mapping that enables us to directly compare video-audio pairs as well as video-text pairs by the cosine similarity. As argued in [1], such comparison is more feasible if we assume there are different levels of semantic granularity for these modalities. To achieve this, we define multi-level projections as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/b55bec1221bbb22d2842cce88674cbef40d822d22b5130812b78534fc3274ef9.jpg",
+        "text": "$$\n\\begin{array} { r l } { z _ { v , v a } = g _ { v \\to v a } ( z _ { \\mathrm { o u t } } ^ { \\mathrm { v i d e o } } ) , \\ } & { z _ { a , v a } = g _ { a \\to v a } ( z _ { \\mathrm { o u t } } ^ { \\mathrm { a u d i o } } ) } \\\\ { z _ { t , v t } = g _ { t \\to v t } ( z _ { \\mathrm { o u t } } ^ { \\mathrm { t e x t } } ) , \\ } & { z _ { v , v t } = g _ { v \\to v t } ( z _ { v , v a } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { v  v a }$ and $g _ { a  v a }$ are the projection heads to respectively map the video and audio Transformers’ outputs to the video-audio common space $\\boldsymbol { S _ { v a } }$ . Moreover, $g _ { t  v t }$ and $g _ { v  v t }$ project the text Transformer’s outputs and the video embedding in the $\\boldsymbol { S } _ { v a }$ space to video-text common space, $\\boldsymbol { S _ { v t } }$ . This multi-level common space projection is depicted in Figure 1 (the rightmost panel). The main intuition behind this hierarchy is that different modalities have different levels of semantic granularity, so we should impose this as an inductive bias in the common space projection. Similar to [1], we use a linear projection for $g _ { a  v a } ( . )$ , $g _ { t  v t } ( . )$ , and $g _ { v  v t } ( . )$ , and a two-layer projection with ReLU in between for $g _ { v  v a } ( . )$ . To ease the training, a batch normalization is used after each linear layer. ",
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+    {
+        "type": "text",
+        "text": "3.4 Multimodal Contrastive Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Inspired by [1, 3, 59], we use Noise Contrastive Estimation (NCE) to align video-audio pairs and Multiple Instance Learning NCE (MIL-NCE) to align video-text pairs. The pairs are composed from different temporal locations in the video-audio-text stream. Positive pairs from two modalities are constructed by sampling their corresponding streams from the same location in the video, and negative pairs are constructed by sampling from any non-matching locations in the video [1]. Concretely, given the common space specified in Section 3, the loss objectives can be written as follows: ",
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+        "type": "equation",
+        "img_path": "images/7c2f0f8f4a01ebe1eac39ffcd4c7dcaff59c601d99a6daab504d10ff546fbcc3.jpg",
+        "text": "$$\n\\mathrm { N C E } ( z _ { v , v a } , z _ { a , v a } ) = - \\log \\left( \\frac { \\exp ( z _ { v , v a } ^ { \\top } z _ { a , v a } / \\tau ) } { \\exp ( z _ { v , v a } ^ { \\top } z _ { a , v a } / \\tau ) + \\sum _ { z ^ { \\prime } \\in \\mathcal { N } } \\exp ( z _ { v , v a } ^ { \\prime \\top } z _ { a , v a } ^ { \\prime } / \\tau ) } \\right) ,\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/184c3de8d8ee0dcc733157d3a1c56997967b05ee53dd7352a9487da389e07e18.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { A } \\mathrm { I L } \\mathrm { - N C E } ( z _ { v , v t } , \\{ z _ { t , v t } \\} ) = - \\log \\left( \\frac { \\sum _ { z _ { k , v t } \\in \\mathcal { P } } \\exp ( z _ { v , v t } ^ { \\top } z _ { t , v t } / \\tau ) } { \\sum _ { z _ { t , v t } \\in \\mathcal { P } } \\exp ( z _ { v , v t } ^ { \\top } z _ { t , v t } / \\tau ) + \\sum _ { z ^ { \\prime } \\in \\mathcal { N } } \\exp ( z _ { v , v t } ^ { \\prime \\top } z _ { t , v t } ^ { \\prime } / \\tau ) } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { N }$ contains all non-matching pairs in a batch. In Equation 5, $\\mathcal { P }$ contains five text clips that are nearest neighbors to the video clip in time. $\\tau$ is a temperature to adjust the softness of the objectives in distinguishing the positive pairs from the negative pairs. ",
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+    {
+        "type": "text",
+        "text": "The overall per-sample objective for training the entire VATT model end-to-end is as follows: ",
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+        "type": "equation",
+        "img_path": "images/a15d86359e82321ef689dcc747011edecd3198b810a077e769152f01899ed49e.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathrm { N C E } ( z _ { v , v a } , z _ { a , v a } ) + \\lambda \\mathrm { M I L } \\mathrm { - N C E } ( z _ { v , v t } , \\{ z _ { t , v t } \\} ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "where $\\lambda$ balances the two losses. The model is optimized based on the back-propagation of the average loss calculated over a batch of samples. ",
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+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we first briefly describe the experimental setup for the pre-training and downstream evaluation, and then present the results and analytic interpretation of VATT in different tasks. We refer the reader to the Appendix for a more detailed description of all experimental settings. ",
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+        "type": "text",
+        "text": "4.1 Experimental Setup ",
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+        "type": "text",
+        "text": "Pre-train: we use a combination of AudioSet [33] and HowTo100M [58] datasets to pre-train VATT— we use only a subset of the HowTo100M dataset in compliance with Youtube’s policies. Following [1], we use video-audio-text triplets from HowTo100M clips while only using video-audio pairs from AudioSet. We sample 32 frames at 10 fps with a spatial size of $2 2 4 \\times 2 2 4$ following a random crop, horizontal flip and color augmentation (details in A.2.1). Accordingly, we sample audio waveforms in sync at $4 8 \\mathrm { k H z }$ . Both video and audio are normalized between [-1,1]. We use patch sizes of $4 \\times 1 6 \\times 1 6$ and 128 for video and raw waveform tokenization, respectively (ablation in A.5). We use one-hot vectors to encode text sequences (capped to 16 tokens) with the vocabulary size of $2 ^ { 1 6 }$ . In all pre-training experiments, we use DropToken with drop rate $50 \\%$ . We train our models using the Adam optimizer [46] with a quarter-period cosine scheduled learning rate from 1e-4 to 5e-5 and 10k warmup steps. Optimization is performed on totally $5 0 0 \\mathrm { k }$ steps with batch size 2048 (512 in exploration experiments). Following the previously established practice [1] for the projection to the common spaces $\\boldsymbol { S _ { v a } }$ and $\\boldsymbol { S _ { v t } }$ , we use $d _ { v a } = 5 1 2$ and $d _ { v t } = 2 5 6$ . We also use the temperature of $\\tau = 0 . 0 7$ and the weight of $\\lambda = 1$ in the loss in Equation 6. We use 4 network sizes in our experiments (details in A.2.2). We use the Medium model (155M parameters) for our modality-agnostic variant (VATT-MA), and 3 variants for the modality-specific video-audio-text backbones: Base-Base-Small (BBS; 197M), Medium-Base-Small (MBS; 264M), and Large-Base-Small (LBS; 415M). Pre-training an MBS VATT with batch size 2048 on 256 TPUs (v3) takes less than 3 days. Pre-training with batch size 512 takes less than 1 day. ",
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+        "text": "Downstream: we evaluate the pre-trained VATT models on 4 major downstream tasks using a total of 10 datasets. We use UCF101 [81], HMDB51 [52], Kinetics-400 [14], Kinetics-600 [15], and Moments in Time [61] for video action recognition. We use ESC50 [66] and AudioSet [33] for audio event classification, and we evaluate the quality of our video-text common space representations by zero-shot text-to-video retrieval on YouCook2 [109] and MSR-VTT [98]. Finally, we evaluate the transferability of the vision backbone by fine-tuning it on ImageNet classification [22]. Since HMDB51, UCF101, and ESC50 are very small datasets compared to the size of our networks, we only use them to train a linear classifier on top of the frozen pre-trained backbones. In our exploration experiments, we report linear classification accuracy and zero-shot video retrieval metrics. We refer to the Appendix for a detailed description of the datasets and the experimental setup. ",
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+    {
+        "type": "text",
+        "text": "4.2 Results ",
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+        "text": "4.2.1 Fine-tuning for video action recognition ",
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+        "type": "text",
+        "text": "We fine-tune VATT’s vision Transformer on Kinetics-400, Kinetics-600, and Moments in Time, three of the arguably most established large-scale datasets for video action recognition. We use the final checkpoints of four pre-train settings for these experiments: three modality-specific variations (LBS, MBS, BBS), and one modality-agnostic (Medium). Table 1 shows the results compared with the state-of-the-art video models. On all three datasets, we achieve higher accuracy than previous works including TimeSFormer [9], a recent effort in fine-tuning the ViT checkpoints obtained by supervised pre-training. In contrast, our pre-training does not rely on any labels curated by humans. To the best of our knowledge, VATT provides the first vision Transformer backbone that is pre-trained from scratch using self-supervision on multimodal videos and achieves state-of-the-art results on video action recognition. It is also worth mentioning that fine-tuning VATT on the most recent Kinetics-700 dataset results in a top-1 accuracy of $7 2 . 7 \\%$ , which outperforms the state-of-the-art top-1 accuracy of $7 2 . 4 \\%$ in [47]. ",
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+        "type": "text",
+        "text": "To further quantify how much the multimodal self-supervised pre-training helps in achieving these numbers, we train a variant from scratch without any pre-training and observe the top-1 and top-5 accuracies of $2 6 . 4 \\%$ and $5 1 . 8 \\%$ on Kinetics-400, respectively. The low accuracies verify the efficacy of our pre-training strategy for VATT. Finally, we find that VATT-MA-Medium, the modality-agnostic backbone shared by the video, audio, and text modalities, is on par with the modality-specific VATTBase when fine-tuned for the video action recognition. This result is encouraging as it indicates the potential of unifying three data modalities by a single Transformer backbone. ",
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+        "type": "table",
+        "img_path": "images/8c187caa118db642d38bd481d9aa692eaa2264b34591f1073deb90f0acf7bf28.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Video action recognition accuracy on Kinetics-400, Kinetics-600, and Moments in Time. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Kinetics-400</td><td colspan=\"2\">Kinetics-600</td><td colspan=\"2\">Moments in Time</td><td rowspan=\"2\">TFLOPS</td></tr><tr><td>METHOD</td><td>ToP-1</td><td>TOP-5</td><td>TOP-1</td><td>TOP-5</td><td>TOP-1</td><td>TOP-5</td></tr><tr><td>I3D[13]</td><td>71.1</td><td>89.3</td><td>71.9</td><td>90.1</td><td>29.5</td><td>56.1</td><td></td></tr><tr><td>R(2+1)D [26]</td><td>72.0</td><td>90.0</td><td>-</td><td>-</td><td>1</td><td>1</td><td>17.5</td></tr><tr><td>bLVNet [27]</td><td>73.5</td><td>91.2</td><td>-</td><td>/</td><td>31.4</td><td>59.3</td><td>0.84</td></tr><tr><td>S3D-G [96]</td><td>74.7</td><td>93.4</td><td>=</td><td></td><td>-</td><td>-</td><td>-</td></tr><tr><td>Oct-I3D+NL [20]</td><td>75.7</td><td>1</td><td>76.0</td><td></td><td>=</td><td>1</td><td>0.84</td></tr><tr><td>D3D[83]</td><td>75.9</td><td>=</td><td>77.9</td><td></td><td></td><td></td><td>=</td></tr><tr><td>I3D+NL [93]</td><td>77.7</td><td>93.3</td><td>-</td><td></td><td></td><td></td><td>10.8</td></tr><tr><td>ip-CSN-152 [87]</td><td>77.8</td><td>92.8</td><td>1</td><td></td><td></td><td>=</td><td>3.3</td></tr><tr><td>AttentionNAS [92]</td><td>-</td><td>1</td><td>79.8</td><td>94.4</td><td>32.5</td><td>60.3</td><td>1.0</td></tr><tr><td>AssembleNet-101 [77]</td><td>-</td><td>=</td><td>-</td><td>-</td><td>34.3</td><td>62.7</td><td>1</td></tr><tr><td>MoViNet-A5 [47]</td><td>78.2</td><td>1</td><td>82.7</td><td>1</td><td>39.1</td><td>-</td><td>0.29</td></tr><tr><td>LGD-3D-101 [69]</td><td>79.4</td><td>94.4</td><td>81.5</td><td>95.6</td><td>-</td><td>=</td><td>-</td></tr><tr><td>SlowFast-R101-NL [30]</td><td>79.8</td><td>93.9</td><td>81.8</td><td>95.1</td><td></td><td></td><td>7.0</td></tr><tr><td>X3D-XL [29]</td><td>79.1</td><td>93.9</td><td>81.9</td><td>95.5</td><td>=</td><td></td><td>1.5</td></tr><tr><td>X3D-XXL [29]</td><td>80.4</td><td>94.6</td><td>1</td><td>-</td><td></td><td></td><td>5.8</td></tr><tr><td>TimeSFormer-L [9]</td><td>80.7</td><td>94.7</td><td>82.2</td><td>95.6</td><td>-</td><td>=</td><td>7.14</td></tr><tr><td>VATT-Base</td><td>79.6</td><td>94.9</td><td>80.5</td><td>95.5</td><td>38.7</td><td>67.5</td><td>9.09</td></tr><tr><td>VATT-Medium</td><td>81.1</td><td>95.6</td><td>82.4</td><td>96.1</td><td>39.5</td><td>68.2</td><td>15.02</td></tr><tr><td>VATT-Large</td><td>82.1</td><td>95.5</td><td>83.6</td><td>96.6</td><td>41.1</td><td>67.7</td><td>29.80</td></tr><tr><td>VATT-MA-Medium</td><td>79.9</td><td>94.9</td><td>80.8</td><td>95.5</td><td>37.8</td><td>65.9</td><td>15.02</td></tr></table>",
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+        "text": "4.2.2 Fine-tuning for audio event classification ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We fine-tune VATT’s audio Transformer on AudioSet, which benchmarks the task of multi-label audio event classification. We use the final checkpoints of two pre-train settings: one modality-specific (BBS), and one modality-agnostic (Medium). Table 2 shows the results compared to state-of-the-art models. Following common practice [34, 48], we report mean Average Precision (mAP), Area Under Curve (AUC), and d-prime (based on AUC) [34]. Our audio Transformer consistently outperforms the existing CNN-based models in all metrics. More interestingly, fine-tuning the modality-agnostic backbone (VATT-MA-Medium) is on par with fine-tuning the modality-specific one (VATT-Base). To the best of our knowledge, VATT is the first Transformer that outperforms CNN-based models in audio event recognition. VATT operates on raw waveforms and does not utilize any handcrafted features. ",
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+    {
+        "type": "text",
+        "text": "4.2.3 Fine-tuning for image classification ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we show that our pipeline is capable of transferring the learned knowledge into another domain by performing the image classification task, even though the models are pre-trained in the multimodal video domain. We fine-tune the vision Transformer in VATT-BBS on ImageNet without any modification to the backbone architecture. Instead, to satisfy the voxel-to-patch layer we replicate the input image 4 times and feed it to the network. The network sees the input as a single-frame video clip and performs spatial self-attention. Table 3 shows the results for fine-tuning the vision Transformer end-to-end on ImageNet. We can see that our pre-training leads to a significant boost in the accuracy compared to training from scratch. We also observe that even though the self-supervised pre-training happens in the video domain, we still achieve competitive results to the supervised pre-training using large-scale image data [25]. ",
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+    {
+        "type": "text",
+        "text": "4.2.4 Zero-shot text-to-video retrieval ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We feed video-text pairs to VATT-MBS, and extract representations in the $\\boldsymbol { S _ { v t } }$ space. We then calculate the similarity between each video-text pair from YouCook2 and MSR-VTT. Given a text query, we rank the videos based on their similarities to the text. We then measure the recall for the correct video in the top-10 videos. We also measure the median of the rank of the correct video. Table 4 compares our video retrieval results to two baselines. In our experiments we observe that the zero-shot retrieval results are heavily affected by the batch size and number of epochs, confirming the observation made in [1]. That said, our model still delivers comparable results to MMV [1] while being pre-trained with a half number of epochs and a half batch size of theirs. We also experiment with a larger batch size 8192 and longer pre-training for 6 epochs, arriving at exactly the same results as MIL-NCE [59] on YouCook2 and the $\\mathbf { R } @ 1 0$ of 29.2 and MedR of 42 on MSR-VTT. We also notice that, probably due to the noisy nature of text transcripts, a sophisticated language model like ours is underrated. As shown in [1], using a simple linear projection would still perform reasonably well. It is worth exploring other, higher-quality text sources in future work. ",
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+    {
+        "type": "table",
+        "img_path": "images/dd9f1aeedf2e59982d3207dd4606042013ed20abb2c91c19bc3014a71ea36cc6.jpg",
+        "table_caption": [
+            "Table 2: Finetuning results for AudioSet event classification. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td>mAP AUC d-prime</td></tr><tr><td>DaiNet [21]</td><td>29.5 95.8 2.437</td></tr><tr><td>LeeNet11 [55]</td><td>26.6 95.3 2.371</td></tr><tr><td>LeeNet24 [55]</td><td>33.6 96.3 2.525</td></tr><tr><td>Res1dNet31 [49]</td><td>36.5 95.8 2.444</td></tr><tr><td>Res1dNet51 [49]</td><td>35.5 94.8 2.295</td></tr><tr><td>Wavegram-CNN [49]</td><td>38.9 96.8 2.612</td></tr><tr><td>VATT-Base</td><td>39.4 97.1 2.895</td></tr><tr><td>VATT-MA-Medium</td><td>39.3 97.0 2.884</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Finetuning results for ImageNet classification. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td>PRE-TRAINING DATA</td><td>TOP-1</td><td>TOP-5</td></tr><tr><td>iGPT-L [16]</td><td>ImageNet</td><td>72.6</td><td>1</td></tr><tr><td>ViT-Base [25]</td><td>JFT</td><td>79.9</td><td>1</td></tr><tr><td>VATT-Base</td><td>-</td><td>64.7</td><td>83.9</td></tr><tr><td>VATT-Base</td><td>HowTo100M</td><td>78.7</td><td>93.9</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/2618937fd285200a1452e39c9e610e04e4a402860d6a2f2b0a49d941f483302a.jpg",
+        "table_caption": [
+            "Table 4: Zero-shot text-to-video retrieval. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">YouCook2 MSR-VTT</td></tr><tr><td>METHOD</td><td>BATCH EPOCH R@10 MedR R@ 10 MedR 27</td><td></td><td></td><td></td></tr><tr><td>MIL-NCE [59]</td><td>8192</td><td>51.2</td><td>10 32.4</td><td>30</td></tr><tr><td>MMV[1]</td><td>4096 8</td><td>45.4</td><td>13 31.1</td><td>38</td></tr><tr><td>VATT-MBS</td><td>2048 4</td><td>45.5</td><td>13 29.7</td><td>49</td></tr><tr><td>VATT-MA-Medium</td><td>2048 4</td><td>40.6</td><td>17</td><td>23.6 67</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4.2.5 Feature visualization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We take our modality-specific and modality-agnostic VATT fine-tuned on Kinetics-400 and visualize their output feature representations using t-SNE. For comparison, we also include the feature visualization of the vision Transformer trained from scratch on Kinetics-400. From Figure 2, we observe that the fine-tuned VATT yields a much better separation than the model trained from scratch. Furthermore, it is worth noting that there is no clear difference between the modality-agnostic features and the modality-specific ones. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We further investigate the VATT backbones without any fine-tuning. We randomly choose 1k video clips from the YouCook2 dataset and store the representations from two points of a pre-trained VATT model. One is after the tokenization layer (input space of the Transformer), and the other is after the common space projection (output space), where the loss is computed. Figure 3-top visualizes the representations, comparing modality-specific VATT to modality-agnostic VATT. Interestingly, we observe that the representations are slightly more mixed together in the modality-agnostic setting compared to the modality-specific ones, implying that the modality-agnostic backbone sees different modalities as different symbols describing the same concept. This is analogous to a unified language model in NLP that supports multiple languages. ",
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+    {
+        "type": "text",
+        "text": "To see how well VATT distinguishes positive video-text pairs from randomly sampled pairs, we calculate pair-wise similarities for all possible pairs and perform a Kernel Density Estimation (KDE) to visualize the distributions of the similarities of the positive pairs vs. negative pairs. We perform this procedure for both input and output spaces of the modality-specific and modality-agnostic backbones. Figure 3-bottom shows the KDE curves of these similarities. We can see that VATT in both settings separates the positive and negative pairs in its output space. This verifies VATT’s efficacy in learning a semantic common space for different modalities, even if we share the backbone across modalities. ",
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+    {
+        "type": "text",
+        "text": "4.2.6 Model Activations ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We measure the average activation of the modality-agnostic VATT when a full multimodal input is fed to the model. More specifically, we sample $1 0 0 \\mathrm { k }$ short video clips from the test split of HowTo100M along with their corresponding audio and text and feed them to the model separately. For each modality, we calculate the average activation of each node at the output of the MLP module, before the residual addition (Figure 1-Transformer Encoder). Figure 4 shows the average activations across all nodes in a Medium-size model. We observe that earlier nodes in the model are activated with the text inputs, while the middle-to-later nodes are activated with video and audio modalities. However, the nodes in the last layers of the network are activated with all modalities almost equally. This might suggest that the model allocates different nodes to certain modalities while reaching the same level of semantic perception for all modalities in the later layers. Such observation encourages further studies on the possibility of utilizing Mixture-of-Experts [79, 28, 76] to increase the model’s capacity for simultaneous multimodal perception. We leave this direction of research for future work. ",
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+    {
+        "type": "image",
+        "img_path": "images/264dcf3b3a980aec5d2a5f2a0fee6ce340c013a3c55bff2abf50990cf4d4afe8.jpg",
+        "image_caption": [
+            "Figure 2: t-SNE visualization of the feature representations extracted by the vision Transformer in different training settings. For better visualization, we show 100 random classes from Kinetics-400. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6f8e58fd69ca0d895846cdad125dcf844cc929bef9f7744a0c560ca0c9eb205.jpg",
+        "image_caption": [
+            "Figure 3: t-SNE visualization and distribution of pair-wise similarities of the input space vs. output space for modality-specific and modality-agnostic backbones when different modalities are fed. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2.7 Effect of DropToken ",
+        "text_level": 1,
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "We introduced a new method to reduce the redundancy in high-resolution data. To study the effect of the proposed DropToken method on downstream applications and the pre-training computation, we perform pre-training by randomly dropping $7 5 \\%$ , $5 0 \\%$ , $2 5 \\%$ , and $0 \\%$ (no drop) of the tokens from the video and audio inputs. Table 5 shows the accuracy of linear classification on HMDB51, UCF101, ESC50 and $\\mathrm { R @ 1 0 }$ on YouCook2 and MSR-VTT vs. the drop rate along with GFLOPs during a forward call. We choose $5 0 \\%$ sampling rate for our large-scale pre-training as it offers a good trade-off between accuracy and computational costs. We then take the final checkpoint of the pre-trained VATT with $5 0 \\%$ DropToken rate and perform fine-tuning on Kinetics-400 at different DropToken rates and at different spatial and temporal resolutions to see how high-resolution inputs coupled with DropToken compare to low-resolution inputs with no tokens dropped during fine-tuning. Table 6 shows the top-1 accuracy on Kinetics-400. We argue against using low-resolution inputs, which is the most common approach to reduce the computational cost during training. Instead, we suggest using high-resolution inputs with DropToken, whose accuracy and training cost are comparable to or better than low-resolution counterparts. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/78326d8ea06d3a0ed997fe60a987577bc3f5e4296c66b02b900d7bc3adee2312.jpg",
+        "image_caption": [
+            "Late Activation for Video & Audio ",
+            "Figure 4: The average node activation across the Modality-Agnostic-Medium VATT while feeding a multimodal video-audio-text triplet to the model. "
+        ],
+        "image_footnote": [],
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+        "table_footnote": [
+            "Table 5: Top-1 accuracy of linear classification and $\\mathrm { R @ 1 0 }$ of video retrieval vs. drop rate vs. inference GFLOPs in the VATT-MBS. "
+        ],
+        "table_body": "<table><tr><td></td><td>DropToken Drop Rate 75%</td></tr><tr><td>Multimodal GFLOPs 188.1</td><td>50% 25% 0% 375.4 574.2 784.8</td></tr><tr><td>HMDB51 62.5</td><td>65.6 66.4</td></tr><tr><td>UCF101</td><td>64.8 85.5 87.2</td></tr><tr><td>84.0 ESC50 78.9</td><td>84.1 84.6</td></tr><tr><td>YouCookII 17.9</td><td>20.7 24.2</td></tr><tr><td>14.1</td><td>14.6 15.1</td></tr></table>",
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+        "table_footnote": [
+            "Table 6: Top-1 accuracy of video action recognition on Kinetics400 using high-resolution inputs coupled with DropToken vs. low-resolution inputs. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Resolution/ FLOPs</td><td colspan=\"4\">DropToken Drop Rate</td></tr><tr><td>75%</td><td>50%</td><td>25%</td><td>0%</td></tr><tr><td>32 × 224 × 224 Inference (GFLOPs)</td><td></td><td></td><td>=</td><td>79.9 548.1</td></tr><tr><td>64×224 ×224 Inference (GFLOPs)</td><td>= -</td><td>=</td><td>= 1</td><td>80.8 1222.1</td></tr><tr><td>32 ×320×320 Inference (GFLOPs)</td><td>79.3 279.8</td><td>80.2 572.5</td><td>80.7 898.9</td><td>81.1 1252.3</td></tr></table>",
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+        "text": "5 Conclusion and Discussion ",
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+        "text": "In this paper, we present a self-supervised multimodal representation learning framework based on Transformers. Our study suggests that Transformers are effective for learning semantic video/audio/text representations — even if one model is shared across modalities — and multimodal self-supervised pre-training is promising for reducing their dependency on large-scale labeled data. We show that DropToken can significantly reduce the pre-training complexity with video and audio modalities and have minor impact on the models’ generalization. We report new records of results on video action recognition and audio event classification and competitive performance on image classification and video retrieval. Having these results, we still see some limitations in our work. Firstly, not all videos have organic audio or speech, while our approach depends on meaningful multimodal correspondences. Besides, the text modality currently consists of speech transcripts, which are noisy and sometimes sparse. Potential negative Societal Impacts are mainly concerned with applications. The models could be biased if one applies our approach to the multimodal videos that are not representative enough. Finally, our method is still demanding in computation, though we managed to avoid the need for human labels. Future work can improve upon these limitations. ",
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+    {
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+        "text": "Acknowledgments and Disclosure of Funding ",
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+        "text": "We would like to thank Min-Hsuan Tsai, Jean-Baptise Alayrac, Andrew Audibert, Yeqing Li, Vidush Mukund, and the TensorFlow team for their help with codes, infrastructure, and insightful discussions. ",
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+            174,
+            883,
+            825,
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            266,
+            106
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+    },
+    {
+        "type": "text",
+        "text": "[1] Jean-Baptiste Alayrac, Adrià Recasens, Rosalia Schneider, Relja Arandjelovic, Jason Rama- ´ puram, Jeffrey De Fauw, Lucas Smaira, Sander Dieleman, and Andrew Zisserman. Selfsupervised multimodal versatile networks. In NeurIPS, 2020. 3, 5, 6, 8, 17, 18, 19, 20 [2] Humam Alwassel, Dhruv Mahajan, Bruno Korbar, Lorenzo Torresani, Bernard Ghanem, and Du Tran. Self-supervised learning by cross-modal audio-video clustering. arXiv preprint arXiv:1911.12667, 2019. 3, 20 [3] Relja Arandjelovic and Andrew Zisserman. Look, listen and learn. In CVPR, 2017. 3, 5 [4] Relja Arandjelovic and Andrew Zisserman. Objects that sound. In ECCV, 2018. 3 [5] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia ´ Schmid. Vivit: A video vision transformer. arXiv preprint arXiv:2103.15691, 2021. 2, 3 [6] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. 5 [7] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR, 2015. 1 [8] Sagie Benaim, Ariel Ephrat, Oran Lang, Inbar Mosseri, William T Freeman, Michael Rubinstein, Michal Irani, and Tali Dekel. Speednet: Learning the speediness in videos. In CVPR,   \n2020. 3 [9] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is space-time attention all you need for video understanding? arXiv preprint arXiv:2102.05095, 2021. 2, 3, 6, 7 [10] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. arXiv preprint arXiv:2005.14165, 2020. 2, 3 [11] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In ECCV, 2020. 3 [12] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. In NeurIPS, 2020.   \n3 [13] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In CVPR, 2017. 7 [14] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In CVPR, 2017. 3, 6, 17 [15] Joao Carreira, Eric Noland, Andras Banki-Horvath, Chloe Hillier, and Andrew Zisserman. A short note about kinetics-600. arXiv preprint arXiv:1808.01340, 2018. 3, 6, 17 [16] Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Heewoo Jun, David Luan, and Ilya Sutskever. Generative pretraining from pixels. In ICML, 2020. 8 [17] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In ICML, 2020. 3 [18] Xinlei Chen, Saining Xie, and Kaiming He. An empirical study of training self-supervised visual transformers. arXiv preprint arXiv:2104.02057, 2021. 3 [19] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed El Kholy, Faisal Ahmed, Zhe Gan, Yu Cheng, and Jingjing Liu. Uniter: Universal image-text representation learning. In ECCV, 2020. 3 [20] Yunpeng Chen, Haoqi Fan, Bing Xu, Zhicheng Yan, Yannis Kalantidis, Marcus Rohrbach, Shuicheng Yan, and Jiashi Feng. Drop an octave: Reducing spatial redundancy in convolutional neural networks with octave convolution. In ICCV, 2019. 7   \n[21] Wei Dai, Chia Dai, Shuhui Qu, Juncheng Li, and Samarjit Das. Very deep convolutional neural networks for raw waveforms. In ICASSP, 2017. 8   \n[22] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009. 2, 3, 6, 17   \n[23] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL, 2019. 2, 3, 4   \n[24] Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In ICCV, 2015. 3   \n[25] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR, 2021. 2, 3, 4, 7, 8   \n[26] Heng Wang Du Tran, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. 2018 ieee. In CVPR, 2017. 7   \n[27] Quanfu Fan, Chun-Fu (Ricarhd) Chen, Hilde Kuehne, Marco Pistoia, and David Cox. More Is Less: Learning Efficient Video Representations by Temporal Aggregation Modules. In NeurIPS. 2019. 7   \n[28] William Fedus, Barret Zoph, and Noam Shazeer. Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity. arXiv preprint arXiv:2101.03961, 2021. 9   \n[29] Christoph Feichtenhofer. X3d: Expanding architectures for efficient video recognition. In CVPR, 2020. 7   \n[30] Christoph Feichtenhofer, Haoqi Fan, Jitendra Malik, and Kaiming He. Slowfast networks for video recognition. In ICCV, 2019. 7, 18   \n[31] Basura Fernando, Hakan Bilen, Efstratios Gavves, and Stephen Gould. Self-supervised video representation learning with odd-one-out networks. In CVPR, 2017. 3   \n[32] Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N Dauphin. Convolutional sequence to sequence learning. In ICML, 2017. 1   \n[33] Jort F. Gemmeke, Daniel P. W. Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R. Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In ICASSP, 2017. 3, 6, 17   \n[34] Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In ICASSP, 2017. 7   \n[35] Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. In ICLR, 2018. 3   \n[36] Rohit Girdhar and Deva Ramanan. Attentional pooling for action recognition. In NeurIPS, 2017. 2   \n[37] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. In NeurIPS, 2020. 3   \n[38] Tengda Han, Weidi Xie, and Andrew Zisserman. Video representation learning by dense predictive coding. In ICCV Workshops, 2019. 3   \n[39] Tengda Han, Weidi Xie, and Andrew Zisserman. Memory-augmented dense predictive coding for video representation learning. In ECCV, 2020. 3 [40] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, 2020. 3 [41] Olivier J Hénaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019. 3 [42] Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415, 2016. 5 [43] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation,   \n1997. 1 [44] Han Hu, Zheng Zhang, Zhenda Xie, and Stephen Lin. Local relation networks for image recognition. In ICCV, 2019. 2 [45] Dahun Kim, Donghyeon Cho, and In So Kweon. Self-supervised video representation learning with space-time cubic puzzles. In AAAI, 2019. 3 [46] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. 6, 18 [47] Dan Kondratyuk, Liangzhe Yuan, Yandong Li, Li Zhang, Mingxing Tan, Matthew Brown, and Boqing Gong. Movinets: Mobile video networks for efficient video recognition. In CVPR,   \n2021. 6, 7 [48] Qiuqiang Kong, Changsong Yu, Yong Xu, Turab Iqbal, Wenwu Wang, and Mark D Plumbley. Weakly labelled audioset tagging with attention neural networks. TASLP, 2019. 7 [49] Qiuqiang Kong, Yin Cao, Turab Iqbal, Yuxuan Wang, Wenwu Wang, and Mark D Plumbley. Panns: Large-scale pretrained audio neural networks for audio pattern recognition. TASLP,   \n2020. 8, 19 [50] Bruno Korbar, Du Tran, and Lorenzo Torresani. Cooperative learning of audio and video models from self-supervised synchronization. NeurIPS, 2018. 3, 20 [51] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NeurIPS, 2012. 1 [52] H. Kuehne, H. Jhuang, E. Garrote, T. Poggio, and T. Serre. HMDB: a large video database for human motion recognition. In ICCV, 2011. 6, 17 [53] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 1998. 1 [54] Hsin-Ying Lee, Jia-Bin Huang, Maneesh Singh, and Ming-Hsuan Yang. Unsupervised representation learning by sorting sequences. In ICCV, 2017. 3 [55] Jongpil Lee, Jiyoung Park, Keunhyoung Luke Kim, and Juhan Nam. Sample-level deep convolutional neural networks for music auto-tagging using raw waveforms. arXiv preprint arXiv:1703.01789, 2017. 8 [56] William Lotter, Gabriel Kreiman, and David Cox. Deep predictive coding networks for video prediction and unsupervised learning. arXiv preprint arXiv:1605.08104, 2016. 3 [57] Huaishao Luo, Lei Ji, Botian Shi, Haoyang Huang, Nan Duan, Tianrui Li, Xilin Chen, and Ming Zhou. Univilm: A unified video and language pre-training model for multimodal understanding and generation. arXiv preprint arXiv:2002.06353, 2020. 3 [58] Antoine Miech, Dimitri Zhukov, Jean-Baptiste Alayrac, Makarand Tapaswi, Ivan Laptev, and Josef Sivic. Howto100m: Learning a text-video embedding by watching hundred million narrated video clips. In ICCV, 2019. 6, 17 ",
+        "bbox": [
+            179,
+            94,
+            828,
+            915
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            178,
+            47,
+            828,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            50,
+            828,
+            921
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[59] Antoine Miech, Jean-Baptiste Alayrac, Lucas Smaira, Ivan Laptev, Josef Sivic, and Andrew Zisserman. End-to-end learning of visual representations from uncurated instructional videos. In CVPR, 2020. 5, 8, 17, 20 ",
+        "bbox": [
+            181,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[60] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013. 4, 18 ",
+        "bbox": [
+            178,
+            142,
+            823,
+            172
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[61] Mathew Monfort, Alex Andonian, Bolei Zhou, Kandan Ramakrishnan, Sarah Adel Bargal, Tom Yan, Lisa Brown, Quanfu Fan, Dan Gutfreund, Carl Vondrick, et al. Moments in time dataset: one million videos for event understanding. TPAMI, 2019. 3, 6, 17 ",
+        "bbox": [
+            181,
+            181,
+            823,
+            226
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[62] Pedro Morgado, Nuno Vasconcelos, and Ishan Misra. Audio-visual instance discrimination with cross-modal agreement. arXiv preprint arXiv:2004.12943, 2020. 3 ",
+        "bbox": [
+            181,
+            234,
+            823,
+            263
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[63] Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In ECCV, 2016. 3 ",
+        "bbox": [
+            181,
+            272,
+            823,
+            303
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[64] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In CVPR, 2016. 3 ",
+        "bbox": [
+            181,
+            310,
+            823,
+            342
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[65] Mandela Patrick, Yuki M Asano, Polina Kuznetsova, Ruth Fong, João F Henriques, Geoffrey Zweig, and Andrea Vedaldi. Multi-modal self-supervision from generalized data transformations. arXiv preprint arXiv:2003.04298, 2020. 20 ",
+        "bbox": [
+            181,
+            349,
+            823,
+            393
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[66] Karol J. Piczak. ESC: Dataset for Environmental Sound Classification. In ACM MM, 2015. 6, 17 ",
+        "bbox": [
+            181,
+            402,
+            825,
+            433
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[67] AJ Piergiovanni, Anelia Angelova, and Michael S Ryoo. Evolving losses for unsupervised video representation learning. In CVPR, 2020. 3, 20 ",
+        "bbox": [
+            178,
+            440,
+            825,
+            470
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[68] Rui Qian, Tianjian Meng, Boqing Gong, Ming-Hsuan Yang, Huisheng Wang, Serge Belongie, and Yin Cui. Spatiotemporal contrastive video representation learning. In CVPR, 2021. 3 [69] Zhaofan Qiu, Ting Yao, Chong-Wah Ngo, Xinmei Tian, and Tao Mei. Learning spatio-temporal representation with local and global diffusion. In CVPR, 2019. 7 ",
+        "bbox": [
+            181,
+            479,
+            825,
+            510
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            179,
+            518,
+            825,
+            547
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[70] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018. 2 ",
+        "bbox": [
+            181,
+            556,
+            823,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[71] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI blog, 2019. 2 ",
+        "bbox": [
+            181,
+            595,
+            821,
+            626
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[72] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. JMLR, 2020. 5 ",
+        "bbox": [
+            183,
+            635,
+            823,
+            676
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[73] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan Bello, Anselm Levskaya, and Jonathon Shlens. Stand-alone self-attention in vision models. arXiv preprint arXiv:1906.05909, 2019. 2 ",
+        "bbox": [
+            181,
+            686,
+            825,
+            728
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[74] Adrià Recasens, Pauline Luc, Jean-Baptiste Alayrac, Luyu Wang, Florian Strub, Corentin Tallec, Mateusz Malinowski, Viorica Patraucean, Florent Altché, Michal Valko, et al. Broaden your views for self-supervised video learning. arXiv preprint arXiv:2103.16559, 2021. 3 ",
+        "bbox": [
+            181,
+            739,
+            825,
+            782
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[75] Steffen Rendle. Factorization machines. In ICDM, 2010. 19 [76] Carlos Riquelme, Joan Puigcerver, Basil Mustafa, Maxim Neumann, Rodolphe Jenatton, André Susano Pinto, Daniel Keysers, and Neil Houlsby. Scaling vision with sparse mixture of experts. arXiv preprint arXiv:2106.05974, 2021. 9 ",
+        "bbox": [
+            181,
+            791,
+            614,
+            808
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            181,
+            816,
+            825,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[77] Michael S Ryoo, AJ Piergiovanni, Mingxing Tan, and Anelia Angelova. Assemblenet: Searching for multi-stream neural connectivity in video architectures. arXiv preprint arXiv:1905.13209, 2019. 7 ",
+        "bbox": [
+            179,
+            868,
+            823,
+            911
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[78] Gilad Sharir, Asaf Noy, and Lihi Zelnik-Manor. An image is worth 16x16 words, what is a video worth? arXiv preprint arXiv:2103.13915, 2021. 3 ",
+        "bbox": [
+            176,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[79] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. arXiv preprint arXiv:1701.06538, 2017. 9 ",
+        "bbox": [
+            183,
+            127,
+            823,
+            171
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[80] Laura Smith, Nikita Dhawan, Marvin Zhang, Pieter Abbeel, and Sergey Levine. Avid: Learning multi-stage tasks via pixel-level translation of human videos. arXiv preprint arXiv:1912.04443, 2019. 20 ",
+        "bbox": [
+            181,
+            179,
+            823,
+            222
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[81] Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. Ucf101: A dataset of 101 human actions classes from videos in the wild. arXiv preprint arXiv:1212.0402, 2012. 6, 17 ",
+        "bbox": [
+            179,
+            229,
+            823,
+            258
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[82] Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In ICML, 2015. 3 ",
+        "bbox": [
+            181,
+            267,
+            823,
+            296
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[83] Jonathan Stroud, David Ross, Chen Sun, Jia Deng, and Rahul Sukthankar. D3d: Distilled 3d networks for video action recognition. In WACV, 2020. 7 ",
+        "bbox": [
+            181,
+            303,
+            825,
+            333
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[84] Chen Sun, Fabien Baradel, Kevin Murphy, and Cordelia Schmid. Learning video representations using contrastive bidirectional transformer. arXiv preprint arXiv:1906.05743, 2019. 3 ",
+        "bbox": [
+            181,
+            340,
+            826,
+            383
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[85] Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. In ECCV, 2020. 3 ",
+        "bbox": [
+            178,
+            391,
+            826,
+            421
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[86] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image transformers & distillation through attention. arXiv preprint arXiv:2012.12877, 2020. 3 ",
+        "bbox": [
+            181,
+            429,
+            825,
+            472
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[87] Du Tran, Heng Wang, Lorenzo Torresani, and Matt Feiszli. Video classification with channelseparated convolutional networks. In ICCV, 2019. 7 ",
+        "bbox": [
+            179,
+            479,
+            825,
+            508
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[88] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017. 1, 2, 3, 4, 5 ",
+        "bbox": [
+            181,
+            517,
+            825,
+            560
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[89] Fei Wang, Mengqing Jiang, Chen Qian, Shuo Yang, Cheng Li, Honggang Zhang, Xiaogang Wang, and Xiaoou Tang. Residual attention network for image classification. In CVPR, 2017. 2 ",
+        "bbox": [
+            181,
+            568,
+            825,
+            609
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[90] Jiangliu Wang, Jianbo Jiao, Linchao Bao, Shengfeng He, Yunhui Liu, and Wei Liu. Selfsupervised spatio-temporal representation learning for videos by predicting motion and appearance statistics. In CVPR, 2019. 3 ",
+        "bbox": [
+            181,
+            618,
+            823,
+            661
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[91] Jiangliu Wang, Jianbo Jiao, and Yun-Hui Liu. Self-supervised video representation learning by pace prediction. In ECCV, 2020. 3 ",
+        "bbox": [
+            179,
+            670,
+            823,
+            699
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[92] Xiaofang Wang, Xuehan Xiong, Maxim Neumann, AJ Piergiovanni, Michael S Ryoo, Anelia Angelova, Kris M Kitani, and Wei Hua. Attentionnas: Spatiotemporal attention cell search for video classification. In ECCV, 2020. 7 ",
+        "bbox": [
+            181,
+            707,
+            823,
+            750
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[93] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In CVPR, 2018. 7 ",
+        "bbox": [
+            179,
+            757,
+            823,
+            786
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[94] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon. Cbam: Convolutional block attention module. In ECCV, 2018. 2 ",
+        "bbox": [
+            179,
+            795,
+            821,
+            824
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[95] Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In CVPR, 2018. 3 ",
+        "bbox": [
+            179,
+            832,
+            821,
+            861
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[96] Saining Xie, Chen Sun, Jonathan Huang, Zhuowen Tu, and Kevin Murphy. Rethinking spatiotemporal feature learning: Speed-accuracy trade-offs in video classification. In ECCV, 2018. 7 ",
+        "bbox": [
+            179,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "[97] Dejing Xu, Jun Xiao, Zhou Zhao, Jian Shao, Di Xie, and Yueting Zhuang. Self-supervised spatiotemporal learning via video clip order prediction. In CVPR, 2019. 3 [98] Jun Xu, Tao Mei, Ting Yao, and Yong Rui. Msr-vtt: A large video description dataset for bridging video and language. In CVPR, 2016. 6, 17 [99] Fuzhi Yang, Huan Yang, Jianlong Fu, Hongtao Lu, and Baining Guo. Learning texture transformer network for image super-resolution. In CVPR, 2020. 3 [100] Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang. Unsupervised embedding learning via invariant and spreading instance feature. In CVPR, 2019. 3 [101] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017. 19 [102] Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In ECCV,   \n2016. 3 [103] Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning by cross-channel prediction. In CVPR, 2017. 3 [104] Xiang Zhang, Junbo Zhao, and Yann Lecun. Character-level convolutional networks for text classification. NeurIPS, 2015. 1 [105] Y Zhang, K Li, K Li, B Zhong, and Y Fu. Residual non-local attention networks for image restoration. In ICLR, 2019. 2 [106] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun. Exploring self-attention for image recognition. In CVPR, 2020. 2 [107] Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip Torr, and Vladlen Koltun. Point transformer. arXiv preprint arXiv:2012.09164, 2020. 3 [108] Sixiao Zheng, Jiachen Lu, Hengshuang Zhao, Xiatian Zhu, Zekun Luo, Yabiao Wang, Yanwei Fu, Jianfeng Feng, Tao Xiang, Philip HS Torr, et al. Rethinking semantic segmentation from a sequence-to-sequence perspective with transformers. arXiv preprint arXiv:2012.15840, 2020.   \n3 ",
+        "bbox": [
+            171,
+            85,
+            828,
+            575
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "[109] Luowei Zhou, Chenliang Xu, and Jason J Corso. Towards automatic learning of procedures from web instructional videos. In AAAI, 2018. 6, 17 ",
+        "bbox": [
+            173,
+            569,
+            821,
+            598
+        ],
+        "page_idx": 15
+    }
+]

parse/train/dvyUaK4neD0/dvyUaK4neD0.md

@@ -0,0 +1,314 @@
+# The Skellam Mechanism for Differentially Private Federated Learning
+
+Naman Agarwal† Peter Kairouz† Ziyu Liu‡⇤ †Google Research ‡Carnegie Mellon University {namanagarwal, kairouz}@google.com, ziyuliu@cs.cmu.edu
+
+# Abstract
+
+We introduce the multi-dimensional Skellam mechanism, a discrete differential privacy mechanism based on the difference of two independent Poisson random variables. To quantify its privacy guarantees, we analyze the privacy loss distribution via a numerical evaluation and provide a sharp bound on the Rényi divergence between two shifted Skellam distributions. While useful in both centralized and distributed privacy applications, we investigate how it can be applied in the context of federated learning with secure aggregation under communication constraints. Our theoretical findings and extensive experimental evaluations demonstrate that the Skellam mechanism provides the same privacy-accuracy trade-offs as the continuous Gaussian mechanism, even when the precision is low. More importantly, Skellam is closed under summation and sampling from it only requires sampling from a Poisson distribution – an efficient routine that ships with all machine learning and data analysis software packages. These features, along with its discrete nature and competitive privacy-accuracy trade-offs, make it an attractive practical alternative to the newly introduced discrete Gaussian mechanism.
