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+# SURROGATE-BASED CONSTRAINED LANGEVIN SAMPLING WITH APPLICATIONS TO OPTIMAL MATERIAL CONFIGURATION DESIGN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We consider the problem of generating configurations that satisfy physical constraints for optimal material nano-pattern design, where multiple (and often conflicting) properties need to be simultaneously satisfied. Consider, for example, the trade-off between thermal resistance, electrical conductivity, and mechanical stability needed to design a nano-porous template with optimal thermoelectric efficiency. To that end, we leverage the posterior regularization framework and show that this constraint satisfaction problem can be formulated as sampling from a Gibbs distribution. The main challenges come from the black-box nature of those physical constraints, since they are obtained via solving highly non-linear PDEs. To overcome those difficulties, we introduce Surrogate-based Constrained Langevin dynamics for black-box sampling. We explore two surrogate approaches. The first approach exploits zero-order approximation of gradients in the Langevin Sampling and we refer to it as Zero-Order Langevin. In practice, this approach can be prohibitive since we still need to often query the expensive PDE solvers. The second approach approximates the gradients in the Langevin dynamics with deep neural networks, allowing us an efficient sampling strategy using the surrogate model. We prove the convergence of those two approaches when the target distribution is log-concave and smooth. We show the effectiveness of both approaches in designing optimal nano-porous material configurations, where the goal is to produce nano-pattern templates with low thermal conductivity and reasonable mechanical stability.
+
+# 1 INTRODUCTION
+
+In many real-world design problems, the optimal design needs to simultaneously satisfy multiple constraints, which can be expensive to estimate. For example, in computational material design, the goal is to come up with material configurations, or samples, satisfying a list of physical constraints that are given by black-box numerical Partial Differential Equations (PDE) solvers. Such solvers (for example, the Boltzmann Transport Equation solver) are often complex, expensive to evaluate, and offer no access to their inner variables or their gradients.
+
+We pose this design-under-constraints problem as sampling from a Gibbs distribution defined on some compact support. The problem of sampling from a distribution with unknown likelihood that can only be point-wise evaluated is called black-box sampling (Chen & Schmeiser, 1998; Neal, 2003). We show in this paper that constrained black-box sampling can be cast as a constrained Langevin dynamics with gradient-free methods. Zero-order optimization via Gaussian smoothing was introduced in Nesterov & Spokoiny (2017) and extended to black-box sampling with Langevin dynamics in Shen et al. (2019). We extend this approach to the constrained setting from a black-box density with compact support.
+
+However, one shortcoming of this approach is that it is computationally very expensive since it requires repeatedly querying PDE solvers in order to get an estimate of the gradient. To alleviate computational issues, we propose Surrogate Model Based Langevin dynamics, that consists of two steps: (i) Learning (using training data) an approximation of the gradient of the potential of the Gibbs distribution. We show that learning the gradient, rather than the potential itself, is important for the mixing of the Langevin dynamics towards the target Gibbs distribution. We devise several objective functions, as well as deep neural-network architectures for parameterizing the approximating function class, for learning the gradient of the potential function. (ii) We then use the surrogate gradient model in the constrained Langevin dynamics in lieu of the black-box potential. Using the surrogate enables more efficient sampling, since it avoids querying the expensive PDE solvers, and obtaining gradients is as efficient as evaluating the functions themselves using automatic differentiation frameworks such as PyTorch or TensorFlow.
+
+To summarize, our main contributions are as follows:
+
+1. We cast the problem of generating samples under constraints in the black-box setting as sampling from a Gibbs distribution.   
+2. We introduce Constrained Zero-Order Langevin Monte Carlo, using projection or proximal methods, and provide the proof of its convergence to the target Gibbs distribution.   
+3. We introduce Surrogate Model Based Projected Langevin Monte Carlo via learning the gradient of the potential of the Gibbs distribution using deep neural networks or reproducing kernel spaces, and prove its convergence to the target distribution when used in conjunction with projection or proximal based methods. We shed the light on the importance of the approximation of the gradient of the potential, and we show how to achieve this using Hermite and Taylor learning.   
+4. We showcase the usability and effectiveness of the proposed methods for the design of nanoporous configurations with improved thermoelectric efficiency. The design consists of finding new configurations with optimized pore locations, such that the resulting configurations have favorable thermal conductivity (i.e., minimal $\kappa$ ) and desired mechanical stability (von Mises Stress $\sigma \leq \tau$ , where $\tau$ is some preset threshold).
+
+# 2 FROM CONSTRAINTS SATISFACTION TO SAMPLING FROM A GIBBS DISTRIBUTION: POSTERIOR REGULARIZATION
+
+In black-box optimization problems (such as the material design under consideration), the goal is to find a posterior distribution $q$ of samples satisfying a list of equality and inequality constraints: $\psi _ { j } ( x ) = y _ { k } , j = 1 \ldots C _ { e }$ , and $\begin{array} { r } { \phi _ { k } ( x ) \le b _ { k } , k = 1 . . . C _ { i } } \end{array}$ where $x \in \Omega$ and $\Omega \doteq \mathbb { R } ^ { d }$ is a bounded domain. We assume a prior distribution $p _ { 0 }$ (whose analytical form is known). The main challenge in black-box optimization is that the functions $\psi _ { j }$ and $\phi _ { k }$ can be only evaluated point-wise, and neither do we have functional forms nor access to their gradients. For example, $\psi$ and $\phi$ might be obtained via aggregating some statistics on the solution of a nonlinear PDE given by a complex solver.
+
+To make the problem of learning under constraints tractable, we choose Lagrangian parameters $\lambda _ { j } > 0$ and obtain the following relaxed objective:
+
+$$
+\operatorname* { m i n } _ { \substack { q , \int _ { \Omega } q ( x ) = 1 } } \mathrm { K L } ( q , p _ { 0 } ) + \sum _ { j = 1 } ^ { C _ { c } } \lambda _ { j } \mathbb { E } _ { x \sim q } ( \psi _ { j } ( x ) - y _ { k } ) ^ { 2 } + \sum _ { k = 1 } ^ { C _ { i } } \lambda _ { k } \mathbb { E } _ { x \sim q } ( \phi _ { k } ( x ) - b _ { k } ) _ { + }
+$$
+
+The formulation in Eq. 1 is similar in spirit to the posterior regularization framework of Ganchev et al. (2010); Hu et al. (2018). However, we highlight two differences: (i) our focus is on constrained settings (where $\Omega$ is bounded), and (ii) we assume a black-box setting. We first obtain:
+
+Lemma 1 (Constraint Satisfaction as Sampling from a Gibbs Distribution). The solution to the distribution learning problem given in Eq. 1 is given by:
+
+$$
+\pi ( x ) = \frac { \exp ( - U ( x ) ) } { Z } \mathbb { 1 } _ { x \in \Omega }
+$$
+
+where $\begin{array} { r } { U ( x ) ~ = ~ - \log p _ { 0 } ( x ) + \sum _ { j = 1 } ^ { C _ { e } } \lambda _ { j } ( \psi _ { j } ( x ) - y _ { k } ) ^ { 2 } + \sum _ { k = 1 } ^ { C _ { i } } \lambda _ { k } ( \phi _ { k } ( x ) - b _ { k } ) _ { + } } \end{array}$ and $Z =$ $\begin{array} { r } { \int _ { x \in \Omega } \exp \left( - U ( x ) \right) d x } \end{array}$ .
+
+Lemma 1 shows that the constraint satisfaction problem formulated in Eq. 1 amounts to sampling from a Gibbs distribution defined on a compact support given in Eq. 2. Sampling from a Gibbs distribution (also known as Boltzmann distribution) has a long history using Langevin dynamics. In the white-box setting when the functions defining the constraints have explicit analytical forms as well as their gradients, Langevin dynamics for Gibbs distribution sampling defined on a compact domain $\Omega$ and their mixing properties were actively studied in Bubeck et al. (2015); Brosse et al. (2017). In the next Section, we provide a more detailed review.
+
+Remark 1 (Relation to Bayesian Optimization). While in Bayesian optimization we are interested in finding a point that satisfies the constraints, in our setting we are interested in finding a distribution of candidate samples that satisfy (black-box) constraints. See (Suzuki et al., 2019) for more details.
+
+Remark 2. For the rest of the paper, we will assume $p _ { 0 }$ to be the uniform distribution on $\Omega$ , which means that its gradients are zero on the support of the domain $\Omega$ . Otherwise, if $p _ { 0 }$ is known and belongs to, for instance, an exponential family or a generative model prior (such as normalizing flows), we can sample from $\pi$ using a mixture of black-box sampling on the constraints $( \psi _ { j } , \phi _ { k } )$ and white-box sampling on $\log ( p _ { 0 } )$ .
+
+# 3 WHITE-BOX SAMPLING: CONSTRAINED LANGEVIN DYNAMICS
+
+We review in this section Langevin dynamics in the unconstrained case $\Omega = \mathbb { R } ^ { d }$ ) and the constrained setting $\smash { ( \Omega \subset \mathbb { R } ^ { d } ) }$ ). Below, $\lVert \cdot \rVert$ denotes the Euclidean norm unless otherwise specified. We are interested in sampling from
+
+$$
+\pi ( x ) = \frac { 1 } { Z } \exp ( - U ( x ) ) \mathbb { 1 } _ { x \in \Omega } ,
+$$
+
+Preliminaries. We give here assumptions, definitions and few preliminary known facts that will be useful later. Those assumptions are commonly used in Langevin sampling analysis (Dalalyan, 2017; Bubeck et al., 2015; Brosse et al., 2017; Durmus et al., 2019).
+
+1. Assumption A: We assume $\Omega$ is a convex such that $0 \in \Omega$ , $\Omega$ contains a Euclidean ball of radius $r$ , and $\Omega$ is contained in a Euclidean ball of radius R. (For example, $\Omega$ might encode box constraints.) The projection onto $\Omega , P _ { \Omega } ( x )$ is defined as follows: for all $x \in \Omega$ , $\begin{array} { r } { P _ { \Omega } ( x ) = \arg \operatorname* { m i n } _ { z \in \Omega } \left\| x - z \right\| ^ { 2 } } \end{array}$ . Let $R = \operatorname* { s u p } _ { x , x ^ { \prime } \in \Omega } | | x - x ^ { \prime } | | < \infty$ .
+
+2. Assumption B: We assume that $U$ is convex, $\beta$ -smooth, and with bounded gradients:
+
+$$
+\begin{array} { r l } & { \left\| \nabla _ { x } U ( x ) - \nabla _ { y } U ( y ) \right\| \leq \beta \left\| x - y \right\| , \quad \forall x , y \in \Omega \mathrm { ~ ( \beta \beta \mathrm { - } s m o o t h n e s s ) } . } \\ & { } \\ & { \left\| \nabla U ( x ) \right\| \leq L , \quad \forall x \in \Omega \mathrm { ~ ( B o u n d e d n e s s ) } . } \end{array}
+$$
+
+The Total Variation (TV) distance between two measures $\mu , \nu$ is defined as follows: $T V ( \mu , \nu ) =$ $\operatorname* { s u p } _ { A } | \mu ( A ) - \nu ( A ) |$ . Pinsker Inequality relates $\mathrm { K L }$ divergence to TV: $T V ( \mu , \nu ) \leq { \sqrt { 2 \mathrm { K L } ( \mu , \nu ) } }$ .
+
+Unconstrained Langevin Dynamics. In the unconstrained case, the goal is to sample from a Gibbs distribution $\pi ( x ) = { \mathrm { e x p } } ( - U ( x ) ) / Z$ that has unbounded support. This sampling can be done via the Langevin Monte Carlo (LMC) algorithm, which is given by the following iteration:
+
+$$
+X _ { k + 1 } = X _ { k } - \eta \nabla _ { x } U ( X _ { k } ) + \sqrt { 2 \lambda \eta } \xi _ { k } , \quad k = 0 \ldots K - 1 \mathrm { ( L M C ) } ,
+$$
+
+where $\xi _ { k } \sim \mathcal { N } ( 0 , I _ { d } )$ , $\eta$ is the learning rate, and $\lambda > 0$ is a variance term.
+
+Constrained Langevin Dynamics. In the constrained case, the goal is to sample from $\pi ( x ) =$ $\exp ( - U ( x ) ) / Z \mathbb { 1 } _ { x \in \Omega } .$ ,. We discuss two variants:
+
+Projected Langevin Dynamics. Similar to projected gradient descent, Bubeck et al. (2015) introduced Projected Langevin Monte Carlo (PLMC) and proved its mixing propreties towards the stationary distribution $\pi$ . PLMC is given by the following iteration :
+
+$$
+X _ { k + 1 } = P _ { \Omega } \left( X _ { k } - \eta \nabla _ { x } U ( X _ { k } ) + \sqrt { 2 \lambda \eta } \xi _ { k } \right) , \quad k = 0 \ldots K - 1 \ \mathrm { ( P L M C ) } ,
+$$
+
+In essence, PLMC consists of a single iteration of LMC, followed by a projection on the set $\Omega$ using the operator $P _ { \Omega }$ .
+
+Proximal Langevin Dynamics. Similar to proximal methods in constrained optimization, Brosse et al. (2017) introduced Proximal LMC (ProxLMC) that uses the iteration:
+
+$$
+X _ { k + 1 } = \left( 1 - { \frac { \eta } { \gamma } } \right) X _ { k } - \eta \nabla _ { x } U ( X _ { k } ) + { \frac { \eta } { \gamma } } P _ { \Omega } ( X _ { k } ) + { \sqrt { 2 \lambda \eta } } \xi _ { k } , k = 0 \ldots K - 1 , \ \mathbf { ( P r o x { L M C } ) }
+$$
+
+where $\eta$ is the step size and $\gamma$ is a regularization parameter. In essence, ProxLMC (Brosse et al., 2017) performs an ordinary LMC on $\bar { U } ^ { \gamma } ( x ) = U ( \bar { x } ) + i _ { \Omega } ^ { \gamma } ( x )$ , where $i _ { \Omega } ^ { \gamma } ( x )$ is the proximal operator:
+
+$$
+i _ { \Omega } ^ { \gamma } ( x ) = \operatorname* { i n f } _ { y } i _ { \Omega } ( x ) + ( 2 \gamma ) ^ { - 1 } \left\| x - y \right\| ^ { 2 } = ( 2 \gamma ) ^ { - 1 } \left\| x - P _ { \Omega } ( x ) \right\| ^ { 2 } ,
+$$
+
+where $i _ { \Omega } ( x ) = 0$ for $x \in \Omega$ and $i _ { \Omega } ( x ) = \infty$ for $x \notin \Omega$ . Therefore, the update in Eq. 6 is a regular Langevin update (as in Eq. 4) with potential gradient $\nabla _ { x } U ^ { \gamma } ( x ) = \nabla _ { x } U ( \boldsymbol { \dot { x } } ) + \gamma ^ { - 1 } \big ( \boldsymbol { \dot { x } } - P _ { \Omega } ( x ) \big )$ .
+
+We denote by and $\mu _ { K } ^ { \mathrm { P r o x L M C } }$ he di ributions of $X _ { K }$ obtained by iterating Eq. 5 and Eq. 6 $\mathbf { A }$ $\mathbf { B }$   
+distribution $\pi$ in the total variation distance. In particular, Bubeck et al. (2015) showed that for   
+$\eta = \tilde { \Theta } ( R ^ { 2 } / K )$ , we obtain:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { P L M C } } , \pi ) \leq \varepsilon \ \mathrm { f o r } \ K = \tilde { \Omega } ( \varepsilon ^ { - 1 2 } d ^ { 1 2 } ) .
+$$
+
+Likewise, Brosse et al. (2017) showed that for $0 < \eta \leq \gamma ( 1 + \beta ^ { 2 } \gamma ^ { 2 } ) ^ { - 1 }$ , we obtain:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { { P r o x L M C } } } , \pi ) \le \varepsilon \ \mathrm { f o r } \ K = \tilde { \Omega } ( \varepsilon ^ { - 6 } d ^ { 5 } ) ,
+$$
+
+where the notation $\alpha _ { n } = \tilde { \Omega } ( \beta _ { n } )$ means that there exists $c \in \mathbb { R } , C > 0$ such that $\alpha _ { n } \geq C \beta _ { n } \log ^ { c } ( \beta _ { n } )$
+
+# 4 CONSTRAINED LANGEVIN DYNAMICS IN THE BLACK-BOX SETTING
+
+We now introduce our variants of constrained LMC for the black-box setting where explicit potential gradients are unavailable. We explore in this paper two strategies for approximating the gradient of $U$ in the black-box setting. In the first strategy, we borrow ideas from derivative-free optimization (in particular, evolutionary search). In the second strategy we learn a surrogate deep model that approximates the gradient of the potential. Below, let $G : \overline { { \Omega } }  \mathbb { R } ^ { d }$ be a vector valued function that approximates the gradient of the potential, $\nabla _ { x } U$ . We make:
+
+Assumption C. The surrogate gradient $G$ satisfies $\mathbb { E } \left\| G ( Y _ { k } ) \right\| ^ { 2 } < \infty , \forall k .$ .
+
+Surrogate Projected Langevin Dynamics. Given $Y _ { 0 }$ , the Surrogate Projected LMC (S-PLMC) replaces the potential gradient $\nabla _ { x } U$ in Eq. 5 with the surrogate gradient $G$ :
+
+$$
+Y _ { k + 1 } = P _ { \Omega } \left( Y _ { k } - \eta \mathbf { G } ( Y _ { k } ) + \sqrt { 2 \lambda \eta } \xi _ { k } \right) , k = 0 \ldots K - 1 \ \mathbf { \Omega } ( \mathbf { S } \mathbf { - } \mathbf { P } \mathbf { L } \mathbf { M } \mathbf { C } )
+$$
+
+Surrogate Proximal Langevin Dynamics. Similarly, the Surrogate Proximal LMC (S-ProxLMC) replaces the unknown potential gradient $\nabla _ { x } U$ in Eq. 6 with the gradient surrogate $G$ :
+
+$$
+Y _ { k + 1 } = \left( 1 - { \frac { \eta } { \gamma } } \right) Y _ { k } - \eta \mathbf { G } ( Y _ { k } ) + { \frac { \eta } { \gamma } } P _ { \Omega } ( Y _ { k } ) + { \sqrt { 2 \lambda \eta } } \xi _ { k } , k = 0 \ldots K - 1 \ { \mathrm { ~ ( S - P r o x { L M C } ) } }
+$$
+
+We now present our main theorems on the approximation properties of surrogate LMC (S-PLMC, and S-ProxLMC). We do so by bounding the total variation distance between the trajectories of the surrogate Langevin dynamics (S-PLMC, and S-ProxLMC) and the true LMC dynamics (PLMC and ProxLMC). Theorem 1 is an application of techniques in Stochastic Differential Equations (SDE) introduced in Dalalyan & Tsybakov (2012) and is mainly based on a variant of Grisanov’s Theorem for change of measures (Lipster & Shiryaev, 2001) and Pinsker’s Inequality that bounds total variation in terms of Kullback-Leibler divergence.
+
+Theorem 1 (S-PLMC and S-ProxLMC Mixing Properties). Under Assumption $\mathbf { C }$ , we have:
+
+1. S-PLMC Convergence. Let $\mu _ { K } ^ { \mathrm { P L M C } }$ be the distribution of the random variable $X _ { K }$ obtained by iterating PLMC Eq. 5, and $\mu _ { K } ^ { \mathrm { S - P L M C } }$ be the distribution of the random variable $Y _ { K }$ obtained by iteration S-PLMC given in Eq. 9. We have:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { { S - P L M C } } } , \mu _ { K } ^ { \mathrm { { P L M C } } } ) \leq \sqrt { \frac { \eta } { \lambda } } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( Y _ { k } ) - \nabla _ { x } U ( Y _ { k } ) \right\| ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+2. S-ProxLMC Convergence. Let µProxLMCK be the distribution of the random variable XK obtained by iterating ProxLMC Eq. 6, and $\mu _ { K } ^ { S - }$ ProxLMC be the distribution of the random variable $Y _ { K }$ obtained by iterating S-ProxLMC given in Eq. 10. We have:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { { S } \cdot P r o x L M C } } , \mu _ { K } ^ { \mathrm { { P r o x L M C } } } ) \leq \sqrt { \frac { \eta } { 2 \lambda } } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( X _ { k } ) - \nabla _ { x } U ( X _ { k } ) \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+From Theorem 1, we see that it suffices to approximate the potential gradient $\nabla _ { x } U ( X )$ (and not the potential $U ( X ) )$ in order to guarantee convergence of surrogate-based Langevin sampling. Using the triangle inequality, and combining Theorem 1 and bounds in Eqs 7 and 8 we obtain:
+
+Theorem 2. (Convergence of Surrogate Constrained LMC to the Gibbs distribution.) Under assumptions $^ { A , B }$ and $C$ we have:
+
+1. Assume in S-PLMC that there exists $\delta > 0$ such that E $\begin{array} { r } { \left\| G ( Y _ { k } ) - \nabla _ { x } U ( Y _ { k } ) \right\| ^ { 2 } \leq \delta , \forall k \geq 0 . } \end{array}$ . Set $\lambda = 1$ , and $-$ where $\alpha = 1 / ( \delta + \beta ^ { 2 } R ^ { 2 } )$ . Then for $K = \tilde { \Omega } ( \varepsilon ^ { - 1 2 } d ^ { 1 2 } )$ , we have:
+
+$$
+T V ( \mu _ { K } ^ { \mathtt { S - P L M C } } , \pi ) \le \varepsilon .
+$$
+
+2. Assume in S-ProxLMC that there exists $\delta > 0$ such that $\begin{array} { r } { \mathbb { E } \left\| G ( X _ { k } ) - \nabla _ { x } U ( X _ { k } ) \right\| ^ { 2 } \leq \delta , \forall k \geq 0 . } \end{array}$ . Set $\lambda = 1$ , and $\begin{array} { r } { \eta = \operatorname* { m i n } ( \gamma ( 1 + \beta ^ { 2 } \gamma ^ { 2 } ) ^ { - 1 } , \frac { 1 } { \delta K ^ { 2 } } ) } \end{array}$ ). Then for $K = \tilde { \Omega } ( \varepsilon ^ { - 6 } d ^ { 5 } )$ we have:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { S - P r o x L M C } } , \pi ) \leq \varepsilon .
+$$
+
+# 5 ZERO-ORDER CONSTRAINED LANGEVIN DYNAMICS
+
+In zero-order optimization (Nesterov & Spokoiny, 2017; Duchi et al., 2015; Ghadimi & Lan, 2013; Shen et al., 2019), one considers the Gaussian smoothed potential $U _ { \nu }$ defined as $U _ { \nu } ( x ) =$ $\mathbb { E } _ { g \sim \mathcal { N } ( 0 , I _ { d } ) } U ( x + \nu g )$ , and its gradient is given by $\begin{array} { r } { \nabla _ { x } U _ { \nu } ( x ) = \mathbb { E } _ { g } \frac { U ( x + \nu g ) - U ( x ) } { \nu } g } \end{array}$ . The following is a Monte Carlo estimate of ∇xUν (x): n
+
+$$
+\hat { G } _ { n } U ( x ) = \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \left( \frac { U ( x + \nu g _ { j } ) - U ( x ) } { \nu } \right) g _ { j } ,
+$$
+
+where $g _ { 1 } , \ldots g _ { n }$ are i.i.d. standard normal vectors.