+
+# 1 Introduction
+
+The Gaussian mechanism is the workhorse for a multitude of differentially private learning algorithms [46, 10, 1]. While simple enough for mathematical reasoning and privacy accounting analyses, its continuous nature presents a number of challenges in practice. For example, it cannot be exactly represented on finite computers, making it prone to numerical errors that can break its privacy guarantees [35]. Moreover, it cannot be used in distributed learning settings with cryptographic multi-party computation primitives involving modular arithmetic, such as secure aggregation [12, 11]. To address these shortcomings, the binomial and (distributed) discrete Gaussian mechanisms were recently introduced [19, 2, 15, 25]. Unfortunately, both have their own drawbacks: the privacy loss for the binomial mechanism can be infinite with a non-zero probability, and the discrete Gaussian: (a) is not closed under summation (i.e. sum of discrete Gaussians is not a discrete Gaussian), complicating analysis in distributed settings and leading to a performance worse than continuous Gaussian in the highly distributed, low-noise regime [25]; (b) requires a sampling algorithm that is not shipped with mainstream machine learning or data analysis software packages, making it difficult for engineers to use it in production settings (naïve implementations may lead to catastrophic privacy errors).
+
+Our contributions To overcome these limitations, we introduce and analyze the multi-dimensional Skellam mechanism, a mechanism based on adding noise distributed according to the difference of two independent Poisson random variables. The Skellam noise is closed under summation (i.e. sums of Skellam random variables is again Skellam distributed) and can be sampled from easily – efficient Poisson samplers are widely available in numerical software packages. Being discrete in nature also means that it can mesh well cryptographic protocols and can lead to communication savings.
+
+To analyze the privacy guarantees of the Skellam mechanism and compare it with other mechanisms, we provide a numerical evaluation of the privacy loss random variable and prove a sharp bound on the Rényi divergence between two shifted Skellam distributions. Our careful analysis shows that for a multi-dimensional query function with $\ell _ { 1 }$ sensitivity $\Delta _ { 1 }$ and $\ell _ { 2 }$ sensitivity $\Delta _ { 2 }$ , the Skellam mechanism with variance $\mu$ achieves $( \alpha , \varepsilon ( \alpha ) )$ Rényi differential privacy (RDP) [36] for $\varepsilon ( \alpha ) \leq$ $\begin{array} { r } { \frac { \alpha \Delta _ { 2 } } { 2 \mu } + \operatorname* { m i n } { \left( \frac { ( 2 \alpha - 1 ) \Delta _ { 2 } + 6 \Delta _ { 1 } } { 4 \mu ^ { 2 } } , \frac { 3 \Delta _ { 1 } } { 2 \mu } \right) } } \end{array}$ (see Theorem 3.5). This implies that the RDP guarantees are at most $1 + O \left( 1 / \mu \right)$ times worse than those of the Gaussian mechanism.
+
+To analyze the performance of the Skellam mechanism in practice, we consider a differentially private and communication constrained federated learning (FL) setting [26] where the noise is added locally to the $d$ -dimensional discretized client updates that are then summed securely via a cryptographic protocol, such as secure aggregation $( { \mathrm { S e c A g g } } )$ [11, 12]. We provide an end-to-end algorithm that appropriately discretizes the data and applies the Skellam mechanism along with modular arithmetic to bound the range of the data and communication costs before applying SecAgg.
+
+We show on distributed mean estimation and two benchmark FL datasets, Federated EMNIST [14] and Stack Overflow [8], that our method can match the performance of the continuous Gaussian baseline under tight privacy and communication budgets, despite using generic RDP amplification via sampling [51] for our approach and the precise RDP analysis for the subsampled Gaussian mechanism [37]. Our method is implemented in TensorFlow Privacy [32] and TensorFlow Federated [24] and will be open-sourced.2 While we mostly focus on FL applications, the Skellam mechanism can also be applied in other contexts of learning and analytics, including centralized settings.
+
+Related work The Skellam mechanism was first introduced in the context of computational differential privacy from lattice-based cryptography [49] and private Bayesian inference [45]. However, the privacy analyses in the prior work do not readily extend to the multi-dimensional case, and they give direct bounds for pure or approximate DP which makes only advanced composition theorems [28, 22] directly applicable in learning settings where the mechanism is applied many times. For example, the guarantees from [49] lead to poor accuracy-privacy trade-offs as demonstrated in Fig. 1. Moreover, we show in Section 3.1 that extending the direct privacy analysis to the multi-dimensional setting is non-trivial because the worst-case neighboring dataset pair is unknown in this case. For these reasons, our tight privacy analysis via a sharp RDP bound makes the Skellam mechanism practical for learning applications for the first time. These guarantees (almost) match those of the Gaussian mechanism and allow us to use generic RDP amplification via subsampling methods [51].
+
+The closest mechanisms to Skellam are the binomial [2, 19] and the discrete Gaussian mechanisms [15, 25]. The binomial mechanism can (asymptotically) match the continuous Gaussian mechanism (when properly scaled). However, it does not achieve Rényi or zero-concentrated DP [36, 13] and has a privacy loss that can be infinite with a non-zero probability, leading to catastrophic privacy failures. The discrete Gaussian mechanism yields Rényi DP and can be applied to distributed settings [25], but it requires a sampling algorithm that is not yet available in data analysis software packages despite being explored in the lattice-based cryptography community (e.g., [43, 18, 38]). The discrete Gaussian is also not closed under summation and the divergence can be large in highly distributed low-noise settings (e.g. quantile estimation [6] and federated analytics [42]), which causes privacy degradation. See the end of Section 4 for more discussion.
+
+# 2 Preliminaries
+
+We begin by providing a formal definition for $( \varepsilon , \delta )$ -differential privacy (DP) [20].
+
+Definition 2.1 (Differential Privacy). For $\varepsilon , \delta \geq 0$ , a randomized mechanism $M$ satisfies $( \varepsilon , \delta )$ -DP if for all neighboring datasets $D , D ^ { \prime }$ and all $s$ in the range of $M$ , we have that
+
+$$
+P \left( M ( D ) \in { \mathcal { S } } \right) \leq e ^ { \varepsilon } P \left( M ( D ^ { \prime } ) \in { \mathcal { S } } \right) + \delta ,
+$$
+
+where $D$ and $D ^ { \prime }$ are neighboring pairs if they can be obtained from each other by adding or removing all the records that belong to a particular user.
+
+In our experiments we consider user-level differential privacy – i.e., $D$ and $D ^ { \prime }$ are neighboring pairs
+
+if one of them can be obtained from the other by adding or removing all the records associated with a single user [33]. This is stronger than the commonly-used notion of item level privacy where, if a user contributes multiple records, only the addition or removal of one record is protected.
+
+We also make use of Rényi differential privacy (RDP) [36] which allows for tight privacy accounting.
+
+Definition 2.2 (Rényi Differential Privacy). A mechanism $M$ satisfies $( \alpha , \varepsilon )$ -RDP if for any two neighboring datasets $D , D ^ { \prime }$ , we have that $D _ { \alpha } ( M ( D ) , M ( D ^ { \prime } ) ) \leq \varepsilon$ where $D _ { \alpha } ( P , Q )$ is the Rényi divergence between $P$ and $Q$ and is given by
+
+$$
+D _ { \alpha } ( P , Q ) \triangleq \frac { 1 } { \alpha - 1 } \log \left( \mathbb { E } _ { x \sim Q } \left[ \left( \frac { P ( x ) } { Q ( x ) } \right) ^ { \alpha } \right] \right) = \frac { 1 } { \alpha - 1 } \log \left( \mathbb { E } _ { x \sim P } \left[ \left( \frac { P ( x ) } { Q ( x ) } \right) ^ { \alpha - 1 } \right] \right) .
+$$
+
+A closely related privacy notion is zero-concentrated DP (zCDP) [21, 13]. In fact, $\scriptstyle { \frac { 1 } { 2 } } \varepsilon ^ { 2 } - z \mathbf { C D P }$ i s equivalent to simultaneously satisfying an infinite family of RDP guarantees, namely $( \alpha , \textstyle { \frac { 1 } { 2 } } \varepsilon ^ { 2 } \alpha )$ - Rényi differential privacy for all $\alpha \in ( 1 , \infty )$ . The following conversion lemma from [13, 15, 7] relates RDP to $( \varepsilon , \delta )$ -DP.
+
+Lemma 2.3. If $M$ satisfies $( \alpha , \varepsilon )$ -RDP, then, for any $\delta > 0$ , $M$ satisfies $( \varepsilon _ { D P } ( \delta ) , \delta )$ -DP, where
+
+$$
+\varepsilon _ { D P } ( \delta ) = \operatorname* { i n f } _ { \alpha > 1 } \varepsilon + \frac { \log ( 1 / \alpha \delta ) } { \alpha - 1 } + \log ( 1 - 1 / \alpha ) .
+$$
+
+For any query function $f$ , we define the $\Delta _ { p }$ sensitivity as $\begin{array} { r } { \operatorname* { m a x } _ { D , D ^ { \prime } } \| f ( D ) - f ( D ^ { \prime } ) \| _ { p } } \end{array}$ , where $D$ and $D ^ { \prime }$ are neighboring pairs differing by adding or removing all the records from a particular user. We also include the RDP guarantees of the discrete Gaussian mechanism (same RDP guarantees as the continuous Gaussian mechanism) to which we compare our method.
+
+Definition 2.4 (The Discrete Gaussian Mechanism [15]). Given an integer-valued query $f ( D ) \in \mathbb { Z } ^ { d }$ and noise variance $\mu _ { ; }$ , the Discrete Gaussian (DGaussian) Mechanism is given by
+
+$$
+f ( D ) + Z , w h e r e Z \sim { \mathcal { N } } _ { \mathbb { Z } } ( 0 , \mu ) ,
+$$
+
+and $\mathcal { N } _ { \mathbb { Z } } ( 0 , \mu )$ denotes the discrete Gaussian distribution defined in Equation $( l )$ of [15]. The discrete Gaussian mechanism achieves $\begin{array} { r } { ( \alpha , \frac { \alpha \Delta _ { 2 } ^ { 2 } } { 2 \mu } ) } \end{array}$ -Rényi $D P .$
+
+# 3 The Skellam Mechanism
+
+We begin by presenting the definition of the Skellam distribution, which is the basis of the Skellam Mechanism for releasing integer ranged multi-dimensional queries.
+
+Definition 3.1 (Skellam Distribution). The multidimensional Skellam distribution $\operatorname { S k } _ { \Delta , \mu }$ over $\mathbb { Z } ^ { d }$ with mean $\Delta \in \mathbb { Z } ^ { d }$ and variance $\mu$ is given with each coordinate $X _ { i }$ distributed independently as
+
+$$
+X _ { i } \sim \mathrm { S k } _ { \Delta _ { i } , \mu } \ w i t h \ P ( X _ { i } = k ) = e ^ { - \mu } I _ { k - \Delta _ { i } } ( \mu ) ,
+$$
+
+for $k \in \mathbb { Z }$ . Here, $I _ { \nu } ( x )$ is the modified Bessel function of the first kind. A key property of Skellam random variables which motivates their use in DP is that they are closed under summation, i.e. let $X _ { 1 } \sim \mathrm { S k } _ { \Delta _ { 1 } , \mu _ { 1 } }$ and $X _ { 2 } \sim \mathrm { S k } _ { \Delta _ { 2 } , \mu _ { 2 } }$ then $X _ { 1 } + X _ { 2 } \sim \mathrm { S k } _ { \Delta _ { 1 } + \Delta _ { 2 } , \mu _ { 1 } + \mu _ { 2 } }$ . This follows from the fact that a Skellam random variable $X$ can be obtained by taking the difference between two independent Poisson random variables with means $\mu$ . 3 We are now ready to introduce the Skellam Mechanism.
+
+Definition 3.2 (The Skellam Mechanism). Given an integer-valued query $f ( D ) \in \mathbb { Z } ^ { d }$ , we define the Skellam Mechanism as
+
+$$
+\begin{array} { r l } & { \qquad \mathrm { S k } _ { 0 , \mu } ( f ( D ) ) = f ( D ) + Z , \ w h e r e \ Z \sim \mathrm { S k } _ { 0 , \mu } , } \\ & { \qquad \cdot o f t h e m e c h a n i s m i s b o u n d e d b y \mathbb { E } \left[ \| \mathrm { S k } _ { 0 , \mu } ( f ( D ) ) - f ( D ) \| _ { 2 } ^ { 2 } \right] \leq d \mu . } \end{array}
+$$
+
+and the total $\ell _ { 2 }$ erro
+
+The Skellam mechanism was first introduced in [49] for the scalar case. As our goal is to apply the Skellam mechanism in the learning context, we have to address the following challenges. (1) Tight privacy compositions: Learning algorithms are iterative in nature and require the application of the DP mechanism many times (often $> 1 0 0 0$ ). The current direct approximate DP analysis in [49] can be combined with advanced composition (AC) theorems [28, 22] but that leads to poor privacy-accuracy trade-offs (see Fig. 1). (2) Privacy analysis for multi-dimensional queries: In learning algorithms, the differentially private queries are multi-dimensional (where the dimension equals the number of model parameters, typically $\geq 1 0 ^ { 6 }$ ). Using composition theorems lead to poor accuracy-privacy trade-offs and a direct extension of approximate DP guarantee [49] for the multi-dimensional case leads to a strong dependence on $\ell _ { 1 }$ sensitivity which is prohibitively large in high dimensions. (3) Data discretization: The gradients are naturally continuous vectors but we would like to apply an integer based mechanism. This requires properly discretizing the data while making sure that the norm of the vectors (sensitivity of the query) is preserved. We will tackle challenges (1) and (2) in the remainder of this section and leave (3) for the next section.
+
+# 3.1 Tight Numerical Accounting via Privacy Loss Distributions
+
+We begin by defining the notion of privacy loss distributions (PLDs).
+
+Definition 3.3 (Privacy Loss Distribution). For a multi-dimensional discrete privacy mechanism $M$ and neighboring datasets $D , D ^ { \prime }$ , for any $x \in \mathbb { Z } ^ { d }$ , we define $f ( x ) =$ $\log \left( { \frac { P ( M ( D ) = x ) } { P ( M ( D ^ { \prime } ) = x ) } } \right)$ . The privacy loss random variable of $M$ at $( D , D ^ { \prime } )$ is $Z _ { D , D ^ { \prime } } =$ $f ( M ( D ) ) \ l { 2 2 } { ] }$ . The privacy loss distribution $( P L D )$ of $M$ , denoted by $\mathrm { P L D } _ { D , D ^ { \prime } }$ , is the distribution of $Z _ { D , D ^ { \prime } }$ .
+
+The PLD of a mechanism $M$ can be used to characterize its $( \varepsilon , \delta )$ -DP guarantees.
+
+![](images/89259f8a4ef4a8b7c950cf0a4b2f3fd06431c154db559e6aac977812e35a7dfb.jpg)  
+Figure 1: Comparing privacy compositions across various mechanisms and accounting methods.
+
+Lemma 3.4. A mechanism $M$ is $( \varepsilon , \delta )$ -DP if and only if $\delta \geq \mathbb { E } _ { Z \sim \mathrm { P L D } _ { D , D ^ { \prime } } } \left[ 1 - e ^ { \varepsilon - Z } \right] _ { + }$ for all neighboring datasets $D , D ^ { \prime }$ where $[ x ] _ { + } = \operatorname* { m a x } ( 0 , x )$ .
+
+When a mechanism $M$ is applied $T$ times on a dataset, the overall PLD of the composed mechanism at $( D , D ^ { \prime } )$ is the $T$ -fold convolution of $\mathrm { P L D } _ { D , D ^ { \prime } }$ [22]. Since discrete convolutions can be computed efficiently using fast Fourier transforms (FFTs) and the expectation in Lemma 3.4 can be numerically approximated, PLDs are attractive for tight numerical accounting [30, 34, 17]. Applying the above to the Skellam mechanism, a direct calculation shows that with $X _ { i }$ are i.i.d. according to $\operatorname { S k } _ { 0 , \mu }$ ,
+
+$$
+Z _ { D , D ^ { \prime } } = \sum _ { i = 1 } ^ { d } \log \left( \frac { I _ { X _ { i } - f ( D ) _ { i } } ( \mu ) } { I _ { X _ { i } - f ( D ^ { \prime } ) _ { i } } ( \mu ) } \right) .
+$$
+
+When $d = 1$ , it suffices to look at $Z = \log ( I _ { X - \Delta } ( \mu ) / I _ { X } ( \mu ) )$ , where $\Delta = \operatorname* { m a x } _ { D , D ^ { \prime } } | f ( D ) - f ( D ^ { \prime } ) |$ and $X \sim \mathrm { S k } _ { 0 , \mu }$ . Since $X$ has a discrete and symmetric probability distribution and the log function is monotonic, the distribution of $Z$ can be easily characterized. This gives us a tight numerical accountant for the Skellam mechanism in the scalar case, which we use to compare it with both the Gaussian and discrete Gaussian mechanisms. Fig. 1 shows this comparison, highlighting the competitiveness of the Skellam mechanism and the problem of combining the direct analysis of [49] with advanced composition (AC) theorems. When $d > 1$ , there are combinatorially many $Z _ { D , D ^ { \prime } }$ ’s that need to be considered, even when the $\ell _ { 2 }$ sensitivity of $f ( D )$ is bounded. The discrete Gaussian mechanism faces a similar issue (see Theorem 15 of [15]). To provide a tight privacy analysis in the multi-dimensional case, we prove a bound on the RDP guarantees of the Skellam mechanism in the next subsection. Fig. 1 and 2 show that our bound is tight and the competitiveness of the Skellam mechanism in high dimensions.
+
+# 3.2 Tight Accounting via Rényi Differential Privacy
+
+The following theorem states our main theoretical result, providing a relatively sharp bound on the RDP properties for the Skellam machanism.
+
+Theorem 3.5. For $\alpha \in \mathbb { Z } , \alpha > 1$ and sensitivity $\Delta \in \mathbb { Z }$ , the Skellam Mechanism is $( \alpha , \varepsilon )$ -RDP with
+
+$$
+\varepsilon ( \alpha ) \leq \frac { \alpha \Delta ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) \Delta ^ { 2 } + 6 \Delta } { 4 \mu ^ { 2 } } , \frac { 3 \Delta } { 2 \mu } \right) ,
+$$
+
+To remind the reader in comparison, the Gaussian mechanism is $( \alpha , \varepsilon )$ -RDP with $\begin{array} { r } { \varepsilon ( \alpha ) = \frac { \alpha \Delta ^ { 2 } } { 2 \mu } } \end{array}$ . The bound we provide is at most $1 + O ( 1 / \mu )$ worse than the bound for the Gaussian, which is negligible for all practical choices of $\mu$ , especially as the privacy requirements increase.4 Next we show a simple corollary which follows via the independent composition of RDP across dimensions.
+
+Corollary 3.6. The multi-dimensional Skellam Mechanism is $( \alpha , \varepsilon )$ -RDP with
+
+$$
+\varepsilon ( \alpha ) \leq \frac { \alpha \Delta _ { 2 } ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) \Delta _ { 2 } ^ { 2 } + 6 \Delta _ { 1 } } { 4 \mu ^ { 2 } } , \frac { 3 \Delta _ { 1 } } { 2 \mu } \right) .
+$$
+
+where $\Delta _ { 1 }$ and $\Delta _ { 2 }$ are the $\ell _ { 1 }$ and $\ell _ { 2 }$ sensitivities respectively.
+
+# 3.2.1 Proof Overview for Theorem 3.5
+
+In this subsection, we provide the proof of Theorem 3.5 assuming a technical bound on the ratios of Bessel functions presented as Lemma 3.7, which is the core of our analysis and may be of independent interest. We provide a proof overview for Lemma 3.7, deferring the full proof to the appendix.
+
+On a macroscopic level, our proof structure mimics the RDP proof for the Gaussian mechanism [36], and the main object of our interest is to bound the following quantity, defined for any $X , \Delta , \alpha$ :
+
+$$
+\Phi _ { X , \alpha , \Delta } ( \mu ) \triangleq \log \left( \frac { I _ { X - \Delta } ( \mu ) } { I _ { X - \alpha \Delta } ( \mu ) } \left( \frac { I _ { X - \Delta } ( \mu ) } { I _ { X } ( \mu ) } \right) ^ { \alpha - 1 } \right) .
+$$
+
+The following lemma states our main bound on this quantity.
+
+Lemma 3.7. For any $X , \alpha \in \mathbb { N } ,$ , with $\alpha > 1$ and $\Delta \in \mathbb { Z } ,$ , we have that for all $\mu \geq 0$
+
+$$
+\Phi _ { X , \alpha , \Delta } ( \mu ) \leq \frac { \alpha ( \alpha - 1 ) \Delta ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) ( \alpha - 1 ) \Delta ^ { 2 } } { 4 \mu ^ { 2 } } + \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu ^ { 2 } } , \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu } \right) .
+$$
+
+Note that in contrast if we consider the analogous notion of $\Phi$ for the Gaussian mechanism (replacing IX(µ) with the Gaussian density eX2/2µ), we readily get the bound ↵(↵1)22µ , which is the same as our bound up to lower order terms. We now provide the proof of Theorem 3.5.
+
+Proof of Theorem 3.5. By RDP definition (2.2), we need to bound the following for any $\Delta , \alpha \geq 1$ ,
+
+$$
+D _ { \alpha } \left( { \mathrm { S k } } _ { \Delta , \mu } , { \mathrm { S k } } _ { 0 , \mu } \right) = \frac { 1 } { \alpha - 1 } { \log \left( \right)} \sum _ { X = - \infty } ^ { \infty } e ^ { - \mu } I _ { X - \Delta } ( \mu ) \left( \frac { I _ { X - \Delta } ( \mu ) } { I _ { X } ( \mu ) } \right) ^ { \alpha - 1 } 
+$$
+
+Now consider the following calculations on the log term:
+
+$$
+\begin{array} { r l } & { \log \left( \displaystyle \sum _ { X = - \infty } ^ { \infty } \frac { I _ { X - \Delta } ( \mu ) } { e ^ { \mu } } \left( \frac { I _ { X - \Delta } ( \mu ) } { I _ { X } ( \mu ) } \right) ^ { \alpha - 1 } \right) = \log \left( \displaystyle \sum _ { X = - \infty } ^ { \infty } \frac { I _ { X - \alpha \Delta } ( \mu ) } { e ^ { \mu } } e ^ { \Phi _ { X , \alpha , \Delta ( \mu ) } } \right) } \\ & { \leq \log \left( \displaystyle \sum _ { X = - \infty } ^ { \infty } e ^ { - \mu } I _ { X - \alpha \Delta } ( \mu ) \right) + \displaystyle \operatorname* { m a x } _ { X \in \mathbb { Z } } \Phi _ { X , \alpha , \Delta ( \mu ) } } \\ & { \leq \frac { \alpha ( \alpha - 1 ) \Delta ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) ( \alpha - 1 ) \Delta ^ { 2 } } { 4 \mu ^ { 2 } } + \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu ^ { 2 } } , \frac { 3 ( \alpha - 1 ) | \Delta | } { 2 \mu } \right) , } \end{array}
+$$
+
+where the inequality follows from Lemma 3.7.
+
+We now provide an overview for the proof of Lemma 3.7 highlighting the crux of the argument. As a first step we collect some known facts regarding Bessel functions. It is known that for $x \geq 0$ and $\nu \in \mathbb { Z }$ , $\nu \geq 0$ , $I _ { \nu } ( x )$ is a decreasing function in $\nu$ , $I _ { - \nu } ( x ) = I _ { \nu } ( x )$ and $\frac { I _ { \nu - 1 } ( \mu ) } { I _ { \nu } ( x ) }$ is an increasing function in $\nu$ [47]. A succession of works consider bounding the ratio of successive Bessel functions $I _ { \nu - 1 } ( x ) / I _ { \nu } ( x )$ , which is a natural quantity to considering the objective in Lemma 3.7. We use the following very tight characterization for this recently proved in [44, Theorem 5].
+
+Lemma 3.8. For any $\nu \geq 1 / 2 , x \geq 0$ define the following function we have that
+
+$$
+\mathrm { a r c s i n h } ( \delta _ { 0 } ( \nu , x ) ) \leq \log ( I _ { \nu - 1 } ( x ) ) - \log ( I _ { \nu } ( x ) ) \leq \mathrm { a r c s i n h } ( \delta _ { 2 } ( \nu , x ) )
+$$
+
+Standard bounds such as those appearing in [5, 49] lead to the following conclusion:
+
+$$
+\mathrm { a r c s i n h } ( ( \nu - 1 / 2 ) / x ) \leq \log ( I _ { \nu - 1 } ( x ) ) - \log ( I _ { \nu } ( x ) ) \leq \mathrm { a r c s i n h } ( \nu / x ) ) .
+$$
+
+While the above bound is significantly easier to work with, it leads to an RDP guarantee of Gaussian $\mathrm { R D P } + O \bigl ( \frac { \Delta } { \mu } \bigr )$ . In high dimensions this manifests as $O ( \frac { \Delta _ { 1 } } { \mu } )$ and overall leads to a constant multiplicative factor over the Gaussian. On the other hand we prove a Gaussian $\mathrm { R D P } + o _ { \mu } ( 1 )$ bound. Our proof of Lemma 3.7 splits into various cases depending on the signs of the quantities involved. We show the derivation for a single case below and defer the full proof to the appendix.
+
+Proof of Lemma 3.7 in the case $X \geq \alpha \Delta$ , $\Delta \geq 0$ . Replacing $Y = X - \alpha \delta$ we get that
+
+$$
+\begin{array} { r l } & { \Phi _ { X , \boldsymbol { \alpha } , \Delta } ( \mu ) = \log \left( \displaystyle \frac { I _ { Y + ( \alpha - 1 ) \Delta } ( \mu ) } { I _ { Y } ( \mu ) } \left( \frac { I _ { Y + ( \alpha - 1 ) \Delta } ( \mu ) } { I _ { Y + \alpha \Delta } ( \mu ) } \right) ^ { \alpha - 1 } \right) } \\ & { \qquad = \displaystyle \sum _ { j = 0 } ^ { \alpha - 2 } \left( \sum _ { i = Y + j \Delta + 1 } ^ { Y + j \Delta + \Delta } \left( \log \left( \frac { I _ { i - 1 } + ( \alpha - 1 - j ) ( \mu ) } { I _ { i + ( \alpha - 1 - j ) \Delta } ( \mu ) } \right) - \log \left( \frac { I _ { i - 1 } ( \mu ) } { I _ { i } ( \mu ) } \right) \right) \right) } \\ & { \qquad \le \displaystyle \sum _ { j = 0 } ^ { \alpha - 2 } \left( \sum _ { i = Y + j \Delta + 1 } ^ { Y + j \Delta + \Delta } \left( \delta _ { 2 } ( i + ( \alpha - 1 - j ) \Delta , \mu ) - \delta _ { 0 } ( i , \mu ) \right) \right) } \\ & { \qquad \le \displaystyle \frac { \alpha ( \alpha - 1 ) \Delta ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { \alpha ( \alpha - 1 ) \Delta ^ { 2 } + 2 ( \alpha - 1 ) \Delta } { 4 \mu ^ { 2 } } , \frac { ( \alpha - 1 ) \Delta } { 2 \mu } \right) , } \end{array}
+$$
+
+where the first inequality follows from Lemma 3.8 and the fact that for all $0 \leq x \leq y$ , $\operatorname { a r c s i n h } ( y ) -$ $\operatorname { a r c s i n h } ( x ) \leq y - x$ and the second inequality follows from Lemma A.1 (provided in the appendix):
+
+$$
+\delta _ { 2 } ( \nu _ { 1 } , x ) - \delta _ { 0 } ( \nu _ { 2 } , x ) \leq \frac { \nu _ { 1 } - \nu _ { 2 } } { x } + \frac { 1 } { 2 x } \operatorname* { m i n } \left( \frac { \nu _ { 1 } - \nu _ { 2 } + 1 } { x } , 1 \right) .
+$$
+
+# 4 Applying the Skellam Mechanism to Federated Learning
+
+With a sharp RDP analysis for the multi-dimensional Skellam mechanism presented in the previous section, we are now ready to apply it to differentially private federated learning. We first outline the general problem setting and then describe our approach under central and distributed DP models.
+
+Problem setting At a high-level, we consider the distributed mean estimation problem. There are $n$ clients each holding a vector $x _ { i }$ in $\mathbb { R } ^ { d }$ such that for all $i$ , the vector norm is bounded as $\| x _ { i } \| _ { 2 } \leq c$ for some $c \geq 0$ . We denote the set of vectors as $\mathcal { X } = \{ x _ { i } \} _ { i = 1 } ^ { n }$ , and the aim is for each client to communicate the vectors $x _ { i }$ to a central server which then aggregates them as $\begin{array} { r } { \widehat { x } = \frac { 1 } { n } \sum _ { i } x _ { i } } \end{array}$ for an external analyst. In federated learning, the client vectors $x _ { i }$ are the model gradients or model deltas (typically $d \geq 1 0 ^ { 6 }$ ) after training on the clients’ local datasets, and this procedure can be repeated for many rounds $T > 1 0 0 0 \ r ,$ . A large $d$ and $T$ thus necessitate accounting methods that provide tight privacy compositions for high-dimensional queries.
+
+We are primarily concerned with three metrics for this procedure and their trade-offs: (1) Privacy: the mean $\widehat { x }$ should be differentially private with a reasonably small $( \varepsilon , \delta )$ ; (2) Error: we wish to minimize the expected $\ell _ { 2 }$ error; and (3) Communication: we wish to minimize the average number of bits communicated per coordinate. Characterizing this trade-off is an important research problem. For example, it has been recently shown [50] that without formal privacy guarantees, the client training data could still be revealed by the model updates $x _ { i }$ ; on the other hand, applying differential privacy [48] to these updates can degrade the final utility.
+
+![](images/8a7da83f72e1327e3ea24b8cddf36aa7549a54ed5a6c18bfacb1f5d9704a83fa.jpg)  
+Figure 2: Benchmarking Skellam on sensitivity-1 queries under various accounting methods. RDP: Rényi DP. PLD: privacy loss distributions. Skellam (Direct): [49]. Gaussian (Analytic): [9]. DGaussian [15] / DDGauss [25]: central / distributed discrete Gaussian. $s$ is the scaling factor applied to both $\Delta$ and $\sigma$ . For Skellam and DDGauss [25], the central noise with std $\sigma$ is split into $n$ shares each applied locally with std $\sigma / { \sqrt { n } }$ ; a large $n$ and small $\sigma$ can thus exacerbate the sum divergence term of DDGauss (left). Left: $\varepsilon \leq 1 0$ . Right: $\varepsilon \leq 1$ .
+
+![](images/c38290ebb9dc27b41d63ab39e366b01c3b59452ece0d54d60bdf13b39554b0e0.jpg)  
+Figure 3: Comparing Skellam and Distributed Discrete Gaussian (DDGauss) on multi-dimensional real-valued queries, rounded to integers with $\beta = e ^ { - 0 . 5 }$ (Prop. 4.2). $s$ is the scaling applied to both $\sigma$ and $\Delta _ { 2 }$ ; a larger $s$ reduces the rounding error and norm inflation. $q$ is the sampling rate. For Skellam and DDGauss [25], the central noise with std $\sigma$ is split into $n$ shares each applied locally with std $\sigma / { \sqrt { n } }$ ; a large $n$ and small $\sigma$ can exacerbate the sum divergence term of DDGauss. Left: Simple setting with $\Delta _ { 2 } = 1$ . Right: FL-like setting for training CNNs on Federated EMNIST.
+
+Skellam for central DP The central DP model refers to adding Skellam noise onto the non-private aggregate $\widehat { x }$ before releasing it to the external analyst. One important consideration is that the model updates in $\mathrm { F L }$ are continuous in nature, while Skellam is a discrete probability distribution. One approach is to appropriately discretize the client updates, e.g., via uniform quantization (which involves scaling the inputs by a factor $s \sim 2 ^ { b }$ for some bit-width $b$ followed by stochastic rounding5 for unbiased estimates), and the server can convert the private aggregate back to real numbers at the end. Note that this allows us to re-parameterize the variance of the added Skellam noise as $s ^ { 2 } \mu$ , giving the following simple corollary based on Cor. 3.6:
+
+Corollary 4.1 (Scaled Skellam Mechanism). With a scaling factor $s \in \mathbb { R } ,$ , the multi-dimensional Skellam Mechanism is $( \alpha , \varepsilon )$ -RDP with
+
+$$
+\varepsilon ( \alpha ) \leq \frac { \alpha \Delta _ { 2 } ^ { 2 } } { 2 \mu } + \operatorname* { m i n } \left( \frac { ( 2 \alpha - 1 ) \Delta _ { 2 } ^ { 2 } } { 4 s ^ { 2 } \mu ^ { 2 } } + \frac { 3 \Delta _ { 1 } } { 2 s ^ { 3 } \mu ^ { 2 } } , \frac { 3 \Delta _ { 1 } } { 2 s \mu } \right) .
+$$
+
+As $s$ increases, the RDP of scaled Skellam rapidly approaches that of Gaussian as the second term above approaches 0, suggesting that under practical regimes with moderate compression bit-width, Skellam should perform competitively compared to Gaussian. Another aspect worth noting is that rounding vector coordinates from reals to integers can inflate the $\ell _ { 2 }$ -sensitivity $\Delta _ { 2 }$ , and thus more noise is required for the same privacy. To this end, we leverage the conditional rounding procedure introduced in [25] to obtain a bounded norm on the scaled and rounded client vector:
+
+Proposition 4.2 (Norm of stochastically rounded vector [25]). Let $\tilde { x }$ be a stochastic rounding of vector $x \in \mathbb { R } ^ { d }$ to the integer grid $\mathbb { Z } ^ { d }$ . Then, for $\beta \in ( 0 , 1 )$ , we have
+
+$$
+\begin{array} { r } { \mathbb { P } \left[ \| \tilde { x } \| _ { 2 } ^ { 2 } \leq \| x \| _ { 2 } ^ { 2 } + d / 4 + \sqrt { 2 \log ( 1 / \beta ) } \cdot \left( \| x \| _ { 2 } + \sqrt { d } / 2 \right) \right] \geq 1 - \beta . } \end{array}
+$$
+
+Conditional rounding is thus defined as retrying the stochastic rounding on $x _ { i }$ until $\| \tilde { x } _ { i } \| _ { 2 } ^ { 2 }$ is within the probabilistic bound above (which also gives the inflated sensitivity $\tilde { \Delta } _ { 2 }$ ). We can then add Skellam noise to the aggregate $\sum _ { i } \tilde { x } _ { i }$ according to $\mathrm { { \bar { \Delta } } } _ { 2 }$ before undoing the quantization (unscaling). Note that a larger scaling $s$ before rounding reduces the norm inflation and the extra noise needed (Fig. 3 right).
+
+Skellam for distributed DP with secure aggregation A stronger notion of privacy in FL can be obtained via the distributed DP model [25] that leverages secure aggregation (SecAgg [12]). The fact that the Skellam distribution is closed under summation allows us to easily extend from central DP to distributed DP. Under this model, the client vectors are quantized as in central DP model, but the Skellam noise is now added locally with variance $\mu / n$ . Then, the noisy client updates are summed via SecAgg ( $b$ bits per coordinate for field size $2 ^ { b }$ ) which only reveals the noisy aggregate to the server. While the local noise might be insufficient for local DP guarantees, the aggregated noise at the server provides privacy and utility comparable to the central DP model, thus removing trust away from the central aggregator. Note that the modulo operations introduced by SecAgg does not impact privacy as it can be viewed as a post-processing of an already differentially private query.