+
+Zero-Order sampling from log-concave densities was recently studied in Shen et al. (2019). We extend it here to the constrained sampling case of log-concave densities with compact support. We define Constrained Zero-Order Projected LMC $\mathbf { Z }$ -PLMC) and Zero-Order Proximal LMC $\mathbf { Z }$ -ProxLMC) by setting $\mathbf { G } ( \mathbf { x } ) = \hat { \mathbf { G } } _ { \mathbf { n } } \mathbf { U } ( \mathbf { x } )$ in Eq. 9 and Eq. 10 respectively.
+
+Lemma 2 (Zero-Order Gradient Approximation(Nesterov & Spokoiny, 2017; Shen et al., 2019)). Under Assumption $B$ , we have for all $x \in \Omega$ :
+
+$$
+\begin{array} { r } { \mathbb { E } _ { g _ { 1 } , \dots , g _ { n } } \left\| \hat { G } _ { n } U ( x ) - \nabla _ { x } U ( x ) \right\| ^ { 2 } \leq \left( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } L \right) ^ { 2 } / n } \end{array}
+$$
+
+Thanks to Lemma 2 that ensures uniform approximation of gradients in expectation, we can apply Theorem 2 and get the following corollary for Z-PLMC and Z-ProxLMC:
+
+Corollary 1 (Zero-order Constrained Langevin approximates the Gibbs distribution). Under Assumptions A and $\pmb { B }$ , let $\delta \in [ 0 , 1 ] ,$ , for $n \geq \left( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } L \right) ^ { 2 } / \delta$ , we have the following bounds in expectation:
+
+1. Set $\lambda = 1$ , and $\eta = \tilde { \Theta } ( \operatorname* { m i n } ( R ^ { 2 } / K , \alpha / K ^ { 2 } ) )$ where $\alpha = 1 / ( \delta + \beta ^ { 2 } R ^ { 2 } )$ . For ${ \cal K } = \tilde { \Omega } ( \varepsilon ^ { - 1 2 } d ^ { 1 2 } ) ;$ we have:
+
+$$
+\mathbb { E } _ { g _ { 1 } , \dots g _ { n } } T V ( \mu _ { K } ^ { \mathrm { Z - P L M C } } , \pi ) \le \varepsilon .
+$$
+
+2. Set $\lambda = 1$ , and $\begin{array} { r } { \eta = \operatorname* { m i n } ( \gamma ( 1 + \beta ^ { 2 } \gamma ^ { 2 } ) ^ { - 1 } , \frac { 1 } { \delta K ^ { 2 } } ) } \end{array}$ . For $K = \tilde { \Omega } ( \varepsilon ^ { - 6 } d ^ { 5 } )$ we have:
+
+$$
+\mathbb { E } _ { g _ { 1 } , \dots g _ { n } } T V ( \mu _ { K } ^ { \mathrm { Z - P r o x L M C } } , \pi ) \le \varepsilon .
+$$
+
+Remark 3. For simplicity, we state the above bound in terms of expectations over the randomness in estimating the gradients. It is possible to get finite-sample bounds using the Vector Bernstein concentration inequality, coupled with covering number estimates of $\Omega$ but omit them due to space.
+
+# 6 SURROGATE MODEL BASED CONSTRAINED LANGEVIN DYNAMICS
+
+Despite its theoretical guarantees, zero-order constrained Langevin (Z-PLMC and Z-ProxLMC) has a prohibitive computation cost as it needs $O ( n K )$ black-box queries (in our case, invocations of a nonlinear PDE solver). To alleviate this issue, we introduce in this Section a neural surrogate model as an alternative to the gradient of the true potential.
+
+6.1 HERMITE LEARNING OF GRADIENTS: JACOBIAN MATCHING OF ZERO-ODER ESTIMATES
+
+From Theorem 2, we saw that in order to guarantee the convergence of constrained Langevin dynamics, we need a good estimate of the gradient of the potential of the Gibbs distribution. Recall that the potential given in Lemma 1 depends on $\psi _ { j }$ and $\phi _ { k }$ , which are scalar outputs of computationally heavy PDE solvers in our material design problem. To avoid this, we propose to train surrogate neural network models approximating each PDE output and their gradients. Concretely, suppose we are given a training set $S$ for a PDE solver for the property $\psi$ (dropping the index $j$ for simplicity):
+
+$$
+S = \{ ( x _ { i } , y _ { i } = \psi ( x _ { i } ) , \tilde { y } _ { i } = \hat { G } _ { n } \psi ( x _ { i } ) ) , x _ { i } \sim \rho _ { \Omega } \mathrm { i . i . d . } , i = 1 , \ldots , N \} ,
+$$
+
+where $\rho \Omega$ is the training distribution and ${ \hat { G } } _ { n } \psi ( . )$ is the zero-order estimate of the gradient of $\psi$ given in Eq. 13. We propose to learn a surrogate model belonging to a function class $\mathcal { H } _ { \theta }$ , $\hat { f } _ { \theta } \in \mathcal { H } _ { \theta }$ , that regresses the value of $\psi$ and matches the zero-order gradient estimates as follows:
+
+$$
+\underset { f _ { \theta } \in \mathcal H _ { \theta } } { \operatorname* { m i n } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \{ ( y _ { i } - f _ { \theta } ( x _ { i } ) ) ^ { 2 } + \| \nabla _ { x } f _ { \theta } ( x _ { i } ) - \tilde { y } _ { i } \| ^ { 2 } \} \mathrm { ( Z \ – H e r m i t e ~ L e a r n i n g ) }
+$$
+
+The problem in Eq. 17 was introduced and analyzed in Shi et al. (2010) where $\mathcal { H } _ { \theta }$ is a ball in a Reproducing Kernel Hilbert Space (RKHS). Following Shi et al. (2010), we refer to this type of learning as Hermite Learning. In the deep learning community, this type of learning is called Jacobian matching and was introduced in Srinivas & Fleuret (2018); Czarnecki et al. (2017) where $\mathcal { H } _ { \theta }$ is a deep neural network parameterized with weights $\theta$ . When $f _ { \theta }$ is a deep network, we can optimize this objective efficiently using common deep learning frameworks (PyTorch, TensorFlow).
+
+(Shi et al., 2010) have shown that when $\mathcal { H } _ { \theta }$ is an RKHS ball and when $\tilde { y } _ { i } = \nabla _ { x } \psi ( x _ { i } )$ are exact gradients, for a sufficiently large training set with $N = O ( 1 / \epsilon ^ { 1 / ( 2 r \zeta ) } )$ (where $r , \zeta$ are exponents in $[ 0 , 1 ]$ that depend on the regularity of the function $\psi$ ). Under the assumption that $\cdot$ we have: $\begin{array} { r } { \int _ { \Omega } \| \dot { \nabla } _ { x } f _ { \theta } ( x ) - \nabla _ { x } \psi ( x ) \| ^ { 2 } \rho _ { \Omega } ( x ) d x \leq \epsilon } \end{array}$ . Since we are using inexact zero-order gradients, we will incur an additional numerical error that is also bounded as shown in Lemma 2.
+
+# 6.2 TAYLOR LEARNING OF GRADIENTS
+
+While Jacobian matching of zero-order gradients is a sound approach, it remains expensive to construct the dataset, as we need for each point to have $2 n + 1$ queries of the PDE solver. We exploit in this section the Taylor learning framework of gradients that was introduced in Mukherjee & Zhou (2006); Mukherjee & Wu (2006), and Wu et al. (2010). In a nutshell, Mukherjee & Zhou (2006) suggests to learn a surrogate potential $f _ { \theta }$ and gradient $G _ { \Lambda }$ that are consistent with the first-order taylor expansion. Given a training set $S = \{ ( x _ { i } , y _ { i } = \psi ( x _ { i } ) ) , x \sim \rho _ { \Omega } , i = 1 \ldots N \}$ , Wu et al. (2010) suggest the following objective:
+
+$$
+\operatorname* { m i n } _ { f _ { \theta } \in \mathcal { H } _ { \theta } , G _ { \Lambda } \in \mathcal { H } _ { \Lambda } ^ { d } } \frac { 1 } { N ^ { 2 } } \sum _ { i , j } w _ { i j } ^ { \sigma } ( y _ { i } - f _ { \theta } ( x _ { j } ) + \langle G _ { \Lambda } ( x _ { i } ) , x _ { j } - x _ { i } \rangle ) ^ { 2 } ( \mathbf { T a y l o r } - 2 ) ,
+$$
+
+where $\begin{array} { r } { w _ { i j } ^ { \sigma } = \exp \left( \frac { - \| x _ { i } - x _ { j } \| ^ { 2 } } { \sigma ^ { 2 } } \right) } \end{array}$ , $\mathcal { H } _ { \theta }$ is an RKHS ball of scalar valued functions, and $\mathcal { H } _ { \Lambda } ^ { d }$ is an RKHS ball of vector valued functions.
+
+Under mild assumptions, Mukherjee & Zhou (2006) shows that we have for $N = O ( 1 / \epsilon ^ { d / 2 } )$ : $\begin{array} { r } { \int _ { \Omega } \left. G _ { \Lambda } ( x ) - \nabla _ { x } \psi ( x ) \right. ^ { 2 } \rho _ { \Omega } ( x ) d x \leq \epsilon } \end{array}$ . We simplify the problem in Eq. 18 and propose the following two objective functions and leverage the deep learning toolkit to parameterize the surrogate $f _ { \theta }$ :
+
+$$
+\operatorname* { m i n } _ { f _ { \theta } \in \mathcal { H } _ { \theta } } \frac { 1 } { N ^ { 2 } } \sum _ { i , j } w _ { i j } ^ { \sigma } ( y _ { i } - f _ { \theta } ( x _ { j } ) + \langle \nabla _ { x } f _ { \theta } ( x _ { i } ) , x _ { j } - x _ { i } \rangle ) ^ { 2 } ( \mathbf { \mathrm { T a y l o r } } \mathbf { - } \mathbf { 1 } ) ,
+$$
+
+$$
+\operatorname* { m i n } _ { f _ { \theta } \in \mathcal { H } _ { \theta } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \{ ( y _ { i } - f _ { \theta } ( x _ { i } ) ) ^ { 2 } + \frac { \lambda } { N ^ { 2 } } \sum _ { i , j } w _ { i j } ^ { \sigma } ( y _ { i } - y _ { j } + \langle \nabla _ { x } f _ { \theta } ( x _ { i } ) , x _ { j } - x _ { i } \rangle ) ^ { 2 } \} , ( \mathbf { T a y l o r - R e g } ) \ .
+$$
+
+The objective in Eq. 19 uses a single surrogate to parameterize the potential and its gradient. The objective in Eq. 20 is similar in spirit to the Jacobian matching formulation in the sense that it adds a regularizer on the gradient of the surrogate to be consistent with the first-order Taylor expansion in local neighborhoods. The advantage of the Taylor learning approach is that we do not need to perform zero-order estimation of gradients to construct the training set and we rely instead on first-order approximation in local neighborhood.
+
+# 6.3 SURROGATE MODEL CONSTRAINED LMC
+
+Consider the surrogate model $f _ { \theta }$ obtained via Hermite Learning (Eq. 17) or via Taylor learning (Eqs 18, 19, 20). We are now ready to define the surrogate model LMC by replacing $\mathbf { G } ( \mathbf { x } ) = \nabla _ { \mathbf { x } } \mathbf { f } _ { \theta } ( \mathbf { x } )$ in the constrained Langevin dynamics in Eqs 9 and 10.
+
+Both Hermite and Taylor learning come with theoretical guarantees when the approximation function space is an RKHS under some mild assumptions on the training distribution and the regularity of the target function $\psi$ . In Hermite learning (Theorem 2 in Shi et al. (2010)) we have: $E _ { x \sim p _ { \Omega } } \left\| \nabla _ { x } f _ { \theta } ( x ) - \nabla _ { x } \psi ( x ) \right\| ^ { 2 } \leq \epsilon$ for sufficiently large training set $N = O ( 1 / \epsilon ^ { 1 / ( 2 \zeta r ) } )$ (where exponents $\zeta , r \in [ 0 , 1 ]$ depend on regularity of $\psi$ ). In Taylor Learning with the objective function given in Eq. 18 (Proposition 7 in Wu et al. (2010) we have: $\begin{array} { r } { \mathbb { E } _ { x \sim \rho _ { \Omega } } \left\| G _ { \Lambda } ( x ) - \nabla _ { x } \psi ( x ) \right\| ^ { 2 } \leq \epsilon } \end{array}$ for $N = O ( 1 / \epsilon ^ { d / 2 } )$ . In order to apply Theorem 2 we need this gradient approximation error to hold in expectation on all intermediate distributions in the Langevin sampling. Hence, we need the following extra-assumption on the training distribution $p _ { \Omega }$ :
+
+Assumption D: Assume we have a learned surrogate $G$ on training distribution $\rho \Omega$ such that $\begin{array} { r } { \mathbb { E } _ { { x } \sim \rho _ { \Omega } } \left\| { G } ( { x } ) - \nabla _ { { x } } { U } ( { x } ) \right\| ^ { 2 } \leq \epsilon } \end{array}$ . Assume $\rho _ { \Omega } ( x ) > 0 , \forall x \in \Omega$ and that it is a dominating measure of Langevin (PLMC, S-PLMC, Prox-LMC, S-ProxLMC ) intermediate distributions $\mu _ { k }$ , i.e. there exists $C > 0$ such that:
+
+$$
+\mu _ { k } ( x ) \leq C \rho _ { \Omega } ( x ) , \forall x \in \Omega , \forall k = 0 , \ldots K - 1 .
+$$
+
+Under Assumption $\mathbf { D }$ , it follows immediately that
+
+$$
+\mathbb { E } \| G ( X _ { k } ) - \nabla U ( X _ { k } ) \| ^ { 2 } = \int _ { \Omega } \| G ( x ) - \nabla U ( x \| ^ { 2 } \frac { \mu _ { k } ( x ) } { \rho _ { \Omega } ( x ) } \rho _ { \Omega } ( x ) \leq C \epsilon
+$$
+
+and hence we can apply Theorem 2 for $\delta = C \epsilon$ , and we obtain $\varepsilon$ -approximation of the target Gibbs distribution in terms of total variation distance.
+
+Remark 4. Assumption $D$ on the $\epsilon$ -approximation of the gradient can be achieved for a large enough training set $N$ , when we use Hermite learning in RKHS under mild assumptions and in Taylor learning. The assumption on the dominance of the training distribution is natural and means that we need a large training set that accounts to what we may encounter in Surrogate LMC iterations.
+
+In what follows we refer to surrogate constrained LMC, as $\mathbf { X }$ -PLMC or $\mathbf { X }$ -ProxLMC where x is one of four suffixes ({Z-Hermite, Taylor-2, Taylor-1, Taylor-Reg}).
+
+# 7 RELATED WORK
+
+Zero-Order Methods. Zero-order optimization with Gaussian smoothing was studied in Nesterov & Spokoiny (2017) and Duchi et al. (2015) in the convex setting. Non-convex zero order optimization was also addressed in Ghadimi & Lan (2013). The closest to our work is the zero-order Langevin Shen et al. (2019) introduced recently for black-box sampling from log concave density. The main difference in our setting, is that the density has a compact support and hence the need to appeal to projected LMC (Bubeck et al., 2015) and Proximal LMC (Brosse et al., 2017). It is worth nothing that Hsieh et al. (2018) introduced recently mirror Langevin sampling that can also be leveraged in our framework.
+
+Gradients and Score functions Estimators. We used the approach of gradient distillation (Srinivas & Fleuret, 2018) and learning gradients of (Wu et al., 2010), since they are convenient for training on different constraints and they come with theoretical guarantees. However, other approaches can be also leveraged such as the score matching approach for learning the gradient of the log likelihood (Hyvärinen, 2005) and other variants appealing to dual embeddings (Dai et al., 2018). Estimating gradients can be also performed using Stein’s method as in (Li & Turner, 2017), or via maintaining a surrogate of the gradient as in Stein descent without gradient (Han & Liu, 2018).
+
+Optimization approaches. Due to space limitation, we restrict the discussion to the optimization methods that are most commonly and recently used for optimal material (or molecule) design. A popular approach to deal with optimization of expensive black-box functions is Bayesian Optimization (BO) (Mockus, 1994; Jones et al., 1998; Frazier, 2018). The standard BO protocol is comprised of estimating the black-box function from data through a probabilistic surrogate model, usually a Gaussian process, and maximizing an acquisition function to decide where to sample next. BO is often performed over a latent space, as in (Gómez-Bombarelli et al., 2018). Hernández-Lobato et al. (2016) proposed an information-theoretic framework for extending BO to address optimization under black-box constraints, which is close to current problem scenario. Genetic Algorithms (GA), a class of meta-heuristic based evolutionary optimization techniques, is another widely used approach for generating (material) samples with desired property (Jennings et al., 2019) and has been also used for handling optimization under constraints (Chehouri et al., 2016). However, GA typically requires a large number of function evaluations, can get stuck in local optima, and does not scale well with complexity. Finally, Zhou et al. (2019) has used deep reinforcement learning technique of Deep Q-networks to optimize molecules under a specific constraint using desired properties as rewards. The advantage of our framework is that we obtain a distribution of optimal configurations (as opposed to a single optimized sample) that does not rely on training on a specific pre-existing dataset and can be further screened and tested for their optimality for the task at hand.
+
+# 8 EXPERIMENTS
+
+In this section, we demonstrate the usability of our black-blox Langevin sampling approach for the design of nano-porous configurations. We first show the performance of the surrogate models in learning the potential function, showcasing the results using four different variants: standard regression, Taylor regularization, Taylor-1 and Taylor-2. We then show how well the surrogate-based Langevin MC generates new samples under the thermal and mechanical constraints. We compare the sample quality on multiple criteria between the surrogate and zero-order approaches with either projection or proximal update step.
+
+Data. We want to learn surrogate models to approximate the gradient of the potential from data. To this end, we generate a dataset of 50K nano-porous structures, each of size $1 0 0 \mathrm { { n m } \times 1 0 0 \mathrm { { n m } } }$ One such example is displayed in Fig. 1. Number of pores is fixed to 10 in this study and each pore is a square with a side length of $1 7 . 3 2 \mathrm { n m }$ . We sample the pore centers uniformly over the unit square and construct the corresponding structure after re-scaling them appropriately. Then, using the solvers OpenBTE (Romano & Grossman, 2015) and Summit $( \mathrm { { \sum } M I T }$ Development Group, 2018), we we collect two datasets: obtain for each structure $x$ $\{ ( x _ { i } , \kappa _ { i } ) \} _ { i = 1 } ^ { N }$ a pair of values: thermal conductivity and $\{ ( x _ { i } , \sigma _ { i } ) \} _ { i = 1 } ^ { N }$ with the same inputs $\kappa$ and von Mises stress $x _ { i }$ ’s and $N = 5 0 K$ $\sigma$ . Finally, samples. More details are given in Appendices B and C on the PDEs and their corresponding solvers.
+
+Features. The pore locations are the natural input features to the surrogate models. Apart from the coordinates, we also derive some other features based on physical intuitions. For example, the distances between pores and the alignment along axes are informative of thermal conductivity (Romano & Grossman, 2016). As such, we compute pore-pore distances along each coordinate axis and add them as additional features.
+
+Surrogate gradient methods. We use feed-forward neural networks to model the surrogates since obtaining gradients for such networks is efficient thanks to automatic differentiation frameworks. We use networks comprised of 4 hidden layers with sizes 128, 72, 64, 32 and apply the same architecture to approximate the gradients for $\kappa$ and $\sigma$ separately. The hidden layers use ReLU activations whereas sigmoid was used at the output layer (after the target output is properly normalized). For the Taylor-2 variant (in Eq. 18), we have an additional output vector of the same size as the input for the gradient prediction. The networks are trained on the corresponding objective functions set up earlier by an
+
+![](images/1b33f4afab0be01eb554ee28d0be82c7bb9d0e45b741425df2ed2d5b4b0daf0a.jpg)  
+Figure 1: Example of nano-porous structures with corresponding heat flux shown using a color gradient. Yellow regions indicate high phonons flux. The thermal conductivity $\kappa$ and von Mises stress $\sigma$ are reported below each structure. The arrows show the moving directions of the pores. (Left) A random sample. (Middle) The sample obtained by Taylor-Reg PMLC starting from the left structure with $\kappa$ constraint. (Right) The sample obtained by Taylor-Reg PMLC with both $\kappa$ and $\sigma$ constraints.
+
+Adam optimizer with learning rate $1 0 ^ { - 4 }$ and decay 1.0. We fine-tune the networks with simple grid-search and select the best models for comparison. Due to the space constraint, we present the results in Appendix A and emphasize that Z-Hermite is not included in the entire comparison but in a small experiment performed with a more lightweight OpenBTE version.
+
+Incorporating constraints and comparison metrics. We demonstrate the usability of our proposed black-box Langevin sampling for the design of nano-configurations under thermal conductivity and mechanical stability constraints that are provided by the corresponding PDE solvers. To compare sampling outcomes, we use the following metrics. We report the minimum value of $\kappa$ and Monte Carlo estimates for both $\kappa$ and $\sigma$ to compare the samples generated by different sampling methods and surrogate models. The Monte Carlo estimates are computed on 20 samples.
+
+Single constraint. Our first task is to design nano-configurations under the thermal conductivity constraint where we want $\kappa$ as low as possible in order to achieve high thermo-electric efficiency. From the posterior regularization formulation Section 2, we pose the constraint satisfaction as sampling from the following Gibbs distribution:
+
+$$
+\pi ( x ) = p _ { 0 } ( x ) \frac { \exp ( - \lambda \kappa ( x ) ^ { 2 } ) } { Z } \mathbb { 1 } _ { x \in [ 0 , 1 ] ^ { 2 0 } }
+$$
+
+where $p _ { 0 } ( x )$ is the uniform distribution over the unit square, which is equivalent to the Poisson process of 10 pores on the square, and $\kappa ( x )$ is the thermal conductivity we want to minimize. Starting from 20 samples initialized from $p _ { 0 } ( x )$ , we run our proposed black-box Langevin MCs and obtain 20 new realizations from the target distribution $\pi ( x )$ . We use four different surrogates (including simple regression, Taylor-Reg, Taylor-1 and zero-order) and each surrogate with either projection or proximal update. We show the summary statistics of these samples in Table 1. The regression-PMLC in the first row and regression-ProxLMC in the fifth represent the sampling where the surrogate model are fitted on solely the mean square error objective. In all methods, we set $\lambda = 1 0 0$ , the step size $\eta = 1 \mathrm { e } { - 3 }$ and the exponential decay rate 0.8. Since keeping track of the true $\kappa$ value is expensive, we stop after $K = 1 0$ iterations. We first observe that the regression-based method (PLMC, ProxLMC) is less effective than the others simply because they do not have an implicit objective for approximating the gradients. Taylor-Reg and Taylor-1 demonstrate its effectiveness in approximating the gradient and are able to achieve lower thermal conductivity. In particular, Taylor-1-ProxLMC and Zero-order-PLMC perform in the similar range in terms of the minimum achieved, but the learned surrogate offers $\mathbf { 1 7 x }$ speed up (per sample) over zero order methods. Due to the space limit, we do not report Taylor-2 results in Table 1, and note that Taylor-2 works in the similar vein as Taylor-1.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Min K</td><td rowspan=1 colspan=1>Mean K</td><td rowspan=1 colspan=1>Per-sam. time (s)</td></tr><tr><td rowspan=1 colspan=1>Regression-PLMC</td><td rowspan=1 colspan=1>0.0757</td><td rowspan=1 colspan=1>0.1206± 0.0480</td><td rowspan=1 colspan=1>1055</td></tr><tr><td rowspan=1 colspan=1>Taylor-Reg-PLMC</td><td rowspan=1 colspan=1>0.0638</td><td rowspan=1 colspan=1>0.1196± 0.0495</td><td rowspan=1 colspan=1>899</td></tr><tr><td rowspan=1 colspan=1>Taylor-1-PLMC</td><td rowspan=1 colspan=1>0.0637</td><td rowspan=1 colspan=1>0.1278±0.0610</td><td rowspan=1 colspan=1>852</td></tr><tr><td rowspan=1 colspan=1>Zero-order-PLMC</td><td rowspan=1 colspan=1>0.0510</td><td rowspan=1 colspan=1>0.1093土：0.0271</td><td rowspan=1 colspan=1>14967</td></tr><tr><td rowspan=1 colspan=1>Regression-ProxLMC</td><td rowspan=1 colspan=1>0.0646</td><td rowspan=1 colspan=1>0.1282土0.0531</td><td rowspan=1 colspan=1>1107</td></tr><tr><td rowspan=1 colspan=1>Taylor-Reg-ProxLMC</td><td rowspan=1 colspan=1>0.0712</td><td rowspan=1 colspan=1>0.1205 ± 0.0455</td><td rowspan=1 colspan=1>899</td></tr><tr><td rowspan=1 colspan=1>Taylor-1-ProxLMC</td><td rowspan=1 colspan=1>0.0575</td><td rowspan=1 colspan=1>0.1297 ± 0.0543</td><td rowspan=1 colspan=1>874</td></tr><tr><td rowspan=1 colspan=1>Zero-order-ProxLMC</td><td rowspan=1 colspan=1>0.0719</td><td rowspan=1 colspan=1>0.1112 ± 0.0363</td><td rowspan=1 colspan=1>14938</td></tr></table>
+
+Table 1: Statistics of 20 new samples obtained by running different surrogate-based Langevin MCs on $\pi$ with the thermal conductivity constraint (Eq. 21). We show the min and mean over the generated samples and the per-sample time. Initialized samples have min $\kappa = 0 . 0 6 1 9$ and mean $\sigma = 0 . 1 2 6 8$ .
+
+Multiple constraints. Achieving the minimal thermal conductivity can be fulfilled without much difficulty (e.g. structures with all pores aligned along the vertical axis), but such structures are often mechanically unstable. In the next step, we study whether adding more (conflicting) constraints helps us design better nano-configurations. Hence, we consider both thermal conductivity $\kappa$ and mechanical stability provided via von Mises stress $\sigma$ . We want a sample $x$ that minimizes $\kappa ( x )$ to achieve high thermo-electric efficiency while maintaining $\sigma ( x )$ less than some threshold (which we explain below). Like the single constraint case, we pose this as sampling from the following Gibbs distribution:
+
+$$
+\pi ( x ) = p _ { 0 } ( x ) \frac { \exp ( - \lambda _ { 1 } \kappa ( x ) ^ { 2 } - \lambda _ { 2 } [ \sigma ( x ) - \tau ] _ { + } ) } { Z } \mathbb { 1 } _ { x \in [ 0 , 1 ] ^ { 2 0 } } ,
+$$
+
+where $p _ { 0 } ( x )$ is the same as above, $\sigma ( x )$ is the von Mises stress and $\tau$ is a threshold on the maximum value of $\sigma$ . With this framework, we relax the inequality constraint to the Hinge loss term on von Mises stress. The results are summarized in Table 2. Note that all the surrogate Langevin MCs are initialized from the same set of 20 samples as above. In this experiment, we set $\tau = 0 . 5$ , $\lambda _ { 1 } = 1 0 0$ , $\lambda _ { 2 } = 1 0$ the step size $\eta = 1 \mathrm { e } { - 3 }$ and the exponential decay rate 0.8. Comparing with Table 1, one can see that not only better $\kappa$ be achieved but also the $\sigma$ can be reduced simultaneously. These results suggest that our approach can effectively sample new configurations under multiple competing constraints. Examples of new nano-configurations are show in Fig. 1 and Appendix A Fig. 5, 6 and 7.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Min K</td><td rowspan=1 colspan=1>Mean K</td><td rowspan=1 colspan=1>Mean σ</td><td rowspan=1 colspan=1>Per-sam. time (s)</td></tr><tr><td rowspan=1 colspan=1>Taylor-Reg-PLMC</td><td rowspan=1 colspan=1>0.0613</td><td rowspan=1 colspan=1>0.1256 ± 0.0538</td><td rowspan=1 colspan=1>0.6590± 0.2261</td><td rowspan=1 colspan=1>952</td></tr><tr><td rowspan=1 colspan=1>Taylor-1-PLMC</td><td rowspan=1 colspan=1>0.0611</td><td rowspan=1 colspan=1>0.1278± 0.0610</td><td rowspan=1 colspan=1>0.6380± 0.1598</td><td rowspan=1 colspan=1>852</td></tr><tr><td rowspan=1 colspan=1>Zero-order-PLMC</td><td rowspan=1 colspan=1>0.0471</td><td rowspan=1 colspan=1>0.1148 ± 0.0475</td><td rowspan=1 colspan=1>0.6511 ± 0.1916</td><td rowspan=1 colspan=1>15677</td></tr><tr><td rowspan=1 colspan=1>Taylor-Reg-ProxLMC</td><td rowspan=1 colspan=1>0.0666</td><td rowspan=1 colspan=1>0.1195 ± 0.0534</td><td rowspan=1 colspan=1>0.6402 ± 0.1464</td><td rowspan=1 colspan=1>856</td></tr><tr><td rowspan=1 colspan=1>Taylor-1-ProxLMC</td><td rowspan=1 colspan=1>0.0548</td><td rowspan=1 colspan=1>0.1298 ± 0.0610</td><td rowspan=1 colspan=1>0.6156 ± 0.1463</td><td rowspan=1 colspan=1>972</td></tr><tr><td rowspan=1 colspan=1>Zero-order-ProxLMC</td><td rowspan=1 colspan=1>0.0354</td><td rowspan=1 colspan=1>0.1080 ± 0.0384</td><td rowspan=1 colspan=1>0.6029 ± 0.1376</td><td rowspan=1 colspan=1>15080</td></tr></table>
+
+Table 2: Summary statistics of 20 new samples obtained by our sampling method on $\pi ( x )$ with $\kappa$ and $\sigma$ constraints Eq. 22. The starting samples are reused from the single constraint case (min $\kappa = 0 . 0 7 5 9$ , mean $\kappa = 0 . 1 2 6 8$ , and mean $\sigma = 0 . 8 1 8 1$ ; note that $\sigma$ can be as high as 16.)
+
+# 9 CONCLUSION
+
+In this paper we introduced Surrogate-Based Constrained Langevin Sampling for black-box sampling from a Gibbs distribution defined on a compact support. We studied two approaches for defining the surrogate: the first through zero-order methods and the second via learning gradient approximations using deep neural networks. We showed the proofs of convergence of the two approaches in the log-concave and smooth case. While zero-order Langevin had prohibitive computational cost, learned surrogate model Langevin enjoy a good tradeoff of lightweight computation and approximation power. We applied our black-box sampling scheme to the problem of nano-material configuration design, where the black box constraints are given by expensive PDE solvers, and showed the efficiency and the promise of our method in finding optimal configurations. Among different approaches for approximating the gradient, the zero-order ones (PLMC, ProxLMC) show overall superior performance, at a prohibitive computational cost. We established that the deep the surrogate (Taylor-1
+
+ProxLMC) is a viable alternative to zero-order methods, achieving reasonable performance, and offering 15x speedup over zero-order methods.
+
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+
+# A SUPPLEMENTAL EXPERIMENTAL RESULTS
+
+Surrogate gradient methods We use feed-forward neural networks to model the surrogates since obtaining gradients for such networks is efficient thanks to automatic differentiation frameworks. We use networks comprised of 4 hidden layers with sizes 128, 72, 64, 32 and apply the same architecture to approximate the gradients for $\kappa$ and $\sigma$ separately. The hidden layers compute ReLU activation whereas sigmoid was used at the output layer (after the target output is properly normalized). For the Taylor-2 variant (in Eq. 18), we have an output vector for the gradient prediction. The networks are trained on the corresponding objective functions set up earlier by Adam optimizer with learning rate $1 0 ^ { - 4 }$ and decay 1.0. We fine-tune the networks with simple grid-search and select the best models for comparison.
+
+As emphasized throughout, our focus is more on approximating the gradient rather than learning the true function. However, we need to somehow evaluate the surrogate models on how well they generalize on a hold-out test set. Like canonical regression problems, we compare the surrogate variants against each other using root mean square error (RMSE) on the test set. Figures 2 and 3 shows the results. The left figure shows RMSE for predicting $\kappa$ and the right one shows RMSE for the von Mises stress $\sigma$ . We can see that the Taylor-Reg generalizes better and also converges faster than Taylor-1 and Taylor-2 to target RMSE for $\kappa$ , while all methods result similarly for $\sigma$ prediction. This is reasonable because the objectives of Taylor-1 and Taylor-2 are not to optimize the mean square error, which we evaluate on here. Figure 3 shows the learning in terms of sample complexity. Again, Taylor-Reg outperforms Taylor-1 and Taylor-2 for $\kappa$ prediction. In contrast, most models work similarly for $\sigma$ regression, particularly when the training size is reduced to $50 \%$ (25K).
+
+![](images/a6ad4fe4e01872fb2c58c72b469e2441c90067e0989fef49ed33aa237f05398e.jpg)  
+Figure 2: Comparison of the surrogate variants in testing RMSE. (Left) prediction accuracy for the thermal conductivity $\kappa$ . (Right) prediction accuracy for mechanical stability $\sigma$ . Note the difference in scale of $\kappa$ and $\sigma$ .
+
+Effectiveness of Z-Hermite learning Notice that Z-Hermite learning is not included in this comparison and as a surrogate model in the black-blox Langevin sammpling in Section 8. The reason is that apart from the usual sample pair $( x _ { i } , y _ { i } )$ , we need the gradient $\tilde { y } _ { i }$ (See Eq. 17). Since we can query the solvers, this gradient can only be estimated using finite difference. For both $\kappa$ and $\sigma$ in our experiment, obtaining such data is extremely expensive. As a consequence, we do not have the full results of the Z-Hermite model. Instead, we ran a separate study to show the effectiveness of Z-Hermite surrogate LMC on a smaller data with a lightweight OpenBTE version (0.9.55). The results in Table 3 shows the working of $\textsf { Z }$ -Hermite learning in learning the gradient of $\kappa ( x )$ . Here, the entropy is based nearest neighbor estimate to demonstrate the diversity of the pore centers in the unit square. With the $( x _ { p } , y _ { p } )$ -coordinates of each pore $p$ , the entropy estimate is given by:
+
+$$
+H = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log ( n \operatorname* { m i n } _ { j \neq i } \| p _ { i } - p _ { j } \| ) + \log 2 + C .
+$$
+
+![](images/79f706b6c72619b01ff8332cf374b1e6dd2aebf363d10a85e6a19ccfde04a5cc.jpg)  
+Figure 3: Comparison of the surrogate models in RMSE on the same test set when the training size is varied. Note the scale difference in the figures due to the different range of values.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Mean K</td><td rowspan=1 colspan=1>Mean entropy</td><td rowspan=1 colspan=1>Per-sam. time (s)</td></tr><tr><td rowspan=1 colspan=1>Zero-order PLMC</td><td rowspan=1 colspan=1>0.0676</td><td rowspan=1 colspan=1>1.960</td><td rowspan=1 colspan=1>3658</td></tr><tr><td rowspan=1 colspan=1>Taylor-1 PLMC</td><td rowspan=1 colspan=1>0.0988</td><td rowspan=1 colspan=1>1.745</td><td rowspan=1 colspan=1>253</td></tr><tr><td rowspan=1 colspan=1>Z-Hermite PLMC</td><td rowspan=1 colspan=1>0.0946</td><td rowspan=1 colspan=1>1.739</td><td rowspan=1 colspan=1>227</td></tr><tr><td rowspan=1 colspan=1>Hybrid (Zero-order + Taylor-1)</td><td rowspan=1 colspan=1>0.0786</td><td rowspan=1 colspan=1>1.867</td><td rowspan=1 colspan=1>2136</td></tr></table>
+
+Table 3: Z-Hermite learning is sample efficient at training time as well as for the Langevin sampling, but collecting the training set is prohibitive. Zero-order and deep surrogate can work in a hybird way and offer a better trade-off between accuracy and computation.
+
+A hybrid algorithm between zero-order and Taylor-1 surrogate We can see in Tables 1, 2 and 3 the trade-off between computation and accuracy of our approach. While zero-order PLMC and ProxLMC can achieve the lowest thermal conductivity, their computational costs are prohibitive. In contrast, deep surrogate models (including Taylor-Reg, Taylor-1) are far more time-efficient but slightly worse in terms of achieving the optimal $\kappa$ . To mitigate the trade-off, we propose a simple hybrid method that combines the best of the zero-order and Taylor-1 surrogate models. The algorithm is shown in Figure A that alternates between using the gradient from the zero-order estimate and the gradient of the deep surrogate depending on whether taking this step would decrease the potential function (i.e. $\kappa _ { - }$ ). We show and compare the achieved $\kappa$ and running time in Table 3. Examples of the samples generated by Zero-order PLMC, Taylor-1 PLMC and the hybrid method are also depicted in Figure 4. The hybrid achieves the thermal conductivity that is lower than Taylor-1 PMLC while running almost $\cdot$ faster than zero-order PLMC. This suggests that the hybrid strategy offers a better trade-off in accuracy and computation. One way to further improve the hybrid is to collect the zero-order gradients while mixing and re-update the surrogate with Z-Hermite learning.
+
+Algorithm 1 A hybrid PLMC algorithm alternating between zero-order and Taylor-1 surrogate gradients.   
+
+<table><tr><td>Train a network fe(x) with Taylor-1 Randomly sample xo from the uniform p(x) Perform a Langevin dynamic step</td></tr><tr><td>for t=1,2,...,K do if κ(x-nVxfe(x))&lt;κ(x) then</td></tr><tr><td>x←P(x-nVxfe(x)+√2nξ)</td></tr><tr><td>else estimate Vκ(x) using zero-order method</td></tr><tr><td>update x ← PΩ(x-n∀κ(x)+ √2nξ)</td></tr><tr><td>end if end for</td></tr><tr><td>Return a new sample x</td></tr></table>
+
+![](images/76b80df98db96b4f61582927af7f5366a61d93519bde044e5ae4ef192563b02c.jpg)  
+Figure 4: Samples from by Zero-order PLMC (left), Taylor-1 PLMC (middle) and the hybrid algorithm of Zero-order and Taylor-1 PLMC (right). All are run with the $\cdot$ constraint.
+
+Additional generated samples We show additional configurations generated by our sampling approach (Taylor-Reg ProxLMC, Taylor-1 ProxLMC and Zero-order ProxLMC) in Fig. 5, 6 and 7.
+
+![](images/4b6120f2882bbc075bd04c38725d333db669e19d658a8e748b58b7a9f82b01b8.jpg)  
+Figure 5: Example of nano-porous structures with corresponding heat flux shown using a color gradient. Yellow regions indicate high phonons flux. The thermal conductivity $\kappa$ and von Mises stress $\sigma$ are reported below each structure. The arrows show the moving directions of the pores. (Left) A random sample. (Middle) The sample obtained by Taylor-Reg ProxLMC starting from the left structure with $\kappa$ constraint. (Right) The sample obtained by Taylor-Reg ProxLMC with both $\kappa$ and $\sigma$ constraints.
+
+![](images/8b032e4bba02e9603fb93e5e9718016febf36d67cb5ddbc70d204f17e6f6caea.jpg)  
+Figure 6: Example of nano-porous structures with corresponding heat flux shown using a color gradient. Yellow regions indicate high phonons flux. The thermal conductivity $\kappa$ and von Mises stress $\sigma$ are reported below each structure. The arrows show the moving directions of the pores. (Left) A random sample. (Middle) The sample obtained by Taylor-1 ProxLMC starting from the left structure with $\kappa$ constraint. (Right) The sample obtained by Taylor-1 ProxLMC with both $\kappa$ and $\sigma$ constraints.
+
+![](images/c365571973178b5b73a0229443cd1c1bf37d0a4299b0b43ab66bc93db9ea2f00.jpg)  
+Figure 7: Example of nano-porous structures with corresponding heat flux shown using a color gradient. Yellow regions indicate high phonons flux. The thermal conductivity $\kappa$ and von Mises stress $\sigma$ are reported below each structure. The arrows show the moving directions of the pores. (Left) A random sample. (Middle) The sample obtained by Zero-order ProxLMC starting from the left structure with $\kappa$ constraint. (Right) The sample obtained by Zero-order ProxLMC with both $\kappa$ and $\sigma$ constraints.
+
+# B BACKGROUND ON MODELING NANOSCALE HEAT TRANSPORT
+
+At the nanoscale, heat transport may exhibit strong ballistic behaviour and a non-diffusive model must be used (Chen, 2005). In this work we use the Boltzmann transport equation under the relaxation time approximation and in the mean-free-path (MFP) formulation (Romano & Grossman, 2015)
+
+$$
+\Lambda \hat { \mathbf { s } } \cdot \nabla T ( \Lambda ) + T ( \Lambda ) = \int \alpha ( \Lambda ^ { \prime } ) \langle T ( \Lambda ^ { \prime } ) \rangle d \Lambda ^ { \prime } ,
+$$
+
+where $T ( \Lambda )$ is the effective temperature associated to phonons with MFP $\Lambda$ and direction $\hat { \mathbf { s } }$ ; the notation $\langle . \rangle$ stands for an angular average. The coefficients $\alpha ( \Lambda ^ { \prime } )$ are given by
+
+$$
+\alpha ( \Lambda ^ { \prime } ) = \frac { K ( \Lambda ^ { \prime } ) } { \Lambda ^ { \prime } } \left[ \int \frac { K ( \Lambda ^ { \prime \prime } ) } { \Lambda ^ { \prime \prime } } d \Lambda ^ { \prime \prime } \right] ^ { - 1 } ,
+$$
+
+where $K ( \Lambda ^ { \prime } )$ is the bulk MFP distribution. In general, such a quantity can span several orders of magnitude; however, for simplicity we assume the gray model, i.e. all phonons travel with the same MFP, $\Lambda _ { 0 }$ . Within this approximation, we have ${ K ( \Lambda ) } = \kappa _ { \mathrm { b u l k } } \delta ( \Lambda - \Lambda _ { 0 } )$ . In this work we choose $\Lambda _ { 0 }$ $= 1 0 \mathrm { n m }$ , namely as large as the unit cell, so that significant phonons size effects occur. With no loss of generality, we set $\kappa _ { \mathrm { b u l k } } = 1 ~ \mathrm { { W m ^ { - 1 } K ^ { - 1 } } }$ . Eq. 23 is an integro-differential PDE, which is solved iteratively for each phonon direction over an unstructured mesh (Romano & Di Carlo, 2011). We apply periodic boundary conditions along the unit cell while imposing a difference of temperature of $\Delta T = 1 \mathrm { ~ K ~ }$ along the $x$ -axis. At the pores’ walls we apply diffusive boundary conditions. Upon convergence, the effective thermal conductivity is computed using Fourier’s law, i.e.
+
+$$
+\kappa _ { \mathrm { e f f } } = - \frac { L } { \Delta T A } \int _ { A } \mathbf { J } \cdot \hat { \mathbf { n } } d S ,
+$$
+
+where ${ \bf J } = ( \kappa _ { \mathrm { b u l k } } / \Lambda _ { 0 } ) \langle T ( \Lambda _ { 0 } ) \hat { \bf s } \rangle \hat { \bf n }$ is the heat flux, $L$ is the size of the unit cell, $A$ is the area of the cold contact (with normal $\hat { \mathbf { n } }$ ). Throughout the text we use the quantity $\kappa = \kappa _ { \mathrm { e f f } } / \kappa _ { \mathrm { b u l k } }$ as a measure of phonon size effects.
+
+# C BACKGROUND ON MODELING MECHANICAL STRESS
+
+We model mechanical stress by using the continuum linear elasticity equations
+
+$$
+{ \frac { \partial } { \partial x _ { j } } } \sigma _ { i j } = f _ { i } ,
+$$
+
+where $f _ { i }$ is the body force (which is zero in this case), and $\sigma _ { i j }$ is the stress tensor. Note that we used the Einstein notation, i.e. repeated indexes are summed over. The strain $\epsilon _ { k l }$ is related to the stress via the fourth-rank tensor elastic constant $C _ { i j k l }$
+
+$$
+\sigma _ { i j } = C _ { i j k l } \epsilon _ { k l } .
+$$
+
+The strain is then related to the displacement $\mathbf { u }$ via
+
+$$
+\epsilon _ { k l } = \frac { 1 } { 2 } \left( \frac { \partial u _ { k } } { \partial x _ { l } } + \frac { \partial u _ { l } } { \partial u _ { k } } \right) .
+$$
+
+We apply periodic boundary conditions along the unit-cell and applied solicitation is a small in-plane expansion. Once the stress tensor is calculated, we compute the von Mises stress as
+
+$$
+\sigma _ { V M } = \sqrt { \frac { 1 } { 2 } \left( \sigma _ { 3 } - \sigma _ { 2 } \right) ^ { 2 } + \left( \sigma _ { 3 } - \sigma _ { 1 } \right) ^ { 2 } + \left( \sigma _ { 2 } - \sigma _ { 1 } \right) ^ { 2 } } ,
+$$
+
+where $\sigma _ { i }$ are the principal stress axis. As a mechanical stability estimator we use $\sigma = \operatorname* { m a x } _ { \mathbf { x } \in D } ( \sigma _ { V M } )$ where $D$ is the simulation domain. To avoid material’s plasticity, $\sigma$ needs to be smaller than the yield stress of a given material. For mechanical simulation we used the SUMIT code $( \mathrm { { \sum } M I T }$ Development Group, 2018).