+
+We remark on several properties of the distributed Skellam compared to the distributed discrete Gaussian (DDGauss [25]). (1) DDGauss is not closed under summation, and the divergence between discrete Gaussians can lead to notable privacy degradation in settings such as quantile estimation [6] and federated analytics [42] with sufficiently large number of clients and small local noises (see also the left side of Fig. 2 and Fig. 3). While scaling mitigates this issue, it also requires additional bit-width which makes Skellam attractive under tight communication constraints. (2) Sampling from Skellam only requires sampling from Poisson, for which efficient implementations are widely available in numerical software packages. While efficient discrete Gaussian sampling has also been explored in the lattice-based cryptography community (e.g., [43, 18, 38]), we believe the accessibility of Skellam samplers would help facilitate the deployment of DP to FL settings with mobile and edge devices. See Appendix D for more discussion. (3) In practice where $s \gg 1$ (dictated by bit-width $b$ ), both Skellam (cf. Cor. 4.1) and DDGauss (with an exponentially small divergence) quickly approaches Gaussian under RDP, and any differences will be negligible (Fig. 3).
+
+# 5 Empirical Evaluation
+
+In this section, we empirically evaluate the Skellam mechanism on two sets of experiments: distributed mean estimation and federated learning. In both cases, we focus on the distributed DP model, but note that the Skellam mechanism can be easily adapted to the central DP setting as discussed in the earlier section. Unless otherwise stated, we use RDP accounting for all experiments due to the high-dimensional data and the ease of composition (Section 3). To obtain $\Delta _ { 1 }$ for Skellam RDP, we note that $\Delta _ { 1 } \leq \Delta _ { 2 } \cdot \operatorname* { m i n } ( \sqrt { d } , \Delta _ { 2 } )$ since $\Delta _ { 1 } \leq \sqrt { d } \Delta _ { 2 }$ in general and $\Delta _ { 1 } \leq \Delta _ { 2 } ^ { 2 }$ for integers.
+
+Under the distributed DP model, we also introduce a random orthogonal transformation [29, 2, 25] before discretizing and aggregating the client vectors (which can be reverted after the aggregation); this makes the vector coordinates sub-Gaussian and helps spread the magnitudes of the vector coordinates across all dimensions, thus reducing the errors from quantization and potential wraparound from SecAgg modulo operations. Moreover, by approximating the sub-exponential tail of the Skellam distribution as sub-Gaussian, we can derive a heuristic for choosing $s$ following [25] based on a bound on the variance $\tilde { \sigma } ^ { 2 }$ of the aggregated signal, as $\tilde { \sigma } ^ { 2 } \le c ^ { 2 } n ^ { 2 } / d + \bar { n } / ( 4 s ^ { 2 } ) + \mu$ . We choose $s$ such that $2 k \tilde { \sigma }$ are bounded within the SecAgg field size $2 ^ { b }$ , where $k$ is a small constant.
+
+Algorithm 1 summarizes the aggregation procedure for the distributed Skellam mechanism via secure
+
+Inputs: Private vector $x _ { i } \in \mathbb { R } ^ { \bar { d } }$ for each client $i$ ; $\ell _ { 2 }$ clip norm $c > 0$ ; Bit-width $b$ ; Target central   
+noise variance $\mu > 0$ ; Number of clients $n$ ; Signal bound multiplier $k > 0$ ; Bias $\beta \in [ 0 , 1 )$ .   
+Shared randomness: $d \times d$ diagonal matrix $D$ with uniformly random $\{ - 1 , + 1 \}$ values, where   
+$d \geq \bar { d }$ is the nearest power of 2.   
+Shared scale: Obtain scaling factor $s$ such that $2 ^ { b } = 2 k \tilde { \sigma } = 2 k \sqrt { c ^ { 2 } n ^ { 2 } / d + n / ( 4 s ^ { 2 } ) + \mu } .$   
+Procedure CLIENTPROCEDURE $\mathbf { \chi } _ { \left( x _ { i } , s , D \right) }$ Clip and scale vector $\hat { x } _ { i } = s \cdot \operatorname* { m i n } ( 1 , c / \| x _ { i } \| _ { 2 } ) \cdot x _ { i }$ , and pad to $\tilde { d }$ dimensions with zeros. Random rotation: $\check { x } _ { i } = \tilde { H } _ { d } D \hat { x } _ { i }$ where $\begin{array} { r } { \tilde { H } _ { d } = \frac { 1 } { \sqrt { d } } H _ { d } } \end{array}$ is the normalized $d \times d$ Hadamard matrix. repeat {conditional stochastic rounding} Stochastically round the coordinates of of ${ \check { x } } _ { i }$ to the integer grid to produce ${ \tilde { x } } _ { i }$ until $\| \widetilde x _ { i } \| _ { 2 } ^ { 2 } \le \operatorname* { m i n } \bigg \{ \Big ( s c + \sqrt d \Big ) ^ { 2 } , s ^ { 2 } c ^ { 2 } + d / 4 + \sqrt { 2 \log ( 1 / \beta ) } \cdot \Big ( s c + \sqrt d / 2 \Big ) \bigg \} .$ . Local noising: Sample noise vector $y _ { i } \in \mathbb { Z } ^ { d }$ where each entry is sampled from $\operatorname { S k } _ { 0 , s ^ { 2 } \mu / n }$ . return $z _ { i } = \tilde { x } _ { i } + y _ { i }$ under the SecAgg protocol with modulo bit-width $b$ .   
+Procedure SERVERPROCEDURE $( z , s , D )$ $\{ z$ is the modular sum of $z _ { i }$ under bit-width $b$ } return $\begin{array} { r } { \bar { x } = \frac { 1 } { s } D \tilde { H } _ { d } ^ { \top } z } \end{array}$ , with $\textstyle { \bar { x } } \approx \sum _ { i } x _ { i } \in \mathbb { R } ^ { d }$ .
+
+![](images/a258818d6f7517c63fee2ce116dc0cb01bea0fe531df9cafcfe5043c2794f7e1.jpg)  
+Figure 4: Distributed mean estimation with the distributed Skellam mechanism.
+
+aggregation as well as the parameters used for the experiments. In summary, we have an $\ell _ { 2 }$ clip norm $c > 0$ ; per-coordinate bit-width $b$ ; target central noise variance $\mu > 0$ ; number of clients $n$ ; signal bound multiplier $k > 0$ ; and rounding bias $\beta \in [ 0 , 1 )$ . We fix $\beta = e ^ { - 1 / 2 }$ for all experiments. Note that the per-coordinate bit-width $b$ is for the aggregated sum as it determines the field size of SecAgg. For federated learning, we also consider the number of rounds $T$ and the total number of clients $N$ (thus the uniform sampling ratio $q = n / N$ at every round). Our experiments are implemented in Python, TensorFlow Privacy [32], and TensorFlow Federated [24]. See also Appendix for additional results and more details on the experimental setup.
+
+# 5.1 Distributed Mean Estimation (DME)
+
+We first consider DME as the generalization of (single round) FL. We randomly generate $n$ client vectors $X = \{ x _ { i } \} _ { i = 1 } ^ { n }$ from the $d$ -dimensional $\ell _ { 2 }$ sphere with radius $c = 1 0$ , and compute the true mean $\begin{array} { r } { \widehat { x } = \frac { 1 } { n } \sum _ { i } ^ { n } x _ { i } } \end{array}$ 2 . We then compute the private estimate of $\widehat { x }$ with the distributed Skellam mechanism (Algorithm 1) as $\bar { x }$ . For a strong baseline, we use the analytic Gaussian mechanism [9] with tight accounting (see also Figure 2). In Figure 4, we plot the MSE as $\| \widehat { x } - \bar { x } \| _ { 2 } ^ { 2 } / d$ with $9 5 \%$ confidence interval (small shaded region) over 10 dataset initializations across different values of $b , d$ , and $n$ . Results demonstrate that Skellam can match Gaussian even with $n = 1 0 0 0 0$ clients as long as the bit-width is sufficient. We emphasize that the communication cost $b$ depends logarithmically on $n$ , and to put numbers into context, Google’s production next-word prediction models [23, 39] use $n \leq 5 0 0$ and the production DP language model [40] uses $n = 2 0 0 0 0$ .
+
+# 5.2 Federated Learning
+
+Setup We evaluate on three public federated datasets with real-world characteristics: Federated EMNIST [16], Shakespeare [31, 14], and Stack Overflow next word prediction (SO-NWP [8]). EMNIST is an image classification dataset for hand-written digits and letters; Shakespeare is a text dataset for next-character-prediction based on the works of William Shakespeare; and SO-NWP is a large-scale text dataset for next-word-prediction based on user questions/answers from stackoverflow.com. We emphasize that all datasets have natural client heterogeneity that are representative of practical FL problems: the images in EMNIST are grouped the writer of the handwritten digits, the lines in Shakespeare are grouped by the speaking role, and the sentences in SO-NWP are grouped by the corresponding Stack Overflow user. We train a small CNN with model size $\bar { d } < 2 ^ { 2 0 }$ for EMNIST and use the recurrent models defined in [41] for Shakespeare and SO-NWP. The hyperparameters for the experiments follow those from [25, 6, 27, 41] and tuning is limited. For EMNIST, we follow [25] and fix $c = 0 . 0 3$ , $n = 1 0 0$ , $T = 1 5 0 0$ , client learning rate $\eta _ { \mathrm { c l i e n t } } = 0 . 3 2$ , server learning rate $\eta _ { \mathrm { s e r v e r } } = 1$ , and client batch size $m = 2 0$ . For Shakespeare, we follow [6] and fix $n = 1 0 0$ , $T = 1 2 0 0$ , $\eta _ { \mathrm { c l i e n t } } = 1$ , $\eta _ { \mathrm { s e r v e r } } = 0 . 3 2$ , and $m = 4$ , and we sweep $c \in \{ 0 . 2 5 , 0 . 5 \}$ . For SO-NWP, we follow [27] and fix $c = 0 . 3$ , $n = 1 0 0$ , $T = 1 6 0 0$ , $\eta _ { \mathrm { c l i e n t } } = 0 . 5$ , and $m = 1 6$ , and we sweep $\eta _ { \mathrm { s e r v e r } } \in \{ 0 . 3 , 1 \}$ and limit max examples per client to 256. In all cases, clients train for 1 epoch on their local datasets, and the client updates are weighted uniformly (as opposed to weighting by number of examples). See Appendix for more results and full details on datasets, models, and hyperparameters.
+
+![](images/17c77904c74691a84afb047deea14f30ee01bb1aff6b1891d87bbfa175d13dd8.jpg)  
+Figure 5: Federated learning with the distributed Skellam mechanism. DDGauss: Distributed Discrete Gaussian [25]. Left / Middle / Right: Test accuracies on EMNIST / Shakespeare / Stack Overflow NWP across different $\varepsilon$ and $b$ . $\delta$ is set to $1 / N , 1 0 ^ { - 6 } , 1 0 ^ { - 6 }$ , respectively. For Shakespeare, privacy is reported with a hypothetical population size $N = 1 0 ^ { 6 }$ .
+
+Results Figure 5 summarizes the FL experiments. For EMNIST and Shakespeare, we report the average test accuracy over the last 100 rounds. For SO-NWP, we report the top-1 accuracy (without padding, out-of-vocab, or begining/end-of-sentence tokens) on the test set. The results indicate that Skellam performs as good as Gaussian despite relying on generic RDP amplification via sampling [51] (cf. Fig. 3) and that Skellam matches DDG consistently under realistic regimes. This bears significant practical relevance given the advantages of Skellam over DDG in real-world deployments.
+
+# 6 Conclusion
+
+We have introduced the multi-dimensional Skellam mechanism for federated learning. We analyzed the Skellam mechanism through the lens of approximate DP, privacy loss distributions, and Rényi divergences, and derived a sharp RDP bound that enables Skellam to match Gaussian and discrete Gaussian in practical settings as demonstrated by our large-scale experiments. Since Skellam is closed under summation and efficient samplers are widely available, it represents an attractive alternative to distributed discrete Gaussian as it easily extends from the central DP model to the distributed DP model. Being a discrete mechanism can also bring potential communication savings over continuous mechanisms and make Skellam less prone to attacks that exploit floating-point arithmetic on digital computers. Some interesting future work includes: (1) our scalar PLD analysis for Skellam suggests room for improvements on our multi-dimensional analysis via a complete PLD characterization, and (2) our results on FL may be further improved via a targeted analysis for RDP amplification via sampling akin to [37]. Overall, this work is situated within the active area of private machine learning and aims at making ML more trustworthy. One potential negative impact is that our method could be (deliberately or inadvertently) misused, such as sampling the wrong noise or using a minuscule scaling factor, to provide non-existent privacy guarantees for real users’ data. We nevertheless believe our results have positive impact as they facilitate the deployment of differential privacy in practice.
+
+# Funding Transparency Statement
+
+The authors were employed at and directly supported by Google. No third party funding was received by any of the authors to pursue this work.
+
+# References
+
+[1] 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. [2] Naman Agarwal, Ananda Theertha Suresh, Felix Xinnan X Yu, Sanjiv Kumar, and Brendan McMahan. cpSGD: Communication-efficient and differentially-private distributed sgd. In Advances in Neural Information Processing Systems, pages 7564–7575, 2018. [3] Maruan Al-Shedivat, Jennifer Gillenwater, Eric Xing, and Afshin Rostamizadeh. Federated learning via posterior averaging: A new perspective and practical algorithms. In ICLR, 2021. [4] Martin R. Albrecht and Michael Walter. dgs, Discrete Gaussians over the Integers. Available at https: //bitbucket.org/malb/dgs, 2018. [5] Donald E Amos. Computation of modified bessel functions and their ratios. Mathematics of Computation, 28(125):239–251, 1974. [6] Galen Andrew, Om Thakkar, H Brendan McMahan, and Swaroop Ramaswamy. Differentially private learning with adaptive clipping. arXiv preprint arXiv:1905.03871, 2019. [7] S. Asoodeh, J. Liao, F. P. Calmon, O. Kosut, and L. Sankar. A better bound gives a hundred rounds: Enhanced privacy guarantees via f-divergences. In 2020 IEEE International Symposium on Information Theory (ISIT), pages 920–925, 2020. [8] The TensorFlow Federated Authors. Tensorflow federated stack overflow dataset, 2019. [9] 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 394–403. PMLR, 2018.   
+[10] Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimization: Efficient algorithms and tight error bounds. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 464–473. IEEE, 2014.   
+[11] James Bell, K. A. Bonawitz, Adrià Gascón, Tancrède Lepoint, and Mariana Raykova. Secure singleserver aggregation with (poly)logarithmic overhead. Cryptology ePrint Archive, Report 2020/704, 2020. https://eprint.iacr.org/2020/704.   
+[12] Keith Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth. Practical secure aggregation for privacy-preserving machine learning. In proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pages 1175–1191, 2017.   
+[13] Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635–658. Springer, 2016.   
+[14] Sebastian Caldas, Sai Meher Karthik Duddu, Peter Wu, Tian Li, Jakub Konecnˇ y, H Brendan McMa- \` han, Virginia Smith, and Ameet Talwalkar. Leaf: A benchmark for federated settings. arXiv preprint arXiv:1812.01097, 2018.   
+[15] Clément Canonne, Gautam Kamath, and Thomas Steinke. The discrete gaussian for differential privacy. In NeurIPS, 2020.   
+[16] Gregory Cohen, Saeed Afshar, Jonathan Tapson, and Andre Van Schaik. Emnist: Extending mnist to handwritten letters. In 2017 International Joint Conference on Neural Networks (IJCNN), pages 2921–2926. IEEE, 2017.   
+[17] Google Differential Privacy Team. Privacy loss distributions. https://github.com/google/ differential-privacy/blob/master/accounting/docs/Privacy_Loss_Distributions.pdf. 2020.   
+[18] Nagarjun C Dwarakanath and Steven D Galbraith. Sampling from discrete gaussians for lattice-based cryptography on a constrained device. Applicable Algebra in Engineering, Communication and Computing, 25(3):159–180, 2014.   
+[19] 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.   
+[20] 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.   
+[21] Cynthia Dwork and Guy N Rothblum. Concentrated differential privacy. arXiv preprint arXiv:1603.01887, 2016.   
+[22] 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.   
+[23] Andrew Hard, Kanishka Rao, Rajiv Mathews, Swaroop Ramaswamy, Françoise Beaufays, Sean Augenstein, Hubert Eichner, Chloé Kiddon, and Daniel Ramage. Federated learning for mobile keyboard prediction. arXiv preprint arXiv:1811.03604, 2018.   
+[24] Alex Ingerman and Krzys Ostrowski. Introducing tensorflow federated, 2019.   
+[25] Peter Kairouz, Ziyu Liu, and Thomas Steinke. The distributed discrete gaussian mechanism for federated learning with secure aggregation. In International Conference on Machine Learning. PMLR, 2021.   
+[26] Peter Kairouz, Brendan McMahan, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   
+[27] Peter Kairouz, Brendan McMahan, Shuang Song, Om Thakkar, Abhradeep Thakurta, and Zheng Xu. Practical and private (deep) learning without sampling or shuffling. arXiv preprint arXiv:2103.00039, 2021.   
+[28] Peter Kairouz, Sewoong Oh, and Pramod Viswanath. The composition theorem for differential privacy. In International conference on machine learning, pages 1376–1385. PMLR, 2015.   
+[29] Jakub Konecnˇ y, H Brendan McMahan, Felix X Yu, Peter Richtárik, Ananda Theertha Suresh, and \` Dave Bacon. Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016.   
+[30] Antti Koskela, Joonas Jälkö, and Antti Honkela. Computing tight differential privacy guarantees using fft. In International Conference on Artificial Intelligence and Statistics, pages 2560–2569. PMLR, 2020.   
+[31] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282, 2017.   
+[32] H Brendan McMahan, Galen Andrew, Ulfar Erlingsson, Steve Chien, Ilya Mironov, Nicolas Papernot, and Peter Kairouz. A general approach to adding differential privacy to iterative training procedures. arXiv preprint arXiv:1812.06210, 2018.   
+[33] H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning differentially private recurrent language models. In ICLR, 2018.   
+[34] 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.   
+[35] Ilya Mironov. On significance of the least significant bits for differential privacy. In Proceedings of the 2012 ACM conference on Computer and communications security, pages 650–661, 2012.   
+[36] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 263–275. IEEE, 2017.   
+[37] Ilya Mironov, Kunal Talwar, and Li Zhang. Rényi differential privacy of the sampled gaussian mechanism. arXiv preprint arXiv:1908.10530, 2019.   
+[38] Thomas Prest, Thomas Ricosset, and Mélissa Rossi. Simple , fast and constant-time gaussian sampling over the integers for falcon. In Second PQC Standardization Conference, 2019.   
+[39] Swaroop Ramaswamy, Rajiv Mathews, Kanishka Rao, and Françoise Beaufays. Federated learning for emoji prediction in a mobile keyboard. arXiv preprint arXiv:1906.04329, 2019.   
+[40] Swaroop Ramaswamy, Om Thakkar, Rajiv Mathews, Galen Andrew, H Brendan McMahan, and Françoise Beaufays. Training production language models without memorizing user data. arXiv preprint arXiv:2009.10031, 2020.   
+[41] Sashank Reddi, Zachary Charles, Manzil Zaheer, Zachary Garrett, Keith Rush, Jakub Konecnˇ y, Sanjiv \` Kumar, and H Brendan McMahan. Adaptive federated optimization. arXiv preprint arXiv:2003.00295, 2020.   
+[42] Google Research. Federated analytics: Collaborative data science without data collection, May 2020.   
+[43] Sujoy Sinha Roy, Frederik Vercauteren, and Ingrid Verbauwhede. High precision discrete gaussian sampling on fpgas. In International Conference on Selected Areas in Cryptography, pages 383–401. Springer, 2013.   
+[44] Diego Ruiz-Antolín and Javier Segura. A new type of sharp bounds for ratios of modified bessel functions. Journal of Mathematical Analysis and Applications, 443(2):1232–1246, 2016.   
+[45] Aaron Schein, Zhiwei Steven Wu, Alexandra Schofield, Mingyuan Zhou, and Hanna Wallach. Locally private bayesian inference for count models. In International Conference on Machine Learning, pages 5638–5648. PMLR, 2019.   
+[46] Shuang Song, Kamalika Chaudhuri, and Anand D Sarwate. Stochastic gradient descent with differentially private updates. In 2013 IEEE Global Conference on Signal and Information Processing, pages 245–248. IEEE, 2013.   
+[47] VR Thiruvenkatachar and TS Nanjundiah. Inequalities concerning bessel functions and orthogonal polynomials. In Proceedings of the Indian Academy of Sciences-Section A, volume 33, page 373. Springer, 1951.   
+[48] Florian Tramèr and Dan Boneh. Differentially private learning needs better features (or much more data). arXiv preprint arXiv:2011.11660, 2020.   
+[49] Filipp Valovich and Francesco Alda. Computational differential privacy from lattice-based cryptography. In International Conference on Number-Theoretic Methods in Cryptology, pages 121–141. Springer, 2017.   
+[50] Hongxu Yin, Arun Mallya, Arash Vahdat, Jose M Alvarez, Jan Kautz, and Pavlo Molchanov. See through gradients: Image batch recovery via gradinversion. arXiv preprint arXiv:2104.07586, 2021.   
+[51] Yuqing Zhu and Yu-Xiang Wang. Poission subsampled rényi differential privacy. In International Conference on Machine Learning, pages 7634–7642. PMLR, 2019.

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+        "type": "text",
+        "text": "The Skellam Mechanism for Differentially Private Federated Learning ",
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+        "text": "Naman Agarwal† Peter Kairouz† Ziyu Liu‡⇤ †Google Research ‡Carnegie Mellon University {namanagarwal, kairouz}@google.com, ziyuliu@cs.cmu.edu ",
+        "bbox": [
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduce the multi-dimensional Skellam mechanism, a discrete differential privacy mechanism based on the difference of two independent Poisson random variables. To quantify its privacy guarantees, we analyze the privacy loss distribution via a numerical evaluation and provide a sharp bound on the Rényi divergence between two shifted Skellam distributions. While useful in both centralized and distributed privacy applications, we investigate how it can be applied in the context of federated learning with secure aggregation under communication constraints. Our theoretical findings and extensive experimental evaluations demonstrate that the Skellam mechanism provides the same privacy-accuracy trade-offs as the continuous Gaussian mechanism, even when the precision is low. More importantly, Skellam is closed under summation and sampling from it only requires sampling from a Poisson distribution – an efficient routine that ships with all machine learning and data analysis software packages. These features, along with its discrete nature and competitive privacy-accuracy trade-offs, make it an attractive practical alternative to the newly introduced discrete Gaussian mechanism. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "The Gaussian mechanism is the workhorse for a multitude of differentially private learning algorithms [46, 10, 1]. While simple enough for mathematical reasoning and privacy accounting analyses, its continuous nature presents a number of challenges in practice. For example, it cannot be exactly represented on finite computers, making it prone to numerical errors that can break its privacy guarantees [35]. Moreover, it cannot be used in distributed learning settings with cryptographic multi-party computation primitives involving modular arithmetic, such as secure aggregation [12, 11]. To address these shortcomings, the binomial and (distributed) discrete Gaussian mechanisms were recently introduced [19, 2, 15, 25]. Unfortunately, both have their own drawbacks: the privacy loss for the binomial mechanism can be infinite with a non-zero probability, and the discrete Gaussian: (a) is not closed under summation (i.e. sum of discrete Gaussians is not a discrete Gaussian), complicating analysis in distributed settings and leading to a performance worse than continuous Gaussian in the highly distributed, low-noise regime [25]; (b) requires a sampling algorithm that is not shipped with mainstream machine learning or data analysis software packages, making it difficult for engineers to use it in production settings (naïve implementations may lead to catastrophic privacy errors). ",
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+        "type": "text",
+        "text": "Our contributions To overcome these limitations, we introduce and analyze the multi-dimensional Skellam mechanism, a mechanism based on adding noise distributed according to the difference of two independent Poisson random variables. The Skellam noise is closed under summation (i.e. sums of Skellam random variables is again Skellam distributed) and can be sampled from easily – efficient Poisson samplers are widely available in numerical software packages. Being discrete in nature also means that it can mesh well cryptographic protocols and can lead to communication savings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To analyze the privacy guarantees of the Skellam mechanism and compare it with other mechanisms, we provide a numerical evaluation of the privacy loss random variable and prove a sharp bound on the Rényi divergence between two shifted Skellam distributions. Our careful analysis shows that for a multi-dimensional query function with $\\ell _ { 1 }$ sensitivity $\\Delta _ { 1 }$ and $\\ell _ { 2 }$ sensitivity $\\Delta _ { 2 }$ , the Skellam mechanism with variance $\\mu$ achieves $( \\alpha , \\varepsilon ( \\alpha ) )$ Rényi differential privacy (RDP) [36] for $\\varepsilon ( \\alpha ) \\leq$ $\\begin{array} { r } { \\frac { \\alpha \\Delta _ { 2 } } { 2 \\mu } + \\operatorname* { m i n } { \\left( \\frac { ( 2 \\alpha - 1 ) \\Delta _ { 2 } + 6 \\Delta _ { 1 } } { 4 \\mu ^ { 2 } } , \\frac { 3 \\Delta _ { 1 } } { 2 \\mu } \\right) } } \\end{array}$ (see Theorem 3.5). This implies that the RDP guarantees are at most $1 + O \\left( 1 / \\mu \\right)$ times worse than those of the Gaussian mechanism. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To analyze the performance of the Skellam mechanism in practice, we consider a differentially private and communication constrained federated learning (FL) setting [26] where the noise is added locally to the $d$ -dimensional discretized client updates that are then summed securely via a cryptographic protocol, such as secure aggregation $( { \\mathrm { S e c A g g } } )$ [11, 12]. We provide an end-to-end algorithm that appropriately discretizes the data and applies the Skellam mechanism along with modular arithmetic to bound the range of the data and communication costs before applying SecAgg. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We show on distributed mean estimation and two benchmark FL datasets, Federated EMNIST [14] and Stack Overflow [8], that our method can match the performance of the continuous Gaussian baseline under tight privacy and communication budgets, despite using generic RDP amplification via sampling [51] for our approach and the precise RDP analysis for the subsampled Gaussian mechanism [37]. Our method is implemented in TensorFlow Privacy [32] and TensorFlow Federated [24] and will be open-sourced.2 While we mostly focus on FL applications, the Skellam mechanism can also be applied in other contexts of learning and analytics, including centralized settings. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Related work The Skellam mechanism was first introduced in the context of computational differential privacy from lattice-based cryptography [49] and private Bayesian inference [45]. However, the privacy analyses in the prior work do not readily extend to the multi-dimensional case, and they give direct bounds for pure or approximate DP which makes only advanced composition theorems [28, 22] directly applicable in learning settings where the mechanism is applied many times. For example, the guarantees from [49] lead to poor accuracy-privacy trade-offs as demonstrated in Fig. 1. Moreover, we show in Section 3.1 that extending the direct privacy analysis to the multi-dimensional setting is non-trivial because the worst-case neighboring dataset pair is unknown in this case. For these reasons, our tight privacy analysis via a sharp RDP bound makes the Skellam mechanism practical for learning applications for the first time. These guarantees (almost) match those of the Gaussian mechanism and allow us to use generic RDP amplification via subsampling methods [51]. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The closest mechanisms to Skellam are the binomial [2, 19] and the discrete Gaussian mechanisms [15, 25]. The binomial mechanism can (asymptotically) match the continuous Gaussian mechanism (when properly scaled). However, it does not achieve Rényi or zero-concentrated DP [36, 13] and has a privacy loss that can be infinite with a non-zero probability, leading to catastrophic privacy failures. The discrete Gaussian mechanism yields Rényi DP and can be applied to distributed settings [25], but it requires a sampling algorithm that is not yet available in data analysis software packages despite being explored in the lattice-based cryptography community (e.g., [43, 18, 38]). The discrete Gaussian is also not closed under summation and the divergence can be large in highly distributed low-noise settings (e.g. quantile estimation [6] and federated analytics [42]), which causes privacy degradation. See the end of Section 4 for more discussion. ",
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+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We begin by providing a formal definition for $( \\varepsilon , \\delta )$ -differential privacy (DP) [20]. ",
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+    {
+        "type": "text",
+        "text": "Definition 2.1 (Differential Privacy). For $\\varepsilon , \\delta \\geq 0$ , a randomized mechanism $M$ satisfies $( \\varepsilon , \\delta )$ -DP if for all neighboring datasets $D , D ^ { \\prime }$ and all $s$ in the range of $M$ , we have that ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be869f9d048df11a4960f08a50ae6f7a76b53d4e308c187484dd7d9996afb720.jpg",
+        "text": "$$\nP \\left( M ( D ) \\in { \\mathcal { S } } \\right) \\leq e ^ { \\varepsilon } P \\left( M ( D ^ { \\prime } ) \\in { \\mathcal { S } } \\right) + \\delta ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $D$ and $D ^ { \\prime }$ are neighboring pairs if they can be obtained from each other by adding or removing all the records that belong to a particular user. ",
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+    {
+        "type": "text",
+        "text": "In our experiments we consider user-level differential privacy – i.e., $D$ and $D ^ { \\prime }$ are neighboring pairs ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "if one of them can be obtained from the other by adding or removing all the records associated with a single user [33]. This is stronger than the commonly-used notion of item level privacy where, if a user contributes multiple records, only the addition or removal of one record is protected. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also make use of Rényi differential privacy (RDP) [36] which allows for tight privacy accounting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 2.2 (Rényi Differential Privacy). A mechanism $M$ satisfies $( \\alpha , \\varepsilon )$ -RDP if for any two neighboring datasets $D , D ^ { \\prime }$ , we have that $D _ { \\alpha } ( M ( D ) , M ( D ^ { \\prime } ) ) \\leq \\varepsilon$ where $D _ { \\alpha } ( P , Q )$ is the Rényi divergence between $P$ and $Q$ and is given by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ee3a3f530e0d859fd11a126a2601a937fdc49e76e537724c03f1bb2a5fc894a.jpg",
+        "text": "$$\nD _ { \\alpha } ( P , Q ) \\triangleq \\frac { 1 } { \\alpha - 1 } \\log \\left( \\mathbb { E } _ { x \\sim Q } \\left[ \\left( \\frac { P ( x ) } { Q ( x ) } \\right) ^ { \\alpha } \\right] \\right) = \\frac { 1 } { \\alpha - 1 } \\log \\left( \\mathbb { E } _ { x \\sim P } \\left[ \\left( \\frac { P ( x ) } { Q ( x ) } \\right) ^ { \\alpha - 1 } \\right] \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A closely related privacy notion is zero-concentrated DP (zCDP) [21, 13]. In fact, $\\scriptstyle { \\frac { 1 } { 2 } } \\varepsilon ^ { 2 } - z \\mathbf { C D P }$ i s equivalent to simultaneously satisfying an infinite family of RDP guarantees, namely $( \\alpha , \\textstyle { \\frac { 1 } { 2 } } \\varepsilon ^ { 2 } \\alpha )$ - Rényi differential privacy for all $\\alpha \\in ( 1 , \\infty )$ . The following conversion lemma from [13, 15, 7] relates RDP to $( \\varepsilon , \\delta )$ -DP. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.3. If $M$ satisfies $( \\alpha , \\varepsilon )$ -RDP, then, for any $\\delta > 0$ , $M$ satisfies $( \\varepsilon _ { D P } ( \\delta ) , \\delta )$ -DP, where ",
+        "bbox": [
+            171,
+            330,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eced28a3cc8b75f2d958122dba881331e91f813b8442eb347818256431eae54d.jpg",
+        "text": "$$\n\\varepsilon _ { D P } ( \\delta ) = \\operatorname* { i n f } _ { \\alpha > 1 } \\varepsilon + \\frac { \\log ( 1 / \\alpha \\delta ) } { \\alpha - 1 } + \\log ( 1 - 1 / \\alpha ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For any query function $f$ , we define the $\\Delta _ { p }$ sensitivity as $\\begin{array} { r } { \\operatorname* { m a x } _ { D , D ^ { \\prime } } \\| f ( D ) - f ( D ^ { \\prime } ) \\| _ { p } } \\end{array}$ , where $D$ and $D ^ { \\prime }$ are neighboring pairs differing by adding or removing all the records from a particular user. We also include the RDP guarantees of the discrete Gaussian mechanism (same RDP guarantees as the continuous Gaussian mechanism) to which we compare our method. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.4 (The Discrete Gaussian Mechanism [15]). Given an integer-valued query $f ( D ) \\in \\mathbb { Z } ^ { d }$ and noise variance $\\mu _ { ; }$ , the Discrete Gaussian (DGaussian) Mechanism is given by ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8fe0efa21754a90b73c38eeb8461682d6522dadef4ae3a9eb4b911bae11eb5b.jpg",
+        "text": "$$\nf ( D ) + Z , w h e r e Z \\sim { \\mathcal { N } } _ { \\mathbb { Z } } ( 0 , \\mu ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            500,
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+            517
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and $\\mathcal { N } _ { \\mathbb { Z } } ( 0 , \\mu )$ denotes the discrete Gaussian distribution defined in Equation $( l )$ of [15]. The discrete Gaussian mechanism achieves $\\begin{array} { r } { ( \\alpha , \\frac { \\alpha \\Delta _ { 2 } ^ { 2 } } { 2 \\mu } ) } \\end{array}$ -Rényi $D P .$ ",
+        "bbox": [
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+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 The Skellam Mechanism ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            578,
+            411,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We begin by presenting the definition of the Skellam distribution, which is the basis of the Skellam Mechanism for releasing integer ranged multi-dimensional queries. ",
+        "bbox": [
+            174,
+            611,
+            821,
+            640
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1 (Skellam Distribution). The multidimensional Skellam distribution $\\operatorname { S k } _ { \\Delta , \\mu }$ over $\\mathbb { Z } ^ { d }$ with mean $\\Delta \\in \\mathbb { Z } ^ { d }$ and variance $\\mu$ is given with each coordinate $X _ { i }$ distributed independently as ",
+        "bbox": [
+            173,
+            643,
+            821,
+            676
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b062ad4a24d146974dd14e1cf0c769a45f3f043b84caa8eb27b1e9c94b2d7cda.jpg",
+        "text": "$$\nX _ { i } \\sim \\mathrm { S k } _ { \\Delta _ { i } , \\mu } \\ w i t h \\ P ( X _ { i } = k ) = e ^ { - \\mu } I _ { k - \\Delta _ { i } } ( \\mu ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            683,
+            656,
+            702
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for $k \\in \\mathbb { Z }$ . Here, $I _ { \\nu } ( x )$ is the modified Bessel function of the first kind. A key property of Skellam random variables which motivates their use in DP is that they are closed under summation, i.e. let $X _ { 1 } \\sim \\mathrm { S k } _ { \\Delta _ { 1 } , \\mu _ { 1 } }$ and $X _ { 2 } \\sim \\mathrm { S k } _ { \\Delta _ { 2 } , \\mu _ { 2 } }$ then $X _ { 1 } + X _ { 2 } \\sim \\mathrm { S k } _ { \\Delta _ { 1 } + \\Delta _ { 2 } , \\mu _ { 1 } + \\mu _ { 2 } }$ . This follows from the fact that a Skellam random variable $X$ can be obtained by taking the difference between two independent Poisson random variables with means $\\mu$ . 3 We are now ready to introduce the Skellam Mechanism. ",
+        "bbox": [
+            173,
+            715,
+            825,
+            786
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.2 (The Skellam Mechanism). Given an integer-valued query $f ( D ) \\in \\mathbb { Z } ^ { d }$ , we define the Skellam Mechanism as ",
+        "bbox": [
+            171,
+            790,
+            823,
+            819
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81e132f498160a47307f457b25508f9e22edb01d29e3ad9af2254edafa152c7f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\qquad \\mathrm { S k } _ { 0 , \\mu } ( f ( D ) ) = f ( D ) + Z , \\ w h e r e \\ Z \\sim \\mathrm { S k } _ { 0 , \\mu } , } \\\\ & { \\qquad \\cdot o f t h e m e c h a n i s m i s b o u n d e d b y \\mathbb { E } \\left[ \\| \\mathrm { S k } _ { 0 , \\mu } ( f ( D ) ) - f ( D ) \\| _ { 2 } ^ { 2 } \\right] \\leq d \\mu . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            825,
+            772,
+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and the total $\\ell _ { 2 }$ erro ",
+        "bbox": [
+            173,
+            853,
+            772,
+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Skellam mechanism was first introduced in [49] for the scalar case. As our goal is to apply the Skellam mechanism in the learning context, we have to address the following challenges. (1) Tight privacy compositions: Learning algorithms are iterative in nature and require the application of the DP mechanism many times (often $> 1 0 0 0$ ). The current direct approximate DP analysis in [49] can be combined with advanced composition (AC) theorems [28, 22] but that leads to poor privacy-accuracy trade-offs (see Fig. 1). (2) Privacy analysis for multi-dimensional queries: In learning algorithms, the differentially private queries are multi-dimensional (where the dimension equals the number of model parameters, typically $\\geq 1 0 ^ { 6 }$ ). Using composition theorems lead to poor accuracy-privacy trade-offs and a direct extension of approximate DP guarantee [49] for the multi-dimensional case leads to a strong dependence on $\\ell _ { 1 }$ sensitivity which is prohibitively large in high dimensions. (3) Data discretization: The gradients are naturally continuous vectors but we would like to apply an integer based mechanism. This requires properly discretizing the data while making sure that the norm of the vectors (sensitivity of the query) is preserved. We will tackle challenges (1) and (2) in the remainder of this section and leave (3) for the next section. ",
+        "bbox": [
+            173,
+            90,
+            826,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Tight Numerical Accounting via Privacy Loss Distributions ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            308,
+            625,
+            324
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We begin by defining the notion of privacy loss distributions (PLDs). ",
+        "bbox": [
+            174,
+            337,
+            452,
+            364
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.3 (Privacy Loss Distribution). For a multi-dimensional discrete privacy mechanism $M$ and neighboring datasets $D , D ^ { \\prime }$ , for any $x \\in \\mathbb { Z } ^ { d }$ , we define $f ( x ) =$ $\\log \\left( { \\frac { P ( M ( D ) = x ) } { P ( M ( D ^ { \\prime } ) = x ) } } \\right)$ . The privacy loss random variable of $M$ at $( D , D ^ { \\prime } )$ is $Z _ { D , D ^ { \\prime } } =$ $f ( M ( D ) ) \\ l { 2 2 } { ] }$ . The privacy loss distribution $( P L D )$ of $M$ , denoted by $\\mathrm { P L D } _ { D , D ^ { \\prime } }$ , is the distribution of $Z _ { D , D ^ { \\prime } }$ . ",
+        "bbox": [
+            173,
+            371,
+            454,
+            508
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The PLD of a mechanism $M$ can be used to characterize its $( \\varepsilon , \\delta )$ -DP guarantees. ",
+        "bbox": [
+            173,
+            522,
+            452,
+            550
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/89259f8a4ef4a8b7c950cf0a4b2f3fd06431c154db559e6aac977812e35a7dfb.jpg",
+        "image_caption": [
+            "Figure 1: Comparing privacy compositions across various mechanisms and accounting methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            464,
+            352,
+            821,
+            493
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.4. A mechanism $M$ is $( \\varepsilon , \\delta )$ -DP if and only if $\\delta \\geq \\mathbb { E } _ { Z \\sim \\mathrm { P L D } _ { D , D ^ { \\prime } } } \\left[ 1 - e ^ { \\varepsilon - Z } \\right] _ { + }$ for all neighboring datasets $D , D ^ { \\prime }$ where $[ x ] _ { + } = \\operatorname* { m a x } ( 0 , x )$ . ",
+        "bbox": [
+            173,
+            558,
+            823,
+            592
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When a mechanism $M$ is applied $T$ times on a dataset, the overall PLD of the composed mechanism at $( D , D ^ { \\prime } )$ is the $T$ -fold convolution of $\\mathrm { P L D } _ { D , D ^ { \\prime } }$ [22]. Since discrete convolutions can be computed efficiently using fast Fourier transforms (FFTs) and the expectation in Lemma 3.4 can be numerically approximated, PLDs are attractive for tight numerical accounting [30, 34, 17]. Applying the above to the Skellam mechanism, a direct calculation shows that with $X _ { i }$ are i.i.d. according to $\\operatorname { S k } _ { 0 , \\mu }$ , ",
+        "bbox": [
+            173,
+            604,
+            825,
+            675
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7cee49270ac343c81fb6c2ed892273dcd89692697fa2b133d5b10d18cbaaea9e.jpg",
+        "text": "$$\nZ _ { D , D ^ { \\prime } } = \\sum _ { i = 1 } ^ { d } \\log \\left( \\frac { I _ { X _ { i } - f ( D ) _ { i } } ( \\mu ) } { I _ { X _ { i } - f ( D ^ { \\prime } ) _ { i } } ( \\mu ) } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            688,
+            622,
+            732
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When $d = 1$ , it suffices to look at $Z = \\log ( I _ { X - \\Delta } ( \\mu ) / I _ { X } ( \\mu ) )$ , where $\\Delta = \\operatorname* { m a x } _ { D , D ^ { \\prime } } | f ( D ) - f ( D ^ { \\prime } ) |$ and $X \\sim \\mathrm { S k } _ { 0 , \\mu }$ . Since $X$ has a discrete and symmetric probability distribution and the log function is monotonic, the distribution of $Z$ can be easily characterized. This gives us a tight numerical accountant for the Skellam mechanism in the scalar case, which we use to compare it with both the Gaussian and discrete Gaussian mechanisms. Fig. 1 shows this comparison, highlighting the competitiveness of the Skellam mechanism and the problem of combining the direct analysis of [49] with advanced composition (AC) theorems. When $d > 1$ , there are combinatorially many $Z _ { D , D ^ { \\prime } }$ ’s that need to be considered, even when the $\\ell _ { 2 }$ sensitivity of $f ( D )$ is bounded. The discrete Gaussian mechanism faces a similar issue (see Theorem 15 of [15]). To provide a tight privacy analysis in the multi-dimensional case, we prove a bound on the RDP guarantees of the Skellam mechanism in the next subsection. Fig. 1 and 2 show that our bound is tight and the competitiveness of the Skellam mechanism in high dimensions. ",
+        "bbox": [
+            173,
+            744,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Tight Accounting via Rényi Differential Privacy ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            90,
+            545,
+            107
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following theorem states our main theoretical result, providing a relatively sharp bound on the RDP properties for the Skellam machanism. ",
+        "bbox": [
+            174,
+            116,