+
+# D BACKGROUND ON STOCHASTIC DIFFERENTIAL EQUATIONS (SDE): CHANGE OF MEASURE AND GRISANOV’S FORMULA
+
+Theorem 3 (Grisanov Theorem, Change of Measure for Brownian Motion (Lipster & Shiryaev, 2001), Theorem 6.3 page 257). Let $( W _ { t } , \mathcal { F } _ { t } )$ be a Wiener process (Brownian motion) and $( \beta _ { t } , \mathcal { F } _ { t } ) a$ random process such that for any $T > 0$
+
+$$
+\int _ { 0 } ^ { T } \left\| \beta _ { t } \right\| ^ { 2 } d t < \infty a . s
+$$
+
+Then the random process : $d { \tilde { W } } _ { t } = d W _ { t } - \beta _ { t } d t$ or written equivalently: $\begin{array} { r } { \tilde { W } _ { t } = W _ { t } - \int _ { 0 } ^ { t } \beta _ { s } d s } \end{array}$ , is a Wiener process with respect to $\mathcal { F } _ { t }$ , $t \in [ 0 , T ]$ . Let $P _ { T } ^ { W } = \mathcal { L } ( W _ { [ 0 , T ] } )$ , and $P _ { T } ^ { \tilde { W } } = \mathcal { L } ( \tilde { W } _ { [ 0 , T ] } )$ the densities are given by: $\begin{array} { r } { \frac { d P _ { T } ^ { \tilde { W } } } { d P _ { T } ^ { W } } = \exp \left( \int _ { 0 } ^ { T } \langle \beta _ { s } , d W _ { s } \rangle - \frac { 1 } { 2 } \int _ { 0 } ^ { T } \left\| \beta _ { s } \right\| ^ { 2 } d s \right) . } \end{array}$ . It follows that:
+
+$$
+K L ( P _ { T } ^ { W } , P _ { T } ^ { \tilde { W } } ) = \frac { 1 } { 2 } \mathbb { E } _ { P _ { T } ^ { W } } \left[ \int _ { 0 } ^ { T } \left. \beta _ { s } \right. ^ { 2 } d s \right]
+$$
+
+Theorem 4 (Grisanov Theorem, Change of Measure for Diffusion Processes, (Lipster & Shiryaev, 2001), ()). Let $( X _ { t } ) _ { t \geq 0 }$ and $( Y _ { t } ) _ { t \geq 0 }$
+
+$$
+\begin{array} { r } { d X _ { t } = \alpha _ { t } ( X ) d t + d W _ { t } } \\ { d Y _ { t } = \beta _ { t } ( Y ) d t + d W _ { t } } \end{array}
+$$
+
+where $X _ { 0 } = Y _ { 0 }$ is an $\mathcal { F } _ { 0 }$ measurable random variable. Suppose that the non-anticipative functionals $\alpha _ { t } ( x )$ and $\beta _ { t } ( x )$ are such that a unique continuous strong solutions exits for both processes. If for any $T > 0$ :
+
+$$
+\int _ { 0 } ^ { T } \| \alpha _ { s } ( X ) \| ^ { 2 } + \| \beta _ { s } ( X ) \| ^ { 2 } d s < \infty ( a . s ) a n d \int _ { 0 } ^ { T } \| \alpha _ { s } ( Y ) \| ^ { 2 } + \| \beta _ { s } ( Y ) \| ^ { 2 } d s < \infty ( a . s ) .
+$$
+
+Let $P _ { T } ^ { X } = \mathcal { L } ( X _ { [ 0 , T ] } )$ , and $P _ { T } ^ { Y } = \mathcal { L } ( Y _ { [ 0 , T ] } )$ .
+
+$$
+\frac { d P _ { T } ^ { Y } } { d P _ { T } ^ { X } } ( X ) = \exp \left( - \int _ { 0 } ^ { T } \langle \alpha _ { s } ( X ) - \beta _ { s } ( X ) , d X _ { s } \rangle + \frac { 1 } { 2 } \int _ { 0 } ^ { T } ( \| \alpha _ { s } ( X ) \| - \| \beta _ { s } ( X ) \| ^ { 2 } ) d s \right) .
+$$
+
+$$
+K L ( P _ { T } ^ { X } , P _ { T } ^ { Y } ) = \frac { 1 } { 2 } \mathbb { E } _ { P _ { T } ^ { X } } \left[ \int _ { 0 } ^ { T } \left. \alpha _ { s } ( X ) - \beta _ { s } ( X ) \right. ^ { 2 } d s \right] .
+$$
+
+# E BACKGROUND ON ZERO-ORDER OPTIMIZATION (GRADIENT-FREE)
+
+Consider the smoothed potential $U _ { \nu }$ defined as follows:
+
+$$
+U _ { \nu } ( x ) = \mathbb { E } _ { g \sim \mathcal { N } ( 0 , I _ { d } ) } U ( x + \nu g )
+$$
+
+its gradient is given by:
+
+$$
+\nabla _ { \boldsymbol { x } } U _ { \nu } ( \boldsymbol { x } ) = \mathbb { E } _ { g } \frac { U ( \boldsymbol { x } + \nu g ) - U ( \boldsymbol { x } ) } { \nu } g ,
+$$
+
+A monte carlo estimate of $\nabla _ { x } U _ { \nu } ( x )$ is:
+
+$$
+\hat { G } _ { n } ( x ) = \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \left( \frac { U ( x + \nu g _ { j } ) - U ( x ) } { \nu } \right) g _ { j } ,
+$$
+
+where $g _ { 1 } , \ldots g _ { n }$ are iid standard Gaussians vectors.
+
+Using known results in zero order optimization under assumptions on smoothness and bounded gradients of the gradients we have for all $x$ ((Nesterov & Spokoiny, 2017; Shen et al., 2019)):
+
+$$
+\begin{array} { r } { \hat { G } _ { 1 } ( x ) - \nabla _ { x } U ( x ) \Big \| ^ { 2 } \leq \Big ( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } \left\| \nabla _ { x } U ( x ) \right\| \Big ) ^ { 2 } \leq \Big ( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } L \Big ) ^ { 2 } } \end{array}
+$$
+
+Finally by independence of $u _ { 1 } , \ldots . u _ { n }$ we have:
+
+$$
+\mathbb { E } _ { g _ { 1 } , \ldots , g _ { n } } \left\| \hat { G } _ { n } ( x ) - \nabla _ { x } U ( x ) \right\| ^ { 2 } \leq \frac { \Big ( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } L \Big ) ^ { 2 } } { n }
+$$
+
+# F PROOFS
+
+Proof of Lemma $I$ . Define the Lagrangian:
+
+$$
+\begin{array} { c } { { \displaystyle { \cal L } ( q , \eta ) = \int _ { \Omega } \log \left( \frac { q ( x ) } { p _ { 0 } ( x ) } \right) q ( x ) d x + \sum _ { j = 1 } ^ { C _ { e } } \lambda _ { j } \int _ { \Omega } ( \psi _ { j } ( x ) - y _ { j } ) ^ { 2 } q ( x ) d x } } \\ { { + \displaystyle { \sum _ { k = 1 } ^ { C _ { i } } \lambda _ { k } \int _ { x \in \Omega } ( \phi _ { k } ( x ) - b _ { k } ) _ { + } q ( x ) d x } + \eta \left( 1 - \displaystyle { \int _ { x \in \Omega } q ( x ) } \right) } } \end{array}
+$$
+
+Setting first order optimality conditions on $q$ , we have for $x \in \Omega$ :
+
+$$
+\log \left( \frac { q ( x ) } { p _ { 0 } ( x ) } \right) + 1 + \sum _ { j = 1 } ^ { C } \lambda _ { j } ( \psi _ { j } ( x ) - y _ { j } ) ^ { 2 } + \sum _ { k = 1 } ^ { C _ { i } } \lambda _ { k } ( \phi _ { k } ( x ) - b _ { k } ) _ { + } - \eta = 0
+$$
+
+Hence we have:
+
+$$
+q ( x ) = p _ { 0 } ( x ) \frac { \exp { \left( - \sum _ { j = 1 } ^ { C _ { c } } \lambda _ { j } ( \psi _ { j } ( x ) - y _ { j } ) ^ { 2 } - \sum _ { k = 1 } ^ { C _ { i } } \lambda _ { k } ( \phi _ { k } ( x ) - b _ { k } ) _ { + } \right) } } { e \exp { - \eta } } , x \in \Omega
+$$
+
+and
+
+$$
+q ( x ) = 0 , x \notin \Omega ,
+$$
+
+First order optimality on $\eta$ give us: $\textstyle \int _ { \Omega } q ( x ) = 1$ , we conclude by setting $e \exp ( - \eta ) = Z$ .
+
+Proof of Theorem $1 \ I$ ) Projected Langevin. Let us define the following continuous processes by interpolation of $X _ { k }$ and $Y _ { K }$ (Piecewise constant):
+
+$$
+d \tilde { X } _ { t } = P _ { \Omega } ( \tilde { U } _ { t } ( \tilde { X } ) d t + \sqrt { 2 \lambda } d W _ { t } )
+$$
+
+where $\begin{array} { r } { \tilde { U } _ { t } ( \tilde { X } ) = - \sum _ { k = 0 } ^ { \infty } \nabla _ { x } U ( \tilde { X } _ { k \eta } ) \mathbb { 1 } _ { t \in [ k \eta , ( k + 1 ) \eta ] } ( t ) } \end{array}$ . Similarly let us define :
+
+$$
+d \tilde { Y } _ { t } = P _ { \Omega } ( G _ { t } ( \tilde { Y } ) d t + \sqrt { 2 \lambda } d W _ { t } )
+$$
+
+where $\begin{array} { r } { G _ { t } ( \tilde { Y } ) = - \sum _ { k = 0 } ^ { \infty } G ( \tilde { Y } _ { k \eta } ) \mathbb { 1 } _ { t \in [ k \eta , ( k + 1 ) \eta ] } ( t ) . } \end{array}$
+
+It is easy to see that we have : $X _ { k } = \tilde { X } _ { k \eta }$ and $Y _ { k } = \tilde { Y } _ { k \eta }$
+
+Let $\pi _ { \tilde { X } } ^ { T }$ and $\pi _ { \tilde { Y } } ^ { T }$ be the distributions of $( \tilde { X } _ { t } ) _ { t \in [ 0 , T ] }$ and $( \tilde { Y } ) _ { t \in [ 0 , T ] }$
+
+Note that :
+
+$$
+d \tilde { Y } _ { t } = P _ { \Omega } \left( \tilde { U } _ { t } ( \tilde { X } _ { t } ) d t + \sqrt { 2 \lambda } ( d W _ { t } + \frac { 1 } { \sqrt { 2 \lambda } } ( G _ { t } ( \tilde { Y } _ { t } ) - \tilde { U } _ { t } ( \tilde { X } _ { t } ) ) d t ) \right)
+$$
+
+Let
+
+$$
+d \tilde { W } _ { t } = d W _ { t } + \frac { 1 } { \sqrt { 2 \lambda } } ( G _ { t } ( \tilde { Y } _ { t } ) - \tilde { U } _ { t } ( \tilde { X } _ { t } ) ) d t
+$$
+
+Hence we have :
+
+$$
+d \tilde { Y } _ { t } = P _ { \Omega } \left( \tilde { U } _ { t } ( \tilde { X } ) + \sqrt { 2 \lambda } d \tilde { W } _ { t } \right) ,
+$$
+
+Assume that $X _ { 0 } = Y _ { 0 }$ there exists $\mathcal { Q }$ such that , $X _ { T } = \mathcal { Q } ( \{ W _ { t } \} _ { t \in [ 0 , T ] } )$ and $Y _ { T } = \mathcal { Q } ( ( \tilde { W } _ { t } ) _ { t \in [ 0 , T ] } )$ . Let $\mu _ { T } ^ { \tilde { X } }$ be the law of $\tilde { X } _ { t \in [ 0 , T ] }$ . Same for $\mu _ { T } ^ { \tilde { Y } }$ . The proof here is similar to the proof of Lemma 8 in (Bubeck et al., 2015). By the data processing inequality we have:
+
+$$
+\mathrm { K L } ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) \leq \mathrm { K L } ( W _ { t \in [ 0 , T ] } , \tilde { W } _ { t \in [ 0 , T ] } ) ,
+$$
+
+Now using Grisanov’s Theorem for change of measure of Brownian Motion (Theorem 3) we have:
+
+$$
+\mathrm { K L } ( W _ { t \in [ 0 , T ] } , \tilde { W } _ { t \in [ 0 , T ] } ) = \frac { 1 } { 4 \lambda } \mathbb { E } \int _ { 0 } ^ { T } | G _ { t } ( \tilde { Y } _ { t } ) - \tilde { U } _ { t } ( \tilde { X } _ { t } ) | ^ { 2 } d t
+$$
+
+Consider $T = K \eta$ , hence we have (with some abuse of notation we drop tilde as $Y _ { k } = \tilde { Y } _ { k \eta } )$ ):
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { K L } ( \mu _ { T } ^ { \vec { X } } , \mu _ { T } ^ { \vec { V } } ) \leq \frac { 1 } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } | G _ { t } ( \tilde { Y } _ { k } ) - \tilde { U } _ { t } ( \tilde { X } _ { t } ) | ^ { 2 } d l } \\ { \displaystyle \quad = \frac { 1 } { 4 \lambda } \mathrm { F } \sum _ { k = 0 } ^ { K - 1 } \int _ { \mathbb { R } ^ { d } } ^ { ( k + 1 ) \eta _ { q } } \| G ( \tilde { Y } _ { k n } ) - \nabla _ { x } U ( X _ { k n } ) \| ^ { 2 } d t } \\ { \displaystyle \quad = \frac { \eta } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| G ( Y _ { k n } ) - \nabla _ { x } U ( X _ { k n } ) \| ^ { 2 } } \\ { \displaystyle \quad = \frac { \eta } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| G ( Y _ { k n } ) - \nabla _ { x } U ( Y _ { k n } ) + \nabla _ { x } U ( Y _ { k n } ) - \nabla _ { x } U ( X _ { k n } ) \| ^ { 2 } } \\ { \displaystyle \quad = \frac { \eta } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| G ( Y _ { k n } ) - \nabla _ { x } U ( Y _ { k n } ) \| ^ { 2 } + \mathbb { E } \| \nabla _ { x } U ( Y _ { k n } ) - \nabla _ { x } U ( X _ { k n } ) \| ^ { 2 } } \\ { \displaystyle \leq \frac { \eta } { 2 \lambda } \sum _ { k = 0 } ^ { K - 1 } \Big ( \mathbb { E } \| G ( Y _ { k n } ) - \nabla _ { x } U ( Y _ { k n } ) \| ^ { 2 } + \mathbb { E } \| \nabla _ { x } U ( Y _ { k n } ) - \nabla _ { x } U ( X _ { k n } ) \| ^ { 2 } \Big ) } \end{array}
+$$
+
+where in the last inequality we used the fact that $| | a - b | | ^ { 2 } \leq 2 ( | | a | | ^ { 2 } + | | b | | ^ { 2 } )$ . Note that we have by smoothness assumption on $U$ :
+
+$$
+\begin{array} { r } { \left\| { \nabla _ { x } U ( Y _ { k \eta } ) } - { \nabla _ { x } U ( X _ { k h } ) } \right\| ^ { 2 } \leq \beta ^ { 2 } \left\| { X _ { k h } - Y _ { k h } } \right\| ^ { 2 } } \end{array}
+$$
+
+Let $R$ be the diameter of $\Omega$ , we can get a bound as follows:
+
+$$
+\begin{array} { r l } {  { \mathbf { K } \mathbf { L } ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) \leq \frac { \eta } { 2 \lambda } ( \underbrace { \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| G ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \| ^ { 2 } } _ { \mathrm { G r a d i e n t ~ a p p r o x i m a t i o n ~ e r o r } } + \beta ^ { 2 } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| X _ { k h } - Y _ { k h } \| ^ { 2 } ) } } \\ & { \leq \frac { \eta } { 2 \lambda } ( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \| G ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \| ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } ) } \end{array}
+$$
+
+Now using Pinsker inequality we have:
+
+$$
+T V ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) ^ { 2 } \leq 2 \mathrm { K L } ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) \leq \frac { \eta } { \lambda } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left. G ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \right. ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right)
+$$
+
+Hence for $T = K \eta$ we have:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { { S - P L M C } } } , \mu _ { K } ^ { \mathrm { { P L M C } } } ) \leq \sqrt { \frac { \eta } { \lambda } } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( Y _ { k } ) - \nabla _ { x } U ( Y _ { k } ) \right\| ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+Proof of Theorem $2$ ) Proximal LMC. Let us define the following continuous processes by interpolation of $X _ { k }$ and $Y _ { K }$ (Piecewise constant):
+
+$$
+d \tilde { X } _ { t } = \tilde { U } _ { t } ( \tilde { X } ) d t + \sqrt { 2 \lambda } d W _ { t }
+$$
+
+where $\begin{array} { r } { \tilde { U } _ { t } ( \tilde { \boldsymbol { X } } ) = - \sum _ { k = 0 } ^ { \infty } ( \nabla _ { \boldsymbol { x } } U ( \tilde { X } _ { k \eta } ) + \frac { 1 } { \gamma } ( \tilde { X } _ { k \eta } - P _ { \Omega } ( \tilde { X } _ { k \eta } ) ) ) \mathbb { 1 } _ { t \in [ k \eta , ( k + 1 ) \eta ] } ( t ) . } \end{array}$ . Similarly let us define :
+
+$$
+d \tilde { Y } _ { t } = G _ { t } ( \tilde { Y } ) d t + \sqrt { 2 \lambda } d W _ { t }
+$$
+
+where $\begin{array} { r } { G _ { t } ( \tilde { Y } ) = - \sum _ { k = 0 } ^ { \infty } ( G ( \tilde { Y } _ { k \eta } ) + \frac { 1 } { \gamma } ( \tilde { Y } _ { k \eta } - P _ { \Omega } ( \tilde { Y } _ { k \eta } ) ) ) \mathbb { 1 } _ { t \in [ k \eta , ( k + 1 ) \eta ] } ( t ) . } \end{array}$ . Now applying Grisanov’s Theorem for diffusions (Theorem 4) we have:
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { K L } ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) = \frac { 1 } { 4 \lambda } \mathbb { E } _ { P _ { T } ^ { X } } \left[ \int _ { 0 } ^ { T } \left\| U _ { t } ( \tilde { X } ) - G _ { t } ( \tilde { X } ) \right\| ^ { 2 } d t \right] } \\ { \displaystyle = \frac { 1 } { 4 \lambda } \mathbb { E } \sum _ { k = 0 } ^ { K - 1 } \int _ { k \eta } ^ { ( k + 1 ) \eta } \left\| G ( \tilde { X } _ { k \eta } ) - \nabla _ { x } U ( \tilde { X } _ { k \eta } ) \right\| ^ { 2 } d t } \\ { \displaystyle \quad = \frac { \eta } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( \tilde { X } _ { k \eta } ) - \nabla _ { x } U ( \tilde { X } _ { k \eta } ) \right\| ^ { 2 } } \\ { \displaystyle \quad = \frac { \eta } { 4 \lambda } \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( X _ { k } ) - \nabla _ { x } U ( X _ { k } ) \right\| ^ { 2 } . } \end{array}
+$$
+
+Now using Pinsker inequality we have:
+
+$$
+T V ( \mu _ { \tilde { X } } ^ { T } , \mu _ { \tilde { Y } } ^ { T } ) ^ { 2 } \le 2 \mathrm { K L } ( \mu _ { \tilde { X } } ^ { T } , \mu _ { \tilde { Y } } ^ { T } ) .
+$$
+
+Hence for $T = K \eta$ we have:
+
+$$
+T V ( \mu _ { K } ^ { \mathrm { { S } \cdot P r o x L M C } } , \mu _ { K } ^ { \mathrm { { P r o x L M C } } } ) \leq \sqrt { \frac { \eta } { 2 \lambda } } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G ( X _ { k } ) - \nabla _ { x } U ( X _ { k } ) \right\| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+Proof of Theorem 2 . S-PLMC. If we set $\lambda = 1$ , $\eta \le \alpha / K ^ { 2 }$ , where $\alpha = 1 / ( \delta + \beta ^ { 2 } R ^ { 2 } )$ , in this Corollary we obtain that : $\begin{array} { r } { T V ( \mu _ { K } ^ { S - P L M C } , \mu _ { K } ^ { P L M C } ) \le \frac { 1 } { \sqrt { K } } } \end{array}$ √1K . Assuming A, B and C we consider $\eta \le \operatorname* { m i n } ( R ^ { 2 } / K , \alpha / K ^ { 2 } )$ , and $K = \tilde { \Omega } ( \varepsilon ^ { - 1 2 } d ^ { 1 2 } )$ . Now using the triangle inequality together with the bounds in Eq.s 7 we have: $T V ( \mu _ { K } ^ { S - \dot { P } L M C } , \pi ) \le T V ( \mu _ { K } ^ { S - P L M C } , \bar { \mu } _ { K } ^ { P L M C } ) + \bar { T } V ( \bar { \mu } ^ { P L M C } , \pi ) \le$ $\varepsilon + { \frac { 1 } { \sqrt { K } } }$ .
+
+S-ProxLMC. We conclude with a similar argument for µS−P roxLMC , π) using Eq.s 8. Considering $\eta = \operatorname* { m i n } ( \gamma ( 1 + \beta ^ { 2 } \gamma ^ { 2 } ) ^ { - 1 } , \frac { 1 } { \delta K ^ { 2 } } )$ , and $K = \tilde { \Omega } ( \varepsilon ^ { - 6 } d ^ { 5 } )$ , we obtain $\begin{array} { r } { ( \varepsilon + \frac { 1 } { \sqrt { K } } ) } \end{array}$ approximation in TV of the target Gibbs distribution.
+
+# Proof of Corollary 1. Z-PLMC: We have:
+
+$$
+T V ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) \leq \sqrt { \frac { \eta } { \lambda } \left( \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left. G _ { n } U ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \right. ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right) }
+$$
+
+Taking the expectation we have:
+
+$$
+\begin{array} { r } { \mathbb { E } _ { g _ { 1 } \dots g _ { n } } T V ( \mu _ { T } ^ { \tilde { X } } , \mu _ { T } ^ { \tilde { Y } } ) \le \mathbb { E } _ { g _ { 1 } \dots g _ { n } } \sqrt { \frac { \eta } { \lambda } \left( \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } \left\| G _ { n } U ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \right\| ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right) } } \\ { \le \sqrt { \displaystyle \frac { \eta } { \lambda } \left( \displaystyle \sum _ { k = 0 } ^ { K - 1 } \mathbb { E } _ { Y } \mathbb { E } _ { g _ { 1 } \dots g _ { n } } \left\| G _ { n } U ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \right\| ^ { 2 } + K \beta ^ { 2 } R ^ { 2 } \right) } } \end{array}
+$$
+
+Note now that we have:
+
+$$
+\begin{array} { r } { \mathbb { E } _ { g _ { 1 } . . . g _ { n } } \left\| G _ { n } U ( Y _ { k \eta } ) - \nabla _ { x } U ( Y _ { k \eta } ) \right\| ^ { 2 } \leq \delta , \forall Y _ { k \eta } . } \end{array}
+$$
+
+For $n \geq \left( \beta \nu ( d + 2 ) ^ { 3 / 2 } + ( d + 1 ) ^ { \frac { 1 } { 2 } } L \right) ^ { 2 } / \delta$ The rest of the proof is an application of Theorem 2.