+            823,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.5. For $\\alpha \\in \\mathbb { Z } , \\alpha > 1$ and sensitivity $\\Delta \\in \\mathbb { Z }$ , the Skellam Mechanism is $( \\alpha , \\varepsilon )$ -RDP with ",
+        "bbox": [
+            174,
+            147,
+            820,
+            162
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d6e48d44863d27a153b9f54d18bd1c60276f3e0b5baad718b64f6689e1cc95d.jpg",
+        "text": "$$\n\\varepsilon ( \\alpha ) \\leq \\frac { \\alpha \\Delta ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { ( 2 \\alpha - 1 ) \\Delta ^ { 2 } + 6 \\Delta } { 4 \\mu ^ { 2 } } , \\frac { 3 \\Delta } { 2 \\mu } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            165,
+            663,
+            200
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To remind the reader in comparison, the Gaussian mechanism is $( \\alpha , \\varepsilon )$ -RDP with $\\begin{array} { r } { \\varepsilon ( \\alpha ) = \\frac { \\alpha \\Delta ^ { 2 } } { 2 \\mu } } \\end{array}$ . The bound we provide is at most $1 + O ( 1 / \\mu )$ worse than the bound for the Gaussian, which is negligible for all practical choices of $\\mu$ , especially as the privacy requirements increase.4 Next we show a simple corollary which follows via the independent composition of RDP across dimensions. ",
+        "bbox": [
+            173,
+            212,
+            825,
+            272
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3.6. The multi-dimensional Skellam Mechanism is $( \\alpha , \\varepsilon )$ -RDP with ",
+        "bbox": [
+            176,
+            275,
+            689,
+            290
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a27f86bf3caba44d5a2085a2d4ac983c5c0c69ba7925ee5c1d128097b60e1420.jpg",
+        "text": "$$\n\\varepsilon ( \\alpha ) \\leq \\frac { \\alpha \\Delta _ { 2 } ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { ( 2 \\alpha - 1 ) \\Delta _ { 2 } ^ { 2 } + 6 \\Delta _ { 1 } } { 4 \\mu ^ { 2 } } , \\frac { 3 \\Delta _ { 1 } } { 2 \\mu } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            292,
+            671,
+            329
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\Delta _ { 1 }$ and $\\Delta _ { 2 }$ are the $\\ell _ { 1 }$ and $\\ell _ { 2 }$ sensitivities respectively. ",
+        "bbox": [
+            173,
+            332,
+            573,
+            347
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 Proof Overview for Theorem 3.5 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            358,
+            449,
+            375
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we provide the proof of Theorem 3.5 assuming a technical bound on the ratios of Bessel functions presented as Lemma 3.7, which is the core of our analysis and may be of independent interest. We provide a proof overview for Lemma 3.7, deferring the full proof to the appendix. ",
+        "bbox": [
+            174,
+            382,
+            826,
+            426
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "On a macroscopic level, our proof structure mimics the RDP proof for the Gaussian mechanism [36], and the main object of our interest is to bound the following quantity, defined for any $X , \\Delta , \\alpha$ : ",
+        "bbox": [
+            173,
+            431,
+            825,
+            460
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d09454290efbfee5c51070ad4c3a17eb4c8aa1cd3d1b384efa8b07ea652ed1cd.jpg",
+        "text": "$$\n\\Phi _ { X , \\alpha , \\Delta } ( \\mu ) \\triangleq \\log \\left( \\frac { I _ { X - \\Delta } ( \\mu ) } { I _ { X - \\alpha \\Delta } ( \\mu ) } \\left( \\frac { I _ { X - \\Delta } ( \\mu ) } { I _ { X } ( \\mu ) } \\right) ^ { \\alpha - 1 } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            463,
+            674,
+            506
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following lemma states our main bound on this quantity. ",
+        "bbox": [
+            171,
+            507,
+            584,
+            522
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.7. For any $X , \\alpha \\in \\mathbb { N } ,$ , with $\\alpha > 1$ and $\\Delta \\in \\mathbb { Z } ,$ , we have that for all $\\mu \\geq 0$ ",
+        "bbox": [
+            179,
+            525,
+            723,
+            539
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d33ee9c683d567b5c982c3e45b44ba005587aa1e84f439b44e232a84aacec516.jpg",
+        "text": "$$\n\\Phi _ { X , \\alpha , \\Delta } ( \\mu ) \\leq \\frac { \\alpha ( \\alpha - 1 ) \\Delta ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { ( 2 \\alpha - 1 ) ( \\alpha - 1 ) \\Delta ^ { 2 } } { 4 \\mu ^ { 2 } } + \\frac { 3 ( \\alpha - 1 ) | \\Delta | } { 2 \\mu ^ { 2 } } , \\frac { 3 ( \\alpha - 1 ) | \\Delta | } { 2 \\mu } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            541,
+            799,
+            577
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that in contrast if we consider the analogous notion of $\\Phi$ for the Gaussian mechanism (replacing IX(µ) with the Gaussian density e\u0000X2/2µ), we readily get the bound ↵(↵\u00001)\u000022µ , which is the same as our bound up to lower order terms. We now provide the proof of Theorem 3.5. ",
+        "bbox": [
+            174,
+            585,
+            823,
+            636
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 3.5. By RDP definition (2.2), we need to bound the following for any $\\Delta , \\alpha \\geq 1$ , ",
+        "bbox": [
+            173,
+            648,
+            820,
+            664
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46c59ed6f622fa23386b47b11e546c5229a09c55a9302bf5d15e5db3f845ce00.jpg",
+        "text": "$$\nD _ { \\alpha } \\left( { \\mathrm { S k } } _ { \\Delta , \\mu } , { \\mathrm { S k } } _ { 0 , \\mu } \\right) = \\frac { 1 } { \\alpha - 1 } { \\log \\left( \\right)} \\sum _ { X = - \\infty } ^ { \\infty } e ^ { - \\mu } I _ { X - \\Delta } ( \\mu ) \\left( \\frac { I _ { X - \\Delta } ( \\mu ) } { I _ { X } ( \\mu ) } \\right) ^ { \\alpha - 1 } \n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            666,
+            746,
+            710
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Now consider the following calculations on the log term: ",
+        "bbox": [
+            173,
+            713,
+            547,
+            727
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a34c1104af70ccc25fc562c8f4f77078cb1190ec63d3c66d1be1465c38b1b3c7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log \\left( \\displaystyle \\sum _ { X = - \\infty } ^ { \\infty } \\frac { I _ { X - \\Delta } ( \\mu ) } { e ^ { \\mu } } \\left( \\frac { I _ { X - \\Delta } ( \\mu ) } { I _ { X } ( \\mu ) } \\right) ^ { \\alpha - 1 } \\right) = \\log \\left( \\displaystyle \\sum _ { X = - \\infty } ^ { \\infty } \\frac { I _ { X - \\alpha \\Delta } ( \\mu ) } { e ^ { \\mu } } e ^ { \\Phi _ { X , \\alpha , \\Delta ( \\mu ) } } \\right) } \\\\ & { \\leq \\log \\left( \\displaystyle \\sum _ { X = - \\infty } ^ { \\infty } e ^ { - \\mu } I _ { X - \\alpha \\Delta } ( \\mu ) \\right) + \\displaystyle \\operatorname* { m a x } _ { X \\in \\mathbb { Z } } \\Phi _ { X , \\alpha , \\Delta ( \\mu ) } } \\\\ & { \\leq \\frac { \\alpha ( \\alpha - 1 ) \\Delta ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { ( 2 \\alpha - 1 ) ( \\alpha - 1 ) \\Delta ^ { 2 } } { 4 \\mu ^ { 2 } } + \\frac { 3 ( \\alpha - 1 ) | \\Delta | } { 2 \\mu ^ { 2 } } , \\frac { 3 ( \\alpha - 1 ) | \\Delta | } { 2 \\mu } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            731,
+            774,
+            856
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the inequality follows from Lemma 3.7. ",
+        "bbox": [
+            174,
+            856,
+            478,
+            871
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now provide an overview for the proof of Lemma 3.7 highlighting the crux of the argument. As a first step we collect some known facts regarding Bessel functions. It is known that for $x \\geq 0$ and $\\nu \\in \\mathbb { Z }$ , $\\nu \\geq 0$ , $I _ { \\nu } ( x )$ is a decreasing function in $\\nu$ , $I _ { - \\nu } ( x ) = I _ { \\nu } ( x )$ and $\\frac { I _ { \\nu - 1 } ( \\mu ) } { I _ { \\nu } ( x ) }$ is an increasing function in $\\nu$ [47]. A succession of works consider bounding the ratio of successive Bessel functions $I _ { \\nu - 1 } ( x ) / I _ { \\nu } ( x )$ , which is a natural quantity to considering the objective in Lemma 3.7. We use the following very tight characterization for this recently proved in [44, Theorem 5]. ",
+        "bbox": [
+            173,
+            90,
+            826,
+            180
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.8. For any $\\nu \\geq 1 / 2 , x \\geq 0$ define the following function we have that ",
+        "bbox": [
+            171,
+            184,
+            700,
+            199
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e82d82d779046fa781ceff63292e3c0e4b1f5e638f07f8035775fca7fdf4df6e.jpg",
+        "text": "$$\n\\mathrm { a r c s i n h } ( \\delta _ { 0 } ( \\nu , x ) ) \\leq \\log ( I _ { \\nu - 1 } ( x ) ) - \\log ( I _ { \\nu } ( x ) ) \\leq \\mathrm { a r c s i n h } ( \\delta _ { 2 } ( \\nu , x ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            205,
+            727,
+            222
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Standard bounds such as those appearing in [5, 49] lead to the following conclusion: ",
+        "bbox": [
+            173,
+            263,
+            727,
+            279
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/963bdaec7bb43b750067bbdf5083f029b624096659fffd2f47549548766c8f05.jpg",
+        "text": "$$\n\\mathrm { a r c s i n h } ( ( \\nu - 1 / 2 ) / x ) \\leq \\log ( I _ { \\nu - 1 } ( x ) ) - \\log ( I _ { \\nu } ( x ) ) \\leq \\mathrm { a r c s i n h } ( \\nu / x ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            284,
+            740,
+            301
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While the above bound is significantly easier to work with, it leads to an RDP guarantee of Gaussian $\\mathrm { R D P } + O \\bigl ( \\frac { \\Delta } { \\mu } \\bigr )$ . In high dimensions this manifests as $O ( \\frac { \\Delta _ { 1 } } { \\mu } )$ and overall leads to a constant multiplicative factor over the Gaussian. On the other hand we prove a Gaussian $\\mathrm { R D P } + o _ { \\mu } ( 1 )$ bound. Our proof of Lemma 3.7 splits into various cases depending on the signs of the quantities involved. We show the derivation for a single case below and defer the full proof to the appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proof of Lemma 3.7 in the case $X \\geq \\alpha \\Delta$ , $\\Delta \\geq 0$ . Replacing $Y = X - \\alpha \\delta$ we get that ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7fc7c1ee3f9d8e42caad97aa81208c7c902c8f581fb6c1cb8da3dacf322b285.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\Phi _ { X , \\boldsymbol { \\alpha } , \\Delta } ( \\mu ) = \\log \\left( \\displaystyle \\frac { I _ { Y + ( \\alpha - 1 ) \\Delta } ( \\mu ) } { I _ { Y } ( \\mu ) } \\left( \\frac { I _ { Y + ( \\alpha - 1 ) \\Delta } ( \\mu ) } { I _ { Y + \\alpha \\Delta } ( \\mu ) } \\right) ^ { \\alpha - 1 } \\right) } \\\\ & { \\qquad = \\displaystyle \\sum _ { j = 0 } ^ { \\alpha - 2 } \\left( \\sum _ { i = Y + j \\Delta + 1 } ^ { Y + j \\Delta + \\Delta } \\left( \\log \\left( \\frac { I _ { i - 1 } + ( \\alpha - 1 - j ) ( \\mu ) } { I _ { i + ( \\alpha - 1 - j ) \\Delta } ( \\mu ) } \\right) - \\log \\left( \\frac { I _ { i - 1 } ( \\mu ) } { I _ { i } ( \\mu ) } \\right) \\right) \\right) } \\\\ & { \\qquad \\le \\displaystyle \\sum _ { j = 0 } ^ { \\alpha - 2 } \\left( \\sum _ { i = Y + j \\Delta + 1 } ^ { Y + j \\Delta + \\Delta } \\left( \\delta _ { 2 } ( i + ( \\alpha - 1 - j ) \\Delta , \\mu ) - \\delta _ { 0 } ( i , \\mu ) \\right) \\right) } \\\\ & { \\qquad \\le \\displaystyle \\frac { \\alpha ( \\alpha - 1 ) \\Delta ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { \\alpha ( \\alpha - 1 ) \\Delta ^ { 2 } + 2 ( \\alpha - 1 ) \\Delta } { 4 \\mu ^ { 2 } } , \\frac { ( \\alpha - 1 ) \\Delta } { 2 \\mu } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the first inequality follows from Lemma 3.8 and the fact that for all $0 \\leq x \\leq y$ , $\\operatorname { a r c s i n h } ( y ) -$ $\\operatorname { a r c s i n h } ( x ) \\leq y - x$ and the second inequality follows from Lemma A.1 (provided in the appendix): ",
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+    {
+        "type": "equation",
+        "img_path": "images/d849ccf675f94f3f49cf967faa1f98df738ec37bda9299fdea5ee92b0622eb64.jpg",
+        "text": "$$\n\\delta _ { 2 } ( \\nu _ { 1 } , x ) - \\delta _ { 0 } ( \\nu _ { 2 } , x ) \\leq \\frac { \\nu _ { 1 } - \\nu _ { 2 } } { x } + \\frac { 1 } { 2 x } \\operatorname* { m i n } \\left( \\frac { \\nu _ { 1 } - \\nu _ { 2 } + 1 } { x } , 1 \\right) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4 Applying the Skellam Mechanism to Federated Learning ",
+        "text_level": 1,
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+        "text": "With a sharp RDP analysis for the multi-dimensional Skellam mechanism presented in the previous section, we are now ready to apply it to differentially private federated learning. We first outline the general problem setting and then describe our approach under central and distributed DP models. ",
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+        "type": "text",
+        "text": "Problem setting At a high-level, we consider the distributed mean estimation problem. There are $n$ clients each holding a vector $x _ { i }$ in $\\mathbb { R } ^ { d }$ such that for all $i$ , the vector norm is bounded as $\\| x _ { i } \\| _ { 2 } \\leq c$ for some $c \\geq 0$ . We denote the set of vectors as $\\mathcal { X } = \\{ x _ { i } \\} _ { i = 1 } ^ { n }$ , and the aim is for each client to communicate the vectors $x _ { i }$ to a central server which then aggregates them as $\\begin{array} { r } { \\widehat { x } = \\frac { 1 } { n } \\sum _ { i } x _ { i } } \\end{array}$ for an external analyst. In federated learning, the client vectors $x _ { i }$ are the model gradients or model deltas (typically $d \\geq 1 0 ^ { 6 }$ ) after training on the clients’ local datasets, and this procedure can be repeated for many rounds $T > 1 0 0 0 \\ r ,$ . A large $d$ and $T$ thus necessitate accounting methods that provide tight privacy compositions for high-dimensional queries. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We are primarily concerned with three metrics for this procedure and their trade-offs: (1) Privacy: the mean $\\widehat { x }$ should be differentially private with a reasonably small $( \\varepsilon , \\delta )$ ; (2) Error: we wish to minimize the expected $\\ell _ { 2 }$ error; and (3) Communication: we wish to minimize the average number of bits communicated per coordinate. Characterizing this trade-off is an important research problem. For example, it has been recently shown [50] that without formal privacy guarantees, the client training data could still be revealed by the model updates $x _ { i }$ ; on the other hand, applying differential privacy [48] to these updates can degrade the final utility. ",
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+        "img_path": "images/8a7da83f72e1327e3ea24b8cddf36aa7549a54ed5a6c18bfacb1f5d9704a83fa.jpg",
+        "image_caption": [
+            "Figure 2: Benchmarking Skellam on sensitivity-1 queries under various accounting methods. RDP: Rényi DP. PLD: privacy loss distributions. Skellam (Direct): [49]. Gaussian (Analytic): [9]. DGaussian [15] / DDGauss [25]: central / distributed discrete Gaussian. $s$ is the scaling factor applied to both $\\Delta$ and $\\sigma$ . For Skellam and DDGauss [25], the central noise with std $\\sigma$ is split into $n$ shares each applied locally with std $\\sigma / { \\sqrt { n } }$ ; a large $n$ and small $\\sigma$ can thus exacerbate the sum divergence term of DDGauss (left). Left: $\\varepsilon \\leq 1 0$ . Right: $\\varepsilon \\leq 1$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/c38290ebb9dc27b41d63ab39e366b01c3b59452ece0d54d60bdf13b39554b0e0.jpg",
+        "image_caption": [
+            "Figure 3: Comparing Skellam and Distributed Discrete Gaussian (DDGauss) on multi-dimensional real-valued queries, rounded to integers with $\\beta = e ^ { - 0 . 5 }$ (Prop. 4.2). $s$ is the scaling applied to both $\\sigma$ and $\\Delta _ { 2 }$ ; a larger $s$ reduces the rounding error and norm inflation. $q$ is the sampling rate. For Skellam and DDGauss [25], the central noise with std $\\sigma$ is split into $n$ shares each applied locally with std $\\sigma / { \\sqrt { n } }$ ; a large $n$ and small $\\sigma$ can exacerbate the sum divergence term of DDGauss. Left: Simple setting with $\\Delta _ { 2 } = 1$ . Right: FL-like setting for training CNNs on Federated EMNIST. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+        "text": "Skellam for central DP The central DP model refers to adding Skellam noise onto the non-private aggregate $\\widehat { x }$ before releasing it to the external analyst. One important consideration is that the model updates in $\\mathrm { F L }$ are continuous in nature, while Skellam is a discrete probability distribution. One approach is to appropriately discretize the client updates, e.g., via uniform quantization (which involves scaling the inputs by a factor $s \\sim 2 ^ { b }$ for some bit-width $b$ followed by stochastic rounding5 for unbiased estimates), and the server can convert the private aggregate back to real numbers at the end. Note that this allows us to re-parameterize the variance of the added Skellam noise as $s ^ { 2 } \\mu$ , giving the following simple corollary based on Cor. 3.6: ",
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+    {
+        "type": "text",
+        "text": "Corollary 4.1 (Scaled Skellam Mechanism). With a scaling factor $s \\in \\mathbb { R } ,$ , the multi-dimensional Skellam Mechanism is $( \\alpha , \\varepsilon )$ -RDP with ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9133e9e1c94c6ece55a387dea136491c7f4bd3a2d9b739e68590c7537641fec8.jpg",
+        "text": "$$\n\\varepsilon ( \\alpha ) \\leq \\frac { \\alpha \\Delta _ { 2 } ^ { 2 } } { 2 \\mu } + \\operatorname* { m i n } \\left( \\frac { ( 2 \\alpha - 1 ) \\Delta _ { 2 } ^ { 2 } } { 4 s ^ { 2 } \\mu ^ { 2 } } + \\frac { 3 \\Delta _ { 1 } } { 2 s ^ { 3 } \\mu ^ { 2 } } , \\frac { 3 \\Delta _ { 1 } } { 2 s \\mu } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As $s$ increases, the RDP of scaled Skellam rapidly approaches that of Gaussian as the second term above approaches 0, suggesting that under practical regimes with moderate compression bit-width, Skellam should perform competitively compared to Gaussian. Another aspect worth noting is that rounding vector coordinates from reals to integers can inflate the $\\ell _ { 2 }$ -sensitivity $\\Delta _ { 2 }$ , and thus more noise is required for the same privacy. To this end, we leverage the conditional rounding procedure introduced in [25] to obtain a bounded norm on the scaled and rounded client vector: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Proposition 4.2 (Norm of stochastically rounded vector [25]). Let $\\tilde { x }$ be a stochastic rounding of vector $x \\in \\mathbb { R } ^ { d }$ to the integer grid $\\mathbb { Z } ^ { d }$ . Then, for $\\beta \\in ( 0 , 1 )$ , we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4e56084f458fd2d83d05336f920c03e385fa8b886ef59bd50bc04a65ba8056df.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left[ \\| \\tilde { x } \\| _ { 2 } ^ { 2 } \\leq \\| x \\| _ { 2 } ^ { 2 } + d / 4 + \\sqrt { 2 \\log ( 1 / \\beta ) } \\cdot \\left( \\| x \\| _ { 2 } + \\sqrt { d } / 2 \\right) \\right] \\geq 1 - \\beta . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Conditional rounding is thus defined as retrying the stochastic rounding on $x _ { i }$ until $\\| \\tilde { x } _ { i } \\| _ { 2 } ^ { 2 }$ is within the probabilistic bound above (which also gives the inflated sensitivity $\\tilde { \\Delta } _ { 2 }$ ). We can then add Skellam noise to the aggregate $\\sum _ { i } \\tilde { x } _ { i }$ according to $\\mathrm { { \\bar { \\Delta } } } _ { 2 }$ before undoing the quantization (unscaling). Note that a larger scaling $s$ before rounding reduces the norm inflation and the extra noise needed (Fig. 3 right). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Skellam for distributed DP with secure aggregation A stronger notion of privacy in FL can be obtained via the distributed DP model [25] that leverages secure aggregation (SecAgg [12]). The fact that the Skellam distribution is closed under summation allows us to easily extend from central DP to distributed DP. Under this model, the client vectors are quantized as in central DP model, but the Skellam noise is now added locally with variance $\\mu / n$ . Then, the noisy client updates are summed via SecAgg ( $b$ bits per coordinate for field size $2 ^ { b }$ ) which only reveals the noisy aggregate to the server. While the local noise might be insufficient for local DP guarantees, the aggregated noise at the server provides privacy and utility comparable to the central DP model, thus removing trust away from the central aggregator. Note that the modulo operations introduced by SecAgg does not impact privacy as it can be viewed as a post-processing of an already differentially private query. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We remark on several properties of the distributed Skellam compared to the distributed discrete Gaussian (DDGauss [25]). (1) DDGauss is not closed under summation, and the divergence between discrete Gaussians can lead to notable privacy degradation in settings such as quantile estimation [6] and federated analytics [42] with sufficiently large number of clients and small local noises (see also the left side of Fig. 2 and Fig. 3). While scaling mitigates this issue, it also requires additional bit-width which makes Skellam attractive under tight communication constraints. (2) Sampling from Skellam only requires sampling from Poisson, for which efficient implementations are widely available in numerical software packages. While efficient discrete Gaussian sampling has also been explored in the lattice-based cryptography community (e.g., [43, 18, 38]), we believe the accessibility of Skellam samplers would help facilitate the deployment of DP to FL settings with mobile and edge devices. See Appendix D for more discussion. (3) In practice where $s \\gg 1$ (dictated by bit-width $b$ ), both Skellam (cf. Cor. 4.1) and DDGauss (with an exponentially small divergence) quickly approaches Gaussian under RDP, and any differences will be negligible (Fig. 3). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 Empirical Evaluation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we empirically evaluate the Skellam mechanism on two sets of experiments: distributed mean estimation and federated learning. In both cases, we focus on the distributed DP model, but note that the Skellam mechanism can be easily adapted to the central DP setting as discussed in the earlier section. Unless otherwise stated, we use RDP accounting for all experiments due to the high-dimensional data and the ease of composition (Section 3). To obtain $\\Delta _ { 1 }$ for Skellam RDP, we note that $\\Delta _ { 1 } \\leq \\Delta _ { 2 } \\cdot \\operatorname* { m i n } ( \\sqrt { d } , \\Delta _ { 2 } )$ since $\\Delta _ { 1 } \\leq \\sqrt { d } \\Delta _ { 2 }$ in general and $\\Delta _ { 1 } \\leq \\Delta _ { 2 } ^ { 2 }$ for integers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Under the distributed DP model, we also introduce a random orthogonal transformation [29, 2, 25] before discretizing and aggregating the client vectors (which can be reverted after the aggregation); this makes the vector coordinates sub-Gaussian and helps spread the magnitudes of the vector coordinates across all dimensions, thus reducing the errors from quantization and potential wraparound from SecAgg modulo operations. Moreover, by approximating the sub-exponential tail of the Skellam distribution as sub-Gaussian, we can derive a heuristic for choosing $s$ following [25] based on a bound on the variance $\\tilde { \\sigma } ^ { 2 }$ of the aggregated signal, as $\\tilde { \\sigma } ^ { 2 } \\le c ^ { 2 } n ^ { 2 } / d + \\bar { n } / ( 4 s ^ { 2 } ) + \\mu$ . We choose $s$ such that $2 k \\tilde { \\sigma }$ are bounded within the SecAgg field size $2 ^ { b }$ , where $k$ is a small constant. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 summarizes the aggregation procedure for the distributed Skellam mechanism via secure ",
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+        "type": "text",
+        "text": "Inputs: Private vector $x _ { i } \\in \\mathbb { R } ^ { \\bar { d } }$ for each client $i$ ; $\\ell _ { 2 }$ clip norm $c > 0$ ; Bit-width $b$ ; Target central   \nnoise variance $\\mu > 0$ ; Number of clients $n$ ; Signal bound multiplier $k > 0$ ; Bias $\\beta \\in [ 0 , 1 )$ .   \nShared randomness: $d \\times d$ diagonal matrix $D$ with uniformly random $\\{ - 1 , + 1 \\}$ values, where   \n$d \\geq \\bar { d }$ is the nearest power of 2.   \nShared scale: Obtain scaling factor $s$ such that $2 ^ { b } = 2 k \\tilde { \\sigma } = 2 k \\sqrt { c ^ { 2 } n ^ { 2 } / d + n / ( 4 s ^ { 2 } ) + \\mu } .$   \nProcedure CLIENTPROCEDURE $\\mathbf { \\chi } _ { \\left( x _ { i } , s , D \\right) }$ Clip and scale vector $\\hat { x } _ { i } = s \\cdot \\operatorname* { m i n } ( 1 , c / \\| x _ { i } \\| _ { 2 } ) \\cdot x _ { i }$ , and pad to $\\tilde { d }$ dimensions with zeros. Random rotation: $\\check { x } _ { i } = \\tilde { H } _ { d } D \\hat { x } _ { i }$ where $\\begin{array} { r } { \\tilde { H } _ { d } = \\frac { 1 } { \\sqrt { d } } H _ { d } } \\end{array}$ is the normalized $d \\times d$ Hadamard matrix. repeat {conditional stochastic rounding} Stochastically round the coordinates of of ${ \\check { x } } _ { i }$ to the integer grid to produce ${ \\tilde { x } } _ { i }$ until $\\| \\widetilde x _ { i } \\| _ { 2 } ^ { 2 } \\le \\operatorname* { m i n } \\bigg \\{ \\Big ( s c + \\sqrt d \\Big ) ^ { 2 } , s ^ { 2 } c ^ { 2 } + d / 4 + \\sqrt { 2 \\log ( 1 / \\beta ) } \\cdot \\Big ( s c + \\sqrt d / 2 \\Big ) \\bigg \\} .$ . Local noising: Sample noise vector $y _ { i } \\in \\mathbb { Z } ^ { d }$ where each entry is sampled from $\\operatorname { S k } _ { 0 , s ^ { 2 } \\mu / n }$ . return $z _ { i } = \\tilde { x } _ { i } + y _ { i }$ under the SecAgg protocol with modulo bit-width $b$ .   \nProcedure SERVERPROCEDURE $( z , s , D )$ $\\{ z$ is the modular sum of $z _ { i }$ under bit-width $b$ } return $\\begin{array} { r } { \\bar { x } = \\frac { 1 } { s } D \\tilde { H } _ { d } ^ { \\top } z } \\end{array}$ , with $\\textstyle { \\bar { x } } \\approx \\sum _ { i } x _ { i } \\in \\mathbb { R } ^ { d }$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/a258818d6f7517c63fee2ce116dc0cb01bea0fe531df9cafcfe5043c2794f7e1.jpg",
+        "image_caption": [
+            "Figure 4: Distributed mean estimation with the distributed Skellam mechanism. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "aggregation as well as the parameters used for the experiments. In summary, we have an $\\ell _ { 2 }$ clip norm $c > 0$ ; per-coordinate bit-width $b$ ; target central noise variance $\\mu > 0$ ; number of clients $n$ ; signal bound multiplier $k > 0$ ; and rounding bias $\\beta \\in [ 0 , 1 )$ . We fix $\\beta = e ^ { - 1 / 2 }$ for all experiments. Note that the per-coordinate bit-width $b$ is for the aggregated sum as it determines the field size of SecAgg. For federated learning, we also consider the number of rounds $T$ and the total number of clients $N$ (thus the uniform sampling ratio $q = n / N$ at every round). Our experiments are implemented in Python, TensorFlow Privacy [32], and TensorFlow Federated [24]. See also Appendix for additional results and more details on the experimental setup. ",
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+    {
+        "type": "text",
+        "text": "5.1 Distributed Mean Estimation (DME) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first consider DME as the generalization of (single round) FL. We randomly generate $n$ client vectors $X = \\{ x _ { i } \\} _ { i = 1 } ^ { n }$ from the $d$ -dimensional $\\ell _ { 2 }$ sphere with radius $c = 1 0$ , and compute the true mean $\\begin{array} { r } { \\widehat { x } = \\frac { 1 } { n } \\sum _ { i } ^ { n } x _ { i } } \\end{array}$ 2 . We then compute the private estimate of $\\widehat { x }$ with the distributed Skellam mechanism (Algorithm 1) as $\\bar { x }$ . For a strong baseline, we use the analytic Gaussian mechanism [9] with tight accounting (see also Figure 2). In Figure 4, we plot the MSE as $\\| \\widehat { x } - \\bar { x } \\| _ { 2 } ^ { 2 } / d$ with $9 5 \\%$ confidence interval (small shaded region) over 10 dataset initializations across different values of $b , d$ , and $n$ . Results demonstrate that Skellam can match Gaussian even with $n = 1 0 0 0 0$ clients as long as the bit-width is sufficient. We emphasize that the communication cost $b$ depends logarithmically on $n$ , and to put numbers into context, Google’s production next-word prediction models [23, 39] use $n \\leq 5 0 0$ and the production DP language model [40] uses $n = 2 0 0 0 0$ . ",
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+    {
+        "type": "text",
+        "text": "5.2 Federated Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setup We evaluate on three public federated datasets with real-world characteristics: Federated EMNIST [16], Shakespeare [31, 14], and Stack Overflow next word prediction (SO-NWP [8]). EMNIST is an image classification dataset for hand-written digits and letters; Shakespeare is a text dataset for next-character-prediction based on the works of William Shakespeare; and SO-NWP is a large-scale text dataset for next-word-prediction based on user questions/answers from stackoverflow.com. We emphasize that all datasets have natural client heterogeneity that are representative of practical FL problems: the images in EMNIST are grouped the writer of the handwritten digits, the lines in Shakespeare are grouped by the speaking role, and the sentences in SO-NWP are grouped by the corresponding Stack Overflow user. We train a small CNN with model size $\\bar { d } < 2 ^ { 2 0 }$ for EMNIST and use the recurrent models defined in [41] for Shakespeare and SO-NWP. The hyperparameters for the experiments follow those from [25, 6, 27, 41] and tuning is limited. For EMNIST, we follow [25] and fix $c = 0 . 0 3$ , $n = 1 0 0$ , $T = 1 5 0 0$ , client learning rate $\\eta _ { \\mathrm { c l i e n t } } = 0 . 3 2$ , server learning rate $\\eta _ { \\mathrm { s e r v e r } } = 1$ , and client batch size $m = 2 0$ . For Shakespeare, we follow [6] and fix $n = 1 0 0$ , $T = 1 2 0 0$ , $\\eta _ { \\mathrm { c l i e n t } } = 1$ , $\\eta _ { \\mathrm { s e r v e r } } = 0 . 3 2$ , and $m = 4$ , and we sweep $c \\in \\{ 0 . 2 5 , 0 . 5 \\}$ . For SO-NWP, we follow [27] and fix $c = 0 . 3$ , $n = 1 0 0$ , $T = 1 6 0 0$ , $\\eta _ { \\mathrm { c l i e n t } } = 0 . 5$ , and $m = 1 6$ , and we sweep $\\eta _ { \\mathrm { s e r v e r } } \\in \\{ 0 . 3 , 1 \\}$ and limit max examples per client to 256. In all cases, clients train for 1 epoch on their local datasets, and the client updates are weighted uniformly (as opposed to weighting by number of examples). See Appendix for more results and full details on datasets, models, and hyperparameters. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/17c77904c74691a84afb047deea14f30ee01bb1aff6b1891d87bbfa175d13dd8.jpg",
+        "image_caption": [
+            "Figure 5: Federated learning with the distributed Skellam mechanism. DDGauss: Distributed Discrete Gaussian [25]. Left / Middle / Right: Test accuracies on EMNIST / Shakespeare / Stack Overflow NWP across different $\\varepsilon$ and $b$ . $\\delta$ is set to $1 / N , 1 0 ^ { - 6 } , 1 0 ^ { - 6 }$ , respectively. For Shakespeare, privacy is reported with a hypothetical population size $N = 1 0 ^ { 6 }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Results Figure 5 summarizes the FL experiments. For EMNIST and Shakespeare, we report the average test accuracy over the last 100 rounds. For SO-NWP, we report the top-1 accuracy (without padding, out-of-vocab, or begining/end-of-sentence tokens) on the test set. The results indicate that Skellam performs as good as Gaussian despite relying on generic RDP amplification via sampling [51] (cf. Fig. 3) and that Skellam matches DDG consistently under realistic regimes. This bears significant practical relevance given the advantages of Skellam over DDG in real-world deployments. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have introduced the multi-dimensional Skellam mechanism for federated learning. We analyzed the Skellam mechanism through the lens of approximate DP, privacy loss distributions, and Rényi divergences, and derived a sharp RDP bound that enables Skellam to match Gaussian and discrete Gaussian in practical settings as demonstrated by our large-scale experiments. Since Skellam is closed under summation and efficient samplers are widely available, it represents an attractive alternative to distributed discrete Gaussian as it easily extends from the central DP model to the distributed DP model. Being a discrete mechanism can also bring potential communication savings over continuous mechanisms and make Skellam less prone to attacks that exploit floating-point arithmetic on digital computers. Some interesting future work includes: (1) our scalar PLD analysis for Skellam suggests room for improvements on our multi-dimensional analysis via a complete PLD characterization, and (2) our results on FL may be further improved via a targeted analysis for RDP amplification via sampling akin to [37]. Overall, this work is situated within the active area of private machine learning and aims at making ML more trustworthy. One potential negative impact is that our method could be (deliberately or inadvertently) misused, such as sampling the wrong noise or using a minuscule scaling factor, to provide non-existent privacy guarantees for real users’ data. We nevertheless believe our results have positive impact as they facilitate the deployment of differential privacy in practice. ",
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+    {
+        "type": "text",
+        "text": "Funding Transparency Statement ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The authors were employed at and directly supported by Google. No third party funding was received by any of the authors to pursue this work. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] 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. [2] Naman Agarwal, Ananda Theertha Suresh, Felix Xinnan X Yu, Sanjiv Kumar, and Brendan McMahan. cpSGD: Communication-efficient and differentially-private distributed sgd. In Advances in Neural Information Processing Systems, pages 7564–7575, 2018. [3] Maruan Al-Shedivat, Jennifer Gillenwater, Eric Xing, and Afshin Rostamizadeh. Federated learning via posterior averaging: A new perspective and practical algorithms. In ICLR, 2021. [4] Martin R. Albrecht and Michael Walter. dgs, Discrete Gaussians over the Integers. Available at https: //bitbucket.org/malb/dgs, 2018. [5] Donald E Amos. Computation of modified bessel functions and their ratios. Mathematics of Computation, 28(125):239–251, 1974. [6] Galen Andrew, Om Thakkar, H Brendan McMahan, and Swaroop Ramaswamy. Differentially private learning with adaptive clipping. arXiv preprint arXiv:1905.03871, 2019. [7] S. Asoodeh, J. Liao, F. P. Calmon, O. Kosut, and L. Sankar. A better bound gives a hundred rounds: Enhanced privacy guarantees via f-divergences. In 2020 IEEE International Symposium on Information Theory (ISIT), pages 920–925, 2020. [8] The TensorFlow Federated Authors. Tensorflow federated stack overflow dataset, 2019. [9] 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 394–403. PMLR, 2018.   \n[10] Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimization: Efficient algorithms and tight error bounds. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 464–473. IEEE, 2014.   \n[11] James Bell, K. A. Bonawitz, Adrià Gascón, Tancrède Lepoint, and Mariana Raykova. Secure singleserver aggregation with (poly)logarithmic overhead. Cryptology ePrint Archive, Report 2020/704, 2020. https://eprint.iacr.org/2020/704.   \n[12] Keith Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth. Practical secure aggregation for privacy-preserving machine learning. In proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pages 1175–1191, 2017.   \n[13] Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635–658. Springer, 2016.   \n[14] Sebastian Caldas, Sai Meher Karthik Duddu, Peter Wu, Tian Li, Jakub Konecnˇ y, H Brendan McMa- \\` han, Virginia Smith, and Ameet Talwalkar. Leaf: A benchmark for federated settings. arXiv preprint arXiv:1812.01097, 2018.   \n[15] Clément Canonne, Gautam Kamath, and Thomas Steinke. The discrete gaussian for differential privacy. In NeurIPS, 2020.   \n[16] Gregory Cohen, Saeed Afshar, Jonathan Tapson, and Andre Van Schaik. Emnist: Extending mnist to handwritten letters. In 2017 International Joint Conference on Neural Networks (IJCNN), pages 2921–2926. IEEE, 2017.   \n[17] Google Differential Privacy Team. Privacy loss distributions. https://github.com/google/ differential-privacy/blob/master/accounting/docs/Privacy_Loss_Distributions.pdf. 2020.   \n[18] Nagarjun C Dwarakanath and Steven D Galbraith. Sampling from discrete gaussians for lattice-based cryptography on a constrained device. Applicable Algebra in Engineering, Communication and Computing, 25(3):159–180, 2014.   \n[19] 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.   \n[20] 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.   \n[21] Cynthia Dwork and Guy N Rothblum. Concentrated differential privacy. arXiv preprint arXiv:1603.01887, 2016.   \n[22] 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.   \n[23] Andrew Hard, Kanishka Rao, Rajiv Mathews, Swaroop Ramaswamy, Françoise Beaufays, Sean Augenstein, Hubert Eichner, Chloé Kiddon, and Daniel Ramage. Federated learning for mobile keyboard prediction. arXiv preprint arXiv:1811.03604, 2018.   \n[24] Alex Ingerman and Krzys Ostrowski. Introducing tensorflow federated, 2019.   \n[25] Peter Kairouz, Ziyu Liu, and Thomas Steinke. The distributed discrete gaussian mechanism for federated learning with secure aggregation. In International Conference on Machine Learning. PMLR, 2021.   \n[26] Peter Kairouz, Brendan McMahan, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   \n[27] Peter Kairouz, Brendan McMahan, Shuang Song, Om Thakkar, Abhradeep Thakurta, and Zheng Xu. Practical and private (deep) learning without sampling or shuffling. arXiv preprint arXiv:2103.00039, 2021.   \n[28] Peter Kairouz, Sewoong Oh, and Pramod Viswanath. The composition theorem for differential privacy. In International conference on machine learning, pages 1376–1385. PMLR, 2015.   \n[29] Jakub Konecnˇ y, H Brendan McMahan, Felix X Yu, Peter Richtárik, Ananda Theertha Suresh, and \\` Dave Bacon. Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016.   \n[30] Antti Koskela, Joonas Jälkö, and Antti Honkela. Computing tight differential privacy guarantees using fft. In International Conference on Artificial Intelligence and Statistics, pages 2560–2569. PMLR, 2020.   \n[31] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282, 2017.   \n[32] H Brendan McMahan, Galen Andrew, Ulfar Erlingsson, Steve Chien, Ilya Mironov, Nicolas Papernot, and Peter Kairouz. A general approach to adding differential privacy to iterative training procedures. arXiv preprint arXiv:1812.06210, 2018.   \n[33] H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning differentially private recurrent language models. In ICLR, 2018.   \n[34] 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.   \n[35] Ilya Mironov. On significance of the least significant bits for differential privacy. In Proceedings of the 2012 ACM conference on Computer and communications security, pages 650–661, 2012.   \n[36] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 263–275. IEEE, 2017.   \n[37] Ilya Mironov, Kunal Talwar, and Li Zhang. Rényi differential privacy of the sampled gaussian mechanism. arXiv preprint arXiv:1908.10530, 2019.   \n[38] Thomas Prest, Thomas Ricosset, and Mélissa Rossi. Simple , fast and constant-time gaussian sampling over the integers for falcon. In Second PQC Standardization Conference, 2019.   \n[39] Swaroop Ramaswamy, Rajiv Mathews, Kanishka Rao, and Françoise Beaufays. Federated learning for emoji prediction in a mobile keyboard. arXiv preprint arXiv:1906.04329, 2019.   \n[40] Swaroop Ramaswamy, Om Thakkar, Rajiv Mathews, Galen Andrew, H Brendan McMahan, and Françoise Beaufays. Training production language models without memorizing user data. arXiv preprint arXiv:2009.10031, 2020.   \n[41] Sashank Reddi, Zachary Charles, Manzil Zaheer, Zachary Garrett, Keith Rush, Jakub Konecnˇ y, Sanjiv \\` Kumar, and H Brendan McMahan. Adaptive federated optimization. arXiv preprint arXiv:2003.00295, 2020.   \n[42] Google Research. Federated analytics: Collaborative data science without data collection, May 2020.   \n[43] Sujoy Sinha Roy, Frederik Vercauteren, and Ingrid Verbauwhede. High precision discrete gaussian sampling on fpgas. In International Conference on Selected Areas in Cryptography, pages 383–401. Springer, 2013.   \n[44] Diego Ruiz-Antolín and Javier Segura. A new type of sharp bounds for ratios of modified bessel functions. Journal of Mathematical Analysis and Applications, 443(2):1232–1246, 2016.   \n[45] Aaron Schein, Zhiwei Steven Wu, Alexandra Schofield, Mingyuan Zhou, and Hanna Wallach. Locally private bayesian inference for count models. In International Conference on Machine Learning, pages 5638–5648. PMLR, 2019.   \n[46] Shuang Song, Kamalika Chaudhuri, and Anand D Sarwate. Stochastic gradient descent with differentially private updates. In 2013 IEEE Global Conference on Signal and Information Processing, pages 245–248. IEEE, 2013.   \n[47] VR Thiruvenkatachar and TS Nanjundiah. Inequalities concerning bessel functions and orthogonal polynomials. In Proceedings of the Indian Academy of Sciences-Section A, volume 33, page 373. Springer, 1951.   \n[48] Florian Tramèr and Dan Boneh. Differentially private learning needs better features (or much more data). arXiv preprint arXiv:2011.11660, 2020.   \n[49] Filipp Valovich and Francesco Alda. Computational differential privacy from lattice-based cryptography. In International Conference on Number-Theoretic Methods in Cryptology, pages 121–141. Springer, 2017.   \n[50] Hongxu Yin, Arun Mallya, Arash Vahdat, Jose M Alvarez, Jan Kautz, and Pavlo Molchanov. See through gradients: Image batch recovery via gradinversion. arXiv preprint arXiv:2104.07586, 2021.   \n[51] Yuqing Zhu and Yu-Xiang Wang. Poission subsampled rényi differential privacy. In International Conference on Machine Learning, pages 7634–7642. PMLR, 2019. ",
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parse/train/heqv8eIweMY/heqv8eIweMY.md

@@ -0,0 +1,305 @@
+# NON-LOCAL GRAPH NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Modern graph neural networks (GNNs) learn node embeddings through multilayer local aggregation and achieve great success in applications on assortative graphs. However, tasks on disassortative graphs usually require non-local aggregation. In addition, we find that local aggregation is even harmful for some disassortative graphs. In this work, we propose a simple yet effective non-local aggregation framework with an efficient attention-guided sorting for GNNs. Based on it, we develop various non-local GNNs. We perform thorough experiments to analyze disassortative graph datasets and evaluate our non-local GNNs. Experimental results demonstrate that our non-local GNNs significantly outperform previous state-of-the-art methods on six benchmark datasets of disassortative graphs, in terms of both model performance and efficiency.
+
+# 1 INTRODUCTION
+
+Graph neural networks (GNNs) process graphs and map each node to an embedding vector (Zhang et al., 2018b; Wu et al., 2019). These node embeddings can be directly used for node-level applications, such as node classification (Kipf & Welling, 2017) and link prediction (Schutt et al., 2017). ¨ In addition, they can be used to learn the graph representation vector with graph pooling (Ying et al., 2018; Zhang et al., 2018a; Lee et al., 2019; Yuan & Ji, 2020), in order to fit graph-level tasks (Yanardag & Vishwanathan, 2015). Many variants of GNNs have been proposed, such as ChebNets (Defferrard et al., 2016), GCNs (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), GATs (Velickovi ˇ c et al., 2018), LGCN (Gao et al., 2018) and GINs (Xu et al., 2019). Their ´ advantages have been shown on various graph datasets and tasks (Errica et al., 2020). However, these GNNs share a multilayer local aggregation framework, which is similar to convolutional neural networks (CNNs) (LeCun et al., 1998) on grid-like data such as images and texts.
+
+In recent years, the importance of non-local aggregation has been demonstrated in many applications in the field of computer vision (Wang et al., 2018; 2020) and natural language processing (Vaswani et al., 2017). In particular, the attention mechanism has been widely explored to achieve non-local aggregation and capture long-range dependencies from distant locations. Basically, the attention mechanism measures the similarity between every pair of locations and enables information to be communicated among distant but similar locations. In terms of graphs, non-local aggregation is also crucial for disassortative graphs, while previous studies of GNNs focus on assortative graph datasets (Section 2.2). In addition, we find that local aggregation is even harmful for some disassortative graphs (Section 4.3). The recently proposed Geom-GCN (Pei et al., 2020) explores to capture longrange dependencies in disassortative graphs. It contains an attention-like step that computes the Euclidean distance between every pair of nodes. However, this step is computationally prohibitive for large-scale graphs, as the computational complexity is quadratic in the number of nodes. In addition, Geom-GCN employs pre-trained node embeddings (Tenenbaum et al., 2000; Nickel & Kiela, 2017; Ribeiro et al., 2017) that are not task-specific, limiting the effectiveness and flexibility.
+
+In this work, we propose a simple yet effective non-local aggregation framework for GNNs. At the heart of the framework lies an efficient attention-guided sorting, which enables non-local aggregation through classic local aggregation operators in general deep learning. The proposed framework can be flexibly used to augment common GNNs with low computational costs. Based on the framework, we build various efficient non-local GNNs. In addition, we perform detailed analysis on existing disassortative graph datasets, and apply different non-local GNNs accordingly. Experimental results show that our non-local GNNs significantly outperform previous state-of-the-art methods on node classification tasks on six benchmark datasets of disassortative graphs.
+
+# 2 BACKGROUND AND RELATED WORK
+
+# 2.1 GRAPH NEURAL NETWORKS
+
+We focus on learning the embedding vector for each node through graph neural networks (GNNs). Most existing GNNs are inspired by convolutional neural networks (CNNs) (LeCun et al., 1998) and follow a local aggregation framework. In general, each layer of GNNs scans every node in the graph and aggregates local information from directly connected nodes, i.e., the 1-hop neighbors.
+
+Specifically, a common layer of GNNs performs a two-step processing similar to the depthwise separable convolution (Chollet, 2017): spatial aggregation and feature transformation. The first step updates each node embedding using embedding vectors of spatially neighboring nodes. For example, GCNs (Kipf & Welling, 2017) and GATs (Velickovi ˇ c et al., 2018) compute a weighted ´ sum of node embeddings within the 1-hop neighborhood, where weights come from the degree of nodes and the interaction between nodes, respectively. GraphSAGE (Hamilton et al., 2017) applies the max pooling, while GINs (Xu et al., 2019) simply sums the node embeddings. The feature transformation step is similar to the $1 \times 1$ convolution, where each node embedding vector is mapped into a new feature space through a shared linear transformation (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovi ˇ c et al., 2018) or multilayer perceptron (MLP) (Xu et al., 2019). Different from ´ these studies, LGCN (Gao et al., 2018) explores to directly apply the regular convolution through top- $k$ ranking.
+
+Nevertheless, each layer of these GNNs only aggregates local information within the 1-hop neighborhood. While stacking multiple layers can theoretically enable communication between nodes across the multi-hop neighborhood, the aggregation is essentially local. In addition, deep GNNs usually suffer from the over-smoothing problem (Xu et al., 2018; Li et al., 2018; Chen et al., 2020).
+
+# 2.2 ASSORTATIVE AND DISASSORTATIVE GRAPHS
+
+There are many kinds of graphs in the literature, such as citation networks (Kipf & Welling, 2017), community networks (Chen et al., 2020), co-occurrence networks (Tang et al., 2009), and webpage linking networks (Rozemberczki et al., 2019). We focus on graph datasets corresponding to the node classification tasks. In particular, we categorize graph datasets into assortative and disassortative ones (Newman, 2002; Ribeiro et al., 2017) according to the node homophily in terms of labels, i.e., how likely nodes with the same label are near each other in the graph.
+
+Assortative graphs refer to those with a high node homophily. Common assortative graph datasets are citation networks and community networks. On the other hand, graphs in disassortative graph datasets contain more nodes that have the same label but are distant from each other. Example disassortative graph datasets are co-occurrence networks and webpage linking networks.
+
+As introduced above, most existing GNNs perform local aggregation only and achieve good performance on assortative graphs (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovi ˇ c et al., 2018; ´ Gao et al., 2018). However, they may fail on disassortative graphs, where informative nodes in the same class tend to be out of the local multi-hop neighborhood and non-local aggregation is needed. Thus, in this work, we explore the non-local GNNs.
+
+# 2.3 ATTENTION MECHANISM
+
+The attention mechanism (Vaswani et al., 2017) has been widely used in GNNs (Velickovi ˇ c et al., ´ 2018; Gao & Ji, 2019; Knyazev et al., 2019) as well as other deep learning models (Yang et al., 2016; Wang et al., 2018; 2020). A typical attention mechanism takes three groups of vectors as inputs, namely the query vector $q$ , key vectors $( k _ { 1 } , k _ { 2 } , \ldots , k _ { n } )$ , value vectors $( v _ { 1 } , v _ { 2 } , \ldots , v _ { n } )$ . Note that key and value vectors have a one-to-one correspondence and can be the same sometimes. The attention mechanism computes the output vector $o$ as
+
+$$
+a _ { i } = \operatorname { A r T E N D } ( q , k _ { i } ) \in \mathbb { R } , i = 1 , 2 , \dotsc , n ; \quad o = \sum _ { i } a _ { i } v _ { i } ,
+$$
+
+where the ATTEND $( \cdot )$ function could be any function that outputs a scalar attention score $a _ { i }$ from the interaction between $q$ and $k _ { i }$ , such as dot product (Gao & Ji, 2019) or even a neural network (Velickovi ˇ c et al., 2018). The definition of the three groups of input vectors depends on the ´ models and applications.
+
+Notably, existing GNNs usually use the attention mechanism for local aggregation (Velickovi ˇ c et al., ´ 2018; Gao & Ji, 2019). Specifically, when aggregating information for node $v$ , the query vector is the embedding vector of $v$ while the key and value vectors come from node embeddings of $v$ ’s directly connected nodes. And the process is iterated for each $v \in V$ . It is worth noting that the attention mechanism can be easily extended for non-local aggregation (Wang et al., 2018; 2020), by letting the key and value vectors correspond to all the nodes in the graph when aggregating information for each node. However, it is computationally prohibitive given large-scale graphs, as iterating it for each node in a graph of $n$ nodes requires $O ( n ^ { 2 } )$ time. In this work, we propose a novel non-local aggregation method that only requires $O ( n \log { n } )$ time.
+
+# 3 THE PROPOSED METHOD
+
+# 3.1 NON-LOCAL AGGREGATION WITH ATTENTION-GUIDED SORTING
+
+We consider a graph $\mathcal { G } = ( V , E )$ , where $V$ is the set of nodes and $E$ is the set of edges. Each edge $e \in E$ connects two nodes so that $E \subseteq V \times V$ . Each node $v \in V$ has a node feature vector $x _ { v } \in \bar { \mathbb { R } ^ { d } }$ . The $k$ -hop neighborhood of $v$ refers to the set of nodes $\mathcal { N } _ { k } ( v )$ that can reach $v$ within $k$ edges. For example, the set of $v$ ’s directly connected nodes is its 1-hop neighborhood $\mathcal { N } _ { 1 } ( v )$ .
+
+Our proposed non-local aggregation framework is composed of three steps, namely local embedding, attention-guided sorting, and non-local aggregation. In the following, we describe them one by one.
+
+Local Embedding: Our proposed framework is built upon a local embedding step that extracts local node embeddings from the node feature vectors. The local embedding step can be as simple as
+
+$$
+z _ { v } = \mathbf { M } \mathbf { L } \mathbf { P } ( x _ { v } ) \in \mathbb { R } ^ { f } , \forall v \in V .
+$$
+
+The $\mathrm { \mathbf { M L P } ( \cdot ) }$ function is a multilayer perceptron (MLP), and $f$ is the dimension of the local node embedding $z _ { v }$ . Note that the $\mathrm { \mathbf { M L P } ( \cdot ) }$ function is shared across all the nodes in the graph. Applying MLP only takes the node itself into consideration without aggregating information from the neighborhood. This property is very important on some disassortative graphs, as shown in Section 4.3.
+
+On the other hand, graph neural networks (GNNs) can be used as the local embedding step as well, so that our proposed framework can be easily employed to augment existing GNNs. As introduced in Section 2.1, modern GNNs perform multilayer local aggregation. Typically, for each node, one layer of a GNN aggregates information from its 1-hop neighborhood. Stacking $L$ such local aggregation layers allows each node to access information that is $L$ hops away. To be specific, the $\ell \cdot$ -th layer of a $L$ -layer GNN $( \ell = 1 , 2 , \ldots , L )$ can be described as
+
+$$
+\begin{array} { r } { z _ { v } ^ { ( \ell ) } = { \mathrm { T R A N S F O R M } } ^ { ( \ell ) } \left( \mathbf { A G G R E G A T E } ^ { ( \ell ) } \left( \{ z _ { u } ^ { ( \ell - 1 ) } : u \in \mathcal { N } _ { 1 } ( v ) \cup v \} \right) \right) \in \mathbb { R } ^ { f } , \forall v \in V , } \end{array}
+$$
+
+where z(0v $z _ { v } ^ { ( 0 ) } = x _ { v }$ , and $z _ { v } ~ = ~ z _ { v } ^ { ( L ) }$ represents the local node embedding. The AGGREGATE $^ { ( \ell ) } ( \cdot )$ and TR $\mathsf { A N S F O R M } ^ { ( \ell ) } ( \cdot )$ functions represent the spatial aggregation and feature transformation step introduced in Section 2.1, respectively. With the above framework, GNNs can capture the node feature information from nodes within a local neighborhood as well as the structural information.
+
+When either MLP or GNNs is used as the local embedding step, the local node embedding $z _ { v }$ only contains local information of a node $v$ . However, $z _ { v }$ can be used to guide non-local aggregation, as distant but informative nodes are likely to have similar node features and local structures. Based on this intuition, we propose the attention-guided sorting to enable the non-local aggregation.
+
+Attention-Guided Sorting: The basic idea of the attention-guided sorting is to learn an ordering of nodes, where distant but informative nodes are put near each other. Specifically, given the local node embedding $z _ { v }$ obtained through the local embedding step, we compute one set of attention scores by
+
+$$
+a _ { v } = \mathrm { A T T E N D } ( c , z _ { v } ) \in \mathbb { R } , \forall v \in V ,
+$$
+
+where $c$ is a calibration vector that is randomly initialized and jointly learned during training (Yang et al., 2016). In this attention operator, $c$ serves as the query vector and $z _ { v }$ are the key vectors. In addition, we also treat $z _ { v }$ as the value vectors. However, unlike the attention mechanism introduced in Section 2.3, we use the attention scores to sort the value vectors instead of computing a weighted sum to aggregating them. Note that originally there is no ordering among nodes in a graph. To be specific, as $a _ { v }$ and $z _ { v }$ have one-to-one correspondence through Equation (4), sorting the attention scores in non-decreasing order into $( a _ { 1 } , a _ { 2 } , \ldots , a _ { n } )$ provides an ordering among nodes, where $n =$ $| V |$ is the number of nodes in the graph. The resulting sequence of local node embeddings can be denoted as $( z _ { 1 } , z _ { 2 } , \ldots , z _ { n } )$ .
+
+The attention process in Equation (4) can be also understood as a projection of local node embeddings onto a 1-dimensional space. The projection depends on the concrete ATTEND $( \cdot )$ function and the calibration vector $c$ . As indicated by its name, the calibration vector $c$ is used to calibrate the 1-dimensional space, in order to push distant but informative nodes close to each other in this space. This goal is fulfilled through the following non-local aggregation step and the training of the calibration vector $c$ , as demonstrated below.
+
+Non-Local Aggregation: We point out that, with the attention-guided sorting, the non-local aggregation can be achieved by convolution, the most common local aggregation operator in deep learning. Specifically, given the sorted sequence of local node embeddings $( z _ { 1 } , z _ { 2 } , \ldots , z _ { n } )$ , we compute
+
+$$
+( \hat { z } _ { 1 } , \hat { z } _ { 2 } , \ldots , \hat { z } _ { n } ) = \mathbf { C } \mathbf { O } \mathbf { N } \mathbf { V } ( z _ { 1 } , z _ { 2 } , \ldots , z _ { n } ) ,
+$$
+
+where the $\mathrm { C o N V } ( \cdot )$ function represents a 1D convolution with appropriate padding. Note that the $\mathrm { C o N V } ( \cdot )$ function can be replaced by a 1D convolutional neural network as long as the number of input and output vectors remains the same.
+
+To see how the $\mathrm { C o N V } ( \cdot )$ function performs non-local aggregation with the attention-guided sorting, we take an example where the $\mathrm { C o N V } ( \cdot )$ function is a 1D convolution of kernel size $2 s + 1$ . In this case, $\hat { z } _ { i }$ is computed from $( z _ { i + s } , \dots , z _ { i - s } )$ , corresponding to the receptive field of the $\operatorname { C o n v } ( \cdot )$ function. As a result, if the attention-guided sorting leads to $( z _ { i + s } , \dots , z _ { i - s } )$ containing nodes that are distant but informative to $z _ { i }$ , the output $\hat { z } _ { i }$ aggregates non-local information. Another view is that we can consider the attention-guided sorting as re-connects nodes in the graph, where $( z _ { i + s } , \dots , z _ { i - s } )$ can be treated as the 1-hop neighborhood of $z _ { i }$ . After the $\operatorname { C o n v } ( \cdot )$ function, $\hat { z } _ { i }$ and $z _ { i }$ are concatenated as the input to a classifier to predict the label of the corresponding node, where both non-local and local dependencies can be captured. In order to enable the end-to-end training of the calibration vector $c$ , we modify Equation (5) into
+
+$$
+( \hat { z } _ { 1 } , \hat { z } _ { 2 } , \ldots , \hat { z } _ { n } ) = \mathbf { C } \mathbf { O } \mathbf { N } \mathbf { v } ( a _ { 1 } z _ { 1 } , a _ { 2 } z _ { 2 } , \ldots , a _ { n } z _ { n } ) ,
+$$
+
+where we multiply the attention score with the corresponding local node embedding. As a result, the calibration vector $c$ receives gradients through the attention scores during training.
+
+The remaining question is how to make sure that the attention-guided sorting pushes distant but informative nodes together. The short answer is that it is not necessary to guarantee this, as the requirement of non-local aggregation depends on the concrete graphs. In fact, our proposed framework grants GNNs the ability of non-local aggregation but lets the end-to-end training process determine whether to use non-local information. The back-propagation from the supervised loss will tune the calibration vector $c$ and encourage $\hat { z } _ { i }$ to capture useful information that is not encoded by $z _ { i }$ . In the case of disassortative graphs, $\hat { z } _ { i }$ usually needs to aggregate information from distant but informative nodes. Hence, the calibration vector $c$ tends to arrange the attention-guided sorting to put distant but informative nodes together, as demonstrated experimentally in Section 4.5. On the other hand, nodes within the local neighborhood are usually much more informative than distant nodes in assortative graphs. In this situation, $\hat { z } _ { i }$ may simply perform local aggregation that is similar to GNNs.
+
+In Section 4, we demonstrate the effectiveness of our proposed non-local aggregation framework on six disassortative graph datasets. In particular, we achieve the state-of-the-art performance on all the datasets with significant improvements over previous methods.
+
+# 3.2 TIME COMPLEXITY ANALYSIS
+
+We perform theoretical analysis of the time complexity of our proposed framework. As discussed in Section 2.3, using the attention mechanism (Vaswani et al., 2017; Wang et al., 2018; 2020) to achieve non-local aggregation requires $O ( n ^ { 2 } )$ time for a graph of $n$ nodes. Essentially, the $O ( n ^ { 2 } )$
+
+time complexity is due to the fact that the ATTEND(·) function needs to be computed between every pair of nodes. In particular, the recently proposed Geom-GCN (Pei et al., 2020) contains a similar non-local aggregation step. For each $v \in V$ , Geom-GCN finds the set of nodes from which the Euclidean distance to $v$ is less than a pre-defined number, where the Euclidean distance between every pair of nodes needs to be computed. As the computation of the the Euclidean distance between two nodes can be understood as the ATTEND(·) function, Geom-GCN has at least $O ( n ^ { 2 } )$ time complexity.
+
+In contrast, our proposed non-local aggregation framework requires only $O ( n \log n )$ time. To see this, note that the ATTEND(·) function in Equation (4) only needs to be computed once, instead of iterating it for each node. As a result, computing the attention scores only takes $O ( n )$ time. Therefore, the time complexity of sorting, i.e. ${ \bar { O } } ( n \log n )$ , dominates the total time complexity of our proposed framework. In Section 4.6, we compare the real running time on different datasets among common GNNs, Geom-GCN, and our non-local GNNs as introduced in the next section.
+
+# 3.3 EFFICIENT NON-LOCAL GRAPH NEURAL NETWORKS
+
+We apply our proposed non-local aggregation framework to build efficient non-local GNNs. Recall that our proposed framework starts with the local embedding step, followed by the attention-guided sorting and the non-local aggregation step.
+
+In particular, the local embedding step can be implemented by either MLP or common GNNs, such as GCNs (Kipf & Welling, 2017) or GATs (Velickovi ˇ c et al., 2018). MLP extracts the local node ´ embedding only from the node feature vector and excludes the information from nodes within the local neighborhood. This property can be helpful on some disassortative graphs, where nodes within the local neighborhood provide more noises than useful information. On other disassortative graphs, informative nodes locate in both local neighborhood and distant locations. In this case, GNNs are more suitable as the local embedding step. Depending on the disassortative graphs in hand, we build different non-local GNNs with either MLP or GNNs as the local embedding step. In Section 4.3, we show that these two categories of disassortative graphs can be distinguished through simple experiments, where we apply different non-local GNNs accordingly. Specifically, the number of layers is set to 2 for both MLP and GNNs.
+
+In terms of the attention-guided sorting, we only need to specify the ATTEND $( \cdot )$ function in Equation (4). In order to make it as efficient as possible, we choose the ATTEND $( \cdot )$ function as
+
+$$
+\begin{array} { r } { a _ { v } = \mathrm { A T T E N D } ( c , z _ { v } ) = c ^ { T } z _ { v } \in \mathbb { R } , \forall v \in V , } \end{array}
+$$
+
+where $c$ is part of the training parameters, as described in Section 3.1.
+
+With the attention-guided sorting, we can implement the non-local aggregation step through convolution, as explained in Section 3.1 and shown in Equation (6). Specifically, $\mathrm { C o N V } ( \cdot )$ function is set as a 2-layer convolutional neural network composed of two 1D convolutions. The kernel size is set to 3 or 5 depending on the datasets. The activation function is ReLU (Krizhevsky et al., 2012).
+
+Finally, we use a linear classifier that takes the concatenation of $\hat { z } _ { i }$ and $z _ { i }$ as inputs and makes prediction for the corresponding node. Depending on the local embedding step, we build three efficient non-local GNNs, namely non-local MLP (NLMLP), non-local GCN (NLGCN), and nonlocal GAT (NLGAT). The models can be end-to-end trained with the classification loss.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS
+
+We perform experiments on six disassortative graph datasets (Rozemberczki et al., 2019; Tang et al., 2009; Pei et al., 2020) (Chameleon, Squirrel, Actor, Cornell, Texas, Wisconsin) and three assortative graph datasets (Sen et al., 2008) (Cora, Citeseer, Pubmed). These datasets are commonly used to evaluate GNNs on node classification tasks (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018; Gao ´ et al., 2018; Pei et al., 2020). We provide detailed descriptions of disassortative graph datasets in Appendix A.1. In order to distinguish assortative and disassortative graph datasets, Pei et al. (2020)
+
+propose a metric to measure the homophily of a graph $\mathcal { G }$ , defined as
+
+$$
+H ( \mathcal { G } ) = \frac { 1 } { | V | } \sum _ { v \in V } \frac { \mathrm { N u m b e r ~ o f ~ } v ^ { \circ } \mathrm { ~ s d i r e c t l y ~ c o n n e c t e d ~ n o d e s ~ w h o ~ h a v e ~ t h e ~ s a m e ~ l a b e l ~ a s ~ } v } { \mathrm { N u m b e r ~ o f ~ } v ^ { \circ } \mathrm { ~ s d i r e c t l y ~ c o n n e c t e d ~ n o d e s } } .
+$$
+
+Intuitively, a large $H ( { \mathcal { G } } )$ indicates an assortative graph, and vice versa. The $H ( { \mathcal { G } } )$ and other statistics are summarized in Table 1.
+
+Table 1: Statistics of the nine datasets used in our experiments. The definition of $H ( { \mathcal { G } } )$ is provided in Section 4.1. $H ( { \mathcal { G } } )$ can be used to distinguish assortative and disassortative graph datasets.   
+
+<table><tr><td colspan="5">Assortative</td><td colspan="5">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>H(G)</td><td>0.83</td><td>0.71</td><td>0.79</td><td>0.25</td><td>0.22</td><td>0.24</td><td>0.11</td><td>0.06</td><td>0.16</td></tr><tr><td>#Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>2277</td><td>5201</td><td>7600</td><td>183</td><td>183</td><td>251</td></tr><tr><td>#Edges</td><td>5429</td><td>4732</td><td>44338</td><td>36101</td><td>217073</td><td>33544</td><td>295</td><td>309</td><td>499</td></tr><tr><td>#Features</td><td>1433</td><td>3703</td><td>500</td><td>2325</td><td>2089</td><td>931</td><td>1703</td><td>1703</td><td>1703</td></tr><tr><td>#Classes</td><td>7</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
+
+In our experiments, we focus on comparing the model performance on disassortative graph datasets, in order to demonstrate the effectiveness of our non-local aggregation framework. The performances on assortative graph datasets are provided for reference, indicating that the proposed framework will not hurt the performance when non-local aggregation is not strongly desired.
+
+# 4.2 BASELINES
+
+We compare our proposed non-local MLP (NLMLP), non-local GCN (NLGCN), and non-local GAT (NLGAT) with various baselines: (1) MLP is the simplest deep learning model. It makes prediction solely based on the node feature vectors, without aggregating any local or non-local information. (2) GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018) are the most common ´ GNNs. As introduced in Section 2.1, they only perform local aggregation. (3) Geom-GCN (Pei et al., 2020) is a recently proposed GNN that can capture long-range dependencies. It is the current stateof-the-art model on several disassortative graph datasets. Geom-GCN requires the use of different node embedding methods, such as Isomap (Tenenbaum et al., 2000), Poincare (Nickel & Kiela, 2017), and struc2vec (Ribeiro et al., 2017). We simply report the best results from Pei et al. (2020) for Geom-GCN and the following two variants without specifying the node embedding method. (4) Geom-GCN- $\mathbf { g }$ (Pei et al., 2020) is a variant of Geom-GCN that performs local aggregation only. It is similar to common GNNs. (5) Geom-GCN-s (Pei et al., 2020) is a variant of Geom-GCN that does not force local aggregation. The designed functionality is similar to our NLMLP.
+
+We implement MLP, GCN, GAT, and our methods using Pytorch (Adam et al., 2017) and Pytorch Geometric (Fey & Lenssen, 2019). As has been discussed1, in fair settings, the results of GCN and GAT differ from those in Pei et al. (2020).
+
+On each dataset, we follow Pei et al. (2020) and randomly split nodes of each class into $60 \%$ , $20 \%$ , and $20 \%$ for training, validation, and testing. The experiments are repeatedly run 10 times with different random splits and the average test accuracy over these 10 runs are reported. Testing is performed when validation accuracy achieves maximum on each run. Apart from the details specified in Section 3.3, we tune the following hyperparameters individually for our proposed models: (1) the number of hidden unit $\in \{ 1 6 , 4 8 , 9 6 \}$ , (2) dropout rate $\in \{ 0 , 0 . 5 , 0 . 8 \}$ , (3) weight decay $\in \{ 0 , 5 \mathrm { e } { - } 4$ , 5e-5, 5e-6}, and (4) learning rate $\in \{ 0 . 0 1 , 0 . 0 5 \}$ .
+
+# 4.3 ANALYSIS OF DISASSORTATIVE GRAPH DATASETS
+
+As discussed in Section 3.3, the disassortative graph datasets can be divided into two categories. Nodes within the local neighborhood provide more noises than useful information in disassortative graphs belonging to the first category. Therefore, local aggregation should be avoided in models on such disassortative graphs. As for the second category, informative nodes locate in both local neighborhood and distant locations. Intuitively, a graph with lower $H ( { \mathcal { G } } )$ is more likely to be in the first category. However, it is not an accurate way to determine the two categories.
+
+Knowing the exact category of a disassortative graph is crucial, as we need to apply non-local GNNs accordingly. As analyzed above, the key difference lies in whether the local aggregation is useful. Hence, we can distinguish two categories of disassortative graphs by comparing the performance between MLP and common GNNs (GCN, GAT) on each of the six disassortative graph datasets.
+
+Table 2: Comparisons between MLP and common GNNs. These analytical experiments are used to determine the two categories of disassortative graph datasets, as introduced in Section 4.3.   
+
+<table><tr><td colspan="5">Assortative</td><td colspan="5">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>76.5±1.3</td><td>73.6±1.9</td><td>87.5±0.4</td><td>48.5±3.0</td><td>31.5±1.4</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>GCN</td><td>88.2±1.2</td><td>75.7±1.3</td><td>88.4±0.6</td><td>67.6±2.4</td><td>54.9±1.9</td><td>30.3±1.6</td><td>54.2±7.3</td><td>61.1±7.0</td><td>59.6±4.5</td></tr><tr><td>GAT</td><td>88.4±1.4</td><td>76.1±1.0</td><td>87.0±0.3</td><td>65.0±3.7</td><td>51.3±2.5</td><td>29.4±1.2</td><td>56.3±4.3</td><td>57.9±6.1</td><td>57.8±4.3</td></tr></table>
+
+The results are summarized in Table 2. We can see that Actor, Cornell, Texas, and Wisconsin fall into the first category, while Chameleon and Squirrel belong to the second category. We add the performance on assortative graph datasets for reference, where the local aggregation is effective so that GNNs tend to outperform MLP.
+
+# 4.4 COMPARISONS WITH BASELINES
+
+According to the insights from Section 4.3, we apply different non-local GNNs according to the category of disassortative graph datasets, and make comparisons with corresponding baselines.
+
+Specifically, we employ NLMLP on Actor, Cornell, Texas, and Wisconsin. The corresponding baselines are MLP, Geom-GCN, and Geom-GCN-s, as Table 2 has shown that GCN and GAT perform much worse than MLP on these datasets. And Geom-GCN- $\mathbf { g }$ is similar to GCN and has worse performance than Geom-GCN-s, which is shown in Ap
+
+Table 3: Comparisons between our NLMLP and strong baselines on the four disassortative graph datasets belonging to the first category as defined in Section 4.3.   
+
+<table><tr><td>Datasets</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>Geom-GCN</td><td>31.6</td><td>60.8</td><td>67.6</td><td>64.1</td></tr><tr><td>Geom-GCN-s</td><td>34.6</td><td>75.4</td><td>73.5</td><td>80.4</td></tr><tr><td>NLMLP</td><td>37.9±1.3</td><td>84.9±5.7</td><td>85.4±3.8</td><td>87.3±4.3</td></tr></table>
+
+pendix A.2. The comparison results are reported in Table 3. While Geom-GCN-s are the previous state-of-the-art GNNs on these datasets (Pei et al., 2020), we find that MLP consistently outperforms Geom-GCN-s by large margins. In particular, although Geom-GCN-s does not explicitly perform local aggregation, it is still outperformed by MLP. A possible explanation is that Geom-GCN-s uses pre-trained node embeddings, which aggregates information from the local neighborhood implicitly. In contrast, our NLMLP is built upon MLP with the proposed non-local aggregation framework, which excludes the local noises and collects useful information from non-local informative nodes. The NLMLP sets the new state-of-the-art performance on these disassortative graph datasets.
+
+On Chameleon and Squirrel that belong to the second category of disassortative graph datasets, we apply NLGCN and NLGAT accordingly. The baselines are GCN, GAT, Geom-GCN, and Geom-GCN- $\mathbf { g }$ . On these datasets, these baselines that explicitly perform local aggregation show advantages over MLP and Geom-GCN-s, as shown in Appendix A.2. As shown in Table 4, our proposed NLGCN achieves the best performance on both datasets. In addition, it is worth noting that our NLGCN and NLGAT are built upon GCN and GAT, respectively. They show improvements over their counterparts, which indicates that the advantages of our proposed non-local aggregation framework are general for common GNNs.
+
+Table 4: Comparisons between our NLGCN, NLGAT and strong baselines on the two disassortative graph datasets belonging to the second category as defined in Section 4.3.   
+
+<table><tr><td>Datasets</td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>GCN</td><td>67.6±2.4</td><td>54.9±1.9</td></tr><tr><td>GAT</td><td>65.0±3.7</td><td>51.3±2.5</td></tr><tr><td>Geom-GCN</td><td>60.9</td><td>38.1</td></tr><tr><td>Geom-GCN-g</td><td>68.0</td><td>46.0</td></tr><tr><td>NLGCN</td><td>70.1±2.9</td><td>59.0±1.2</td></tr><tr><td>NLGAT</td><td>65.7±1.4</td><td>56.8±2.5</td></tr></table>
+
+We provide the results of all the models on all datasets in Appendix A.2 for reference.
+
+# 4.5 ANALYSIS OF THE ATTENTION-GUIDED SORTING
+
+We analyze the results of the attention-guided sorting in our proposed framework, in order to show that our non-local GNNs indeed perform non-local aggregation.
+
+Suppose the attention-guided sorting leads to the sorted sequence $( z _ { 1 } , z _ { 2 } , \ldots , z _ { n } )$ , which goes through a convolution or CNN into $\left( \hat { z } _ { 1 } , \hat { z } _ { 2 } , \dots , \hat { z } _ { n } \right)$ . As discussed in Section 3.1, we can consider the sequence $( z _ { 1 } , z _ { 2 } , \ldots , z _ { n } )$ as a re-connected graph $\hat { \mathcal G }$ , where we treat nodes within the receptive field of $\hat { z } _ { i }$ as directly connected to $z _ { i }$ , i.e. $z _ { i }$ ’s 1-hop neighborhood. The information within this new 1-hop neighborhood will be aggregated. If our non-local GNNs indeed perform non-local aggregation, the homophily of the re-connected graph should be larger than the original graph. Therefore, we compute $H ( \hat { \mathcal G } )$ for each dataset to verify this statement. Following Section 4.4, we apply NLMLP on Actor, Cornell, Texas, and Wisconsin and NLGCN on Chameleon and Squirrel.
+
+Figure 1 compares $H ( \hat { \mathcal G } )$ with $H ( { \mathcal { G } } )$ for each dataset. We can observe that $H ( \hat { \mathcal G } )$ is much larger than $H ( { \mathcal { G } } )$ , indicating that distant but informative nodes are near each other in the re-connected graph $\hat { \mathcal G }$ . We also provide the visualizations of the sorted sequence for Cornell and Texas. We can see that nodes with the same label tend to be clustered together. These facts indicate that our non-local GNNs perform non-local aggregation with the attention-guided sorting.
+
+![](images/ce6b80695989c7253e5e2d1b0da6b98cab5728cae3949ffc7571a9f7cafbd3a1.jpg)  
+Figure 1: (a) Comparisons of the homophily between the original graph and the re-connected graph given by our NLGCN on Chameleon and Squirrel. (b) Comparisons of the homophily between the original graph and the re-connected graph given by our NLMLP on Actor, Cornell, Texas, and Wisconsin. (c) Visualization of sorted node sequence after the attention-guided sorting for Cornell and Texas. The colors denote node labels. Details are explained in Section 4.5.
+
+# 4.6 EFFICIENCY COMPARISONS
+
+As analyzed in Section 3.2, our proposed nonlocal aggregation framework is more efficient than previous methods based on the original attention mechanism, such as Geom-GCN (Pei et al., 2020). Concretely, our method requires only $O ( n \log n )$ computation time in contrast to $O ( n ^ { 2 } )$ . In this section, we compare the real running time to verify our analysis. Specifi
+
+Table 5: Comparisons in terms of real running time (milliseconds).   
+
+<table><tr><td></td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>GCN</td><td>22.2 (1.0×)</td><td>14.3 (1.0×)</td></tr><tr><td>GAT</td><td>33.2 (1.5x)</td><td>163.3 (11.4×)</td></tr><tr><td>Geom-GCN</td><td>3615.0 (163.1×)</td><td>10430.0 (727.3×)</td></tr><tr><td>NLGCN</td><td>26.3 (1.2×)</td><td>39.6 (2.8×)</td></tr></table>
+
+cally, we compare NLGCN with Geom-GCN as well as GCN and GAT. For Geom-GCN, we use the code provided in Pei et al. (2020). Each model is trained for 500 epochs on each dataset and the average training time per epoch is reported.
+
+The results are shown in Table 5. Although our NLGCN is built upon GCN, it is just slightly slower than GCN and faster than GAT, showing the efficiency of our non-local aggregation framework. On the other hand, Geom-GCN is significantly slower due to the fact that it has $\bar { O ( n ^ { 2 } ) }$ time complexity.
+
+# 5 CONCLUSION
+
+In this work, we propose a simple yet effective non-local aggregation framework for GNNs. The core of the framework is an efficient attention-guided sorting, which enables non-local aggregation through convolution. The proposed framework can be easily used to build non-local GNNs with low computational costs. We perform thorough experiments on node classification tasks to evaluate our proposed method. In particular, we experimentally analyze existing disassortative graph datasets and apply different non-local GNNs accordingly. The results show that our non-local GNNs significantly outperform previous state-of-the-art methods on six benchmark datasets of disassortative graphs, in terms of both accuracy and speed.
+
+# REFERENCES
+
+Paszke Adam, Gross Sam, Chintala Soumith, Chanan Gregory, Yang Edward, D Zachary, Lin Zeming, Desmaison Alban, Antiga Luca, and Lerer Adam. Automatic differentiation in pytorch. In Proceedings of Neural Information Processing Systems Autodiff Workshop, 2017.
+
+Deli Chen, Yankai Lin, Wei Li, Peng Li, Jie Zhou, and Xu Sun. Measuring and relieving the oversmoothing problem for graph neural networks from the topological view. In Thirty-Fourth AAAI Conference on Artificial Intelligence, 2020.
+
+Franc¸ois Chollet. Xception: Deep learning with depthwise separable convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1251–1258, 2017.
+
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks ¨ on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pp. 3844–3852, 2016.
+
+Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. In International Conference on Learning Representations, 2020.
+
+Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019.
+
+Hongyang Gao and Shuiwang Ji. Graph representation learning via hard and channel-wise attention networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 741–749, 2019.