+
+Z-ProxLMC. A similar argument holds.

parse/train/Nl7VO_Y7K4Q/Nl7VO_Y7K4Q.md

@@ -0,0 +1,369 @@
+# Learning Policies with Zero or Bounded Constraint Violation for Constrained MDPs
+
+Tao Liu∗ Texas A&M University tliu@tamu.edu
+
+Ruida Zhou∗ Texas A&M University ruida@tamu.edu
+
+Dileep Kalathil Texas A&M University dileep.kalathil@tamu.edu
+
+P. R. Kumar Texas A&M University prk@tamu.edu
+
+Chao Tian Texas A&M University chao.tian@tamu.edu
+
+# Abstract
+
+We address the issue of safety in reinforcement learning. We pose the problem in an episodic framework of a constrained Markov decision process. Existing√ results have shown that it is possible to achieve a reward regret of $\tilde { \mathcal { O } } ( \sqrt { K } )$ while allowing an $\tilde { \mathcal { O } } ( \sqrt { K } )$ constraint violation in $K$ episodes. A critical question that arises is whether it is possible to keep the constraint violation even smaller. We show that when a strictly safe policy is known, then one can confine the system to zero constraint violation with arbitrarily high probability while keeping the reward regret of order $\tilde { \mathcal { O } } ( \sqrt { K } )$ . The algorithm which does so employs the principle of optimistic pessimism in the face of uncertainty to achieve safe exploration. When no strictly safe policy is known, though one is known to exist, then it is possible to restrict the system to bounded constraint violation with arbitrarily high probability. This is shown to be realized by a primal-dual algorithm with an optimistic primal estimate and a pessimistic dual update.
+
+# 1 Introduction
+
+Reinforcement learning (RL) addresses the problem of learning an optimal control policy that maximizes the expected cumulative reward while interacting with an unknown environment [25]. Standard RL algorithms typically focus only on maximizing a single objective. However, in many real-world applications, the control policy learned by an RL algorithm has to additionally satisfy stringent safety constraints [9, 4]. For example, an autonomous vehicle may need to reach its destination in the minimum possible time without violating safety constraints such as crossing the middle of the road. The Constrained Markov Decision Process (CMDP) [2, 12] formalism, where one seeks to maximize a reward while satisfying safety constraints, is a standard approach for modeling the necessary safety criteria of a control problem via constraints on cumulative costs.
+
+Several policy-gradient-based algorithms have been proposed to solve CMDPs. Lagrangian-based methods [26, 24, 21, 16] formulate the CMDP problem as a saddle-point problem and optimize it via primal-dual methods, while Constrained Policy Optimization [1, 29] (inspired by the trust region policy optimization [23]) computes new dual variables from scratch at each update to maintain constraints during learning. Although these algorithms provide ways to learn an optimal policy, performance guarantees about reward regret, safety violation or sample complexity are rare.
+
+One class of RL algorithms for which performance guarantees are available follow the principle√ of Optimism in the Face of Uncertainty (OFU) [8, 10, 22], and provide an $\tilde { \mathcal { O } } ( \sqrt { K } )$ guarantee for the reward regret, where √ $K$ is the number of episodes. However, these algorithms also have $\tilde { \mathcal { O } } ( \sqrt { K } )$ safety violations. Such significant violation of the safety constraints during learning may be unacceptable in many safety-critical real-world applications such as the control of autonomous vehicles or power systems. These applications demand a class of safe RL algorithms that can provably guarantee safety during learning. With this goal in mind, we aim to answer the following open theoretical question in this paper:
+
+Table 1: Regret and constraint violation comparisons for algorithms on episodic CMDPs   
+
+<table><tr><td>Algorithm</td><td>Regret2 Constraint violation ²</td></tr><tr><td>O（H√ OPDOP [8] OptCMDP[10] 3 O(H² OptCMDP-bonus [10] 3 O(H²√ OptDual-CMDP [10] 3 (H²IS|3IAK)</td><td>[S|2|A|K) (H3√SPIA|K) S|3|A|K) (H²√S|3IAK) s|3|A|K) (H²S|3AK) (H²√s3 A|K)</td></tr><tr><td>OptPrimalDual-CMDP[10] 3 C-UCRL [30] 4</td><td>(H²√Is|3|A|K) (H²√Is|3|A|K) （T 0</td></tr><tr><td>OptPess-LP 0（ H T-C0 6 OptPess-PrimalDual H</td><td>/S|3|A|K) 0 v √S|3|A|K) 0(1) 5</td></tr></table>
+
+Can we design safe RL algorithms that can achieve an $\tilde { \mathcal { O } } ( \sqrt { K } )$ regret with respect to the performance objective, while guaranteeing zero or bounded safety constraint violation with arbitrarily high probability?
+
+We answer the above question affirmatively by proposing two algorithms and establishing their stringent safety performance during learning. Our focus is on the tabular episodic constrained RL setting (unknown transition probabilities, rewards, and costs). The key idea behind both algorithms is a concept used earlier for safe exploration in constrained bandits [20, 17], which we call “Optimistic Pessimism in the Face of Uncertainty (OPFU)” here. The optimistic aspect incentivizes the algorithm for using exploration policies that can visit new state-action pairs, while the pessimistic aspect disincentivizes the algorithm from using exploration policies that can violate safety constraints. By carefully balancing optimism and pessimism, the proposed algorithms guarantee zero or bounded√ safety constraint violation during learning while achieving an $\tilde { \mathcal { O } } ( \sqrt { K } )$ regret with respect to the reward objective.
+
+The two algorithms address two different classes of the safe learning problem: whether a strictly safe policy is known a priori or not. The resulting exploration strategies are very different in the two cases.
+
+1. OptPess-LP Algorithm: This algorithm assumes the prior knowledge of a strictly safe policy. It ensures zero safety constraint violation during learning with high probability and utilizes the linear programming (LP) approach for solving a CMDP problem. The algorithm achieves a reward regret of O˜( H3τ−c0 $\begin{array} { r } { \tilde { \mathcal { O } } ( \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | { \cal S } | ^ { 3 } | { \cal A } | K } ) } \end{array}$ with respect to the performance objective, where $H$ is the number of steps per episode, $\tau$ is the given constraint on safety violation, $c ^ { 0 }$ is the known safety constraint value of a strictly safe policy $\pi ^ { 0 }$ , and $| S |$ and $| { \cal A } |$ are the number of states and actions respectively.
+
+2. OptPess-PrimalDual Algorithm: This algorithm addresses the case where no strictly safe policy, but a feasible strictly safe cost is known. By allowing a bounded (in $K$ ) safety cost, it opens up space for exploration. The OptPess-PrimalDual algorithm avoids linear programming and its attendant complexity and exploits the primal-dual approach for solving a CMDP problem. The proposed approach improves the computational tractability, while ensuring a bounded safety constraint violation during learning and a reward regret of $\begin{array} { r l r } {  { \tilde { \mathcal { O } } ( \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | \boldsymbol { S } | ^ { 3 } | \boldsymbol { A } | K } ) } } \end{array}$ with respect to the objective.
+
+Compared with the other methods listed in Table 1, though the proposed algorithms have an additional $H / ( \tau - c ^ { 0 } )$ or $\sqrt { | \boldsymbol { S } | } / ( \tau - c ^ { 0 } )$ factor in the regret bounds, they are able to reduce the constraint violation to zero or constant with high probability. This improvement in safety can be extremely important for many mission-critical applications.
+
+# 1.1 Related work
+
+The problem of learning an optimal control policy that satisfies safety constraints has been studied both in the RL setting and the multi-armed bandits setting.
+
+Constrained RL: Several policy-gradient algorithms have seen success in practice [26, 24, 21, 16, 1, 29]. Also of interest are works which utilize Gaussian processes to model the transition probabilities and value functions [5, 27, 15, 7].
+
+Several algorithms with provable guarantees are closely related to our work. Zheng et al. [30] considered the constrained RL problem in an infinite horizon setting with unknown reward and cost functions. The approach is similar to the UCRL2 algorithm [13] and achieves a sub-linear reward regret $\tilde { \mathcal { O } } ( T ^ { \frac { 3 } { 4 } } )$ while satisfying constraints with high probability during learning. In contrast to this work, we consider the setting of unknown transition probabilities. Efroni et al [10] focused on the episodic setting of unknown non-stationary transitions over a finite horizon, attaining both a reward regret and a constraint violation of $\tilde { \mathcal { O } } ( H ^ { 2 } \sqrt { | { \cal S } | ^ { 3 } | { \cal A } | K } )$ . Ding et al. [8] studied an episodic setting with linear function approximation (suitable for large state space cases), and proposed algorithms that√ can achieve $\tilde { \mathcal O } ( d H ^ { 3 } \sqrt { K } )$ for both the regret and the constraint violation (where $d$ is the dimension of the feature mapping). The regret analysis in [8] can be easily extended to the tabular case, yielding $\tilde { \mathcal { O } } ( H ^ { 3 } \sqrt { | { \cal S } | ^ { 2 } | \mathcal { A } | K } ) ^ { - }$ .
+
+Constrained Multi-Armed Bandits: Multi-armed bandit problems are special cases of MDPs, with both the number of states as well as the episode length being one. Linear bandits with constraints (satisfied with high probability) have been investigated in different settings. One setting, referred to as conservative bandits [28, 14, 11], requires the cumulative reward to remain above a fixed percentage of the cumulative reward of a given baseline policy. Another setting is where each arm is associated with two unknown distributions (similar to our setting), generating reward and cost signals respectively [3, 20, 17, 18].
+
+# 2 Problem formulation
+
+A finite-horizon constrained non-stationary MDP model is defined as a tuple $\begin{array} { r l } { M } & { { } = } \end{array}$ $( \mathcal S , \mathcal A , H , P , r , c , \tau , \mu )$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $H$ is the number of steps in each episode, $r : S \times A  [ 0 , 1 ]$ is the unknown reward function of interest, $c : S \times A  [ 0 , 1 ]$ is the unknown safety cost function used to model the constraint violation, $\tau \in ( 0 , H ]$ is the given constant used to define the safety constraint, and $\mu$ is the known initial distribution of the state. $P . ( \cdot | s , a ) \in \Delta _ { s } ^ { H } , \forall s \in S , \forall a \in \mathcal { A }$ , where $\Delta _ { \mathcal { S } }$ is the $| S |$ -dimensional probability simplex, and $P _ { h } ( s ^ { \prime } | s , a )$ is the unknown transition probability that the next state is $s ^ { \prime }$ when action $a$ is taken for state $s$ at step $h$ .
+
+Some further notation is necessary to define the problem. A (randomized Markov) policy is defined by a map $\pi : S \times [ H ] \to \Delta _ { { \cal A } }$ , with $\pi _ { h } ( a | s )$ being the probability of taking action $a$ in state $s$ at time step $h$ . With $S _ { t }$ and $A _ { t }$ representing the state and the action at time $t$ respectively, let
+
+$$
+V _ { h } ^ { \pi } ( s ; g , P ) : = \mathbb { E } _ { P , \pi } \left[ \sum _ { t = h } ^ { H } g ( S _ { t } , A _ { t } ) | S _ { h } = s \right] , \quad \forall s \in \mathcal { S }
+$$
+
+denote the expected cumulative value with respect to a function $g : { \mathcal { S } } \times { \mathcal { A } } \to \mathbb { R } _ { + }$ under $P$ for a policy $\pi$ over a time interval $[ h , h + 1 , \ldots , H ]$ . With slightly abuse of notation, we use $V _ { 1 } ^ { \pi } ( \mu ; g , P )$ to denote $\mathbb { E } _ { S _ { 1 } \sim \mu } [ V _ { 1 } ^ { \pi } ( S _ { 1 } ; g , \dot { P } ) ]$ .
+
+In the formulation below there are a total of $K$ episodes with $H$ steps each. Each episode $k \in [ K ]$ begins with an initial probability distribution $\mu$ for $S _ { 1 }$ . Then, the agent determines a randomized Markov policy $\pi ^ { k }$ for that episode based on the information gathered from the previous episodes, and executes it. At time step $h$ during the execution of the $k$ -th episode, after taking action $A _ { h } ^ { k }$ at state $S _ { h } ^ { k }$ , the agent receives a noisy reward and cost of $R _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) = r _ { h } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) + \xi _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ; r )$ and $\begin{array} { r } { \dot { C } _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) = c _ { h } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) + \xi _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ; c ) } \end{array}$ , respectively.
+
+Assumption 2.1 (Sub-Gaussian noise). For all $h \in [ H ] , k \in [ K ]$ , the reward and cost noise random variables are conditionally independent zero-mean $1 / 2$ -sub-Gaussian, i.e., $\mathbb { E } [ \xi _ { h } ^ { k } | \mathcal { F } _ { k - 1 } ] = 0 ;$ , $\mathbb { E } [ \exp ( \lambda \xi _ { h } ^ { k } ) | \mathcal { F } _ { k - 1 } ] \le \exp ( \lambda ^ { 2 } / 4 ) , $ $\forall \lambda \in \mathbb { R }$ . Here $\mathcal { F } _ { k }$ is the $\sigma$ -algebra generated by the random variables up to episode $k$ .
+
+Let $\pi ^ { * }$ denote the optimal policy of the following CMDP model:
+
+$$
+\operatorname* { m a x } _ { \pi } \quad V _ { 1 } ^ { \pi } ( \mu ; r , P ) \qquad \mathrm { s . t . } \quad V _ { 1 } ^ { \pi } ( \mu ; c , P ) \leq \tau .
+$$
+
+A policy $\pi$ is said to be strictly safe if $V _ { 1 } ^ { \pi } ( \mu ; c , P ) < \tau$ .
+
+Assumption 2.2. There exists a strictly safe policy $\pi ^ { 0 }$ with ${ V _ { 1 } ^ { \pi } } ^ { 0 } ( \mu ; c , P ) = c ^ { 0 } < \tau .$ .
+
+There are two important cases of the safe learning problem.
+
+Zero constraint violation case: The agent has prior knowledge of a strictly safe policy $\pi ^ { 0 }$ and its safety cost value $c ^ { 0 } : = V _ { 1 } ^ { \pi ^ { 0 } } ( \mu ; c , P )$ . The agent wishes to attain a sublinear (in $K$ ) cumulative regret,
+
+$$
+R e g ( K ; r ) : = \sum _ { k = 1 } ^ { K } \Big ( { V _ { 1 } ^ { \pi } } ^ { * } ( \mu ; r , P ) - { V _ { 1 } ^ { \pi } } ^ { k } ( \mu ; r , P ) \Big ) ,
+$$
+
+while incurring zero constraint violation with at least a specified high probability $( 1 - \delta )$ , i.e.,
+
+$$
+\begin{array} { r } { \mathbb { P } \left( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; c , P ) \leq \tau , \forall k \in [ K ] \right) \geq 1 - \delta . } \end{array}
+$$
+
+Bounded constraint violation case: The agent knows that there exists a strictly safe policy with a known safety cost value $c ^ { 0 }$ , but does not know any strictly safe policy. It aims to achieve a cumulative regret (2) that grows sublinearly with $K$ , while ensuring that the regret of constraint violation,
+
+$$
+R e g ( K ; c ) : = \left( \sum _ { k = 1 } ^ { K } \left( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; c , P ) - \tau \right) \right) _ { + } \left( \mathrm { w h e r e } \left( a \right) _ { + } : = \operatorname * { m a x } \{ a , 0 \} \right) ,
+$$
+
+satisfies $\operatorname* { s u p } _ { K } R e g ( K ; c ) < + \infty$ with at least a specified high probability $( 1 - \delta )$ .
+
+Remarks. For the zero constraint violation case, the assumption of knowing $c ^ { 0 }$ can be relaxed, as shown in Appendix E.
+
+# 3 Zero constraint violation case
+
+We start by considering the zero constraint violation case. On one hand, to balance the explorationexploitation trade-off, we employ an optimistic estimate of the reward function, as embodied in the OFU principle. On the other hand, to maintain absolute safety with high probability during the exploration, we employ a pessimistic estimate of the safety cost. Such an OPFU principle was previously discussed in constrained bandits [3] and we adapt it to the unknown CMDP setting.
+
+At each episode $k$ , we begin by forming empirical estimates of the transition probabilities, the reward function, and the cost function from step $h$ of all previous episodes:
+
+$$
+\begin{array} { r l } & { \hat { P } _ { h } ^ { k } ( s ^ { \prime } | s , a ) : = \frac { \sum _ { k ^ { \prime } = 1 } ^ { k - 1 } \mathbb { 1 } \left( S _ { h } ^ { k ^ { \prime } } = s , A _ { h } ^ { k ^ { \prime } } = a , S _ { h + 1 } ^ { k ^ { \prime } } = s ^ { \prime } \right) } { N _ { h } ^ { k } ( s , a ) \vee 1 } , } \\ & { \hat { g } _ { h } ^ { k } ( s , a ) : = \frac { \sum _ { k ^ { \prime } = 1 } ^ { k - 1 } \mathbb { 1 } \left( S _ { h } ^ { k ^ { \prime } } = s , A _ { h } ^ { k ^ { \prime } } = a \right) \left( g _ { h } ( s , a ) + \xi _ { h } ^ { k } ( s , a ; g ) \right) } { N _ { h } ^ { k } ( s , a ) \vee 1 } , \quad \mathrm { ~ f o r ~ } g = r , c , } \end{array}
+$$
+
+where $a \vee b : = \operatorname* { m a x } \{ a , b \}$ , and
+
+Next we fix some $\delta \in ( 0 , 1 )$ and form a common (for notational simplicity) confidence radius $\beta _ { h } ^ { k } ( s , a )$ for the transition probabilities, the rewards, and the costs,
+
+$$
+\beta _ { h } ^ { k } ( s , a ) : = \sqrt { \frac { 1 } { N _ { h } ^ { k } ( s , a ) \vee 1 } Z } , \quad \mathrm { w h e r e ~ } Z : = \log ( 1 6 | S | ^ { 2 } | A | H K / \delta ) .
+$$
+
+At each episode $k$ , we define the optimistically biased reward estimate as
+
+$$
+\bar { r } _ { h } ^ { k } ( s , a ) : = \hat { r } _ { h } ^ { k } ( s , a ) + \alpha _ { r } \beta _ { h } ^ { k } ( s , a ) , \quad \forall ( s , a , h ) \in \mathcal { S } \times \mathcal { A } \times [ H ] ,
+$$
+
+where the scaling factor is
+
+$$
+\alpha _ { r } : = 1 + | S | H + \frac { 4 H ( 1 + | S | H ) } { \tau - c ^ { 0 } } .
+$$
+
+To guarantee safe exploration, define the pessimistically biased safety cost estimate at episode $k$ as
+
+$$
+\begin{array} { r } { \underline { { \mathsf { c } } } _ { h } ^ { k } ( s , a ) : = \hat { c } _ { h } ^ { k } ( s , a ) + ( 1 + H | S | ) \beta _ { h } ^ { k } ( s , a ) , \quad \forall ( s , a , h ) \in S \times { \cal A } \times [ H ] . } \end{array}
+$$
+
+The policy we execute at episode $k$ is chosen from a “pessimistically safe" policy set $\Pi ^ { k }$ , defined as
+
+$$
+\Pi ^ { k } : = \left\{ \begin{array} { l l }  { \{ \pi ^ { 0 } \} } & { { \mathrm { i f ~ } V _ { 1 } ^ { \pi ^ { 0 } } ( \mu ; \underline { { { c } } } ^ { k } , \hat { P } ^ { k } ) \geq ( \tau + c ^ { 0 } ) / 2 , } } \\  { \{ \pi : V _ { 1 } ^ { \pi } ( \mu ; \underline { { { c } } } ^ { k } , \hat { P } ^ { k } ) \leq \tau \} } & { { \mathrm { o t h e r w i s e . } } } \end{array} \right.
+$$
+
+We simply use the strictly safe policy $\pi ^ { 0 }$ until $V _ { 1 } ^ { \pi ^ { 0 } } ( \mu ; \underline { { { \mathrm { c } } } } ^ { k } , \hat { P } ^ { k } ) < ( \tau + c ^ { 0 } ) / 2$ , which is a sufficient condition to guarantee that the set $\{ \pi : V _ { 1 } ^ { \pi } ( \mu ; \underline { { { \mathrm { c } } } } ^ { k } , \hat { P } ^ { k } ) \leq \tau \}$ is non-empty. Within this set, we choose the optimistically best reward earning policy $\pi ^ { k }$ , which can be solved by linear programming with $\Theta ( | S | | A | H )$ decision variables and constraints [10]. The resulting Optimistic Pessimism-based Linear Programming (OptPess-LP) algorithm is presented below:
+
+# Algorithm 1: OptPess-LP
+
+Input: $K , \delta , \pi ^ { 0 } , c ^ { 0 } , \tau$ ;   
+Initialization: $N _ { h } ^ { 1 } ( s , a ) = 0$ , $\forall ( s , a , h ) \in \mathcal { S } \times \mathcal { A } \times [ H ] ;$ ;   
+for $k = 1 , 2 , \ldots , K$ do Update empirical model (i.e., $\hat { P } ^ { k } , \hat { r } ^ { k } , \hat { c } ^ { k } )$ as in Equations (3)-(5); Update $\bar { r } ^ { k } , \mathrm { c } ^ { k }$ , and $\Pi ^ { k }$ as in Equations (6)-(9); Calculate $\pi ^ { k } \in \arg \operatorname* { m a x } _ { \pi \in \Pi ^ { k } } V _ { 1 } ^ { \pi } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } )$ ; Execute $\pi ^ { k }$ and collect a trajectory $( S _ { h } ^ { k } , A _ { h } ^ { k } , R _ { h } ^ { k } , C _ { h } ^ { k } )$ , $\forall h \in [ H ]$ ; Update counters $N _ { h } ^ { k + 1 } ( S _ { h } ^ { k } , A _ { h } ^ { k } )$ , $\forall h \in [ H ]$ ;
+
+end
+
+Theorem 3.1 (Regret and constraint violation bounds for OptPess-LP). Fix any $\delta \in ( 0 , 1 )$ . With probability at least $( 1 - \delta )$ , OptPess- $. L P$ has zero constraint violation with
+
+$$
+R e g ^ { \theta P L P } ( K ; r ) = \tilde { \mathcal { O } } \left( \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | \mathcal { S } | ^ { 3 } | \mathcal { A } | K } + \frac { H ^ { 5 } | \mathcal { S } | ^ { 3 } | \mathcal { A } | } { ( \tau - c ^ { 0 } ) ^ { 2 } \wedge ( \tau - c ^ { 0 } ) } \right) , w h e r e \ a \wedge b : = \operatorname* { m i n } \{ a , b \} .