+
+Hongyang Gao, Zhengyang Wang, and Shuiwang Ji. Large-scale learnable graph convolutional networks. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1416–1424, 2018.
+
+Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pp. 1024–1034, 2017.
+
+Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations, 2017.
+
+Boris Knyazev, Graham W Taylor, and Mohamed Amer. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105, 2012.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In International Conference on Machine Learning, pp. 3734–3743, 2019.
+
+Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
+
+Mark EJ Newman. Assortative mixing in networks. Physical review letters, 89(20):208701, 2002.
+
+Maximillian Nickel and Douwe Kiela. Poincare embeddings for learning hierarchical representa- ´ tions. In Advances in Neural Information Processing Systems, pp. 6338–6347, 2017.
+
+Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-GCN: Geometric graph convolutional networks. In International Conference on Learning Representations, 2020.
+
+Leonardo FR Ribeiro, Pedro HP Saverese, and Daniel R Figueiredo. struc2vec: Learning node representations from structural identity. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 385–394, 2017.
+
+Benedek Rozemberczki, Carl Allen, and Rik Sarkar. Multi-scale attributed node embedding. arXiv preprint arXiv:1909.13021, 2019.
+
+Kristof Schutt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre ¨ Tkatchenko, and Klaus-Robert Muller. Schnet: A continuous-filter convolutional neural network ¨ for modeling quantum interactions. In Advances in Neural Information Processing Systems, pp. 991–1001, 2017.
+
+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93–93, 2008.
+
+Jie Tang, Jimeng Sun, Chi Wang, and Zi Yang. Social influence analysis in large-scale networks. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 807–816, 2009.
+
+Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319–2323, 2000.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017.
+
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. In International Conference on Learning Representations, 2018.
+
+Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7794–7803, 2018.
+
+Zhengyang Wang, Na Zou, Dinggang Shen, and Shuiwang Ji. Non-local U-Nets for biomedical image segmentation. In Thirty-Fourth AAAI Conference on Artificial Intelligence, 2020.
+
+Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019.
+
+Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In International Conference on Machine Learning, pp. 5449–5458, 2018.
+
+Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019.
+
+Pinar Yanardag and SVN Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1365–1374. ACM, 2015.
+
+Zichao Yang, Diyi Yang, Chris Dyer, Xiaodong He, Alex Smola, and Eduard Hovy. Hierarchical attention networks for document classification. In Proceedings of the 2016 Conference of the North American chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 1480–1489, 2016.
+
+Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems, pp. 4800–4810, 2018.
+
+Hao Yuan and Shuiwang Ji. StructPool: Structured graph pooling via conditional random fields. In International Conference on Learning Representations, 2020.
+
+Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018a.
+
+Si Zhang, Hanghang Tong, Jiejun Xu, and Ross Maciejewski. Graph convolutional networks: Algorithms, applications and open challenges. In International Conference on Computational Social Networks, pp. 79–91. Springer, 2018b.
+
+# A APPENDIX
+
+A.1 DETAILS OF DISASSORTATIVE GRAPH DATASETS
+
+Here are the details of disassortative graph datasets used in our experiments:
+
+• Chameleon and Squirrel are Wikipedia networks (Rozemberczki et al., 2019) where nodes represent web pages from Wikipedia and edges indicate mutual links between pages. Node feature vectors are bag-of-word representation of informative nouns in the corresponding pages. Each node is labeled with one of five classes according to the number of the average monthly traffic of the web page.   
+• Actor is an actor co-occurrence network, where nodes denote actors and edges indicate cooccurrence on the same web page from Wikipedia. It is extracted from the film-directoractor-writer network proposed by Tang et al. (Tang et al., 2009). Node feature vectors are bag-of-word representation of keywords in the actors’ Wikipedia pages. Each node is labeled with one of five classes according to the topic of the actor’s Wikipedia page. Cornell, Texas, and Wisconsin come from the WebKB dataset collected by Carnegie Mellon University. Nodes represent web pages and edges denote hyperlinks between them. Node feature vectors are bag-of-word representation of the corresponding web pages. Each node is labeled with one of student, project, course, staff, and faculty.
+
+# A.2 FULL EXPERIMENTAL RESULTS
+
+Table 6: Comparisons between our NLMLP, NLGCN, NLGAT and baselines on all the nine datasets.   
+
+<table><tr><td></td><td colspan="3">Assortative</td><td colspan="6">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>76.5±1.3</td><td>73.6±1.9</td><td>87.5±0.4</td><td>48.5±3.0</td><td>31.5±1.4</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>GCN</td><td>88.2±1.2</td><td>75.7±1.3</td><td>88.4±0.6</td><td>67.6±2.4</td><td>54.9±1.9</td><td>30.3±1.6</td><td>54.2±7.3</td><td>61.1±7.0</td><td>59.6±4.5</td></tr><tr><td>GAT</td><td>88.4±1.4</td><td>76.1±1.0</td><td>87.0±0.3</td><td>65.0±3.7</td><td>51.3±2.5</td><td>29.4±1.2</td><td>56.3±4.3</td><td>57.9±6.1</td><td>57.8±4.3</td></tr><tr><td>Geom-GCN</td><td>85.3</td><td>78.0</td><td>90.1</td><td>60.9</td><td>38.1</td><td>31.6</td><td>60.8</td><td>67.6</td><td>64.1</td></tr><tr><td>Geom-GCN-g</td><td>87.0</td><td>80.6</td><td>90.7</td><td>68.0</td><td>46.0</td><td>32.0</td><td>67.0</td><td>73.1</td><td>69.4</td></tr><tr><td>Geom-GCN-s</td><td>73.3</td><td>72.2</td><td>87.0</td><td>61.6</td><td>38.0</td><td>34.6</td><td>75.4</td><td>73.5</td><td>80.4</td></tr><tr><td>NLMLP</td><td>76.9±1.8</td><td>73.4±1.9</td><td>88.2±0.5</td><td>50.7±2.2</td><td>33.7±1.5</td><td>37.9±1.3</td><td>84.9±5.7</td><td>85.4±3.8</td><td>87.3±4.3</td></tr><tr><td>NLGCN</td><td>88.1±1.0</td><td>75.2±1.4</td><td>89.0±0.5</td><td>70.1±2.9</td><td>59.0±1.2</td><td>31.6±1.0</td><td>57.6±5.5</td><td>65.5±6.6</td><td>60.2±5.3</td></tr><tr><td>NLGAT</td><td>88.5±1.8</td><td>76.2±1.6</td><td>88.2±0.3</td><td>65.7±1.4</td><td>56.8±2.5</td><td>29.5±1.3</td><td>54.7±7.6</td><td>62.6±7.1</td><td>56.9±7.3</td></tr></table>

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+    {
+        "type": "text",
+        "text": "NON-LOCAL GRAPH NEURAL NETWORKS ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Modern graph neural networks (GNNs) learn node embeddings through multilayer local aggregation and achieve great success in applications on assortative graphs. However, tasks on disassortative graphs usually require non-local aggregation. In addition, we find that local aggregation is even harmful for some disassortative graphs. In this work, we propose a simple yet effective non-local aggregation framework with an efficient attention-guided sorting for GNNs. Based on it, we develop various non-local GNNs. We perform thorough experiments to analyze disassortative graph datasets and evaluate our non-local GNNs. Experimental results demonstrate that our non-local GNNs significantly outperform previous state-of-the-art methods on six benchmark datasets of disassortative graphs, in terms of both model performance and efficiency. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) process graphs and map each node to an embedding vector (Zhang et al., 2018b; Wu et al., 2019). These node embeddings can be directly used for node-level applications, such as node classification (Kipf & Welling, 2017) and link prediction (Schutt et al., 2017). ¨ In addition, they can be used to learn the graph representation vector with graph pooling (Ying et al., 2018; Zhang et al., 2018a; Lee et al., 2019; Yuan & Ji, 2020), in order to fit graph-level tasks (Yanardag & Vishwanathan, 2015). Many variants of GNNs have been proposed, such as ChebNets (Defferrard et al., 2016), GCNs (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), GATs (Velickovi ˇ c et al., 2018), LGCN (Gao et al., 2018) and GINs (Xu et al., 2019). Their ´ advantages have been shown on various graph datasets and tasks (Errica et al., 2020). However, these GNNs share a multilayer local aggregation framework, which is similar to convolutional neural networks (CNNs) (LeCun et al., 1998) on grid-like data such as images and texts. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In recent years, the importance of non-local aggregation has been demonstrated in many applications in the field of computer vision (Wang et al., 2018; 2020) and natural language processing (Vaswani et al., 2017). In particular, the attention mechanism has been widely explored to achieve non-local aggregation and capture long-range dependencies from distant locations. Basically, the attention mechanism measures the similarity between every pair of locations and enables information to be communicated among distant but similar locations. In terms of graphs, non-local aggregation is also crucial for disassortative graphs, while previous studies of GNNs focus on assortative graph datasets (Section 2.2). In addition, we find that local aggregation is even harmful for some disassortative graphs (Section 4.3). The recently proposed Geom-GCN (Pei et al., 2020) explores to capture longrange dependencies in disassortative graphs. It contains an attention-like step that computes the Euclidean distance between every pair of nodes. However, this step is computationally prohibitive for large-scale graphs, as the computational complexity is quadratic in the number of nodes. In addition, Geom-GCN employs pre-trained node embeddings (Tenenbaum et al., 2000; Nickel & Kiela, 2017; Ribeiro et al., 2017) that are not task-specific, limiting the effectiveness and flexibility. ",
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+        "type": "text",
+        "text": "In this work, we propose a simple yet effective non-local aggregation framework for GNNs. At the heart of the framework lies an efficient attention-guided sorting, which enables non-local aggregation through classic local aggregation operators in general deep learning. The proposed framework can be flexibly used to augment common GNNs with low computational costs. Based on the framework, we build various efficient non-local GNNs. In addition, we perform detailed analysis on existing disassortative graph datasets, and apply different non-local GNNs accordingly. Experimental results show that our non-local GNNs significantly outperform previous state-of-the-art methods on node classification tasks on six benchmark datasets of disassortative graphs. ",
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+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
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+        "text": "2.1 GRAPH NEURAL NETWORKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We focus on learning the embedding vector for each node through graph neural networks (GNNs). Most existing GNNs are inspired by convolutional neural networks (CNNs) (LeCun et al., 1998) and follow a local aggregation framework. In general, each layer of GNNs scans every node in the graph and aggregates local information from directly connected nodes, i.e., the 1-hop neighbors. ",
+        "bbox": [
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+    {
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+        "text": "Specifically, a common layer of GNNs performs a two-step processing similar to the depthwise separable convolution (Chollet, 2017): spatial aggregation and feature transformation. The first step updates each node embedding using embedding vectors of spatially neighboring nodes. For example, GCNs (Kipf & Welling, 2017) and GATs (Velickovi ˇ c et al., 2018) compute a weighted ´ sum of node embeddings within the 1-hop neighborhood, where weights come from the degree of nodes and the interaction between nodes, respectively. GraphSAGE (Hamilton et al., 2017) applies the max pooling, while GINs (Xu et al., 2019) simply sums the node embeddings. The feature transformation step is similar to the $1 \\times 1$ convolution, where each node embedding vector is mapped into a new feature space through a shared linear transformation (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovi ˇ c et al., 2018) or multilayer perceptron (MLP) (Xu et al., 2019). Different from ´ these studies, LGCN (Gao et al., 2018) explores to directly apply the regular convolution through top- $k$ ranking. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Nevertheless, each layer of these GNNs only aggregates local information within the 1-hop neighborhood. While stacking multiple layers can theoretically enable communication between nodes across the multi-hop neighborhood, the aggregation is essentially local. In addition, deep GNNs usually suffer from the over-smoothing problem (Xu et al., 2018; Li et al., 2018; Chen et al., 2020). ",
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+        "text": "2.2 ASSORTATIVE AND DISASSORTATIVE GRAPHS ",
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+        "text": "There are many kinds of graphs in the literature, such as citation networks (Kipf & Welling, 2017), community networks (Chen et al., 2020), co-occurrence networks (Tang et al., 2009), and webpage linking networks (Rozemberczki et al., 2019). We focus on graph datasets corresponding to the node classification tasks. In particular, we categorize graph datasets into assortative and disassortative ones (Newman, 2002; Ribeiro et al., 2017) according to the node homophily in terms of labels, i.e., how likely nodes with the same label are near each other in the graph. ",
+        "bbox": [
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+        "text": "Assortative graphs refer to those with a high node homophily. Common assortative graph datasets are citation networks and community networks. On the other hand, graphs in disassortative graph datasets contain more nodes that have the same label but are distant from each other. Example disassortative graph datasets are co-occurrence networks and webpage linking networks. ",
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+        "text": "As introduced above, most existing GNNs perform local aggregation only and achieve good performance on assortative graphs (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovi ˇ c et al., 2018; ´ Gao et al., 2018). However, they may fail on disassortative graphs, where informative nodes in the same class tend to be out of the local multi-hop neighborhood and non-local aggregation is needed. Thus, in this work, we explore the non-local GNNs. ",
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+        "text": "2.3 ATTENTION MECHANISM ",
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+        "text": "The attention mechanism (Vaswani et al., 2017) has been widely used in GNNs (Velickovi ˇ c et al., ´ 2018; Gao & Ji, 2019; Knyazev et al., 2019) as well as other deep learning models (Yang et al., 2016; Wang et al., 2018; 2020). A typical attention mechanism takes three groups of vectors as inputs, namely the query vector $q$ , key vectors $( k _ { 1 } , k _ { 2 } , \\ldots , k _ { n } )$ , value vectors $( v _ { 1 } , v _ { 2 } , \\ldots , v _ { n } )$ . Note that key and value vectors have a one-to-one correspondence and can be the same sometimes. The attention mechanism computes the output vector $o$ as ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/b474c40bd326284c924a1abd87c01f1d426d66b77744887dbbf4c5a679d8b040.jpg",
+        "text": "$$\na _ { i } = \\operatorname { A r T E N D } ( q , k _ { i } ) \\in \\mathbb { R } , i = 1 , 2 , \\dotsc , n ; \\quad o = \\sum _ { i } a _ { i } v _ { i } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the ATTEND $( \\cdot )$ function could be any function that outputs a scalar attention score $a _ { i }$ from the interaction between $q$ and $k _ { i }$ , such as dot product (Gao & Ji, 2019) or even a neural network (Velickovi ˇ c et al., 2018). The definition of the three groups of input vectors depends on the ´ models and applications. ",
+        "bbox": [
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+        "page_idx": 1
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Notably, existing GNNs usually use the attention mechanism for local aggregation (Velickovi ˇ c et al., ´ 2018; Gao & Ji, 2019). Specifically, when aggregating information for node $v$ , the query vector is the embedding vector of $v$ while the key and value vectors come from node embeddings of $v$ ’s directly connected nodes. And the process is iterated for each $v \\in V$ . It is worth noting that the attention mechanism can be easily extended for non-local aggregation (Wang et al., 2018; 2020), by letting the key and value vectors correspond to all the nodes in the graph when aggregating information for each node. However, it is computationally prohibitive given large-scale graphs, as iterating it for each node in a graph of $n$ nodes requires $O ( n ^ { 2 } )$ time. In this work, we propose a novel non-local aggregation method that only requires $O ( n \\log { n } )$ time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 THE PROPOSED METHOD ",
+        "text_level": 1,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3.1 NON-LOCAL AGGREGATION WITH ATTENTION-GUIDED SORTING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider a graph $\\mathcal { G } = ( V , E )$ , where $V$ is the set of nodes and $E$ is the set of edges. Each edge $e \\in E$ connects two nodes so that $E \\subseteq V \\times V$ . Each node $v \\in V$ has a node feature vector $x _ { v } \\in \\bar { \\mathbb { R } ^ { d } }$ . The $k$ -hop neighborhood of $v$ refers to the set of nodes $\\mathcal { N } _ { k } ( v )$ that can reach $v$ within $k$ edges. For example, the set of $v$ ’s directly connected nodes is its 1-hop neighborhood $\\mathcal { N } _ { 1 } ( v )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our proposed non-local aggregation framework is composed of three steps, namely local embedding, attention-guided sorting, and non-local aggregation. In the following, we describe them one by one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Local Embedding: Our proposed framework is built upon a local embedding step that extracts local node embeddings from the node feature vectors. The local embedding step can be as simple as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8a11e15476544a4afed76cb79fa8c6dafd341e9b634b58fdcb0eff568ac0a465.jpg",
+        "text": "$$\nz _ { v } = \\mathbf { M } \\mathbf { L } \\mathbf { P } ( x _ { v } ) \\in \\mathbb { R } ^ { f } , \\forall v \\in V .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The $\\mathrm { \\mathbf { M L P } ( \\cdot ) }$ function is a multilayer perceptron (MLP), and $f$ is the dimension of the local node embedding $z _ { v }$ . Note that the $\\mathrm { \\mathbf { M L P } ( \\cdot ) }$ function is shared across all the nodes in the graph. Applying MLP only takes the node itself into consideration without aggregating information from the neighborhood. This property is very important on some disassortative graphs, as shown in Section 4.3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On the other hand, graph neural networks (GNNs) can be used as the local embedding step as well, so that our proposed framework can be easily employed to augment existing GNNs. As introduced in Section 2.1, modern GNNs perform multilayer local aggregation. Typically, for each node, one layer of a GNN aggregates information from its 1-hop neighborhood. Stacking $L$ such local aggregation layers allows each node to access information that is $L$ hops away. To be specific, the $\\ell \\cdot$ -th layer of a $L$ -layer GNN $( \\ell = 1 , 2 , \\ldots , L )$ can be described as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b48de0bf328f73f117e87989ac12115c538c410f8dc5fa41f92845417b7b4915.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { v } ^ { ( \\ell ) } = { \\mathrm { T R A N S F O R M } } ^ { ( \\ell ) } \\left( \\mathbf { A G G R E G A T E } ^ { ( \\ell ) } \\left( \\{ z _ { u } ^ { ( \\ell - 1 ) } : u \\in \\mathcal { N } _ { 1 } ( v ) \\cup v \\} \\right) \\right) \\in \\mathbb { R } ^ { f } , \\forall v \\in V , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            199,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where z(0v $z _ { v } ^ { ( 0 ) } = x _ { v }$ , and $z _ { v } ~ = ~ z _ { v } ^ { ( L ) }$ represents the local node embedding. The AGGREGATE $^ { ( \\ell ) } ( \\cdot )$ and TR $\\mathsf { A N S F O R M } ^ { ( \\ell ) } ( \\cdot )$ functions represent the spatial aggregation and feature transformation step introduced in Section 2.1, respectively. With the above framework, GNNs can capture the node feature information from nodes within a local neighborhood as well as the structural information. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When either MLP or GNNs is used as the local embedding step, the local node embedding $z _ { v }$ only contains local information of a node $v$ . However, $z _ { v }$ can be used to guide non-local aggregation, as distant but informative nodes are likely to have similar node features and local structures. Based on this intuition, we propose the attention-guided sorting to enable the non-local aggregation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Attention-Guided Sorting: The basic idea of the attention-guided sorting is to learn an ordering of nodes, where distant but informative nodes are put near each other. Specifically, given the local node embedding $z _ { v }$ obtained through the local embedding step, we compute one set of attention scores by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a591c3aadd5b33363ac4f7e6f75381ab26f31c2f70d3e2ebe7c167c8b9f831ea.jpg",
+        "text": "$$\na _ { v } = \\mathrm { A T T E N D } ( c , z _ { v } ) \\in \\mathbb { R } , \\forall v \\in V ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $c$ is a calibration vector that is randomly initialized and jointly learned during training (Yang et al., 2016). In this attention operator, $c$ serves as the query vector and $z _ { v }$ are the key vectors. In addition, we also treat $z _ { v }$ as the value vectors. However, unlike the attention mechanism introduced in Section 2.3, we use the attention scores to sort the value vectors instead of computing a weighted sum to aggregating them. Note that originally there is no ordering among nodes in a graph. To be specific, as $a _ { v }$ and $z _ { v }$ have one-to-one correspondence through Equation (4), sorting the attention scores in non-decreasing order into $( a _ { 1 } , a _ { 2 } , \\ldots , a _ { n } )$ provides an ordering among nodes, where $n =$ $| V |$ is the number of nodes in the graph. The resulting sequence of local node embeddings can be denoted as $( z _ { 1 } , z _ { 2 } , \\ldots , z _ { n } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The attention process in Equation (4) can be also understood as a projection of local node embeddings onto a 1-dimensional space. The projection depends on the concrete ATTEND $( \\cdot )$ function and the calibration vector $c$ . As indicated by its name, the calibration vector $c$ is used to calibrate the 1-dimensional space, in order to push distant but informative nodes close to each other in this space. This goal is fulfilled through the following non-local aggregation step and the training of the calibration vector $c$ , as demonstrated below. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Non-Local Aggregation: We point out that, with the attention-guided sorting, the non-local aggregation can be achieved by convolution, the most common local aggregation operator in deep learning. Specifically, given the sorted sequence of local node embeddings $( z _ { 1 } , z _ { 2 } , \\ldots , z _ { n } )$ , we compute ",
+        "bbox": [
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+            306,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8fb6c1a92814a03de58cfabb6055433cd5f843e1226ab1be7152e5ddde52dd24.jpg",
+        "text": "$$\n( \\hat { z } _ { 1 } , \\hat { z } _ { 2 } , \\ldots , \\hat { z } _ { n } ) = \\mathbf { C } \\mathbf { O } \\mathbf { N } \\mathbf { V } ( z _ { 1 } , z _ { 2 } , \\ldots , z _ { n } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            362,
+            635,
+            378
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the $\\mathrm { C o N V } ( \\cdot )$ function represents a 1D convolution with appropriate padding. Note that the $\\mathrm { C o N V } ( \\cdot )$ function can be replaced by a 1D convolutional neural network as long as the number of input and output vectors remains the same. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To see how the $\\mathrm { C o N V } ( \\cdot )$ function performs non-local aggregation with the attention-guided sorting, we take an example where the $\\mathrm { C o N V } ( \\cdot )$ function is a 1D convolution of kernel size $2 s + 1$ . In this case, $\\hat { z } _ { i }$ is computed from $( z _ { i + s } , \\dots , z _ { i - s } )$ , corresponding to the receptive field of the $\\operatorname { C o n v } ( \\cdot )$ function. As a result, if the attention-guided sorting leads to $( z _ { i + s } , \\dots , z _ { i - s } )$ containing nodes that are distant but informative to $z _ { i }$ , the output $\\hat { z } _ { i }$ aggregates non-local information. Another view is that we can consider the attention-guided sorting as re-connects nodes in the graph, where $( z _ { i + s } , \\dots , z _ { i - s } )$ can be treated as the 1-hop neighborhood of $z _ { i }$ . After the $\\operatorname { C o n v } ( \\cdot )$ function, $\\hat { z } _ { i }$ and $z _ { i }$ are concatenated as the input to a classifier to predict the label of the corresponding node, where both non-local and local dependencies can be captured. In order to enable the end-to-end training of the calibration vector $c$ , we modify Equation (5) into ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/391144d88014d2e90f025ee9f1a413f852cc55d85da7c08cadd98bc9da1ebfda.jpg",
+        "text": "$$\n( \\hat { z } _ { 1 } , \\hat { z } _ { 2 } , \\ldots , \\hat { z } _ { n } ) = \\mathbf { C } \\mathbf { O } \\mathbf { N } \\mathbf { v } ( a _ { 1 } z _ { 1 } , a _ { 2 } z _ { 2 } , \\ldots , a _ { n } z _ { n } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            577,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we multiply the attention score with the corresponding local node embedding. As a result, the calibration vector $c$ receives gradients through the attention scores during training. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The remaining question is how to make sure that the attention-guided sorting pushes distant but informative nodes together. The short answer is that it is not necessary to guarantee this, as the requirement of non-local aggregation depends on the concrete graphs. In fact, our proposed framework grants GNNs the ability of non-local aggregation but lets the end-to-end training process determine whether to use non-local information. The back-propagation from the supervised loss will tune the calibration vector $c$ and encourage $\\hat { z } _ { i }$ to capture useful information that is not encoded by $z _ { i }$ . In the case of disassortative graphs, $\\hat { z } _ { i }$ usually needs to aggregate information from distant but informative nodes. Hence, the calibration vector $c$ tends to arrange the attention-guided sorting to put distant but informative nodes together, as demonstrated experimentally in Section 4.5. On the other hand, nodes within the local neighborhood are usually much more informative than distant nodes in assortative graphs. In this situation, $\\hat { z } _ { i }$ may simply perform local aggregation that is similar to GNNs. ",
+        "bbox": [
+            173,
+            636,
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+            790
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Section 4, we demonstrate the effectiveness of our proposed non-local aggregation framework on six disassortative graph datasets. In particular, we achieve the state-of-the-art performance on all the datasets with significant improvements over previous methods. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 TIME COMPLEXITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            426,
+            869
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We perform theoretical analysis of the time complexity of our proposed framework. As discussed in Section 2.3, using the attention mechanism (Vaswani et al., 2017; Wang et al., 2018; 2020) to achieve non-local aggregation requires $O ( n ^ { 2 } )$ time for a graph of $n$ nodes. Essentially, the $O ( n ^ { 2 } )$ ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "time complexity is due to the fact that the ATTEND(·) function needs to be computed between every pair of nodes. In particular, the recently proposed Geom-GCN (Pei et al., 2020) contains a similar non-local aggregation step. For each $v \\in V$ , Geom-GCN finds the set of nodes from which the Euclidean distance to $v$ is less than a pre-defined number, where the Euclidean distance between every pair of nodes needs to be computed. As the computation of the the Euclidean distance between two nodes can be understood as the ATTEND(·) function, Geom-GCN has at least $O ( n ^ { 2 } )$ time complexity. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In contrast, our proposed non-local aggregation framework requires only $O ( n \\log n )$ time. To see this, note that the ATTEND(·) function in Equation (4) only needs to be computed once, instead of iterating it for each node. As a result, computing the attention scores only takes $O ( n )$ time. Therefore, the time complexity of sorting, i.e. ${ \\bar { O } } ( n \\log n )$ , dominates the total time complexity of our proposed framework. In Section 4.6, we compare the real running time on different datasets among common GNNs, Geom-GCN, and our non-local GNNs as introduced in the next section. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 EFFICIENT NON-LOCAL GRAPH NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            311,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We apply our proposed non-local aggregation framework to build efficient non-local GNNs. Recall that our proposed framework starts with the local embedding step, followed by the attention-guided sorting and the non-local aggregation step. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In particular, the local embedding step can be implemented by either MLP or common GNNs, such as GCNs (Kipf & Welling, 2017) or GATs (Velickovi ˇ c et al., 2018). MLP extracts the local node ´ embedding only from the node feature vector and excludes the information from nodes within the local neighborhood. This property can be helpful on some disassortative graphs, where nodes within the local neighborhood provide more noises than useful information. On other disassortative graphs, informative nodes locate in both local neighborhood and distant locations. In this case, GNNs are more suitable as the local embedding step. Depending on the disassortative graphs in hand, we build different non-local GNNs with either MLP or GNNs as the local embedding step. In Section 4.3, we show that these two categories of disassortative graphs can be distinguished through simple experiments, where we apply different non-local GNNs accordingly. Specifically, the number of layers is set to 2 for both MLP and GNNs. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In terms of the attention-guided sorting, we only need to specify the ATTEND $( \\cdot )$ function in Equation (4). In order to make it as efficient as possible, we choose the ATTEND $( \\cdot )$ function as ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/faaf0a0ebb8be6bab23005493566d1679dbabc28271967dfa2f0448ccaea0578.jpg",
+        "text": "$$\n\\begin{array} { r } { a _ { v } = \\mathrm { A T T E N D } ( c , z _ { v } ) = c ^ { T } z _ { v } \\in \\mathbb { R } , \\forall v \\in V , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            587,
+            645,
+            606
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $c$ is part of the training parameters, as described in Section 3.1. ",
+        "bbox": [
+            176,
+            614,
+            627,
+            631
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With the attention-guided sorting, we can implement the non-local aggregation step through convolution, as explained in Section 3.1 and shown in Equation (6). Specifically, $\\mathrm { C o N V } ( \\cdot )$ function is set as a 2-layer convolutional neural network composed of two 1D convolutions. The kernel size is set to 3 or 5 depending on the datasets. The activation function is ReLU (Krizhevsky et al., 2012). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, we use a linear classifier that takes the concatenation of $\\hat { z } _ { i }$ and $z _ { i }$ as inputs and makes prediction for the corresponding node. Depending on the local embedding step, we build three efficient non-local GNNs, namely non-local MLP (NLMLP), non-local GCN (NLGCN), and nonlocal GAT (NLGAT). The models can be end-to-end trained with the classification loss. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments on six disassortative graph datasets (Rozemberczki et al., 2019; Tang et al., 2009; Pei et al., 2020) (Chameleon, Squirrel, Actor, Cornell, Texas, Wisconsin) and three assortative graph datasets (Sen et al., 2008) (Cora, Citeseer, Pubmed). These datasets are commonly used to evaluate GNNs on node classification tasks (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018; Gao ´ et al., 2018; Pei et al., 2020). We provide detailed descriptions of disassortative graph datasets in Appendix A.1. In order to distinguish assortative and disassortative graph datasets, Pei et al. (2020) ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "propose a metric to measure the homophily of a graph $\\mathcal { G }$ , defined as ",
+        "bbox": [
+            173,
+            103,
+            617,
+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f51483db18b69e45ee5cb1bdf920e4275999bf93f4b55e06b11abc398b6a9e2d.jpg",
+        "text": "$$\nH ( \\mathcal { G } ) = \\frac { 1 } { | V | } \\sum _ { v \\in V } \\frac { \\mathrm { N u m b e r ~ o f ~ } v ^ { \\circ } \\mathrm { ~ s d i r e c t l y ~ c o n n e c t e d ~ n o d e s ~ w h o ~ h a v e ~ t h e ~ s a m e ~ l a b e l ~ a s ~ } v } { \\mathrm { N u m b e r ~ o f ~ } v ^ { \\circ } \\mathrm { ~ s d i r e c t l y ~ c o n n e c t e d ~ n o d e s } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            122,
+            784,
+            164
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, a large $H ( { \\mathcal { G } } )$ indicates an assortative graph, and vice versa. The $H ( { \\mathcal { G } } )$ and other statistics are summarized in Table 1. ",
+        "bbox": [
+            174,
+            170,
+            825,
+            199
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d420ef39c5c73af3ad974fc65b9cc1a7dfe750cf1d8cdc0dc1ef6366eb5be15.jpg",
+        "table_caption": [
+            "Table 1: Statistics of the nine datasets used in our experiments. The definition of $H ( { \\mathcal { G } } )$ is provided in Section 4.1. $H ( { \\mathcal { G } } )$ can be used to distinguish assortative and disassortative graph datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"5\">Assortative</td><td colspan=\"5\">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>H(G)</td><td>0.83</td><td>0.71</td><td>0.79</td><td>0.25</td><td>0.22</td><td>0.24</td><td>0.11</td><td>0.06</td><td>0.16</td></tr><tr><td>#Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>2277</td><td>5201</td><td>7600</td><td>183</td><td>183</td><td>251</td></tr><tr><td>#Edges</td><td>5429</td><td>4732</td><td>44338</td><td>36101</td><td>217073</td><td>33544</td><td>295</td><td>309</td><td>499</td></tr><tr><td>#Features</td><td>1433</td><td>3703</td><td>500</td><td>2325</td><td>2089</td><td>931</td><td>1703</td><td>1703</td><td>1703</td></tr><tr><td>#Classes</td><td>7</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>",
+        "bbox": [
+            176,
+            233,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we focus on comparing the model performance on disassortative graph datasets, in order to demonstrate the effectiveness of our non-local aggregation framework. The performances on assortative graph datasets are provided for reference, indicating that the proposed framework will not hurt the performance when non-local aggregation is not strongly desired. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 BASELINES",
+        "text_level": 1,
+        "bbox": [
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+            294,
+            426
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare our proposed non-local MLP (NLMLP), non-local GCN (NLGCN), and non-local GAT (NLGAT) with various baselines: (1) MLP is the simplest deep learning model. It makes prediction solely based on the node feature vectors, without aggregating any local or non-local information. (2) GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018) are the most common ´ GNNs. As introduced in Section 2.1, they only perform local aggregation. (3) Geom-GCN (Pei et al., 2020) is a recently proposed GNN that can capture long-range dependencies. It is the current stateof-the-art model on several disassortative graph datasets. Geom-GCN requires the use of different node embedding methods, such as Isomap (Tenenbaum et al., 2000), Poincare (Nickel & Kiela, 2017), and struc2vec (Ribeiro et al., 2017). We simply report the best results from Pei et al. (2020) for Geom-GCN and the following two variants without specifying the node embedding method. (4) Geom-GCN- $\\mathbf { g }$ (Pei et al., 2020) is a variant of Geom-GCN that performs local aggregation only. It is similar to common GNNs. (5) Geom-GCN-s (Pei et al., 2020) is a variant of Geom-GCN that does not force local aggregation. The designed functionality is similar to our NLMLP. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We implement MLP, GCN, GAT, and our methods using Pytorch (Adam et al., 2017) and Pytorch Geometric (Fey & Lenssen, 2019). As has been discussed1, in fair settings, the results of GCN and GAT differ from those in Pei et al. (2020). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "On each dataset, we follow Pei et al. (2020) and randomly split nodes of each class into $60 \\%$ , $20 \\%$ , and $20 \\%$ for training, validation, and testing. The experiments are repeatedly run 10 times with different random splits and the average test accuracy over these 10 runs are reported. Testing is performed when validation accuracy achieves maximum on each run. Apart from the details specified in Section 3.3, we tune the following hyperparameters individually for our proposed models: (1) the number of hidden unit $\\in \\{ 1 6 , 4 8 , 9 6 \\}$ , (2) dropout rate $\\in \\{ 0 , 0 . 5 , 0 . 8 \\}$ , (3) weight decay $\\in \\{ 0 , 5 \\mathrm { e } { - } 4$ , 5e-5, 5e-6}, and (4) learning rate $\\in \\{ 0 . 0 1 , 0 . 0 5 \\}$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 ANALYSIS OF DISASSORTATIVE GRAPH DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+            790,
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+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 3.3, the disassortative graph datasets can be divided into two categories. Nodes within the local neighborhood provide more noises than useful information in disassortative graphs belonging to the first category. Therefore, local aggregation should be avoided in models on such disassortative graphs. As for the second category, informative nodes locate in both local neighborhood and distant locations. Intuitively, a graph with lower $H ( { \\mathcal { G } } )$ is more likely to be in the first category. However, it is not an accurate way to determine the two categories. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Knowing the exact category of a disassortative graph is crucial, as we need to apply non-local GNNs accordingly. As analyzed above, the key difference lies in whether the local aggregation is useful. Hence, we can distinguish two categories of disassortative graphs by comparing the performance between MLP and common GNNs (GCN, GAT) on each of the six disassortative graph datasets. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/86d33a506352f46b8f1c13c8e951e67662986ff2ba86c3cd041ca784ee52f2c2.jpg",
+        "table_caption": [
+            "Table 2: Comparisons between MLP and common GNNs. These analytical experiments are used to determine the two categories of disassortative graph datasets, as introduced in Section 4.3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"5\">Assortative</td><td colspan=\"5\">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>76.5±1.3</td><td>73.6±1.9</td><td>87.5±0.4</td><td>48.5±3.0</td><td>31.5±1.4</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>GCN</td><td>88.2±1.2</td><td>75.7±1.3</td><td>88.4±0.6</td><td>67.6±2.4</td><td>54.9±1.9</td><td>30.3±1.6</td><td>54.2±7.3</td><td>61.1±7.0</td><td>59.6±4.5</td></tr><tr><td>GAT</td><td>88.4±1.4</td><td>76.1±1.0</td><td>87.0±0.3</td><td>65.0±3.7</td><td>51.3±2.5</td><td>29.4±1.2</td><td>56.3±4.3</td><td>57.9±6.1</td><td>57.8±4.3</td></tr></table>",
+        "bbox": [
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+            193,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results are summarized in Table 2. We can see that Actor, Cornell, Texas, and Wisconsin fall into the first category, while Chameleon and Squirrel belong to the second category. We add the performance on assortative graph datasets for reference, where the local aggregation is effective so that GNNs tend to outperform MLP. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 COMPARISONS WITH BASELINES",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "According to the insights from Section 4.3, we apply different non-local GNNs according to the category of disassortative graph datasets, and make comparisons with corresponding baselines. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we employ NLMLP on Actor, Cornell, Texas, and Wisconsin. The corresponding baselines are MLP, Geom-GCN, and Geom-GCN-s, as Table 2 has shown that GCN and GAT perform much worse than MLP on these datasets. And Geom-GCN- $\\mathbf { g }$ is similar to GCN and has worse performance than Geom-GCN-s, which is shown in Ap",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/24ef7297a2f91c7539a229afe0944011ca88548bcce92cbb567d1ae519ac7eda.jpg",
+        "table_caption": [
+            "Table 3: Comparisons between our NLMLP and strong baselines on the four disassortative graph datasets belonging to the first category as defined in Section 4.3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>Geom-GCN</td><td>31.6</td><td>60.8</td><td>67.6</td><td>64.1</td></tr><tr><td>Geom-GCN-s</td><td>34.6</td><td>75.4</td><td>73.5</td><td>80.4</td></tr><tr><td>NLMLP</td><td>37.9±1.3</td><td>84.9±5.7</td><td>85.4±3.8</td><td>87.3±4.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "pendix A.2. The comparison results are reported in Table 3. While Geom-GCN-s are the previous state-of-the-art GNNs on these datasets (Pei et al., 2020), we find that MLP consistently outperforms Geom-GCN-s by large margins. In particular, although Geom-GCN-s does not explicitly perform local aggregation, it is still outperformed by MLP. A possible explanation is that Geom-GCN-s uses pre-trained node embeddings, which aggregates information from the local neighborhood implicitly. In contrast, our NLMLP is built upon MLP with the proposed non-local aggregation framework, which excludes the local noises and collects useful information from non-local informative nodes. The NLMLP sets the new state-of-the-art performance on these disassortative graph datasets. ",
+        "bbox": [
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+            530,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On Chameleon and Squirrel that belong to the second category of disassortative graph datasets, we apply NLGCN and NLGAT accordingly. The baselines are GCN, GAT, Geom-GCN, and Geom-GCN- $\\mathbf { g }$ . On these datasets, these baselines that explicitly perform local aggregation show advantages over MLP and Geom-GCN-s, as shown in Appendix A.2. As shown in Table 4, our proposed NLGCN achieves the best performance on both datasets. In addition, it is worth noting that our NLGCN and NLGAT are built upon GCN and GAT, respectively. They show improvements over their counterparts, which indicates that the advantages of our proposed non-local aggregation framework are general for common GNNs. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/a6a5374d67e9f62f6f1b60f34085100edca0423710e09da059800500bf6487fd.jpg",
+        "table_caption": [
+            "Table 4: Comparisons between our NLGCN, NLGAT and strong baselines on the two disassortative graph datasets belonging to the second category as defined in Section 4.3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>GCN</td><td>67.6±2.4</td><td>54.9±1.9</td></tr><tr><td>GAT</td><td>65.0±3.7</td><td>51.3±2.5</td></tr><tr><td>Geom-GCN</td><td>60.9</td><td>38.1</td></tr><tr><td>Geom-GCN-g</td><td>68.0</td><td>46.0</td></tr><tr><td>NLGCN</td><td>70.1±2.9</td><td>59.0±1.2</td></tr><tr><td>NLGAT</td><td>65.7±1.4</td><td>56.8±2.5</td></tr></table>",
+        "bbox": [
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+            715,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We provide the results of all the models on all datasets in Appendix A.2 for reference. ",
+        "bbox": [
+            176,
+            835,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 ANALYSIS OF THE ATTENTION-GUIDED SORTING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            868,
+            557,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We analyze the results of the attention-guided sorting in our proposed framework, in order to show that our non-local GNNs indeed perform non-local aggregation. ",
+        "bbox": [
+            176,
+            895,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Suppose the attention-guided sorting leads to the sorted sequence $( z _ { 1 } , z _ { 2 } , \\ldots , z _ { n } )$ , which goes through a convolution or CNN into $\\left( \\hat { z } _ { 1 } , \\hat { z } _ { 2 } , \\dots , \\hat { z } _ { n } \\right)$ . As discussed in Section 3.1, we can consider the sequence $( z _ { 1 } , z _ { 2 } , \\ldots , z _ { n } )$ as a re-connected graph $\\hat { \\mathcal G }$ , where we treat nodes within the receptive field of $\\hat { z } _ { i }$ as directly connected to $z _ { i }$ , i.e. $z _ { i }$ ’s 1-hop neighborhood. The information within this new 1-hop neighborhood will be aggregated. If our non-local GNNs indeed perform non-local aggregation, the homophily of the re-connected graph should be larger than the original graph. Therefore, we compute $H ( \\hat { \\mathcal G } )$ for each dataset to verify this statement. Following Section 4.4, we apply NLMLP on Actor, Cornell, Texas, and Wisconsin and NLGCN on Chameleon and Squirrel. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            219
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 compares $H ( \\hat { \\mathcal G } )$ with $H ( { \\mathcal { G } } )$ for each dataset. We can observe that $H ( \\hat { \\mathcal G } )$ is much larger than $H ( { \\mathcal { G } } )$ , indicating that distant but informative nodes are near each other in the re-connected graph $\\hat { \\mathcal G }$ . We also provide the visualizations of the sorted sequence for Cornell and Texas. We can see that nodes with the same label tend to be clustered together. These facts indicate that our non-local GNNs perform non-local aggregation with the attention-guided sorting. ",
+        "bbox": [
+            173,
+            227,
+            825,
+            301
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce6b80695989c7253e5e2d1b0da6b98cab5728cae3949ffc7571a9f7cafbd3a1.jpg",
+        "image_caption": [
+            "Figure 1: (a) Comparisons of the homophily between the original graph and the re-connected graph given by our NLGCN on Chameleon and Squirrel. (b) Comparisons of the homophily between the original graph and the re-connected graph given by our NLMLP on Actor, Cornell, Texas, and Wisconsin. (c) Visualization of sorted node sequence after the attention-guided sorting for Cornell and Texas. The colors denote node labels. Details are explained in Section 4.5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            314,
+            825,
+            434
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.6 EFFICIENCY COMPARISONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            405,
+            545
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As analyzed in Section 3.2, our proposed nonlocal aggregation framework is more efficient than previous methods based on the original attention mechanism, such as Geom-GCN (Pei et al., 2020). Concretely, our method requires only $O ( n \\log n )$ computation time in contrast to $O ( n ^ { 2 } )$ . In this section, we compare the real running time to verify our analysis. Specifi",
+        "bbox": [
+            174,
+            556,
+            482,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/064d517a71a9c49f24a37bb7d6b59187e5303dcc0736c9198398ea0a6721eea1.jpg",
+        "table_caption": [
+            "Table 5: Comparisons in terms of real running time (milliseconds). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>GCN</td><td>22.2 (1.0×)</td><td>14.3 (1.0×)</td></tr><tr><td>GAT</td><td>33.2 (1.5x)</td><td>163.3 (11.4×)</td></tr><tr><td>Geom-GCN</td><td>3615.0 (163.1×)</td><td>10430.0 (727.3×)</td></tr><tr><td>NLGCN</td><td>26.3 (1.2×)</td><td>39.6 (2.8×)</td></tr></table>",
+        "bbox": [
+            498,
+            585,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "cally, we compare NLGCN with Geom-GCN as well as GCN and GAT. For Geom-GCN, we use the code provided in Pei et al. (2020). Each model is trained for 500 epochs on each dataset and the average training time per epoch is reported. ",
+        "bbox": [
+            176,
+            667,
+            823,
+            710
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Table 5. Although our NLGCN is built upon GCN, it is just slightly slower than GCN and faster than GAT, showing the efficiency of our non-local aggregation framework. On the other hand, Geom-GCN is significantly slower due to the fact that it has $\\bar { O ( n ^ { 2 } ) }$ time complexity. ",
+        "bbox": [
+            176,
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+            760
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            780,
+            318,
+            796
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a simple yet effective non-local aggregation framework for GNNs. The core of the framework is an efficient attention-guided sorting, which enables non-local aggregation through convolution. The proposed framework can be easily used to build non-local GNNs with low computational costs. We perform thorough experiments on node classification tasks to evaluate our proposed method. In particular, we experimentally analyze existing disassortative graph datasets and apply different non-local GNNs accordingly. The results show that our non-local GNNs significantly outperform previous state-of-the-art methods on six benchmark datasets of disassortative graphs, in terms of both accuracy and speed. ",
+        "bbox": [
+            174,
+            811,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            285,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Paszke Adam, Gross Sam, Chintala Soumith, Chanan Gregory, Yang Edward, D Zachary, Lin Zeming, Desmaison Alban, Antiga Luca, and Lerer Adam. Automatic differentiation in pytorch. In Proceedings of Neural Information Processing Systems Autodiff Workshop, 2017. ",
+        "bbox": [
+            174,
+            126,
+            821,
+            170
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Deli Chen, Yankai Lin, Wei Li, Peng Li, Jie Zhou, and Xu Sun. Measuring and relieving the oversmoothing problem for graph neural networks from the topological view. In Thirty-Fourth AAAI Conference on Artificial Intelligence, 2020. ",
+        "bbox": [
+            176,
+            178,
+            821,
+            222
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Franc¸ois Chollet. Xception: Deep learning with depthwise separable convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1251–1258, 2017. ",
+        "bbox": [
+            173,
+            231,
+            821,
+            261
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks ¨ on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pp. 3844–3852, 2016. ",
+        "bbox": [
+            173,
+            268,
+            826,
+            313
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. In International Conference on Learning Representations, 2020. ",
+        "bbox": [
+            173,
+            321,
+            825,
+            364
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019. ",
+        "bbox": [
+            171,
+            375,
+            821,
+            405
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hongyang Gao and Shuiwang Ji. Graph representation learning via hard and channel-wise attention networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 741–749, 2019. ",
+        "bbox": [
+            174,
+            414,
+            823,
+            457
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hongyang Gao, Zhengyang Wang, and Shuiwang Ji. Large-scale learnable graph convolutional networks. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1416–1424, 2018. ",
+        "bbox": [
+            173,
+            465,
+            823,
+            510
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pp. 1024–1034, 2017. ",
+        "bbox": [
+            173,
+            518,
+            820,
+            549
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations, 2017. ",
+        "bbox": [
+            174,
+            556,
+            820,
+            587
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Boris Knyazev, Graham W Taylor, and Mohamed Amer. Understanding attention and generalization in graph neural networks. In Advances in Neural Information Processing Systems, pp. 4204–4214, 2019. ",
+        "bbox": [
+            174,
+            595,
+            825,
+            638
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105, 2012. ",
+        "bbox": [
+            174,
+            648,
+            825,
+            691
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. ",
+        "bbox": [
+            171,
+            700,
+            823,
+            731
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In International Conference on Machine Learning, pp. 3734–3743, 2019. ",
+        "bbox": [
+            171,
+            739,
+            823,
+            770
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018. ",
+        "bbox": [
+            173,
+            779,
+            823,
+            809
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mark EJ Newman. Assortative mixing in networks. Physical review letters, 89(20):208701, 2002. ",
+        "bbox": [
+            173,
+            818,
+            816,
+            833
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Maximillian Nickel and Douwe Kiela. Poincare embeddings for learning hierarchical representa- ´ tions. In Advances in Neural Information Processing Systems, pp. 6338–6347, 2017. ",
+        "bbox": [
+            171,
+            842,
+            823,
+            872
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-GCN: Geometric graph convolutional networks. In International Conference on Learning Representations, 2020. ",
+        "bbox": [
+            174,
+            881,
+            823,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Leonardo FR Ribeiro, Pedro HP Saverese, and Daniel R Figueiredo. struc2vec: Learning node representations from structural identity. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 385–394, 2017. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Benedek Rozemberczki, Carl Allen, and Rik Sarkar. Multi-scale attributed node embedding. arXiv preprint arXiv:1909.13021, 2019. ",
+        "bbox": [
+            171,
+            156,
+            823,
+            185
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kristof Schutt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre ¨ Tkatchenko, and Klaus-Robert Muller. Schnet: A continuous-filter convolutional neural network ¨ for modeling quantum interactions. In Advances in Neural Information Processing Systems, pp. 991–1001, 2017. ",
+        "bbox": [
+            173,
+            195,
+            825,
+            251
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93–93, 2008. ",
+        "bbox": [
+            173,
+            262,
+            823,
+            291
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jie Tang, Jimeng Sun, Chi Wang, and Zi Yang. Social influence analysis in large-scale networks. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 807–816, 2009. ",
+        "bbox": [
+            174,
+            301,
+            821,
+            343
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319–2323, 2000. ",
+        "bbox": [
+            169,
+            354,
+            823,
+            383
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017. ",
+        "bbox": [
+            174,
+            392,
+            823,
+            436
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. In International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            174,
+            445,
+            825,
+            488
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7794–7803, 2018. ",
+        "bbox": [
+            173,
+            500,
+            826,
+            541
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhengyang Wang, Na Zou, Dinggang Shen, and Shuiwang Ji. Non-local U-Nets for biomedical image segmentation. In Thirty-Fourth AAAI Conference on Artificial Intelligence, 2020. ",
+        "bbox": [
+            171,
+            551,
+            823,
+            582
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019. ",
+        "bbox": [
+            171,
+            590,
+            823,
+            621
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In International Conference on Machine Learning, pp. 5449–5458, 2018. ",
+        "bbox": [
+            176,
+            631,
+            823,
+            672
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            173,
+            683,
+            821,
+            712
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Pinar Yanardag and SVN Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1365–1374. ACM, 2015. ",
+        "bbox": [
+            173,
+            722,
+            825,
+            765
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zichao Yang, Diyi Yang, Chris Dyer, Xiaodong He, Alex Smola, and Eduard Hovy. Hierarchical attention networks for document classification. In Proceedings of the 2016 Conference of the North American chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 1480–1489, 2016. ",
+        "bbox": [
+            173,
+            775,
+            825,
+            832
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems, pp. 4800–4810, 2018. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            885
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hao Yuan and Shuiwang Ji. StructPool: Structured graph pooling via conditional random fields. In International Conference on Learning Representations, 2020. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018a. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Si Zhang, Hanghang Tong, Jiejun Xu, and Ross Maciejewski. Graph convolutional networks: Algorithms, applications and open challenges. In International Conference on Computational Social Networks, pp. 79–91. Springer, 2018b. ",
+        "bbox": [
+            176,
+            155,
+            825,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DETAILS OF DISASSORTATIVE GRAPH DATASETS ",
+        "bbox": [
+            174,
+            133,
+            558,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here are the details of disassortative graph datasets used in our experiments: ",
+        "bbox": [
+            174,
+            160,
+            671,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Chameleon and Squirrel are Wikipedia networks (Rozemberczki et al., 2019) where nodes represent web pages from Wikipedia and edges indicate mutual links between pages. Node feature vectors are bag-of-word representation of informative nouns in the corresponding pages. Each node is labeled with one of five classes according to the number of the average monthly traffic of the web page.   \n• Actor is an actor co-occurrence network, where nodes denote actors and edges indicate cooccurrence on the same web page from Wikipedia. It is extracted from the film-directoractor-writer network proposed by Tang et al. (Tang et al., 2009). Node feature vectors are bag-of-word representation of keywords in the actors’ Wikipedia pages. Each node is labeled with one of five classes according to the topic of the actor’s Wikipedia page. Cornell, Texas, and Wisconsin come from the WebKB dataset collected by Carnegie Mellon University. Nodes represent web pages and edges denote hyperlinks between them. Node feature vectors are bag-of-word representation of the corresponding web pages. Each node is labeled with one of student, project, course, staff, and faculty. ",
+        "bbox": [
+            214,
+            185,
+            825,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 FULL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            407,
+            437,
+            421
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/e295d12ff87ce3272ad2fcb82c45c6b1c469ffb07c56ed89a9873332edefbc72.jpg",
+        "table_caption": [
+            "Table 6: Comparisons between our NLMLP, NLGCN, NLGAT and baselines on all the nine datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Assortative</td><td colspan=\"6\">Disassortative</td></tr><tr><td>Datasets</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Chameleon</td><td>Squirrel</td><td>Actor</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td></tr><tr><td>MLP</td><td>76.5±1.3</td><td>73.6±1.9</td><td>87.5±0.4</td><td>48.5±3.0</td><td>31.5±1.4</td><td>35.1±0.8</td><td>81.6±6.3</td><td>81.3±7.1</td><td>84.9±5.3</td></tr><tr><td>GCN</td><td>88.2±1.2</td><td>75.7±1.3</td><td>88.4±0.6</td><td>67.6±2.4</td><td>54.9±1.9</td><td>30.3±1.6</td><td>54.2±7.3</td><td>61.1±7.0</td><td>59.6±4.5</td></tr><tr><td>GAT</td><td>88.4±1.4</td><td>76.1±1.0</td><td>87.0±0.3</td><td>65.0±3.7</td><td>51.3±2.5</td><td>29.4±1.2</td><td>56.3±4.3</td><td>57.9±6.1</td><td>57.8±4.3</td></tr><tr><td>Geom-GCN</td><td>85.3</td><td>78.0</td><td>90.1</td><td>60.9</td><td>38.1</td><td>31.6</td><td>60.8</td><td>67.6</td><td>64.1</td></tr><tr><td>Geom-GCN-g</td><td>87.0</td><td>80.6</td><td>90.7</td><td>68.0</td><td>46.0</td><td>32.0</td><td>67.0</td><td>73.1</td><td>69.4</td></tr><tr><td>Geom-GCN-s</td><td>73.3</td><td>72.2</td><td>87.0</td><td>61.6</td><td>38.0</td><td>34.6</td><td>75.4</td><td>73.5</td><td>80.4</td></tr><tr><td>NLMLP</td><td>76.9±1.8</td><td>73.4±1.9</td><td>88.2±0.5</td><td>50.7±2.2</td><td>33.7±1.5</td><td>37.9±1.3</td><td>84.9±5.7</td><td>85.4±3.8</td><td>87.3±4.3</td></tr><tr><td>NLGCN</td><td>88.1±1.0</td><td>75.2±1.4</td><td>89.0±0.5</td><td>70.1±2.9</td><td>59.0±1.2</td><td>31.6±1.0</td><td>57.6±5.5</td><td>65.5±6.6</td><td>60.2±5.3</td></tr><tr><td>NLGAT</td><td>88.5±1.8</td><td>76.2±1.6</td><td>88.2±0.3</td><td>65.7±1.4</td><td>56.8±2.5</td><td>29.5±1.3</td><td>54.7±7.6</td><td>62.6±7.1</td><td>56.9±7.3</td></tr></table>",
+        "bbox": [
+            176,
+            449,
+            821,
+            590
+        ],
+        "page_idx": 11
+    }
+]

parse/train/o2tx_m7hK3t/o2tx_m7hK3t.md

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+# Missing Data Infill with Automunge
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Missing data is a fundamental obstacle in the practice of data science. This paper   
+2 surveys a few conventions for imputation as available in the Automunge open   
+3 source python library platform for tabular data preprocessing, including “ML infill”   
+4 in which auto ML models are trained for target features from partitioned extracts   
+5 of a training set. A series of validation experiments were performed to benchmark   
+6 imputation scenarios towards downstream model performance, in which it was   
+7 found for the given benchmark sets that in many cases ML infill outperformed for   
+8 both numeric and categoric target features, and was otherwise at minimum within   
+9 noise distributions of the other imputation scenarios. Evidence also suggested   
+10 supplementing ML infill with the addition of support columns with boolean integer   
+11 markers signaling presence of infill was usually beneficial to downstream model   
+12 performance. We consider these results sufficient to recommend defaulting to   
+13 ML infill for tabular learning, and further recommend supplementing imputations   
+14 with support columns signaling presence of infill, each as can be prepared with   
+15 push-button operation in the Automunge library. Our contributions include an   
+16 auto ML derived missing data imputation library for tabular learning in the python   
+17 ecosystem, fully integrated into a preprocessing platform with an extensive library   
+18 of feature transformations, with a novel production friendly implementation that   
+19 bases imputation models on a designated train set for consistent basis towards   
+20 additional data.
+
+# 21 1 Introduction
+
+22 Missing data is a fundamental obstacle for data science practitioners. Missing data refers to feature   
+23 sets in which a portion of entries do not have samples recorded, which may interfere with model   
+24 training and/or inference. In some cases, the missing entries may be randomly distributed within the   
+25 samples of a feature set, a scenario known as missing at random. In other cases, certain segments of a   
+26 feature set’s distribution may have a higher prevalence of missing data than other portions, a scenario   
+27 known as missing not at random. In some cases, the presence of missing data may even correlate   
+28 with label set properties, resulting in a kind of data leakage for a supervised training operation.   
+29 In a tabular data set (that is a data set aggregated as a 2D matrix of feature set columns and collected   
+30 sample rows), missing data may be represented by a few conventions. A common one is for missing   
+31 entries to be received as a NaN value, which is a special numeric data type representing “not a   
+32 number”. Some dataframe libraries may have other special data types for this purpose. In another   
+33 configuration, missing data may be represented by some particular value (like a string configuration)   
+34 associated with a feature set.   
+35 When a tabular data set with missing values present is intended to serve as a target for supervised   
+36 training, machine learning (ML) libraries may require as a prerequisite some kind of imputation   
+37 to ensure the set has all valid entries, which for most libraries means all numeric entries (although   
+38 there are some libraries that accept designated categoric feature sets in their string representations).   
+39 Conventions for imputation may follow a variety of options to target numeric or categoric feature sets   
+40 [Table 1], many of which apply a uniform infill value, which may either be arbitrary or derived as a   
+41 function of other entries in the feature set.   
+42 Other, more sophisticated conventions for infill may derive an imputation value as a function of   
+43 corresponding samples of the other features. For example, one of many learning algorithms (like   
+44 random forest, gradient boosting, neural networks, etc.) may be trained for a target feature where   
+45 the populated entries in that feature are treated as labels and surrounding features sub-aggregated   
+46 as features for the imputation model, and where the model may serve as either a classification or   
+47 regression operation based on properties of the target feature.   
+48 This paper is to document a series of validation experiments that were performed to compare   
+49 downstream model performance as a result of a few of these different infill conventions. We crafted a   
+50 contrived set of scenarios representing paradigms like missing at random or missing not at random   
+51 as injected in either a numeric or categoric target feature selected for influence toward downstream   
+52 model performance. Along the way we will offer a brief introduction to the Automunge library for   
+53 tabular data preprocessing, particularly those aspects of the library associated with missing data infill.   
+54 The results of these experiments summarized below may serve as a validation of defaulting to ML   
+55 infill for tabular learning even when faced with different types of missing data, and further defaulting   
+56 to supplementing imputations with support columns signaling presence of infill.   
+57 Our contributions include an auto ML derived missing data imputation library for tabular learning   
+58 in the python ecosystem, fully integrated into a preprocessing platform with an extensive library of   
+59 feature transformations, extending the ML imputation capabilities of R libraries like MissForest [1]   
+60 to a more production friendly implementation that bases imputation models on a designated train set   
+61 for consistent basis towards additional data.
+
+Table 1: Imputation Conventions   
+
+<table><tr><td>Imputation Value</td><td>Numeric</td><td>Categoric</td></tr><tr><td>mean</td><td></td><td></td></tr><tr><td>median</td><td>√ √</td><td></td></tr><tr><td>mode</td><td></td><td>√</td></tr><tr><td>adjacent cell</td><td>【</td><td>←</td></tr><tr><td>arbitrary (e.g. O or 1)</td><td>√</td><td>√</td></tr><tr><td>distinct activation</td><td></td><td>√</td></tr><tr><td>ML infill</td><td>「</td><td>√</td></tr></table>
+
+# 62 2 Automunge
+
+63 Automunge [2], put simply, is a python library platform for preparing tabular data for machine   
+64 learning, built on top of the Pandas dataframe library [3] and open sourced under a GNU GPL   
+65 v3.0 license. The interface is channeled through two master functions: automunge(.) for the initial   
+66 preparation of training data, and postmunge(.) for subsequent efficient preparation of additional “test”   
+67 data on the train set basis. In addition to returning transformed data, the automunge(.) function also   
+68 populates and returns a compact dictionary recording all of the steps and parameters of transformations   
+69 and imputations, which dictionary may then serve as a key for consistently preparing additional data   
+70 in the postmunge(.) function on the train set basis.   
+71 Under automation the automunge(.) function performs an evaluation of feature set properties to   
+72 derive appropriate simple feature engineering transformations that may serve to normalize numeric   
+73 sets and binarize (or hash) categoric sets. A user may also apply custom transformations, or even   
+74 custom sets of transformations, assigned to distinct columns. Such transformations may be sourced   
+75 from an extensive internal library, or even may be custom defined. The resulting transformed data log   
+76 the applied stages of derivations by way of suffix appenders on the returned column headers.   
+77 Missing data imputation is handled automatically in the library, where each transformation applied   
+78 includes a default imputation convention to serve as a precursor to imputation model training, one   
+79 that may also be overridden for use of other conventions by assignment.   
+80 Included in the library of infill options is an auto ML solution we refer to as ML infill, in which a   
+81 distinct model is trained for each target feature and saved in the returned dictionary for a consistent   
+82 imputation basis of subsequent data in the postmunge(.) function. The model architecture defaults to   
+83 random forest [4] by Scikit-Learn [5], and other auto ML library options are also supported.   
+84 The ML infill implementation works by first collecting a ‘NArw’ support column for each received   
+85 feature set containing boolean integer markers (1’s and 0’s) with activations corresponding to entries   
+86 with missing or improperly formatted data. The types of data to be considered improperly formatted   
+87 are tailored to the root transformation category to be applied to the column, where for example   
+88 for a numeric transform non-numeric entries may be subject to infill, or for a categoric transform   
+89 invalid entries may just be special data types like NaN or None. Other transforms may have other   
+90 configurations, for example a power law transform may only accept positive numeric entries, or an   
+91 integer transform may only accept integer entries.   
+92 This NArw support column can then be used to perform a target feature specific partitioning of the   
+93 training data for use to train a ML infill model [Fig 1]. The partitioning segregates rows between those   
+94 corresponding to missing data in the target feature verses those rows with valid entries, with the target   
+95 feature valid entries to serve as labels for a supervised training and the other corresponding features’   
+96 samples to serve as training data. Feature samples corresponding to the target feature missing data   
+97 are grouped for an inference operation. Note that for cases where a transformation set has prepared   
+98 a target input feature in multiple configurations, those derivations other than the target feature are   
+99 omitted from the partitions to avoid data leakage. A similar partitioning is performed for test data   
+100 sets for ML infill imputation, although in this case only the rows corresponding to entries of missing   
+101 data in the target feature are utilized for inference. As a further variation available for any of the   
+102 imputation methods, the NArw support columns may themselves be appended to the returned data   
+103 sets as a signal to training of entries that were subject to infill.
+
+# ML infill
+
+![](images/bb34845c2aaec3c73fa5b0742eeb30fb09a32a62c1b9fa20a3e9d52652f5b4bf.jpg)  
+Figure 1: ML Infill partitioning
+
+104 There is a categorization associated with each preprocessing transformation category to determine   
+105 the type of ML infill training operation, for example a target feature set derived from a transform that   
+106 returns a numeric form may be a target for a regression operation or a target feature set derived from   
+107 a transform that returns an ordinal encoding may be a target for a classification operation. In some   
+108 cases a target feature may be composed of a set of more than one column, like in the case of a set   
+109 returned from a one-hot encoding. For cases where a learner library does not accept some particular   
+110 form of encoding as valid labels there is a conversion of the target feature set for training and an   
+111 inverse conversion after any inference, for example it may be necessary to convert a binarized target   
+112 feature set to one-hot encoding or ordinal encoding for use as labels in different auto ML frameworks.   
+113 As may be particularly beneficial in cases with high prevalence of missing data across features, the   
+114 sequential training of feature imputation models may be iterated through repeated rounds of imputa  
+115 tions. For instance in the first round of model trainings and imputations the models’ performance   
+116 may be slightly degraded by high prevalence of missing data populated with the initial transformation   
+117 function imputation conventions in surrounding features, but after that first round of imputations a   
+118 second iteration of model trainings may have slight improvement of performance due to the presence   
+119 of ML infill imputations, and similarly ML infill may benefit from any additional iterations of model   
+120 trainings and imputations. In each iteration the sequence of imputations between columns are applied   
+121 in an order from features with highest prevalence of missing data to least. The library defaults to a   
+122 single round of imputations, with the option to specify an additional iteration quantity.   
+123 The final trained models for each target feature, as derived from properties of a designated train set   
+124 passed to the automunge(.) function, are collectively saved and returned to the user in a dictionary   
+125 that may serve as a key for consistent imputation basis to additional data in the postmunge(.) function,   
+126 with such dictionary also serving as a key for any applied preprocessing transformations.
+
+# 127 3 Preprocessing
+
+128 The utility of the library extends well beyond missing data infill. Automunge is intended as a platform   
+129 for all of the tabular learning steps following receipt of tidy data [6] (meaning one column per   
+130 feature and one row per sample) and immediately preceding the application of machine learning. We   
+131 found that by integrating the imputations directly into a preprocessing library, benefits included that   
+132 imputations can be applied to returned multi-column categoric representations like one-hot encodings   
+133 or binarized encodings, can account for potential data leakage between redundantly encoded feature   
+134 sets, and can accept raw data as input as may include string encoded and date-time entries with only   
+135 the minimal requirement of data received in a tidy form.   
+136 Under automation, Automunge normalizes numeric sets by z-score normalization and binarizes   
+137 categoric sets (where binarize refers to a multi-column boolean integer representation where each   
+138 categoric unique entry is represented by a distinct set of zero, one, or more simultaneous activations).   
+139 We have a separate kind of binarization for categoric sets with two unique entries, which returns a   
+140 single boolean integer encoded column (available as a single column by not having a distinct encoding   
+141 set for missing data which is instead grouped with the most common entry). High cardinality categoric   
+142 sets with unique entry count above a configurable heuristic threshold are instead applied with a hashing   
+143 trick transform [7, 8], and for highest cardinality approaching all unique entries features are given a   
+144 parsed hashing [9] which accesses distinct words found within entries. Further automated encodings   
+145 are available for date-time sets in which entries are segregated by time scale and subject to separate   
+146 sets of sine and cosine transforms at periodicity of time scale and additionally supplemented by   
+147 binned activations for business hours, weekdays, and holidays. Designated label sets are treated a   
+148 little differently, where numeric sets are left un-normalized and categoric sets are ordinal encoded (a   
+149 single column of integer activations). All of the defaults under automation are custom configurable.   
+150 A user need not defer to automation. There is a built in extensive library of feature transformations to   
+151 choose from. Numeric features may be assigned to any range of transformations, normalizations, and   
+152 bin aggregations [10]. Sequential numeric features may be supplemented by proxies for derivatives   
+153 [10]. Categoric features may be subject to encodings like ordinal, one-hot, binarization, hashing, or   
+154 even parsed categoric encoding [11] with an increased information retention in comparison to one-hot   
+155 encoding by a vectorization as a function of grammatical structure shared between entries. Categoric   
+156 sets may be collectively aggregated into a single common binarization. Categoric labels may have   
+157 label smoothing applied [12], or fitted smoothing where null values are fit to class distributions. Data   
+158 augmentation transformations [10] may be applied which make use of noise injection, including   
+159 several variants for both numeric and categoric features. Sets of transformations to be directed at a   
+160 target feature can be assembled which include generations and branches of derivations by making use   
+161 of our “family tree primitives” [13], as can be used to redundantly encode a feature set in multiple   
+162 configurations of varying information content. Such transformation sets may be accessed from those   
+163 predefined in an internal library for simple assignment or alternatively may be custom configured.   
+164 Even the transformation functions themselves may be custom defined with only minimal requirements   
+165 of simple data structures. Through application statistics of the features are recorded to facilitate   
+166 detection of distribution drift. Inversion is available to recover the original form of data found   
+167 preceding transformations, as may be useful to recover the original form of labels after inference.   
+168 Or of course if the data is received already numerically encoded the library can simply be applied as   
+169 a tool for missing data infill.
+
+# 170 4 Code Demonstration
+
+171 Jupyter notebook install and imports are as follows:
+
+!pip install Automunge from Automunge import $^ *$ am $=$ AutoMunge()
+
+176 The automunge(.) function accepts as input a Pandas dataframe or tabular Numpy array of training   
+177 data and optionally also corresponding test data. If any of the sets include a label column that header   
+178 should be designated, similarly with any index header or list of headers to exclude from the ML infill   
+179 basis. For Numpy, headers are the index integer and labels should be positioned as final column.   
+180 import pandas as pd   
+181 df_train $=$ pd.read_csv('train.csv')   
+182 df_test $=$ pd.read_csv('test.csv')   
+183 labels_column $=$ '<labels_column_header>'   
+184 trainID_column $=$ '<ID_column_header>'   
+185 These data sets can be passed to automunge(.) to automatically encode and impute. The function   
+186 returns 10 sets (9 dataframes and 1 dictionary) which in some cases may be empty based on parameter   
+187 settings, we suggest the following optional naming convention. The final set, the “postprocess_dict”,   
+188 is the key for consistently preparing additional data in postmunge(.). Note that if a validation set   
+189 is desired it can be partitioned from df_train with valpercent and prepared on the train set basis.   
+190 Shuffling is on by default for train data and off by default for test data, the associated parameter   
+191 is shown for reference. Here we demonstrate with the assigncat parameter assigning the root   
+192 category of a transformation set to some target column which will override the default transform   
+193 under automation. We also demonstrate with the assigninfill parameter assigning an alternate   
+194 infill convention to a column. The ML infill and NArw column aggregation are on by default, their   
+195 associated activation parameters are shown for reference. Note that if the data is already numerically   
+196 encoded and user just desires infill, they can pass parameter powertransform $=$ 'infill'.   
+197 train, train_ID, labels, \   
+198 val, val_ID, val_labels, \   
+199 test, test_ID, test_labels, \   
+200 postprocess_dict $=$ \   
+201 am.automunge(df_train,   
+202 df_test $=$ df_test,   
+203 labels_column $=$ labels_column,   
+204 trainID_column $=$ trainID_column,   
+205 valpercent $\ c = \ 0 . 2$ ,   
+206 shuffletrain $=$ True,   
+207 assigncat $=$ {'or23' : ['<parsed_categoric_target_column>'] },   
+208 assigninfill $=$ {'modeinfill' : ['<infill_target_column>'] },   
+209 MLinfill $=$ True,   
+210 NArw_marker $=$ True)   
+211 A list of columns returned from some particular input feature can be accessed with   
+212 postprocess_dict['column_map']['<input_feature_header>']. A report classifying the   
+213 returned column types (such as continuous, boolean, ordinal, onehot, binary, etc.) and their groupings   
+214 can be accessed with postprocess_dict['columntype_report'].   
+215 If the returned train set is to be used for training a model that may go into production, the postpro  
+216 cess_dict should be saved externally, such as with the pickle library.
+
+217 We can then prepare additional data on the train set basis with postmunge(.).
+
+218 test, test_ID, test_labels, \   
+219 postreports_dict $=$ \   
+220 am.postmunge(postprocess_dict,   
+221 df_test)   
+223 The R ecosystem has long enjoyed access to missing data imputation libraries that apply learned   
+224 models to predict infill based on other features in a set, such as MissForest [1] and mice [14], where   
+225 MissForest differs from mice as a deterministic imputation built on top of random forest and mice   
+226 applies chained equations with pooled linear models and sampling from a conditional distribution.   
+227 One of the limitations of these libraries are that the algorithms must be run through both training   
+228 and inference for each separate data set, as may be required if test data is not available at time of   
+229 training, which practice may not be amenable to production environments. Automunge on the other   
+230 hand bases imputations on a designated train set, returning from application a collected dictionary of   
+231 feature set specific models that can then be applied as a key for consistently preparing additional data   
+232 on the train set basis.   
+233 Automunge’s ML infill also differs from these R libraries by providing multiple auto ML options   
+234 for imputation models. We are continuing to build out a range that currently includes Catboost [15],   
+235 AutoGluon [16], and FLAML [17] libraries. Our default configuration is built on top of Scikit-Learn   
+236 [5] random forest [4] models and may be individually tuned to each target feature with grid or random   
+237 search by passing fit parameters to ML infill as lists or distributions.   
+238 There are of course several other variants of machine learning derived imputations that have been   
+239 demonstrated elsewhere. Imputations from generative adversarial networks [18] may improve   
+240 performance compared to ML infill (at a cost of complexity). Gaussian copula imputation [19] has   
+241 a benefit of being able to estimate uncertainty of imputations. There are even imputation solutions   
+242 built around causal graphical models [20]. Towards the other end of complexity spectrum, $\mathbf { k }$ -Nearest   
+243 Neighbor imputation [21] for continuous data is available in common frameworks like Scikit-Learn.   
+244 Being built on top of the Pandas library, there is an inherent limitation that Automunge operations are   
+245 capped at in-memory scale data sets. Other dataframe libraries like Spark [22] have the ability to   
+246 operate on distributed datasets. We believe this is not a major limitation because the in memory scale   
+247 is only associated with datasets passed to automunge(.) to serve as the basis for transformations and   
+248 imputations. Once the basis has been established, transformations to any scale of data can be applied   
+249 by passing partitions to the postmunge(.) function. We expect there may be potential to parallelize   
+250 such an operation with a library like Dask [23] or Ray [24], such an implementation is currently   
+251 intended as a future direction of research.   
+252 Another limitation associated with Pandas dataframes is that operations take place on the CPU. There   
+253 are emerging dataframe platforms like Rapids [25] which are capable of GPU accelerated operations,   
+254 which may particularly be of benefit when you take account for the elimination of a handoff step   
+255 between main and GPU memory to implement training. Although the Pandas aspects of Automunge   
+256 are CPU bound, the range of auto ML libraries incorporated are in some cases capable of GPU   
+257 training for ML infill.   
+258 There will always be a simplicity advantage to deep learning libraries like Tensorflow [26] or PyTorch   
+259 [27] which can integrate preprocessing as a layer directly into a model’s architecture, eliminating the   
+260 need to consider preprocessing in inference. We believe the single added inference step of passing   
+261 data to the postmunge(.) function is an acceptable tradeoff because by keeping the preprocessing   
+262 operations separate it facilitates a ML framework agnostic tabular preprocessing platform.
+
+# 263 6 Experiments
+
+264 Some experiments were performed to evaluate efficacy of a few different imputation methods in   
+265 different scenarios of missing data. To amplify the impact of imputations, each of two data sets   
+266 were pared down to a reduced set of the top 15 features based on an Automunge feature importance   
+267 evaluation [11] by shuffle permutation [28]. (This step had the side benefit of reducing the training   
+268 durations of experiments.) The top ranked importance categoric and numeric features were selected   
+269 to separately serve as targets for injections of missing data, with such injections simulating scenarios   
+270 of both missing at random and missing not at random.   
+271 To simulate cases of missing not at random, and also again to amplify the impact of imputation, the   
+272 target features were evaluated to determine the most influential segments of the features’ distributions   
+3 [29], which for the target categoric features was one of the activations and for the target numeric   
+4 features turned out to be the far right tail for both benchmark data sets.   
+75 Further variations were aggregated associated with either the ratio of full feature or ratio of distribution   
+76 segments injected with missing data, ranging from no injections to full replacement.
+
+Finally, for each of these scenarios, variations were assembled associated with the type of infill applied by Automunge, including scenarios for defaults (mean imputation for numeric or distinct activations for categoric), imputation with mode, adjacent cell, and ML infill. The ML infill scenario was applied making use of the CatBoost library to take advantage of GPU acceleration.
+
+Having prepared the data in each of these scenarios with an automunge(.) call, the final step was to train a downstream model to evaluate impact, again here with the CatBoost library. The performance metric applied was root mean squared error for the regression applications. Each scenario was repeated 68 or more times with the metrics averaged to de-noise the results.
+
+Finally, the ML infill scenarios were repeated again with the addition of the NArw support columns to supplement the target features.
+
+# 7 Results
+
+The results of the various scenarios are presented [Fig 2, 3, 4, 5]. Here the y axis are the performance metrics and the x axis the ratio of entries with missing data injections, which were given as $\{ 0 , 0 . 1$ , 0.33, 0.67, 1.0}, where in the 0.0 case no missing data was injected and with 1.0 the entire feature or feature segment was injected. Because the 0.0 cases had equivalent entries between infill types, their spread across the four infill scenarios are a good approximation for the noise inherent in the learning algorithm. An additional source of noise for the other ratios was from the stochasticity of injections, with a distinct set for each trial. Consistent with common sense, as the injection ratio was ramped up the trend across infill scenarios was a degradation of the performance metric.
+
+296 We did find that with increased repetitions incorporated the spread of the averaged performance   
+97 metrics were tightened, leading us to repeat the experiments at increased scale for some improved   
+98 statistical significance.   
+299 For the missing at random injections [Fig 2, 3], ML infill was at or near top performance across both   
+300 data sets, although the spread between imputations was not extremely pronounced. In most of the   
+301 setups, mode imputation and adjacent cell trended as reduced performance in comparison to ML infill   
+302 or the default imputations (mean for numeric sets and distinct activation set for categoric).   
+303 For not at random injections to the right tail of numeric sets [Fig 4], it appears that ML infill had a   
+304 pronounced benefit to the Ames Housing data set [30], especially as the injection ratio increased,   
+305 and more of an intermediate performance to the Allstate Claims data set [31]. We speculate that ML   
+306 infill had some degree of variability across these demonstrations due to correlations (or lack thereof)   
+307 between the target feature and the other features, without which ML infill may struggle to establish a   
+308 basis for inference. In the final scenario of not at random injections to categoric [Fig 5] we believe   
+309 default performed well because it served as a direct replacement for the single missing activation.   
+310 An additional comparable series of injections were conducted with ML infill and the added difference   
+311 of appending the NArw support columns corresponding to the target columns for injections. Again   
+312 these NArw support columns are the boolean integer markers for presence of infill in the corresponding   
+313 entries which support the partitioning of sets for ML infill. The expectation was that by using these   
+314 markers to signal to the training operation which of the entries were subjected to infill, there would   
+315 be some benefit to downstream model performance. For many of the scenarios the visible impact was   
+316 that supplementing with the NArw support column improved the ML infill performance, demonstrated   
+317 here for missing at random [Fig 6, 7] and missing not at random [Fig 8, 9] with the other imputation   
+318 scenarios shown again for context.
+
+![](images/45c437556a465aa5dc992177208d5c53bf30020d2b467d6f1f92dd24cb6acdd9.jpg)
+
+![](images/3ba119c2a2554c3cdf89111a3b461ebcff079999f52b0f5eb6bb9d17af79fb44.jpg)  
+Figure 2: Missing at Random - Numeric Target Feature   
+Figure 3: Missing at Random - Categoric Target Feature
+
+![](images/c69e1a3ca7729f1a95208f71e4c3c48b610ababb0a82ecd3534cae240ce0af34.jpg)
+
+![](images/2e5c4483c28923780baf40482fc32a9c59d7db6f49afdf17a0a8b270e056e358.jpg)  
+Figure 4: Not at Random - Numeric Target Feature   
+Figure 5: Not at Random - Categoric Target Feature
+
+# 319 8 Discussion
+
+320 One of the primary goals of this experiment was to validate the efficacy of ML infill as evidenced   
+321 by improvements to downstream model performance. For the Ames Housing benchmark data set,   
+322 there was a notable demonstration of ML infill benefiting model performance in the scenario of   
+323 the numeric target column with not at random injections at increased injection ratios, and also to   
+324 a lesser extent with missing at random injections. We speculate an explanation for this advantage   
+325 towards the numeric target columns may partly be attributed to the fact that the downstream model   
+326 was also a regression application, so that the other features selected for label correlation may by   
+327 proxy have correlations with the target numeric feature. The corollary is that the more mundane   
+328 performance of ML infill toward the categoric target columns may be a result of these having less   
+329 correspondence with the surrounding features. The fact that even in these cases the ML infill still fell   
+330 within noise distribution of the other imputation scenarios we believe presents a reasonable argument   
+331 for defaulting to ML infill for tabular learning.   
+332 Note that as another argument for defaulting to ML infill as opposed to static imputations is that the   
+333 imputation model may serve as a hedge against imperfections in subsequent data streams, particularly   
+334 if one of the features experiences downtime in a streaming application for instance.   
+335 The other key finding of the experiment was the pronounced benefit to downstream model performance   
+336 when including the NArw support column in the returned data set as a supplement to ML infill. This   
+337 finding was consistent with our intuition, which was that increased information retention about infill   
+338 points should help model performance. Note there is some small tradeoff, as the added training set   
+339 dimensionality may increase training time. Another benefit to including NArw support columns may   
+340 be for interpretability in inspection of imputations. We recommend including the NArw support   
+341 columns for model training based on these findings, with the one caveat that care should be taken to   
+342 avoid inclusion in the data leakage scenario where there is some kind of correlation between presence   
+343 of missing data and label set properties that won’t be present in production.
+
+![](images/0c384e6a5f96e198ee72be9f57a7385afc203e769b148b0878d5758bd62952fc.jpg)
+
+![](images/bc182dcaceae281bb6433789415fe76d81d91e725ab77b9753c448ed934a1503.jpg)  
+Figure 6: NArw comparison - Missing at Random - Numeric Target Feature   
+Missing at Random - Categoric
+
+![](images/f4b089490ed5243731c3f3894da72fdb294238ef724fe3446307b330ecf4aaf0.jpg)  
+Figure 7: NArw comparison - Missing at Random - Categoric Target Feature
+
+![](images/95e8ec6c42422a2c737704e8fa765d1fe879e400f4ced62623905130f0712432.jpg)  
+Figure 8: NArw comparison - Not at Random - Numeric Target Feature   
+t Random-Categoric   
+Figure 9: NArw comparison - Not at Random - Categoric Target Feature
+
+# 344 9 Conclusion
+
+45 Automunge offers a push-button solution to preparing tabular data for ML, with automated data   
+46 cleaning operations like normalizations, binarizations, and auto ML derived missing data imputation   
+47 aka ML infill. Transformations and imputations are fit to properties of a designated train set, and with   
+48 application of automunge(.) a compact dictionary is returned recording transformation parameters   
+49 and trained imputation models, which dictionary may then serve as a key for consistently preparing   
+50 additional data on the train set basis with postmunge(.).   