+$$
+
+Theorem 3.1 shows that it is possible to achieve sublinear regret in $K$ , while simultaneously incurring no constraint violation with arbitrarily high probability. The proof is sketched in Section 5.1, with detailed proofs presented in Appendix B.
+
+# 4 Bounded constraint violation case
+
+Without prior knowledge of a strictly safe policy $\pi ^ { 0 }$ , we may not be able to guarantee zero constraint violation with high probability. However, by relaxing the requirement to bounded (in $K _ { \cdot }$ ) safety constraint violation, we can incorporate more exploration and design a more efficient algorithm by a primal-dual approach. It is inspired by the design of the pessimistic term in constrained bandits [17].
+
+Different from traditional optimistic dual analysis methods in [10], we introduce an additive pessimistic term $\epsilon _ { k }$ at each episode $k$ in the original optimization problem (1), i.e.,
+
+$$
+\operatorname* { m a x } _ { \pi } \quad V _ { 1 } ^ { \pi } ( \mu ; r , P ) \qquad \mathrm { s . t . } \quad V _ { 1 } ^ { \pi } ( \mu ; c , P ) + \epsilon _ { k } \leq \tau .
+$$
+
+The pessimistic term restrains the constraint violation and will be progressively decreased as learning proceeds.
+
+The CMDP problem in (10) may however not have any feasible solution. To overcome this, we consider the Lagrangian,
+
+$$
+L ^ { k } ( \pi , \lambda ) : = V _ { 1 } ^ { \pi } ( \mu ; r , P ) + \lambda ( \tau - \epsilon _ { k } - V _ { 1 } ^ { \pi } ( \mu ; c , P ) ) ,
+$$
+
+for which, given any Lagrange multiplier $\lambda$ , we can always solve for the optimizer $\operatorname* { m a x } _ { \pi } L ^ { k } ( \pi , \lambda )$ by dynamic programming.
+
+With the introduction of the additive term $\epsilon _ { k }$ , the dual variable (denoted by $\lambda ^ { k }$ ) governed by the subgradient algorithm grows faster, which enhances safety constraints in the next episode.
+
+In order to guarantee sufficient optimism of rewards and costs, we integrate the uncertainty of transitions into rewards and costs, i.e.,
+
+$$
+\begin{array} { r l } & { \tilde { r } _ { h } ^ { k } ( s , a ) : = \hat { r } _ { h } ^ { k } ( s , a ) + \beta ( n _ { k } ( s , a , h ) ) + H | S | \beta ( n _ { k } ( s , a , h ) ) , \quad \forall ( s , a , h ) \in S \times A \times [ H ] , } \\ & { \tilde { c } _ { h } ^ { k } ( s , a ) : = \hat { c } _ { h } ^ { k } ( s , a ) - \beta ( n _ { k } ( s , a , h ) ) - H | S | \beta ( n _ { k } ( s , a , h ) ) , \quad \forall ( s , a , h ) \in S \times A \times [ H ] . } \end{array}
+$$
+
+Note that in contrast to the zero constraint violation case, we optimistically estimate the safety cost function, with the pessimism only governed by $\epsilon _ { k }$ .
+
+We employ truncated value functions due to the additional uncertainties from transitions, i.e.,
+
+$$
+\hat { V } _ { 1 } ^ { \pi } ( \mu ; \hat { r } ^ { k } , \hat { P } ^ { k } ) : = \operatorname* { m i n } \{ H , V _ { 1 } ^ { \pi } ( \mu ; \hat { r } ^ { k } , \hat { P } ^ { k } ) \} , \hat { V } _ { 1 } ^ { \pi } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) : = \operatorname* { m a x } \{ 0 , V _ { 1 } ^ { \pi } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) \} .
+$$
+
+For the policy update of the primal variable (denoted by $\pi ^ { k }$ ), we can apply standard dynamic programming by viewing $\begin{array} { r } { \tilde { r } _ { h } ^ { k } ( s , a ) - \frac { \lambda ^ { k } } { \eta ^ { k } } ( \tilde { c } _ { h } ^ { k } ( s , a ) - \tau ) } \end{array}$ as the reward function. Specifically, we apply backward induction to solve for the optimal policy:
+
+$$
+Q _ { h } ^ { k } ( s , a ) = \tilde { r } _ { h } ^ { k } ( s , a ) - \frac { \lambda ^ { k } } { \eta ^ { k } } ( \tilde { c } _ { h } ^ { k } ( s , a ) - \tau ) + \sum _ { s ^ { \prime } \in S } \hat { P } _ { h } ^ { k } ( s ^ { \prime } | s , a ) \operatorname* { m a x } _ { a ^ { \prime } } Q _ { h + 1 } ^ { k } ( s ^ { \prime } , a ^ { \prime } ) , \quad \forall h \in [ H ]
+$$
+
+with $Q _ { H + 1 } ^ { k } ( s , a ) = 0 , \forall ( s , a ) \in \mathcal { S } \times \mathcal { A }$ . Then, $\pi _ { h } ^ { k } \in \arg \operatorname* { m a x } _ { a } Q _ { h } ^ { k } ( s , a )$ , which is computationally efficient (as efficient as policy-gradient-based algorithms in the tabular case).
+
+The resulting Optimistic Pessimism-based Primal-Dual (OptPess-PrimalDual) algorithm is shown in Algorithm 2. It chooses $\begin{array} { r } { \epsilon _ { k } : = 5 H ^ { 2 } \sqrt { | { \cal S } | ^ { 3 } | A | } ( \log \frac { k } { \delta ^ { \prime } } + 1 ) / \sqrt { k \log \frac { k } { \delta ^ { \prime } } } } \end{array}$ , $\delta ^ { \prime } = \delta / ( 1 6 | S | ^ { 2 } | A | H )$ , the scaling parameter $\eta ^ { k } : = ( \tau - c ^ { 0 } ) H \sqrt { k }$ in the primal policy update, and, for convenience, a step size of 1 for the dual update.
+
+Theorem 4.1 (Regret and constraint violation bounds for OptPess-PrimalDual). Fix any $\delta \in ( 0 , 1 )$ . Then,
+
+$$
+\begin{array} { l } { { R e g ^ { O P P D } ( K ; r ) = \tilde { \mathcal { O } } \left( \displaystyle \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | \mathcal { S } | ^ { 3 } | \mathcal { A } | K } + \displaystyle \frac { H ^ { 5 } | \mathcal { S } | ^ { 3 } | \mathcal { A } | } { ( \tau - c ^ { 0 } ) ^ { 2 } } \right) , } } \\ { { R e g ^ { O P P D } ( K ; c ) = \mathcal { O } \left( C ^ { \prime \prime } ( H - \tau ) + H ^ { 2 } \sqrt { | \mathcal { S } | ^ { 3 } | \mathcal { A } | C ^ { \prime \prime } } \right) = \mathcal { O } ( 1 ) , } } \end{array}
+$$
+
+where $\begin{array} { r } { C ^ { \prime \prime } = \mathcal { O } ( \frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \tau - c ^ { 0 } ) ^ { 2 } } \log \frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \tau - c ^ { 0 } ) ^ { 2 } \delta ^ { \prime } } ) } \end{array}$ is a coefficient independent of $K$
+
+Theorem 4.1 shows that it is possible to achieve an $\tilde { \mathcal { O } } ( \sqrt { K } )$ reward regret, while only allowing bounded constraint violation with arbitrarily high probability. Detailed proofs are presented in Section 5.2 and Appendix $\textrm { C }$ .
+
+Input: $K , \delta , c ^ { 0 } , \tau$ ;   
+Initialization: $N _ { h } ^ { 1 } ( s , a ) = 0 , \forall ( s , a , h ) \in \mathcal { S } \times \mathcal { A } \times [ H ] , \lambda ^ { 1 } = 0 ;$   
+for $k = 1 , 2 , \dots , \overset { \cdot } { K }$ do S $\begin{array} { r } { \mathfrak { x } \mathfrak { t } \mathfrak { \epsilon } _ { k } = 5 H ^ { 2 } \sqrt { | \mathcal { S } | ^ { 3 } | \mathcal { A } | } ( \log \frac { k } { \delta ^ { \prime } } + 1 ) / \sqrt { k \log \frac { k } { \delta ^ { \prime } } } , \delta ^ { \prime } = \delta / ( 1 6 | \mathcal { S } | ^ { 2 } | \mathcal { A } | H ) , } \end{array}$ $\eta ^ { k } = ( \tau - c ^ { 0 } ) H \sqrt { k } ;$ ; Update empirical model (i.e., $\hat { P } ^ { k } , \hat { r } ^ { k } , \hat { c } ^ { k } )$ as in Equations (3)-(5); Update $\tilde { r } ^ { k }$ , $\tilde { c } ^ { k }$ as in Equation (11); $\begin{array} { r l } & { \dot { ( P o l i c y ~ U p d a t e ) } \pi ^ { k } \in \overset { \cdot } { \arg \operatorname* { m a x } } _ { \pi \in \Pi } \hat { V } _ { 1 } ^ { \pi } ( \mu ; \tilde { r } ^ { k } , \hat { P } ^ { k } ) - \frac { \lambda ^ { k } } { \eta ^ { k } } \left( \hat { V } _ { 1 } ^ { \pi } ( \mu ; \tilde { c } ^ { k } , \hat { P } ^ { k } ) - \tau \right) ; } \\ & { ( D u a l ~ U p d a t e ) ~ \lambda ^ { k + 1 } = \Bigl ( \lambda ^ { k } + \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \tilde { c } ^ { k } , \hat { P } ^ { k } ) + \epsilon _ { k } - \tau \Bigr ) _ { + } ; } \end{array}$ Execute $\pi ^ { k }$ and collect a trajectory $( S _ { h } ^ { k } , A _ { h } ^ { k } , R _ { h } ^ { k } , C _ { h } ^ { k } )$ , $\forall h \in [ H ]$ ; Update counters $N _ { h } ^ { k + 1 } ( S _ { h } ^ { k } , A _ { h } ^ { k } )$ , $\forall h \in [ H ]$ ;   
+end
+
+# 5 Regret and constraint violation analysis
+
+We sketch the key steps in the proofs of Theorems 3.1 and 4.1. Full details are relegated to Appendices B and C, respectively. Note that our analysis and results are conditioned on the same high probability event (specifically defined in Appendix A), which occurs with probability at least $( 1 - \delta )$ .
+
+# 5.1 Analysis of OptPess-LP (Algorithm 1)
+
+Constraint violation analysis The zero constraint violation of OptPess-LP follows from the following property of the pessimistic policy set $\Pi ^ { k }$ :
+
+Lemma 5.1. With probability at least $( 1 - \delta )$ , for any $k \in [ K ]$ and policy $\pi \in \Pi ^ { k }$ , $V _ { 1 } ^ { \pi } ( \mu ; c , P ) \leq \tau$
+
+Regret of reward analysis When the parameters $c , P$ are not well estimated, there may not exist any policy $\pi$ such that $V _ { 1 } ^ { \pi } ( \mu ; \underline { { { \mathrm { c } } } } ^ { k } , \hat { P } ^ { k } ) \le \tau$ . Hence, as defined in (9), $\Pi ^ { k }$ is a singleton set $\{ \pi ^ { 0 } \}$ , and accordingly $\pi ^ { 0 }$ is executed, even though $\pi ^ { 0 }$ is not safe for $( \underline { { \mathrm { c } } } ^ { k } , \hat { P } ^ { k } )$ . It subsequently takes several episodes of exploration using $\pi ^ { 0 }$ until it becomes strictly safe for $( \underline { { \mathrm { c } } } ^ { k } , \hat { P } ^ { k } )$ . At that time, policies close enough to $\pi ^ { 0 }$ , of which there are infinitely many, are also safe for $( \underline { { \mathsf { c } } } ^ { k } , \hat { P } ^ { k } )$ , and so $| \Pi ^ { k } | = + \infty$ . At this point the learning algorithm can proceed to enhance reward while maintaining safety with respect to $( \underline { { \mathrm { c } } } ^ { k } , \hat { P } ^ { k } )$ .
+
+To analyze the algorithm, we decompose the reward regret as follows:
+
+$$
+\begin{array} { l } { { \displaystyle { R e g ^ { \mathbf { o p L P } } ( K ; r ) = \sum _ { k = 1 } ^ { K } \mathbb { 1 } \left( | \Pi ^ { k } | = 1 \right) \left( V _ { 1 } ^ { \pi ^ { * } } ( \mu ; r , P ) - V _ { 1 } ^ { \pi ^ { 0 } } ( \mu ; r , P ) \right) } } } \\ { { \displaystyle { ~ + \sum _ { k = 1 } ^ { K } \mathbb { 1 } \left( | \Pi ^ { k } | > 1 \right) \left( V _ { 1 } ^ { \pi ^ { * } } ( \mu ; r , P ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } ) \right) } } } \\ { { \displaystyle { ~ + \sum _ { k = 1 } ^ { K } \mathbb { 1 } \left( | \Pi ^ { k } | > 1 \right) \left( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; r , P ) \right) } . } } \end{array}
+$$
+
+To bound the first term on the right-hand side (RHS) of (13), we have the following lemma which gives an upper bound on the number of episodes for exploration by policy $\pi ^ { 0 }$ :
+
+Lemma 5.2. With probability at least $( 1 ~ - ~ \delta )$ $) , ~ \sum _ { k = 1 } ^ { K } \mathbb { 1 } ( | \Pi ^ { k } | ~ = ~ 1 ) ~ \le ~ C ^ { \prime } ,$ , where $\begin{array} { r l } { C ^ { \prime } } & { { } = } \end{array}$ $\tilde { \mathcal { O } } ( H ^ { 4 } | S | ^ { 3 } | A | / ( ( \tau - c ^ { 0 } ) ^ { 2 } \wedge ( \tau - c ^ { 0 } ) ) )$ .
+
+Turning to the second term on the RHS of (13), we first note that $\pi ^ { * }$ may not be in $\Pi ^ { k }$ . To ensure that the term $V _ { 1 } ^ { \pi ^ { * } } ( \mu ; r , P ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } )$ is nevertheless non-positive even when $\pi ^ { * } \notin \Pi ^ { k }$ , we set $\alpha _ { r }$ to the large value shown in (7). This increases $\bar { r } ^ { k }$ , and hence also $V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } )$ . In addition, we show that there is a policy $\hat { \pi } ^ { k }$ that attains the same reward as the (non-Markov) probabilistic mixed policy, $\tilde { \pi } ^ { k } : = B _ { \gamma _ { k } } \pi ^ { \bar { * } } + ( \mathrm { i } - B _ { \gamma _ { k } } ) \pi ^ { 0 }$ , where $B _ { \gamma _ { k } }$ is a Bernoulli distributed random variable with mean $\gamma _ { k }$ for $\gamma _ { k } \in [ 0 , 1 ]$ . $\gamma _ { k }$ will be chosen as the largest coefficient such that $V _ { 1 } ^ { \tilde { \pi } ^ { k } } ( \mu ; \underline { { { c } } } ^ { k } , \hat { P } ^ { k } ) \le \tau$ . This latter policy in turn has a larger reward than $\pi ^ { 0 }$ since it is a mixture with $\pi ^ { * }$ , yielding the following:
+
+Lemma 5.3. With probability at least $( 1 - \delta )$
+
+$$
+\sum _ { k = 1 } ^ { K } \mathbb { 1 } ( | \Pi ^ { k } | > 1 ) \left( V _ { 1 } ^ { \pi ^ { * } } ( \mu ; r , P ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } ) \right) \leq 0 .
+$$
+
+Finally, concerning the third term on the RHS of (13), akin to the closed-loop identifiability property [6, 19], while $\hat { P } ^ { k }$ may not converge to $P$ , the difference in the rewards $V _ { 1 } ^ { \pi ^ { k } } ( \bar { \mu ; } \bar { r } ^ { k } , \hat { P } ^ { k } ) - \bar { V _ { 1 } ^ { \pi ^ { k } } } \bar { ( \mu ; \boldsymbol { r } , P ) }$ grows sublinearly in $k$ since the same policy $\pi ^ { k }$ is used in both values:
+
+Lemma 5.4. With probability at least $( 1 - \delta )$ ,
+
+$$
+\sum _ { k = 1 } ^ { K } \mathbb { 1 } ( | \Pi ^ { k } | > 1 ) \left( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; \bar { r } ^ { k } , \hat { P } ^ { k } ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; r , P ) \right) = \tilde { \mathcal { O } } \left( \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | { \cal S } | ^ { 3 } | { \cal A } | K } + \frac { H ^ { 5 } | { \cal S } | ^ { 3 } | { \cal A } | } { \tau - c ^ { 0 } } \right) .
+$$
+
+Combining Lemmas 5.2, 5.3, and 5.4 yields Theorem 3.1.
+
+# 5.2 Analysis of OptPess-PrimalDual (Algorithm 2)
+
+In this section, we outline the steps in the proof of Theorem 4.1 by analyzing regret and constraint violation of OptPess-PrimalDual respectively.
+
+Recall $\begin{array} { r } { \epsilon _ { k } = 5 H ^ { 2 } \sqrt { | { \cal S } | ^ { 3 } | { \cal A } | } ( \log \frac { k } { \delta ^ { \prime } } + 1 ) / \sqrt { k \log \frac { k } { \delta ^ { \prime } } } } \end{array}$ , where $\delta ^ { \prime } = \delta / ( 1 6 | S | ^ { 2 } | A | H )$ . The existence of a feasible solution to (10) can be guaranteed if $\epsilon _ { k } \leq \tau - c ^ { 0 }$ . Let $C ^ { \prime \prime }$ be the smallest value such that $\forall k \geq$ $C ^ { \prime \prime } , \epsilon _ { k } \le ( \tau - c ^ { 0 } ) / 2$ . Then the perturbed optimization problem (y simple calculation, one can verify that $k \geq C ^ { \prime \prime }$ $\begin{array} { r } { C ^ { \prime \prime } = \mathcal { O } ( \frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \tau - c ^ { 0 } ) ^ { 2 } } \log \frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \tau - c ^ { 0 } ) ^ { 2 } \delta ^ { \prime } } ) } \end{array}$ Notice that $\epsilon _ { k }$ is a function not depending on , and so is the coefficient $C ^ { \prime \prime }$ .
+
+Constraint violation analysis The bounded constraint violation of OptPess-PrimalDual is established as follows. We first decompose the constraint violation as
+
+$$
+\begin{array} { l } { { \displaystyle R e g ^ { \mathbf { o p p D } } ( K ; c ) = ( \sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; c , P ) - \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) ) + \sum _ { k = 1 } ^ { K } ( \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) - \tau ) ) _ { + } } } \\ { { \displaystyle \qquad \leq ( \sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; c , P ) - \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) ) + \lambda ^ { K + 1 } - \sum _ { k = 1 } ^ { K } \epsilon _ { k } ) _ { + } . \qquad ( 1 \otimes ( \tau ^ { k } ) ^ { K } ( \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } \otimes \tau ^ { k } ) _ { - \tau ^ { k } } } } \end{array}
+$$
+
+The first summation term and $\lambda ^ { K + 1 }$ in (14) can be bounded as follows:
+
+Lemma 5.5. Recall $\delta ^ { \prime } = \delta / ( 1 6 | S | ^ { 2 } | A | H )$ , with probability at least $( 1 - \delta )$ ,
+
+$$
+\sum _ { k = 1 } ^ { K } \left( V _ { 1 } ^ { \pi ^ { k } } ( \mu ; c , P ) - \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \hat { c } ^ { k } , \hat { P } ^ { k } ) \right) \leq 8 H ^ { 2 } \sqrt { | \mathcal { S } | ^ { 3 } | A | K \log \frac { K } { \delta ^ { \prime } } } + \mathcal { O } ( P o l y L o g ( K ) ) .
+$$
+
+Lemma 5.6. For any $k \geq C ^ { \prime \prime }$ , with probability at least $( 1 - \delta )$ ,
+
+$$
+\lambda ^ { k } \leq \frac { 1 } { \zeta } \ln \frac { 1 1 \nu _ { \mathrm { m a x } } ^ { 2 } } { 3 \rho ^ { 2 } } + C ^ { \prime \prime } ( H - \tau ) + \sum _ { u = 1 } ^ { C ^ { \prime \prime } } \epsilon _ { u } + H + \frac { 4 ( H ^ { 2 } + \epsilon _ { k } ^ { 2 } + \eta ^ { k } H ) } { \tau - c ^ { 0 } } ,
+$$
+
+where $\rho = ( \tau - c ^ { 0 } ) / 4$ , $\nu _ { \mathrm { m a x } } = H$ , $\zeta = \rho / ( \nu _ { \mathrm { m a x } } ^ { 2 } + \nu _ { \mathrm { m a x } } \rho / 3 )$ .
+
+To guarantee bounded violation, we ensure that $\sum _ { k = 1 } ^ { K } \epsilon _ { k }$ in (14) can cancel the nt terms in the $( 1 - \delta )$ is bounded as
+
+$$
+R e g ^ { \mathbf { 0 p p D } } ( K ; c ) = { \mathcal { O } } \left( C ^ { \prime \prime } H + H ^ { 2 } { \sqrt { | S | ^ { 3 } | A | C ^ { \prime \prime } \log \left( C ^ { \prime \prime } / \delta ^ { \prime } \right) } } \right) .
+$$
+
+Regret of reward analysis For episode $k$ with $k \geq C ^ { \prime \prime }$ , let $\pi ^ { \epsilon _ { k } , * }$ be the optimal policy for (10), which is well-defined by the definition of $C ^ { \prime \prime }$ . We decompose the reward regret as
+
+$$
+\begin{array} { l } { { \displaystyle { 2 e g ^ { { \bf 0 } { \bf p } { \bf p } { \bf p } } } ( K ; r ) = \sum _ { k = 1 } ^ { C ^ { \prime \prime } } \left( { V _ { 1 } } ^ { \pi ^ { * } } ( \mu ; r , P ) - { V _ { 1 } ^ { \pi ^ { k } } } ( \mu ; r , P ) \right) } \ ~ } \\ { { \displaystyle ~ + \sum _ { k = C ^ { \prime \prime } } ^ { K } \left( { V _ { 1 } ^ { \pi ^ { * } } } ( \mu ; r , P ) - { V _ { 1 } ^ { \pi ^ { * } , k } } ^ { * } ( \mu ; r , P ) \right) + \sum _ { k = C ^ { \prime \prime } } ^ { K } \left( { V _ { 1 } ^ { \pi ^ { * } k , * } } ( \mu ; r , P ) - { \hat { V } _ { 1 } ^ { \pi ^ { * } k , * } } ( \mu ; \hat { r } ^ { k } , \hat { P } ^ { k } ) \right) } \ ~ } \\ { { \displaystyle ~ + \sum _ { k = C ^ { \prime \prime } } ^ { K } \left( { \hat { V } _ { 1 } ^ { \pi ^ { * } k , * } } \left( \mu ; { \hat { r } ^ { k } } , \hat { P } ^ { k } \right) - { \hat { V } _ { 1 } ^ { \pi ^ { k } } } ( \mu ; { \hat { r } ^ { k } } , \hat { P } ^ { k } ) \right) + \sum _ { k = C ^ { \prime \prime } } ^ { K } \left( { \hat { V } _ { 1 } ^ { \pi ^ { k } } } ( \mu ; { \hat { r } ^ { k } } , \hat { P } ^ { k } ) - { V _ { 1 } ^ { \pi ^ { k } } } ( \mu ; r , P ) \right) } . } \end{array}
+$$
+
+We upper bound each term on the RHS of (15). Since $V _ { 1 } ^ { \pi } ( \mu ; r , P ) \in [ 0 , H ]$ for any policy $\pi$ , the first term is upper bounded by $H C ^ { \prime \prime }$ . The second and third terms can be bounded by the following two lemmas:
+
+Lemma 5.7. With probability at least $( 1 - \delta )$
+
+$$
+\sum _ { k = C ^ { \prime \prime } } ^ { K } \left( V _ { 1 } ^ { \pi ^ { * } } ( \mu ; r , P ) - V _ { 1 } ^ { \pi ^ { \epsilon _ { k } , * } } ( \mu ; r , P ) \right) \le \sum _ { k = C ^ { \prime \prime } } ^ { K } \frac { \epsilon _ { k } H } { \tau - c ^ { 0 } } = \tilde { \mathcal { O } } \left( \frac { H ^ { 3 } } { \tau - c ^ { 0 } } \sqrt { | \mathcal { S } | ^ { 3 } | \mathcal { A } | K } \right) .