+351 We hope that these experiments may serve as a kind of validation of defaulting to ML infill with   
+352 supplemented NArw support columns in tabular learning for users of the Automunge library, as   
+353 even if in our experiments the material benefits towards downstream model performance were not   
+354 demonstrated for all target feature scenarios, in other cases there did not appear to be any material   
+355 penalty. Note that ML infill can be activated for push-button operation by the automunge(.) parameter   
+356 MLinfill True and the NArw support columns included by parameter NArw_marker $\backsimeq$ True. Based   
+357 on these findings these two parameter settings are now cast as defaults for the Automunge platform.
+
+# 358 Acknowledgments
+
+A thank you owed to those facilitators behind Stack Overflow, Python, Numpy, Scipy Stats, PyPI, GitHub, Colaboratory, Anaconda, VSCode, and Jupyter. Special thanks to Scikit-Learn and Pandas.
+
+# References
+
+362 [1] Daniel J. Stekhoven, Peter Bühlmann. MissForest - nonparametric missing value imputation for mixed-type   
+363 data (2011) arXiv:1105.0828   
+364 [2] Author(s) (2021) Automunge, GitHub repository (Please see supplemental material)   
+365 [3] W. McKinney. Data structures for statistical computing in python. Proceedings of the 9th Python in Science   
+366 Conference, pages 51–56, 2010.   
+367 [4] L. Breiman. Random Forests. Machine Learning, 45(1), 2001.   
+368 [5] Pedregosa et al., Scikit-learn: Machine Learning in Python, JMLR 12, pp. 2825-2830, 2011.   
+369 [6] H. Wickham. Tidy data. Journal of Statistical Software, 59(10), 2014.   
+370 [7] John Moody. Fast Learning in Multi-Resolution Hierarchies. NIPS Proceedings, 1989   
+371 [8] Kilian Weinberger, Anirban Dasgupta, John Langford, Alex Smola, Josh Attenberg. Feature Hashing for   
+372 Large Scale Multitask Learning. ICML Proceedings, 2009   
+373 [9] Author(s) Hashed Categoric Encodings with Automunge (2020) (Please see preprint in supplemental material)   
+374 [10] Author(s) Numeric Encoding Options with Automunge (2020) (Please see preprint in supplemental material)   
+375 [11] Author(s) Parsed Categoric Encodings with Automunge (2020) (Please see preprint in supplemental   
+376 material)   
+377 [12] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, Zbigniew Wojna. Rethinking the Inception   
+378 Architecture for Computer Vision. IEEE conference on computer vision and pattern recognition, 2016   
+379 [13] Author(s) Specification of Derivations with Automunge (2020) (Please see preprint in supplemental   
+380 material)   
+381 [14] Stef van Buuren, Karin Groothuis-Oudshoorn. mice: Multivariate Imputation by Chained Equations in R   
+382 (2011) https://www.jstatsoft.org/article/view/v045i03   
+383 [15] Anna Veronika Dorogush, Vasily Ershov, Andrey Gulin. CatBoost: gradient boosting with categorical   
+384 features support (2018) arXiv:1810.11363   
+385 [16] Nick Erickson, Jonas Mueller, Alexander Shirkov, Hang Zhang, Pedro Larroy, Mu Li, and Alexander Smola.   
+386 AutoGluon-Tabular: Robust and Accurate AutoML for Structured Data (2020) arxiv:2003.06505   
+387 [17] Chi Wang, Qingyun Wu, Markus Weimer, Erkang Zhu. FLAML: A Fast and Lightweight AutoML Library   
+388 (2019) arXiv:1911.04706   
+389 [18] Jinsung Yoon, James Jordon, Mihaela van der Schaar. GAIN: Missing Data Imputation using Generative   
+390 Adversarial Nets (2018 International Conference of Machine Learning), arXiv:1806.02920   
+391 [19] Yuxuan Zhao, Madeleine Udell. Missing Value Imputation for Mixed Data via Gaussian Copula (KDD   
+392 2020), arXiv:1910.12845   
+393 [20] K. Mohan, J. Pearl. Graphical Models for Processing Missing Data (2019), arXiv:1801.03583   
+394 [21] Olga Troyanskaya, Michael Cantor, Gavin Sherlock, Pat Brown, Trevor Hastie, Robert Tibshirani, David   
+395 Botstein and Russ B. Altman. Missing value estimation methods for DNA microarrays, BIOINFORMATICS   
+396 Vol. 17 no. 6, 2001 Pages 520-525.   
+397 [22] Matei Zaharia, Reynold S. Xin, Patrick Wendell, Tathagata Das, Michael Armbrust, Ankur Dave, Xiangrui   
+398 Meng, Josh Rosen, Shivaram Venkataraman, Michael J. Franklin, Ali Ghodsi, Joseph Gonzalez, Scott Shenker,   
+399 Ion Stoica. Apache Spark: a unified engine for big data processing. Communications of the ACM, 59(11), 2016   
+400 [23] Dask Development Team. Dask: Library for dynamic task scheduling (2016) https://dask.org   
+401 [24] Philipp Moritz, Robert Nishihara, Stephanie Wang, Alexey Tumanov, Richard Liaw, Eric Liang, Melih   
+402 Elibol, Zongheng Yang, William Paul, Michael I. Jordan, Ion Stoica. Ray: A Distributed Framework for   
+403 Emerging AI Applications. 13th USENIX Symposium on Operating Systems Design and Implementation   
+404 (2018), arXiv:1712.05889
+
+[26] Abadi, Martín, Barham P, Chen J, Chen Z, Davis A, Dean J, et al. Tensorflow: A system for large-scale machine learning. 12th USENIX Symposium on Operating Systems Design and Implementation (2016) p. 265–83.   
+[27] Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Kopf, Andreas and Yang, Edward and DeVito, Zachary and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith. PyTorch: An Imperative Style, High-Performance Deep Learning Library. NeurIPS Proceedings, 2019   
+[28] Terrence Parr, Kerem Turgutlu, Christopher Csiszar, and Jeremy Howard. Beware default random forest importances. Explained.ai (blog), 2018. https://explained.ai/rf-importance/.   
+[29] Author(s) Automunge Influence (2020) (Please see preprint in supplemental material)   
+[30] Dean De Cock. Ames, Iowa: Alternative to the Boston Housing Data as an End of Semester Regression Project, Journal of Statistics Education, Volume 19, Number 3 (2011)   
+[31] Kaggle: Allstate Claims Severity, https://www.kaggle.com/c/allstate-claims-severity
+
+# 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] Please see discussions in section 5 Related Work   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] A Broader Impacts discussion is provided as Appendix C   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Please see jupyter notebooks provided with supplemental material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We noted that missing data injections were random for each trial. Performance was evaluated on a $2 5 \%$ validation split. We used hyperparameter defaults for learning.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We noted that since scenarios for $0 \%$ injection are comparable between imputation methods, their spread may serve as a proxy for noise inherent in the operation.   
+(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] Our experiments did not require significant compute.
+
+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]   
+(b) Did you mention the license of the assets? [Yes] We note licenses of supporting packages in the read me document included in the github repository folder within the supplemental material.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "1 Missing data is a fundamental obstacle in the practice of data science. This paper   \n2 surveys a few conventions for imputation as available in the Automunge open   \n3 source python library platform for tabular data preprocessing, including “ML infill”   \n4 in which auto ML models are trained for target features from partitioned extracts   \n5 of a training set. A series of validation experiments were performed to benchmark   \n6 imputation scenarios towards downstream model performance, in which it was   \n7 found for the given benchmark sets that in many cases ML infill outperformed for   \n8 both numeric and categoric target features, and was otherwise at minimum within   \n9 noise distributions of the other imputation scenarios. Evidence also suggested   \n10 supplementing ML infill with the addition of support columns with boolean integer   \n11 markers signaling presence of infill was usually beneficial to downstream model   \n12 performance. We consider these results sufficient to recommend defaulting to   \n13 ML infill for tabular learning, and further recommend supplementing imputations   \n14 with support columns signaling presence of infill, each as can be prepared with   \n15 push-button operation in the Automunge library. Our contributions include an   \n16 auto ML derived missing data imputation library for tabular learning in the python   \n17 ecosystem, fully integrated into a preprocessing platform with an extensive library   \n18 of feature transformations, with a novel production friendly implementation that   \n19 bases imputation models on a designated train set for consistent basis towards   \n20 additional data. ",
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+        "text": "21 1 Introduction ",
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+        "text": "22 Missing data is a fundamental obstacle for data science practitioners. Missing data refers to feature   \n23 sets in which a portion of entries do not have samples recorded, which may interfere with model   \n24 training and/or inference. In some cases, the missing entries may be randomly distributed within the   \n25 samples of a feature set, a scenario known as missing at random. In other cases, certain segments of a   \n26 feature set’s distribution may have a higher prevalence of missing data than other portions, a scenario   \n27 known as missing not at random. In some cases, the presence of missing data may even correlate   \n28 with label set properties, resulting in a kind of data leakage for a supervised training operation.   \n29 In a tabular data set (that is a data set aggregated as a 2D matrix of feature set columns and collected   \n30 sample rows), missing data may be represented by a few conventions. A common one is for missing   \n31 entries to be received as a NaN value, which is a special numeric data type representing “not a   \n32 number”. Some dataframe libraries may have other special data types for this purpose. In another   \n33 configuration, missing data may be represented by some particular value (like a string configuration)   \n34 associated with a feature set.   \n35 When a tabular data set with missing values present is intended to serve as a target for supervised   \n36 training, machine learning (ML) libraries may require as a prerequisite some kind of imputation   \n37 to ensure the set has all valid entries, which for most libraries means all numeric entries (although   \n38 there are some libraries that accept designated categoric feature sets in their string representations).   \n39 Conventions for imputation may follow a variety of options to target numeric or categoric feature sets   \n40 [Table 1], many of which apply a uniform infill value, which may either be arbitrary or derived as a   \n41 function of other entries in the feature set.   \n42 Other, more sophisticated conventions for infill may derive an imputation value as a function of   \n43 corresponding samples of the other features. For example, one of many learning algorithms (like   \n44 random forest, gradient boosting, neural networks, etc.) may be trained for a target feature where   \n45 the populated entries in that feature are treated as labels and surrounding features sub-aggregated   \n46 as features for the imputation model, and where the model may serve as either a classification or   \n47 regression operation based on properties of the target feature.   \n48 This paper is to document a series of validation experiments that were performed to compare   \n49 downstream model performance as a result of a few of these different infill conventions. We crafted a   \n50 contrived set of scenarios representing paradigms like missing at random or missing not at random   \n51 as injected in either a numeric or categoric target feature selected for influence toward downstream   \n52 model performance. Along the way we will offer a brief introduction to the Automunge library for   \n53 tabular data preprocessing, particularly those aspects of the library associated with missing data infill.   \n54 The results of these experiments summarized below may serve as a validation of defaulting to ML   \n55 infill for tabular learning even when faced with different types of missing data, and further defaulting   \n56 to supplementing imputations with support columns signaling presence of infill.   \n57 Our contributions include an auto ML derived missing data imputation library for tabular learning   \n58 in the python ecosystem, fully integrated into a preprocessing platform with an extensive library of   \n59 feature transformations, extending the ML imputation capabilities of R libraries like MissForest [1]   \n60 to a more production friendly implementation that bases imputation models on a designated train set   \n61 for consistent basis towards additional data. ",
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+            "Table 1: Imputation Conventions "
+        ],
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+        "table_body": "<table><tr><td>Imputation Value</td><td>Numeric</td><td>Categoric</td></tr><tr><td>mean</td><td></td><td></td></tr><tr><td>median</td><td>√ √</td><td></td></tr><tr><td>mode</td><td></td><td>√</td></tr><tr><td>adjacent cell</td><td>【</td><td>←</td></tr><tr><td>arbitrary (e.g. O or 1)</td><td>√</td><td>√</td></tr><tr><td>distinct activation</td><td></td><td>√</td></tr><tr><td>ML infill</td><td>「</td><td>√</td></tr></table>",
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+        "text": "62 2 Automunge ",
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+        "text": "63 Automunge [2], put simply, is a python library platform for preparing tabular data for machine   \n64 learning, built on top of the Pandas dataframe library [3] and open sourced under a GNU GPL   \n65 v3.0 license. The interface is channeled through two master functions: automunge(.) for the initial   \n66 preparation of training data, and postmunge(.) for subsequent efficient preparation of additional “test”   \n67 data on the train set basis. In addition to returning transformed data, the automunge(.) function also   \n68 populates and returns a compact dictionary recording all of the steps and parameters of transformations   \n69 and imputations, which dictionary may then serve as a key for consistently preparing additional data   \n70 in the postmunge(.) function on the train set basis.   \n71 Under automation the automunge(.) function performs an evaluation of feature set properties to   \n72 derive appropriate simple feature engineering transformations that may serve to normalize numeric   \n73 sets and binarize (or hash) categoric sets. A user may also apply custom transformations, or even   \n74 custom sets of transformations, assigned to distinct columns. Such transformations may be sourced   \n75 from an extensive internal library, or even may be custom defined. The resulting transformed data log   \n76 the applied stages of derivations by way of suffix appenders on the returned column headers.   \n77 Missing data imputation is handled automatically in the library, where each transformation applied   \n78 includes a default imputation convention to serve as a precursor to imputation model training, one   \n79 that may also be overridden for use of other conventions by assignment.   \n80 Included in the library of infill options is an auto ML solution we refer to as ML infill, in which a   \n81 distinct model is trained for each target feature and saved in the returned dictionary for a consistent   \n82 imputation basis of subsequent data in the postmunge(.) function. The model architecture defaults to   \n83 random forest [4] by Scikit-Learn [5], and other auto ML library options are also supported.   \n84 The ML infill implementation works by first collecting a ‘NArw’ support column for each received   \n85 feature set containing boolean integer markers (1’s and 0’s) with activations corresponding to entries   \n86 with missing or improperly formatted data. The types of data to be considered improperly formatted   \n87 are tailored to the root transformation category to be applied to the column, where for example   \n88 for a numeric transform non-numeric entries may be subject to infill, or for a categoric transform   \n89 invalid entries may just be special data types like NaN or None. Other transforms may have other   \n90 configurations, for example a power law transform may only accept positive numeric entries, or an   \n91 integer transform may only accept integer entries.   \n92 This NArw support column can then be used to perform a target feature specific partitioning of the   \n93 training data for use to train a ML infill model [Fig 1]. The partitioning segregates rows between those   \n94 corresponding to missing data in the target feature verses those rows with valid entries, with the target   \n95 feature valid entries to serve as labels for a supervised training and the other corresponding features’   \n96 samples to serve as training data. Feature samples corresponding to the target feature missing data   \n97 are grouped for an inference operation. Note that for cases where a transformation set has prepared   \n98 a target input feature in multiple configurations, those derivations other than the target feature are   \n99 omitted from the partitions to avoid data leakage. A similar partitioning is performed for test data   \n100 sets for ML infill imputation, although in this case only the rows corresponding to entries of missing   \n101 data in the target feature are utilized for inference. As a further variation available for any of the   \n102 imputation methods, the NArw support columns may themselves be appended to the returned data   \n103 sets as a signal to training of entries that were subject to infill. ",
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+        "text": "ML infill ",
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+        "image_caption": [
+            "Figure 1: ML Infill partitioning "
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+        "text": "104 There is a categorization associated with each preprocessing transformation category to determine   \n105 the type of ML infill training operation, for example a target feature set derived from a transform that   \n106 returns a numeric form may be a target for a regression operation or a target feature set derived from   \n107 a transform that returns an ordinal encoding may be a target for a classification operation. In some   \n108 cases a target feature may be composed of a set of more than one column, like in the case of a set   \n109 returned from a one-hot encoding. For cases where a learner library does not accept some particular   \n110 form of encoding as valid labels there is a conversion of the target feature set for training and an   \n111 inverse conversion after any inference, for example it may be necessary to convert a binarized target   \n112 feature set to one-hot encoding or ordinal encoding for use as labels in different auto ML frameworks.   \n113 As may be particularly beneficial in cases with high prevalence of missing data across features, the   \n114 sequential training of feature imputation models may be iterated through repeated rounds of imputa  \n115 tions. For instance in the first round of model trainings and imputations the models’ performance   \n116 may be slightly degraded by high prevalence of missing data populated with the initial transformation   \n117 function imputation conventions in surrounding features, but after that first round of imputations a   \n118 second iteration of model trainings may have slight improvement of performance due to the presence   \n119 of ML infill imputations, and similarly ML infill may benefit from any additional iterations of model   \n120 trainings and imputations. In each iteration the sequence of imputations between columns are applied   \n121 in an order from features with highest prevalence of missing data to least. The library defaults to a   \n122 single round of imputations, with the option to specify an additional iteration quantity.   \n123 The final trained models for each target feature, as derived from properties of a designated train set   \n124 passed to the automunge(.) function, are collectively saved and returned to the user in a dictionary   \n125 that may serve as a key for consistent imputation basis to additional data in the postmunge(.) function,   \n126 with such dictionary also serving as a key for any applied preprocessing transformations. ",
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+        "text": "127 3 Preprocessing ",
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+        "text": "128 The utility of the library extends well beyond missing data infill. Automunge is intended as a platform   \n129 for all of the tabular learning steps following receipt of tidy data [6] (meaning one column per   \n130 feature and one row per sample) and immediately preceding the application of machine learning. We   \n131 found that by integrating the imputations directly into a preprocessing library, benefits included that   \n132 imputations can be applied to returned multi-column categoric representations like one-hot encodings   \n133 or binarized encodings, can account for potential data leakage between redundantly encoded feature   \n134 sets, and can accept raw data as input as may include string encoded and date-time entries with only   \n135 the minimal requirement of data received in a tidy form.   \n136 Under automation, Automunge normalizes numeric sets by z-score normalization and binarizes   \n137 categoric sets (where binarize refers to a multi-column boolean integer representation where each   \n138 categoric unique entry is represented by a distinct set of zero, one, or more simultaneous activations).   \n139 We have a separate kind of binarization for categoric sets with two unique entries, which returns a   \n140 single boolean integer encoded column (available as a single column by not having a distinct encoding   \n141 set for missing data which is instead grouped with the most common entry). High cardinality categoric   \n142 sets with unique entry count above a configurable heuristic threshold are instead applied with a hashing   \n143 trick transform [7, 8], and for highest cardinality approaching all unique entries features are given a   \n144 parsed hashing [9] which accesses distinct words found within entries. Further automated encodings   \n145 are available for date-time sets in which entries are segregated by time scale and subject to separate   \n146 sets of sine and cosine transforms at periodicity of time scale and additionally supplemented by   \n147 binned activations for business hours, weekdays, and holidays. Designated label sets are treated a   \n148 little differently, where numeric sets are left un-normalized and categoric sets are ordinal encoded (a   \n149 single column of integer activations). All of the defaults under automation are custom configurable.   \n150 A user need not defer to automation. There is a built in extensive library of feature transformations to   \n151 choose from. Numeric features may be assigned to any range of transformations, normalizations, and   \n152 bin aggregations [10]. Sequential numeric features may be supplemented by proxies for derivatives   \n153 [10]. Categoric features may be subject to encodings like ordinal, one-hot, binarization, hashing, or   \n154 even parsed categoric encoding [11] with an increased information retention in comparison to one-hot   \n155 encoding by a vectorization as a function of grammatical structure shared between entries. Categoric   \n156 sets may be collectively aggregated into a single common binarization. Categoric labels may have   \n157 label smoothing applied [12], or fitted smoothing where null values are fit to class distributions. Data   \n158 augmentation transformations [10] may be applied which make use of noise injection, including   \n159 several variants for both numeric and categoric features. Sets of transformations to be directed at a   \n160 target feature can be assembled which include generations and branches of derivations by making use   \n161 of our “family tree primitives” [13], as can be used to redundantly encode a feature set in multiple   \n162 configurations of varying information content. Such transformation sets may be accessed from those   \n163 predefined in an internal library for simple assignment or alternatively may be custom configured.   \n164 Even the transformation functions themselves may be custom defined with only minimal requirements   \n165 of simple data structures. Through application statistics of the features are recorded to facilitate   \n166 detection of distribution drift. Inversion is available to recover the original form of data found   \n167 preceding transformations, as may be useful to recover the original form of labels after inference.   \n168 Or of course if the data is received already numerically encoded the library can simply be applied as   \n169 a tool for missing data infill. ",
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+        "text": "170 4 Code Demonstration ",
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+        "text": "171 Jupyter notebook install and imports are as follows: ",
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+        "text": "!pip install Automunge from Automunge import $^ *$ am $=$ AutoMunge() ",
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+        "text": "176 The automunge(.) function accepts as input a Pandas dataframe or tabular Numpy array of training   \n177 data and optionally also corresponding test data. If any of the sets include a label column that header   \n178 should be designated, similarly with any index header or list of headers to exclude from the ML infill   \n179 basis. For Numpy, headers are the index integer and labels should be positioned as final column.   \n180 import pandas as pd   \n181 df_train $=$ pd.read_csv('train.csv')   \n182 df_test $=$ pd.read_csv('test.csv')   \n183 labels_column $=$ '<labels_column_header>'   \n184 trainID_column $=$ '<ID_column_header>'   \n185 These data sets can be passed to automunge(.) to automatically encode and impute. The function   \n186 returns 10 sets (9 dataframes and 1 dictionary) which in some cases may be empty based on parameter   \n187 settings, we suggest the following optional naming convention. The final set, the “postprocess_dict”,   \n188 is the key for consistently preparing additional data in postmunge(.). Note that if a validation set   \n189 is desired it can be partitioned from df_train with valpercent and prepared on the train set basis.   \n190 Shuffling is on by default for train data and off by default for test data, the associated parameter   \n191 is shown for reference. Here we demonstrate with the assigncat parameter assigning the root   \n192 category of a transformation set to some target column which will override the default transform   \n193 under automation. We also demonstrate with the assigninfill parameter assigning an alternate   \n194 infill convention to a column. The ML infill and NArw column aggregation are on by default, their   \n195 associated activation parameters are shown for reference. Note that if the data is already numerically   \n196 encoded and user just desires infill, they can pass parameter powertransform $=$ 'infill'.   \n197 train, train_ID, labels, \\   \n198 val, val_ID, val_labels, \\   \n199 test, test_ID, test_labels, \\   \n200 postprocess_dict $=$ \\   \n201 am.automunge(df_train,   \n202 df_test $=$ df_test,   \n203 labels_column $=$ labels_column,   \n204 trainID_column $=$ trainID_column,   \n205 valpercent $\\ c = \\ 0 . 2$ ,   \n206 shuffletrain $=$ True,   \n207 assigncat $=$ {'or23' : ['<parsed_categoric_target_column>'] },   \n208 assigninfill $=$ {'modeinfill' : ['<infill_target_column>'] },   \n209 MLinfill $=$ True,   \n210 NArw_marker $=$ True)   \n211 A list of columns returned from some particular input feature can be accessed with   \n212 postprocess_dict['column_map']['<input_feature_header>']. A report classifying the   \n213 returned column types (such as continuous, boolean, ordinal, onehot, binary, etc.) and their groupings   \n214 can be accessed with postprocess_dict['columntype_report'].   \n215 If the returned train set is to be used for training a model that may go into production, the postpro  \n216 cess_dict should be saved externally, such as with the pickle library. ",
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+        "text": "217 We can then prepare additional data on the train set basis with postmunge(.). ",
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+        "text": "218 test, test_ID, test_labels, \\   \n219 postreports_dict $=$ \\   \n220 am.postmunge(postprocess_dict,   \n221 df_test)   \n223 The R ecosystem has long enjoyed access to missing data imputation libraries that apply learned   \n224 models to predict infill based on other features in a set, such as MissForest [1] and mice [14], where   \n225 MissForest differs from mice as a deterministic imputation built on top of random forest and mice   \n226 applies chained equations with pooled linear models and sampling from a conditional distribution.   \n227 One of the limitations of these libraries are that the algorithms must be run through both training   \n228 and inference for each separate data set, as may be required if test data is not available at time of   \n229 training, which practice may not be amenable to production environments. Automunge on the other   \n230 hand bases imputations on a designated train set, returning from application a collected dictionary of   \n231 feature set specific models that can then be applied as a key for consistently preparing additional data   \n232 on the train set basis.   \n233 Automunge’s ML infill also differs from these R libraries by providing multiple auto ML options   \n234 for imputation models. We are continuing to build out a range that currently includes Catboost [15],   \n235 AutoGluon [16], and FLAML [17] libraries. Our default configuration is built on top of Scikit-Learn   \n236 [5] random forest [4] models and may be individually tuned to each target feature with grid or random   \n237 search by passing fit parameters to ML infill as lists or distributions.   \n238 There are of course several other variants of machine learning derived imputations that have been   \n239 demonstrated elsewhere. Imputations from generative adversarial networks [18] may improve   \n240 performance compared to ML infill (at a cost of complexity). Gaussian copula imputation [19] has   \n241 a benefit of being able to estimate uncertainty of imputations. There are even imputation solutions   \n242 built around causal graphical models [20]. Towards the other end of complexity spectrum, $\\mathbf { k }$ -Nearest   \n243 Neighbor imputation [21] for continuous data is available in common frameworks like Scikit-Learn.   \n244 Being built on top of the Pandas library, there is an inherent limitation that Automunge operations are   \n245 capped at in-memory scale data sets. Other dataframe libraries like Spark [22] have the ability to   \n246 operate on distributed datasets. We believe this is not a major limitation because the in memory scale   \n247 is only associated with datasets passed to automunge(.) to serve as the basis for transformations and   \n248 imputations. Once the basis has been established, transformations to any scale of data can be applied   \n249 by passing partitions to the postmunge(.) function. We expect there may be potential to parallelize   \n250 such an operation with a library like Dask [23] or Ray [24], such an implementation is currently   \n251 intended as a future direction of research.   \n252 Another limitation associated with Pandas dataframes is that operations take place on the CPU. There   \n253 are emerging dataframe platforms like Rapids [25] which are capable of GPU accelerated operations,   \n254 which may particularly be of benefit when you take account for the elimination of a handoff step   \n255 between main and GPU memory to implement training. Although the Pandas aspects of Automunge   \n256 are CPU bound, the range of auto ML libraries incorporated are in some cases capable of GPU   \n257 training for ML infill.   \n258 There will always be a simplicity advantage to deep learning libraries like Tensorflow [26] or PyTorch   \n259 [27] which can integrate preprocessing as a layer directly into a model’s architecture, eliminating the   \n260 need to consider preprocessing in inference. We believe the single added inference step of passing   \n261 data to the postmunge(.) function is an acceptable tradeoff because by keeping the preprocessing   \n262 operations separate it facilitates a ML framework agnostic tabular preprocessing platform. ",
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+        "type": "text",
+        "text": "263 6 Experiments ",
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+        "text": "264 Some experiments were performed to evaluate efficacy of a few different imputation methods in   \n265 different scenarios of missing data. To amplify the impact of imputations, each of two data sets   \n266 were pared down to a reduced set of the top 15 features based on an Automunge feature importance   \n267 evaluation [11] by shuffle permutation [28]. (This step had the side benefit of reducing the training   \n268 durations of experiments.) The top ranked importance categoric and numeric features were selected   \n269 to separately serve as targets for injections of missing data, with such injections simulating scenarios   \n270 of both missing at random and missing not at random.   \n271 To simulate cases of missing not at random, and also again to amplify the impact of imputation, the   \n272 target features were evaluated to determine the most influential segments of the features’ distributions   \n3 [29], which for the target categoric features was one of the activations and for the target numeric   \n4 features turned out to be the far right tail for both benchmark data sets.   \n75 Further variations were aggregated associated with either the ratio of full feature or ratio of distribution   \n76 segments injected with missing data, ranging from no injections to full replacement. ",
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+        "text": "Finally, for each of these scenarios, variations were assembled associated with the type of infill applied by Automunge, including scenarios for defaults (mean imputation for numeric or distinct activations for categoric), imputation with mode, adjacent cell, and ML infill. The ML infill scenario was applied making use of the CatBoost library to take advantage of GPU acceleration. ",
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+        "text": "Having prepared the data in each of these scenarios with an automunge(.) call, the final step was to train a downstream model to evaluate impact, again here with the CatBoost library. The performance metric applied was root mean squared error for the regression applications. Each scenario was repeated 68 or more times with the metrics averaged to de-noise the results. ",
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+        "text": "Finally, the ML infill scenarios were repeated again with the addition of the NArw support columns to supplement the target features. ",
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+        "text": "7 Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The results of the various scenarios are presented [Fig 2, 3, 4, 5]. Here the y axis are the performance metrics and the x axis the ratio of entries with missing data injections, which were given as $\\{ 0 , 0 . 1$ , 0.33, 0.67, 1.0}, where in the 0.0 case no missing data was injected and with 1.0 the entire feature or feature segment was injected. Because the 0.0 cases had equivalent entries between infill types, their spread across the four infill scenarios are a good approximation for the noise inherent in the learning algorithm. An additional source of noise for the other ratios was from the stochasticity of injections, with a distinct set for each trial. Consistent with common sense, as the injection ratio was ramped up the trend across infill scenarios was a degradation of the performance metric. ",
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+        "text": "296 We did find that with increased repetitions incorporated the spread of the averaged performance   \n97 metrics were tightened, leading us to repeat the experiments at increased scale for some improved   \n98 statistical significance.   \n299 For the missing at random injections [Fig 2, 3], ML infill was at or near top performance across both   \n300 data sets, although the spread between imputations was not extremely pronounced. In most of the   \n301 setups, mode imputation and adjacent cell trended as reduced performance in comparison to ML infill   \n302 or the default imputations (mean for numeric sets and distinct activation set for categoric).   \n303 For not at random injections to the right tail of numeric sets [Fig 4], it appears that ML infill had a   \n304 pronounced benefit to the Ames Housing data set [30], especially as the injection ratio increased,   \n305 and more of an intermediate performance to the Allstate Claims data set [31]. We speculate that ML   \n306 infill had some degree of variability across these demonstrations due to correlations (or lack thereof)   \n307 between the target feature and the other features, without which ML infill may struggle to establish a   \n308 basis for inference. In the final scenario of not at random injections to categoric [Fig 5] we believe   \n309 default performed well because it served as a direct replacement for the single missing activation.   \n310 An additional comparable series of injections were conducted with ML infill and the added difference   \n311 of appending the NArw support columns corresponding to the target columns for injections. Again   \n312 these NArw support columns are the boolean integer markers for presence of infill in the corresponding   \n313 entries which support the partitioning of sets for ML infill. The expectation was that by using these   \n314 markers to signal to the training operation which of the entries were subjected to infill, there would   \n315 be some benefit to downstream model performance. For many of the scenarios the visible impact was   \n316 that supplementing with the NArw support column improved the ML infill performance, demonstrated   \n317 here for missing at random [Fig 6, 7] and missing not at random [Fig 8, 9] with the other imputation   \n318 scenarios shown again for context. ",
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+            "Figure 2: Missing at Random - Numeric Target Feature ",
+            "Figure 3: Missing at Random - Categoric Target Feature "
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+            "Figure 4: Not at Random - Numeric Target Feature ",
+            "Figure 5: Not at Random - Categoric Target Feature "
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+        "type": "text",
+        "text": "319 8 Discussion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "320 One of the primary goals of this experiment was to validate the efficacy of ML infill as evidenced   \n321 by improvements to downstream model performance. For the Ames Housing benchmark data set,   \n322 there was a notable demonstration of ML infill benefiting model performance in the scenario of   \n323 the numeric target column with not at random injections at increased injection ratios, and also to   \n324 a lesser extent with missing at random injections. We speculate an explanation for this advantage   \n325 towards the numeric target columns may partly be attributed to the fact that the downstream model   \n326 was also a regression application, so that the other features selected for label correlation may by   \n327 proxy have correlations with the target numeric feature. The corollary is that the more mundane   \n328 performance of ML infill toward the categoric target columns may be a result of these having less   \n329 correspondence with the surrounding features. The fact that even in these cases the ML infill still fell   \n330 within noise distribution of the other imputation scenarios we believe presents a reasonable argument   \n331 for defaulting to ML infill for tabular learning.   \n332 Note that as another argument for defaulting to ML infill as opposed to static imputations is that the   \n333 imputation model may serve as a hedge against imperfections in subsequent data streams, particularly   \n334 if one of the features experiences downtime in a streaming application for instance.   \n335 The other key finding of the experiment was the pronounced benefit to downstream model performance   \n336 when including the NArw support column in the returned data set as a supplement to ML infill. This   \n337 finding was consistent with our intuition, which was that increased information retention about infill   \n338 points should help model performance. Note there is some small tradeoff, as the added training set   \n339 dimensionality may increase training time. Another benefit to including NArw support columns may   \n340 be for interpretability in inspection of imputations. We recommend including the NArw support   \n341 columns for model training based on these findings, with the one caveat that care should be taken to   \n342 avoid inclusion in the data leakage scenario where there is some kind of correlation between presence   \n343 of missing data and label set properties that won’t be present in production. ",
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+        "image_caption": [
+            "Figure 6: NArw comparison - Missing at Random - Numeric Target Feature ",
+            "Missing at Random - Categoric "
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+        "image_caption": [
+            "Figure 7: NArw comparison - Missing at Random - Categoric Target Feature "
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+        "image_caption": [
+            "Figure 8: NArw comparison - Not at Random - Numeric Target Feature ",
+            "t Random-Categoric ",
+            "Figure 9: NArw comparison - Not at Random - Categoric Target Feature "
+        ],
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+    {
+        "type": "text",
+        "text": "344 9 Conclusion ",
+        "text_level": 1,
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+        "text": "45 Automunge offers a push-button solution to preparing tabular data for ML, with automated data   \n46 cleaning operations like normalizations, binarizations, and auto ML derived missing data imputation   \n47 aka ML infill. Transformations and imputations are fit to properties of a designated train set, and with   \n48 application of automunge(.) a compact dictionary is returned recording transformation parameters   \n49 and trained imputation models, which dictionary may then serve as a key for consistently preparing   \n50 additional data on the train set basis with postmunge(.).   \n351 We hope that these experiments may serve as a kind of validation of defaulting to ML infill with   \n352 supplemented NArw support columns in tabular learning for users of the Automunge library, as   \n353 even if in our experiments the material benefits towards downstream model performance were not   \n354 demonstrated for all target feature scenarios, in other cases there did not appear to be any material   \n355 penalty. Note that ML infill can be activated for push-button operation by the automunge(.) parameter   \n356 MLinfill True and the NArw support columns included by parameter NArw_marker $\\backsimeq$ True. Based   \n357 on these findings these two parameter settings are now cast as defaults for the Automunge platform. ",
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+    {
+        "type": "text",
+        "text": "358 Acknowledgments ",
+        "text_level": 1,
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+    {
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+        "text": "A thank you owed to those facilitators behind Stack Overflow, Python, Numpy, Scipy Stats, PyPI, GitHub, Colaboratory, Anaconda, VSCode, and Jupyter. Special thanks to Scikit-Learn and Pandas. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            167,
+            266,
+            184
+        ],
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+    },
+    {
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+        "text": "362 [1] Daniel J. Stekhoven, Peter Bühlmann. MissForest - nonparametric missing value imputation for mixed-type   \n363 data (2011) arXiv:1105.0828   \n364 [2] Author(s) (2021) Automunge, GitHub repository (Please see supplemental material)   \n365 [3] W. McKinney. Data structures for statistical computing in python. Proceedings of the 9th Python in Science   \n366 Conference, pages 51–56, 2010.   \n367 [4] L. Breiman. Random Forests. Machine Learning, 45(1), 2001.   \n368 [5] Pedregosa et al., Scikit-learn: Machine Learning in Python, JMLR 12, pp. 2825-2830, 2011.   \n369 [6] H. Wickham. Tidy data. Journal of Statistical Software, 59(10), 2014.   \n370 [7] John Moody. Fast Learning in Multi-Resolution Hierarchies. NIPS Proceedings, 1989   \n371 [8] Kilian Weinberger, Anirban Dasgupta, John Langford, Alex Smola, Josh Attenberg. Feature Hashing for   \n372 Large Scale Multitask Learning. ICML Proceedings, 2009   \n373 [9] Author(s) Hashed Categoric Encodings with Automunge (2020) (Please see preprint in supplemental material)   \n374 [10] Author(s) Numeric Encoding Options with Automunge (2020) (Please see preprint in supplemental material)   \n375 [11] Author(s) Parsed Categoric Encodings with Automunge (2020) (Please see preprint in supplemental   \n376 material)   \n377 [12] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, Zbigniew Wojna. Rethinking the Inception   \n378 Architecture for Computer Vision. IEEE conference on computer vision and pattern recognition, 2016   \n379 [13] Author(s) Specification of Derivations with Automunge (2020) (Please see preprint in supplemental   \n380 material)   \n381 [14] Stef van Buuren, Karin Groothuis-Oudshoorn. mice: Multivariate Imputation by Chained Equations in R   \n382 (2011) https://www.jstatsoft.org/article/view/v045i03   \n383 [15] Anna Veronika Dorogush, Vasily Ershov, Andrey Gulin. CatBoost: gradient boosting with categorical   \n384 features support (2018) arXiv:1810.11363   \n385 [16] Nick Erickson, Jonas Mueller, Alexander Shirkov, Hang Zhang, Pedro Larroy, Mu Li, and Alexander Smola.   \n386 AutoGluon-Tabular: Robust and Accurate AutoML for Structured Data (2020) arxiv:2003.06505   \n387 [17] Chi Wang, Qingyun Wu, Markus Weimer, Erkang Zhu. FLAML: A Fast and Lightweight AutoML Library   \n388 (2019) arXiv:1911.04706   \n389 [18] Jinsung Yoon, James Jordon, Mihaela van der Schaar. GAIN: Missing Data Imputation using Generative   \n390 Adversarial Nets (2018 International Conference of Machine Learning), arXiv:1806.02920   \n391 [19] Yuxuan Zhao, Madeleine Udell. Missing Value Imputation for Mixed Data via Gaussian Copula (KDD   \n392 2020), arXiv:1910.12845   \n393 [20] K. Mohan, J. Pearl. Graphical Models for Processing Missing Data (2019), arXiv:1801.03583   \n394 [21] Olga Troyanskaya, Michael Cantor, Gavin Sherlock, Pat Brown, Trevor Hastie, Robert Tibshirani, David   \n395 Botstein and Russ B. Altman. Missing value estimation methods for DNA microarrays, BIOINFORMATICS   \n396 Vol. 17 no. 6, 2001 Pages 520-525.   \n397 [22] Matei Zaharia, Reynold S. Xin, Patrick Wendell, Tathagata Das, Michael Armbrust, Ankur Dave, Xiangrui   \n398 Meng, Josh Rosen, Shivaram Venkataraman, Michael J. Franklin, Ali Ghodsi, Joseph Gonzalez, Scott Shenker,   \n399 Ion Stoica. Apache Spark: a unified engine for big data processing. Communications of the ACM, 59(11), 2016   \n400 [23] Dask Development Team. Dask: Library for dynamic task scheduling (2016) https://dask.org   \n401 [24] Philipp Moritz, Robert Nishihara, Stephanie Wang, Alexey Tumanov, Richard Liaw, Eric Liang, Melih   \n402 Elibol, Zongheng Yang, William Paul, Michael I. Jordan, Ion Stoica. Ray: A Distributed Framework for   \n403 Emerging AI Applications. 13th USENIX Symposium on Operating Systems Design and Implementation   \n404 (2018), arXiv:1712.05889 ",
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+        "text": "[26] Abadi, Martín, Barham P, Chen J, Chen Z, Davis A, Dean J, et al. Tensorflow: A system for large-scale machine learning. 12th USENIX Symposium on Operating Systems Design and Implementation (2016) p. 265–83.   \n[27] Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Kopf, Andreas and Yang, Edward and DeVito, Zachary and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith. PyTorch: An Imperative Style, High-Performance Deep Learning Library. NeurIPS Proceedings, 2019   \n[28] Terrence Parr, Kerem Turgutlu, Christopher Csiszar, and Jeremy Howard. Beware default random forest importances. Explained.ai (blog), 2018. https://explained.ai/rf-importance/.   \n[29] Author(s) Automunge Influence (2020) (Please see preprint in supplemental material)   \n[30] Dean De Cock. Ames, Iowa: Alternative to the Boston Housing Data as an End of Semester Regression Project, Journal of Statistics Education, Volume 19, Number 3 (2011)   \n[31] Kaggle: Allstate Claims Severity, https://www.kaggle.com/c/allstate-claims-severity ",
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