+$$
+
+Lemma 5.8. With probability at least $\begin{array} { r l } & { ( 1 - \delta ) , \sum _ { k = C ^ { \prime \prime } } ^ { K } \Big ( V _ { 1 } ^ { \pi ^ { \epsilon _ { k } , * } } ( \mu ; r , P ) - \hat { V } _ { 1 } ^ { \pi ^ { \epsilon _ { k } , * } } ( \mu ; \tilde { r } ^ { k } , \hat { P } ^ { k } ) \Big ) \leq 0 . } \end{array}$
+
+The pivotal step is to leverage optimism of $\pi ^ { k }$ to further decompose the fourth term on the RHS of (15), and utilize the projected dual update to transfer it into the form of $\lambda ^ { k } ( \lambda ^ { k } - \lambda ^ { k + 1 } )$ . The following lemmas provide high probability bounds for the remaining two terms on the RHS of (15):
+
+Lemma 5.9. With probability at least $( 1 - \delta )$
+
+$$
+\sum _ { k = C ^ { \prime \prime } } ^ { K } \left( \hat { V } _ { 1 } ^ { \pi ^ { \epsilon _ { k } , * } } ( \mu ; \tilde { r } ^ { k } , \hat { P } ^ { k } ) - \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \tilde { r } ^ { k } , \hat { P } ^ { k } ) \right) = \tilde { \mathcal { O } } \left( \frac { H } { \tau - c ^ { 0 } } \sqrt { K } \right) .
+$$
+
+Lemma 5.10. With probability at least $( 1 - \delta )$ ,
+
+$$
+\sum _ { k = C ^ { \prime \prime } } ^ { K } \left( \hat { V } _ { 1 } ^ { \pi ^ { k } } ( \mu ; \tilde { r } ^ { k } , \hat { P } ^ { k } ) - V _ { 1 } ^ { \pi ^ { k } } ( \mu ; r , P ) \right) = \tilde { \mathcal { O } } \left( H ^ { 2 } \sqrt { | \mathcal { S } | ^ { 3 } | \hat { A } | K } + H ^ { 4 } | \mathcal { S } | ^ { 3 } | \hat { A } | \right) .
+$$
+
+Applying Lemmas 5.7, 5.8, 5.9, and 5.10 yields Theorem 4.1.
+
+# 6 Concluding remarks
+
+We present two optimistic pessimism-based algorithms that maintain stringent safety constraints (either zero or bounded safety constraint violation) for unknown CMDPs with high probability, while still attaining an $\tilde { \mathcal { O } } ( \sqrt { K } )$ regret of reward. The algorithms employ, respectively, a pessimistically safe policy set $\Pi ^ { k }$ or an additional pessimistic term $\epsilon _ { k }$ into the safety constraint. The first algorithm, OptPess-LP, guarantees zero violation with high probability by solving a linear programming with $\Theta ( | S | | A | H )$ decision variables, while the second algorithm, OptPess-PrimalDual, is as efficient as policy-gradient-based algorithms in the tabular case, but violates constraints during initial episodes. A possible future direction for exploration is the application of the above OPFU principle in model-free algorithms.
+
+# Acknowledgement
+
+P. R. Kumar’s work is partially supported by US National Science Foundation under CMMI-2038625, HDR Tripods CCF-1934904; US Office of Naval Research under N00014-21-1-2385; US ARO under W911NF1810331, W911NF2120064; and U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number DEEE0009031. The views expressed herein and conclusions contained in this document are those of the authors and should not be interpreted as representing the views or official policies, either expressed or implied, of the U.S. NSF, ONR, ARO, Department of Energy or the United States Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
+
+Dileep Kalathil gratefully acknowledges funding from the U.S. National Science Foundation (NSF) grants NSF-CRII- CPS-1850206 and NSF-CAREER-EPCN-2045783.
+
+# References
+
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+[3] S. Amani. Linear stochastic bandits under safety constraints. NeurIPS 2019, 2019.   
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+    {
+        "type": "text",
+        "text": "Learning Policies with Zero or Bounded Constraint Violation for Constrained MDPs ",
+        "text_level": 1,
+        "bbox": [
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+            122,
+            810,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tao Liu∗ Texas A&M University tliu@tamu.edu ",
+        "bbox": [
+            204,
+            222,
+            357,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ruida Zhou∗ Texas A&M University ruida@tamu.edu ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dileep Kalathil Texas A&M University dileep.kalathil@tamu.edu ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "P. R. Kumar Texas A&M University prk@tamu.edu ",
+        "bbox": [
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+            285,
+            439,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chao Tian Texas A&M University chao.tian@tamu.edu ",
+        "bbox": [
+            557,
+            285,
+            714,
+            327
+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+            363,
+            535,
+            378
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We address the issue of safety in reinforcement learning. We pose the problem in an episodic framework of a constrained Markov decision process. Existing√ results have shown that it is possible to achieve a reward regret of $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ while allowing an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ constraint violation in $K$ episodes. A critical question that arises is whether it is possible to keep the constraint violation even smaller. We show that when a strictly safe policy is known, then one can confine the system to zero constraint violation with arbitrarily high probability while keeping the reward regret of order $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ . The algorithm which does so employs the principle of optimistic pessimism in the face of uncertainty to achieve safe exploration. When no strictly safe policy is known, though one is known to exist, then it is possible to restrict the system to bounded constraint violation with arbitrarily high probability. This is shown to be realized by a primal-dual algorithm with an optimistic primal estimate and a pessimistic dual update. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            599,
+            310,
+            617
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) addresses the problem of learning an optimal control policy that maximizes the expected cumulative reward while interacting with an unknown environment [25]. Standard RL algorithms typically focus only on maximizing a single objective. However, in many real-world applications, the control policy learned by an RL algorithm has to additionally satisfy stringent safety constraints [9, 4]. For example, an autonomous vehicle may need to reach its destination in the minimum possible time without violating safety constraints such as crossing the middle of the road. The Constrained Markov Decision Process (CMDP) [2, 12] formalism, where one seeks to maximize a reward while satisfying safety constraints, is a standard approach for modeling the necessary safety criteria of a control problem via constraints on cumulative costs. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Several policy-gradient-based algorithms have been proposed to solve CMDPs. Lagrangian-based methods [26, 24, 21, 16] formulate the CMDP problem as a saddle-point problem and optimize it via primal-dual methods, while Constrained Policy Optimization [1, 29] (inspired by the trust region policy optimization [23]) computes new dual variables from scratch at each update to maintain constraints during learning. Although these algorithms provide ways to learn an optimal policy, performance guarantees about reward regret, safety violation or sample complexity are rare. ",
+        "bbox": [
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+            762,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One class of RL algorithms for which performance guarantees are available follow the principle√ of Optimism in the Face of Uncertainty (OFU) [8, 10, 22], and provide an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ guarantee for the reward regret, where √ $K$ is the number of episodes. However, these algorithms also have $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ safety violations. Such significant violation of the safety constraints during learning may be unacceptable in many safety-critical real-world applications such as the control of autonomous vehicles or power systems. These applications demand a class of safe RL algorithms that can provably guarantee safety during learning. With this goal in mind, we aim to answer the following open theoretical question in this paper: ",
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+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/735952b1f9c29590e3231513d95d041c2d90779d8774f871fadc0ba467e94f1d.jpg",
+        "table_caption": [
+            "Table 1: Regret and constraint violation comparisons for algorithms on episodic CMDPs "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>Regret2 Constraint violation ²</td></tr><tr><td>O（H√ OPDOP [8] OptCMDP[10] 3 O(H² OptCMDP-bonus [10] 3 O(H²√ OptDual-CMDP [10] 3 (H²IS|3IAK)</td><td>[S|2|A|K) (H3√SPIA|K) S|3|A|K) (H²√S|3IAK) s|3|A|K) (H²S|3AK) (H²√s3 A|K)</td></tr><tr><td>OptPrimalDual-CMDP[10] 3 C-UCRL [30] 4</td><td>(H²√Is|3|A|K) (H²√Is|3|A|K) （T 0</td></tr><tr><td>OptPess-LP 0（ H T-C0 6 OptPess-PrimalDual H</td><td>/S|3|A|K) 0 v √S|3|A|K) 0(1) 5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
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+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Can we design safe RL algorithms that can achieve an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ regret with respect to the performance objective, while guaranteeing zero or bounded safety constraint violation with arbitrarily high probability? ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We answer the above question affirmatively by proposing two algorithms and establishing their stringent safety performance during learning. Our focus is on the tabular episodic constrained RL setting (unknown transition probabilities, rewards, and costs). The key idea behind both algorithms is a concept used earlier for safe exploration in constrained bandits [20, 17], which we call “Optimistic Pessimism in the Face of Uncertainty (OPFU)” here. The optimistic aspect incentivizes the algorithm for using exploration policies that can visit new state-action pairs, while the pessimistic aspect disincentivizes the algorithm from using exploration policies that can violate safety constraints. By carefully balancing optimism and pessimism, the proposed algorithms guarantee zero or bounded√ safety constraint violation during learning while achieving an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ regret with respect to the reward objective. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The two algorithms address two different classes of the safe learning problem: whether a strictly safe policy is known a priori or not. The resulting exploration strategies are very different in the two cases. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. OptPess-LP Algorithm: This algorithm assumes the prior knowledge of a strictly safe policy. It ensures zero safety constraint violation during learning with high probability and utilizes the linear programming (LP) approach for solving a CMDP problem. The algorithm achieves a reward regret of O˜( H3τ−c0 $\\begin{array} { r } { \\tilde { \\mathcal { O } } ( \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | { \\cal S } | ^ { 3 } | { \\cal A } | K } ) } \\end{array}$ with respect to the performance objective, where $H$ is the number of steps per episode, $\\tau$ is the given constraint on safety violation, $c ^ { 0 }$ is the known safety constraint value of a strictly safe policy $\\pi ^ { 0 }$ , and $| S |$ and $| { \\cal A } |$ are the number of states and actions respectively. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. OptPess-PrimalDual Algorithm: This algorithm addresses the case where no strictly safe policy, but a feasible strictly safe cost is known. By allowing a bounded (in $K$ ) safety cost, it opens up space for exploration. The OptPess-PrimalDual algorithm avoids linear programming and its attendant complexity and exploits the primal-dual approach for solving a CMDP problem. The proposed approach improves the computational tractability, while ensuring a bounded safety constraint violation during learning and a reward regret of $\\begin{array} { r l r } {  { \\tilde { \\mathcal { O } } ( \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | \\boldsymbol { S } | ^ { 3 } | \\boldsymbol { A } | K } ) } } \\end{array}$ with respect to the objective. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Compared with the other methods listed in Table 1, though the proposed algorithms have an additional $H / ( \\tau - c ^ { 0 } )$ or $\\sqrt { | \\boldsymbol { S } | } / ( \\tau - c ^ { 0 } )$ factor in the regret bounds, they are able to reduce the constraint violation to zero or constant with high probability. This improvement in safety can be extremely important for many mission-critical applications. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1.1 Related work ",
+        "text_level": 1,
+        "bbox": [
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+            210,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The problem of learning an optimal control policy that satisfies safety constraints has been studied both in the RL setting and the multi-armed bandits setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Constrained RL: Several policy-gradient algorithms have seen success in practice [26, 24, 21, 16, 1, 29]. Also of interest are works which utilize Gaussian processes to model the transition probabilities and value functions [5, 27, 15, 7]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Several algorithms with provable guarantees are closely related to our work. Zheng et al. [30] considered the constrained RL problem in an infinite horizon setting with unknown reward and cost functions. The approach is similar to the UCRL2 algorithm [13] and achieves a sub-linear reward regret $\\tilde { \\mathcal { O } } ( T ^ { \\frac { 3 } { 4 } } )$ while satisfying constraints with high probability during learning. In contrast to this work, we consider the setting of unknown transition probabilities. Efroni et al [10] focused on the episodic setting of unknown non-stationary transitions over a finite horizon, attaining both a reward regret and a constraint violation of $\\tilde { \\mathcal { O } } ( H ^ { 2 } \\sqrt { | { \\cal S } | ^ { 3 } | { \\cal A } | K } )$ . Ding et al. [8] studied an episodic setting with linear function approximation (suitable for large state space cases), and proposed algorithms that√ can achieve $\\tilde { \\mathcal O } ( d H ^ { 3 } \\sqrt { K } )$ for both the regret and the constraint violation (where $d$ is the dimension of the feature mapping). The regret analysis in [8] can be easily extended to the tabular case, yielding $\\tilde { \\mathcal { O } } ( H ^ { 3 } \\sqrt { | { \\cal S } | ^ { 2 } | \\mathcal { A } | K } ) ^ { - }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Constrained Multi-Armed Bandits: Multi-armed bandit problems are special cases of MDPs, with both the number of states as well as the episode length being one. Linear bandits with constraints (satisfied with high probability) have been investigated in different settings. One setting, referred to as conservative bandits [28, 14, 11], requires the cumulative reward to remain above a fixed percentage of the cumulative reward of a given baseline policy. Another setting is where each arm is associated with two unknown distributions (similar to our setting), generating reward and cost signals respectively [3, 20, 17, 18]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Problem formulation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A finite-horizon constrained non-stationary MDP model is defined as a tuple $\\begin{array} { r l } { M } & { { } = } \\end{array}$ $( \\mathcal S , \\mathcal A , H , P , r , c , \\tau , \\mu )$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $H$ is the number of steps in each episode, $r : S \\times A  [ 0 , 1 ]$ is the unknown reward function of interest, $c : S \\times A  [ 0 , 1 ]$ is the unknown safety cost function used to model the constraint violation, $\\tau \\in ( 0 , H ]$ is the given constant used to define the safety constraint, and $\\mu$ is the known initial distribution of the state. $P . ( \\cdot | s , a ) \\in \\Delta _ { s } ^ { H } , \\forall s \\in S , \\forall a \\in \\mathcal { A }$ , where $\\Delta _ { \\mathcal { S } }$ is the $| S |$ -dimensional probability simplex, and $P _ { h } ( s ^ { \\prime } | s , a )$ is the unknown transition probability that the next state is $s ^ { \\prime }$ when action $a$ is taken for state $s$ at step $h$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Some further notation is necessary to define the problem. A (randomized Markov) policy is defined by a map $\\pi : S \\times [ H ] \\to \\Delta _ { { \\cal A } }$ , with $\\pi _ { h } ( a | s )$ being the probability of taking action $a$ in state $s$ at time step $h$ . With $S _ { t }$ and $A _ { t }$ representing the state and the action at time $t$ respectively, let ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56a550a3a43ec2c89a19cd73ef4933f4127325d21a26f614e15b5e07e1540402.jpg",
+        "text": "$$\nV _ { h } ^ { \\pi } ( s ; g , P ) : = \\mathbb { E } _ { P , \\pi } \\left[ \\sum _ { t = h } ^ { H } g ( S _ { t } , A _ { t } ) | S _ { h } = s \\right] , \\quad \\forall s \\in \\mathcal { S }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "denote the expected cumulative value with respect to a function $g : { \\mathcal { S } } \\times { \\mathcal { A } } \\to \\mathbb { R } _ { + }$ under $P$ for a policy $\\pi$ over a time interval $[ h , h + 1 , \\ldots , H ]$ . With slightly abuse of notation, we use $V _ { 1 } ^ { \\pi } ( \\mu ; g , P )$ to denote $\\mathbb { E } _ { S _ { 1 } \\sim \\mu } [ V _ { 1 } ^ { \\pi } ( S _ { 1 } ; g , \\dot { P } ) ]$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the formulation below there are a total of $K$ episodes with $H$ steps each. Each episode $k \\in [ K ]$ begins with an initial probability distribution $\\mu$ for $S _ { 1 }$ . Then, the agent determines a randomized Markov policy $\\pi ^ { k }$ for that episode based on the information gathered from the previous episodes, and executes it. At time step $h$ during the execution of the $k$ -th episode, after taking action $A _ { h } ^ { k }$ at state $S _ { h } ^ { k }$ , the agent receives a noisy reward and cost of $R _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) = r _ { h } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) + \\xi _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ; r )$ and $\\begin{array} { r } { \\dot { C } _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) = c _ { h } ( S _ { h } ^ { k } , A _ { h } ^ { k } ) + \\xi _ { h } ^ { k } ( S _ { h } ^ { k } , A _ { h } ^ { k } ; c ) } \\end{array}$ , respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2.1 (Sub-Gaussian noise). For all $h \\in [ H ] , k \\in [ K ]$ , the reward and cost noise random variables are conditionally independent zero-mean $1 / 2$ -sub-Gaussian, i.e., $\\mathbb { E } [ \\xi _ { h } ^ { k } | \\mathcal { F } _ { k - 1 } ] = 0 ;$ , $\\mathbb { E } [ \\exp ( \\lambda \\xi _ { h } ^ { k } ) | \\mathcal { F } _ { k - 1 } ] \\le \\exp ( \\lambda ^ { 2 } / 4 ) , $ $\\forall \\lambda \\in \\mathbb { R }$ . Here $\\mathcal { F } _ { k }$ is the $\\sigma$ -algebra generated by the random variables up to episode $k$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $\\pi ^ { * }$ denote the optimal policy of the following CMDP model: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98627bae777ed6971b2ecd47446fc3cba442aac325eb8926b369ffbb8555951b.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\quad V _ { 1 } ^ { \\pi } ( \\mu ; r , P ) \\qquad \\mathrm { s . t . } \\quad V _ { 1 } ^ { \\pi } ( \\mu ; c , P ) \\leq \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A policy $\\pi$ is said to be strictly safe if $V _ { 1 } ^ { \\pi } ( \\mu ; c , P ) < \\tau$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2.2. There exists a strictly safe policy $\\pi ^ { 0 }$ with ${ V _ { 1 } ^ { \\pi } } ^ { 0 } ( \\mu ; c , P ) = c ^ { 0 } < \\tau .$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There are two important cases of the safe learning problem. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Zero constraint violation case: The agent has prior knowledge of a strictly safe policy $\\pi ^ { 0 }$ and its safety cost value $c ^ { 0 } : = V _ { 1 } ^ { \\pi ^ { 0 } } ( \\mu ; c , P )$ . The agent wishes to attain a sublinear (in $K$ ) cumulative regret, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/caa634fcca0f120ab20a75b9488a43c1b5d730e0831558c71b43b7398135520b.jpg",
+        "text": "$$\nR e g ( K ; r ) : = \\sum _ { k = 1 } ^ { K } \\Big ( { V _ { 1 } ^ { \\pi } } ^ { * } ( \\mu ; r , P ) - { V _ { 1 } ^ { \\pi } } ^ { k } ( \\mu ; r , P ) \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            412,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "while incurring zero constraint violation with at least a specified high probability $( 1 - \\delta )$ , i.e., ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7418194e63094bee69f37f4ae38d8391e4a1c04980a11d32f5bd7cbc1ac3951.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; c , P ) \\leq \\tau , \\forall k \\in [ K ] \\right) \\geq 1 - \\delta . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Bounded constraint violation case: The agent knows that there exists a strictly safe policy with a known safety cost value $c ^ { 0 }$ , but does not know any strictly safe policy. It aims to achieve a cumulative regret (2) that grows sublinearly with $K$ , while ensuring that the regret of constraint violation, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21a2e141a24513af7caf8e3046c776a80f8a2c36c6e495ced9309676a82a990e.jpg",
+        "text": "$$\nR e g ( K ; c ) : = \\left( \\sum _ { k = 1 } ^ { K } \\left( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; c , P ) - \\tau \\right) \\right) _ { + } \\left( \\mathrm { w h e r e } \\left( a \\right) _ { + } : = \\operatorname * { m a x } \\{ a , 0 \\} \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "satisfies $\\operatorname* { s u p } _ { K } R e g ( K ; c ) < + \\infty$ with at least a specified high probability $( 1 - \\delta )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remarks. For the zero constraint violation case, the assumption of knowing $c ^ { 0 }$ can be relaxed, as shown in Appendix E. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Zero constraint violation case ",
+        "text_level": 1,
+        "bbox": [
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+            689,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We start by considering the zero constraint violation case. On one hand, to balance the explorationexploitation trade-off, we employ an optimistic estimate of the reward function, as embodied in the OFU principle. On the other hand, to maintain absolute safety with high probability during the exploration, we employ a pessimistic estimate of the safety cost. Such an OPFU principle was previously discussed in constrained bandits [3] and we adapt it to the unknown CMDP setting. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At each episode $k$ , we begin by forming empirical estimates of the transition probabilities, the reward function, and the cost function from step $h$ of all previous episodes: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84511d73305eb285007f2abe5039ae219a241fff305d13da2b2021f539cbc111.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { P } _ { h } ^ { k } ( s ^ { \\prime } | s , a ) : = \\frac { \\sum _ { k ^ { \\prime } = 1 } ^ { k - 1 } \\mathbb { 1 } \\left( S _ { h } ^ { k ^ { \\prime } } = s , A _ { h } ^ { k ^ { \\prime } } = a , S _ { h + 1 } ^ { k ^ { \\prime } } = s ^ { \\prime } \\right) } { N _ { h } ^ { k } ( s , a ) \\vee 1 } , } \\\\ & { \\hat { g } _ { h } ^ { k } ( s , a ) : = \\frac { \\sum _ { k ^ { \\prime } = 1 } ^ { k - 1 } \\mathbb { 1 } \\left( S _ { h } ^ { k ^ { \\prime } } = s , A _ { h } ^ { k ^ { \\prime } } = a \\right) \\left( g _ { h } ( s , a ) + \\xi _ { h } ^ { k } ( s , a ; g ) \\right) } { N _ { h } ^ { k } ( s , a ) \\vee 1 } , \\quad \\mathrm { ~ f o r ~ } g = r , c , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $a \\vee b : = \\operatorname* { m a x } \\{ a , b \\}$ , and ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next we fix some $\\delta \\in ( 0 , 1 )$ and form a common (for notational simplicity) confidence radius $\\beta _ { h } ^ { k } ( s , a )$ for the transition probabilities, the rewards, and the costs, ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/c3bfb6613ace6746f101d1afb76aaf401621e100cc001d9fab248bc6a3ec5457.jpg",
+        "text": "$$\n\\beta _ { h } ^ { k } ( s , a ) : = \\sqrt { \\frac { 1 } { N _ { h } ^ { k } ( s , a ) \\vee 1 } Z } , \\quad \\mathrm { w h e r e ~ } Z : = \\log ( 1 6 | S | ^ { 2 } | A | H K / \\delta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At each episode $k$ , we define the optimistically biased reward estimate as ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db836e49f6ab69c401ca0f5dc4df2477bb4563a1dd2384bfbab18e57a878da45.jpg",
+        "text": "$$\n\\bar { r } _ { h } ^ { k } ( s , a ) : = \\hat { r } _ { h } ^ { k } ( s , a ) + \\alpha _ { r } \\beta _ { h } ^ { k } ( s , a ) , \\quad \\forall ( s , a , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the scaling factor is ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27477da4893aca2e3265aec1595a94a476cfe97a398782b7b6ae3bd7727a9fa9.jpg",
+        "text": "$$\n\\alpha _ { r } : = 1 + | S | H + \\frac { 4 H ( 1 + | S | H ) } { \\tau - c ^ { 0 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To guarantee safe exploration, define the pessimistically biased safety cost estimate at episode $k$ as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac820defeb5f9685bbf289d05b453687340de61c343ee1a8799560818610ad39.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underline { { \\mathsf { c } } } _ { h } ^ { k } ( s , a ) : = \\hat { c } _ { h } ^ { k } ( s , a ) + ( 1 + H | S | ) \\beta _ { h } ^ { k } ( s , a ) , \\quad \\forall ( s , a , h ) \\in S \\times { \\cal A } \\times [ H ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The policy we execute at episode $k$ is chosen from a “pessimistically safe\" policy set $\\Pi ^ { k }$ , defined as ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0384f6e64ca8f2e07e04491dedc27b13dbcf13a47a90f9c631ae99bd5c126d2a.jpg",
+        "text": "$$\n\\Pi ^ { k } : = \\left\\{ \\begin{array} { l l }  { \\{ \\pi ^ { 0 } \\} } & { { \\mathrm { i f ~ } V _ { 1 } ^ { \\pi ^ { 0 } } ( \\mu ; \\underline { { { c } } } ^ { k } , \\hat { P } ^ { k } ) \\geq ( \\tau + c ^ { 0 } ) / 2 , } } \\\\  { \\{ \\pi : V _ { 1 } ^ { \\pi } ( \\mu ; \\underline { { { c } } } ^ { k } , \\hat { P } ^ { k } ) \\leq \\tau \\} } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            438,
+            727,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We simply use the strictly safe policy $\\pi ^ { 0 }$ until $V _ { 1 } ^ { \\pi ^ { 0 } } ( \\mu ; \\underline { { { \\mathrm { c } } } } ^ { k } , \\hat { P } ^ { k } ) < ( \\tau + c ^ { 0 } ) / 2$ , which is a sufficient condition to guarantee that the set $\\{ \\pi : V _ { 1 } ^ { \\pi } ( \\mu ; \\underline { { { \\mathrm { c } } } } ^ { k } , \\hat { P } ^ { k } ) \\leq \\tau \\}$ is non-empty. Within this set, we choose the optimistically best reward earning policy $\\pi ^ { k }$ , which can be solved by linear programming with $\\Theta ( | S | | A | H )$ decision variables and constraints [10]. The resulting Optimistic Pessimism-based Linear Programming (OptPess-LP) algorithm is presented below: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: OptPess-LP ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Input: $K , \\delta , \\pi ^ { 0 } , c ^ { 0 } , \\tau$ ;   \nInitialization: $N _ { h } ^ { 1 } ( s , a ) = 0$ , $\\forall ( s , a , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ] ;$ ;   \nfor $k = 1 , 2 , \\ldots , K$ do Update empirical model (i.e., $\\hat { P } ^ { k } , \\hat { r } ^ { k } , \\hat { c } ^ { k } )$ as in Equations (3)-(5); Update $\\bar { r } ^ { k } , \\mathrm { c } ^ { k }$ , and $\\Pi ^ { k }$ as in Equations (6)-(9); Calculate $\\pi ^ { k } \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi ^ { k } } V _ { 1 } ^ { \\pi } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } )$ ; Execute $\\pi ^ { k }$ and collect a trajectory $( S _ { h } ^ { k } , A _ { h } ^ { k } , R _ { h } ^ { k } , C _ { h } ^ { k } )$ , $\\forall h \\in [ H ]$ ; Update counters $N _ { h } ^ { k + 1 } ( S _ { h } ^ { k } , A _ { h } ^ { k } )$ , $\\forall h \\in [ H ]$ ; ",
+        "bbox": [
+            173,
+            609,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "end ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 (Regret and constraint violation bounds for OptPess-LP). Fix any $\\delta \\in ( 0 , 1 )$ . With probability at least $( 1 - \\delta )$ , OptPess- $. L P$ has zero constraint violation with ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39c2a3f8f0a54ccc7aab7bc20153ba435a009d1d5d1e886750048a448a95a3de.jpg",
+        "text": "$$\nR e g ^ { \\theta P L P } ( K ; r ) = \\tilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | K } + \\frac { H ^ { 5 } | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | } { ( \\tau - c ^ { 0 } ) ^ { 2 } \\wedge ( \\tau - c ^ { 0 } ) } \\right) , w h e r e \\ a \\wedge b : = \\operatorname* { m i n } \\{ a , b \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            809,
+            825,
+            844
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 shows that it is possible to achieve sublinear regret in $K$ , while simultaneously incurring no constraint violation with arbitrarily high probability. The proof is sketched in Section 5.1, with detailed proofs presented in Appendix B. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Bounded constraint violation case ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            488,
+            107
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Without prior knowledge of a strictly safe policy $\\pi ^ { 0 }$ , we may not be able to guarantee zero constraint violation with high probability. However, by relaxing the requirement to bounded (in $K _ { \\cdot }$ ) safety constraint violation, we can incorporate more exploration and design a more efficient algorithm by a primal-dual approach. It is inspired by the design of the pessimistic term in constrained bandits [17]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Different from traditional optimistic dual analysis methods in [10], we introduce an additive pessimistic term $\\epsilon _ { k }$ at each episode $k$ in the original optimization problem (1), i.e., ",
+        "bbox": [
+            169,
+            180,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3caa0ed73b684d8d2c0911f93e516e3c18a4ed50ce97bc038bd11c47fa5f4d82.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\quad V _ { 1 } ^ { \\pi } ( \\mu ; r , P ) \\qquad \\mathrm { s . t . } \\quad V _ { 1 } ^ { \\pi } ( \\mu ; c , P ) + \\epsilon _ { k } \\leq \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            210,
+            671,
+            233
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The pessimistic term restrains the constraint violation and will be progressively decreased as learning proceeds. ",
+        "bbox": [
+            176,
+            239,
+            823,
+            270
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The CMDP problem in (10) may however not have any feasible solution. To overcome this, we consider the Lagrangian, ",
+        "bbox": [
+            173,
+            273,
+            825,
+            303
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd921b2b9af270c36f989399c99ed4e084f268122f6cf78f5957c599eba36308.jpg",
+        "text": "$$\nL ^ { k } ( \\pi , \\lambda ) : = V _ { 1 } ^ { \\pi } ( \\mu ; r , P ) + \\lambda ( \\tau - \\epsilon _ { k } - V _ { 1 } ^ { \\pi } ( \\mu ; c , P ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            304,
+            678,
+            323
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for which, given any Lagrange multiplier $\\lambda$ , we can always solve for the optimizer $\\operatorname* { m a x } _ { \\pi } L ^ { k } ( \\pi , \\lambda )$ by dynamic programming. ",
+        "bbox": [
+            178,
+            325,
+            823,
+            354
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "With the introduction of the additive term $\\epsilon _ { k }$ , the dual variable (denoted by $\\lambda ^ { k }$ ) governed by the subgradient algorithm grows faster, which enhances safety constraints in the next episode. ",
+        "bbox": [
+            174,
+            359,
+            823,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to guarantee sufficient optimism of rewards and costs, we integrate the uncertainty of transitions into rewards and costs, i.e., ",
+        "bbox": [
+            171,
+            393,
+            825,
+            422
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f11364cad7657181d9649d907223e096ed586b871299863f442984fb144029a9.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { r } _ { h } ^ { k } ( s , a ) : = \\hat { r } _ { h } ^ { k } ( s , a ) + \\beta ( n _ { k } ( s , a , h ) ) + H | S | \\beta ( n _ { k } ( s , a , h ) ) , \\quad \\forall ( s , a , h ) \\in S \\times A \\times [ H ] , } \\\\ & { \\tilde { c } _ { h } ^ { k } ( s , a ) : = \\hat { c } _ { h } ^ { k } ( s , a ) - \\beta ( n _ { k } ( s , a , h ) ) - H | S | \\beta ( n _ { k } ( s , a , h ) ) , \\quad \\forall ( s , a , h ) \\in S \\times A \\times [ H ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            422,
+            782,
+            464
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that in contrast to the zero constraint violation case, we optimistically estimate the safety cost function, with the pessimism only governed by $\\epsilon _ { k }$ . ",
+        "bbox": [
+            171,
+            468,
+            825,
+            498
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We employ truncated value functions due to the additional uncertainties from transitions, i.e., ",
+        "bbox": [
+            178,
+            503,
+            784,
+            518
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05aaf3bc50c422bb0d2ffcb07e7aeeb80c6a7141f3b6312fafcc17ae62c9ccb3.jpg",
+        "text": "$$\n\\hat { V } _ { 1 } ^ { \\pi } ( \\mu ; \\hat { r } ^ { k } , \\hat { P } ^ { k } ) : = \\operatorname* { m i n } \\{ H , V _ { 1 } ^ { \\pi } ( \\mu ; \\hat { r } ^ { k } , \\hat { P } ^ { k } ) \\} , \\hat { V } _ { 1 } ^ { \\pi } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) : = \\operatorname* { m a x } \\{ 0 , V _ { 1 } ^ { \\pi } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            518,
+            777,
+            539
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the policy update of the primal variable (denoted by $\\pi ^ { k }$ ), we can apply standard dynamic programming by viewing $\\begin{array} { r } { \\tilde { r } _ { h } ^ { k } ( s , a ) - \\frac { \\lambda ^ { k } } { \\eta ^ { k } } ( \\tilde { c } _ { h } ^ { k } ( s , a ) - \\tau ) } \\end{array}$ as the reward function. Specifically, we apply backward induction to solve for the optimal policy: ",
+        "bbox": [
+            174,
+            547,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e869209efb6ea9bff41128de1033d67fb898e2f044c8a4396edd25df3b8a4f37.jpg",
+        "text": "$$\nQ _ { h } ^ { k } ( s , a ) = \\tilde { r } _ { h } ^ { k } ( s , a ) - \\frac { \\lambda ^ { k } } { \\eta ^ { k } } ( \\tilde { c } _ { h } ^ { k } ( s , a ) - \\tau ) + \\sum _ { s ^ { \\prime } \\in S } \\hat { P } _ { h } ^ { k } ( s ^ { \\prime } | s , a ) \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { h + 1 } ^ { k } ( s ^ { \\prime } , a ^ { \\prime } ) , \\quad \\forall h \\in [ H ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            199,
+            597,
+            799,
+            637
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with $Q _ { H + 1 } ^ { k } ( s , a ) = 0 , \\forall ( s , a ) \\in \\mathcal { S } \\times \\mathcal { A }$ . Then, $\\pi _ { h } ^ { k } \\in \\arg \\operatorname* { m a x } _ { a } Q _ { h } ^ { k } ( s , a )$ , which is computationally efficient (as efficient as policy-gradient-based algorithms in the tabular case). ",
+        "bbox": [
+            173,
+            638,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The resulting Optimistic Pessimism-based Primal-Dual (OptPess-PrimalDual) algorithm is shown in Algorithm 2. It chooses $\\begin{array} { r } { \\epsilon _ { k } : = 5 H ^ { 2 } \\sqrt { | { \\cal S } | ^ { 3 } | A | } ( \\log \\frac { k } { \\delta ^ { \\prime } } + 1 ) / \\sqrt { k \\log \\frac { k } { \\delta ^ { \\prime } } } } \\end{array}$ , $\\delta ^ { \\prime } = \\delta / ( 1 6 | S | ^ { 2 } | A | H )$ , the scaling parameter $\\eta ^ { k } : = ( \\tau - c ^ { 0 } ) H \\sqrt { k }$ in the primal policy update, and, for convenience, a step size of 1 for the dual update. ",
+        "bbox": [
+            173,
+            672,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 (Regret and constraint violation bounds for OptPess-PrimalDual). Fix any $\\delta \\in ( 0 , 1 )$ . Then, ",
+        "bbox": [
+            171,
+            743,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0c36e89b169464f43b24634293426dcbba3f4e1d05e741866784d38cccbe2c1f.jpg",
+        "text": "$$\n\\begin{array} { l } { { R e g ^ { O P P D } ( K ; r ) = \\tilde { \\mathcal { O } } \\left( \\displaystyle \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | K } + \\displaystyle \\frac { H ^ { 5 } | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | } { ( \\tau - c ^ { 0 } ) ^ { 2 } } \\right) , } } \\\\ { { R e g ^ { O P P D } ( K ; c ) = \\mathcal { O } \\left( C ^ { \\prime \\prime } ( H - \\tau ) + H ^ { 2 } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | C ^ { \\prime \\prime } } \\right) = \\mathcal { O } ( 1 ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            772,
+            715,
+            835
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { C ^ { \\prime \\prime } = \\mathcal { O } ( \\frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \\tau - c ^ { 0 } ) ^ { 2 } } \\log \\frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \\tau - c ^ { 0 } ) ^ { 2 } \\delta ^ { \\prime } } ) } \\end{array}$ is a coefficient independent of $K$ ",
+        "bbox": [
+            174,
+            835,
+            661,
+            859
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 shows that it is possible to achieve an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ reward regret, while only allowing bounded constraint violation with arbitrarily high probability. Detailed proofs are presented in Section 5.2 and Appendix $\\textrm { C }$ . ",
+        "bbox": [
+            174,
+            867,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: $K , \\delta , c ^ { 0 } , \\tau$ ;   \nInitialization: $N _ { h } ^ { 1 } ( s , a ) = 0 , \\forall ( s , a , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ] , \\lambda ^ { 1 } = 0 ;$   \nfor $k = 1 , 2 , \\dots , \\overset { \\cdot } { K }$ do S $\\begin{array} { r } { \\mathfrak { x } \\mathfrak { t } \\mathfrak { \\epsilon } _ { k } = 5 H ^ { 2 } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | } ( \\log \\frac { k } { \\delta ^ { \\prime } } + 1 ) / \\sqrt { k \\log \\frac { k } { \\delta ^ { \\prime } } } , \\delta ^ { \\prime } = \\delta / ( 1 6 | \\mathcal { S } | ^ { 2 } | \\mathcal { A } | H ) , } \\end{array}$ $\\eta ^ { k } = ( \\tau - c ^ { 0 } ) H \\sqrt { k } ;$ ; Update empirical model (i.e., $\\hat { P } ^ { k } , \\hat { r } ^ { k } , \\hat { c } ^ { k } )$ as in Equations (3)-(5); Update $\\tilde { r } ^ { k }$ , $\\tilde { c } ^ { k }$ as in Equation (11); $\\begin{array} { r l } & { \\dot { ( P o l i c y ~ U p d a t e ) } \\pi ^ { k } \\in \\overset { \\cdot } { \\arg \\operatorname* { m a x } } _ { \\pi \\in \\Pi } \\hat { V } _ { 1 } ^ { \\pi } ( \\mu ; \\tilde { r } ^ { k } , \\hat { P } ^ { k } ) - \\frac { \\lambda ^ { k } } { \\eta ^ { k } } \\left( \\hat { V } _ { 1 } ^ { \\pi } ( \\mu ; \\tilde { c } ^ { k } , \\hat { P } ^ { k } ) - \\tau \\right) ; } \\\\ & { ( D u a l ~ U p d a t e ) ~ \\lambda ^ { k + 1 } = \\Bigl ( \\lambda ^ { k } + \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\tilde { c } ^ { k } , \\hat { P } ^ { k } ) + \\epsilon _ { k } - \\tau \\Bigr ) _ { + } ; } \\end{array}$ Execute $\\pi ^ { k }$ and collect a trajectory $( S _ { h } ^ { k } , A _ { h } ^ { k } , R _ { h } ^ { k } , C _ { h } ^ { k } )$ , $\\forall h \\in [ H ]$ ; Update counters $N _ { h } ^ { k + 1 } ( S _ { h } ^ { k } , A _ { h } ^ { k } )$ , $\\forall h \\in [ H ]$ ;   \nend ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Regret and constraint violation analysis ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            351,
+            537,
+            369
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We sketch the key steps in the proofs of Theorems 3.1 and 4.1. Full details are relegated to Appendices B and C, respectively. Note that our analysis and results are conditioned on the same high probability event (specifically defined in Appendix A), which occurs with probability at least $( 1 - \\delta )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Analysis of OptPess-LP (Algorithm 1) ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Constraint violation analysis The zero constraint violation of OptPess-LP follows from the following property of the pessimistic policy set $\\Pi ^ { k }$ : ",
+        "bbox": [
+            174,
+            465,
+            825,
+            494
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.1. With probability at least $( 1 - \\delta )$ , for any $k \\in [ K ]$ and policy $\\pi \\in \\Pi ^ { k }$ , $V _ { 1 } ^ { \\pi } ( \\mu ; c , P ) \\leq \\tau$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Regret of reward analysis When the parameters $c , P$ are not well estimated, there may not exist any policy $\\pi$ such that $V _ { 1 } ^ { \\pi } ( \\mu ; \\underline { { { \\mathrm { c } } } } ^ { k } , \\hat { P } ^ { k } ) \\le \\tau$ . Hence, as defined in (9), $\\Pi ^ { k }$ is a singleton set $\\{ \\pi ^ { 0 } \\}$ , and accordingly $\\pi ^ { 0 }$ is executed, even though $\\pi ^ { 0 }$ is not safe for $( \\underline { { \\mathrm { c } } } ^ { k } , \\hat { P } ^ { k } )$ . It subsequently takes several episodes of exploration using $\\pi ^ { 0 }$ until it becomes strictly safe for $( \\underline { { \\mathrm { c } } } ^ { k } , \\hat { P } ^ { k } )$ . At that time, policies close enough to $\\pi ^ { 0 }$ , of which there are infinitely many, are also safe for $( \\underline { { \\mathsf { c } } } ^ { k } , \\hat { P } ^ { k } )$ , and so $| \\Pi ^ { k } | = + \\infty$ . At this point the learning algorithm can proceed to enhance reward while maintaining safety with respect to $( \\underline { { \\mathrm { c } } } ^ { k } , \\hat { P } ^ { k } )$ . ",
+        "bbox": [
+            173,
+            525,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To analyze the algorithm, we decompose the reward regret as follows: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2fb19d35978c9d1082e1ba9291aede521b3f39e79913fdfcd79f59c5e8dc3984.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { R e g ^ { \\mathbf { o p L P } } ( K ; r ) = \\sum _ { k = 1 } ^ { K } \\mathbb { 1 } \\left( | \\Pi ^ { k } | = 1 \\right) \\left( V _ { 1 } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - V _ { 1 } ^ { \\pi ^ { 0 } } ( \\mu ; r , P ) \\right) } } } \\\\ { { \\displaystyle { ~ + \\sum _ { k = 1 } ^ { K } \\mathbb { 1 } \\left( | \\Pi ^ { k } | > 1 \\right) \\left( V _ { 1 } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } ) \\right) } } } \\\\ { { \\displaystyle { ~ + \\sum _ { k = 1 } ^ { K } \\mathbb { 1 } \\left( | \\Pi ^ { k } | > 1 \\right) \\left( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; r , P ) \\right) } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            660,
+            735,
+            794
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To bound the first term on the right-hand side (RHS) of (13), we have the following lemma which gives an upper bound on the number of episodes for exploration by policy $\\pi ^ { 0 }$ : ",
+        "bbox": [
+            173,
+            803,
+            823,
+            832
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.2. With probability at least $( 1 ~ - ~ \\delta )$ $) , ~ \\sum _ { k = 1 } ^ { K } \\mathbb { 1 } ( | \\Pi ^ { k } | ~ = ~ 1 ) ~ \\le ~ C ^ { \\prime } ,$ , where $\\begin{array} { r l } { C ^ { \\prime } } & { { } = } \\end{array}$ $\\tilde { \\mathcal { O } } ( H ^ { 4 } | S | ^ { 3 } | A | / ( ( \\tau - c ^ { 0 } ) ^ { 2 } \\wedge ( \\tau - c ^ { 0 } ) ) )$ . ",
+        "bbox": [
+            173,
+            834,
+            821,
+            871
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Turning to the second term on the RHS of (13), we first note that $\\pi ^ { * }$ may not be in $\\Pi ^ { k }$ . To ensure that the term $V _ { 1 } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } )$ is nevertheless non-positive even when $\\pi ^ { * } \\notin \\Pi ^ { k }$ , we set $\\alpha _ { r }$ to the large value shown in (7). This increases $\\bar { r } ^ { k }$ , and hence also $V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } )$ . In addition, we show that there is a policy $\\hat { \\pi } ^ { k }$ that attains the same reward as the (non-Markov) probabilistic mixed policy, $\\tilde { \\pi } ^ { k } : = B _ { \\gamma _ { k } } \\pi ^ { \\bar { * } } + ( \\mathrm { i } - B _ { \\gamma _ { k } } ) \\pi ^ { 0 }$ , where $B _ { \\gamma _ { k } }$ is a Bernoulli distributed random variable with mean $\\gamma _ { k }$ for $\\gamma _ { k } \\in [ 0 , 1 ]$ . $\\gamma _ { k }$ will be chosen as the largest coefficient such that $V _ { 1 } ^ { \\tilde { \\pi } ^ { k } } ( \\mu ; \\underline { { { c } } } ^ { k } , \\hat { P } ^ { k } ) \\le \\tau$ . This latter policy in turn has a larger reward than $\\pi ^ { 0 }$ since it is a mixture with $\\pi ^ { * }$ , yielding the following: ",
+        "bbox": [
+            173,
+            878,
+            823,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            88,
+            826,
+            166
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.3. With probability at least $( 1 - \\delta )$ ",
+        "bbox": [
+            174,
+            170,
+            478,
+            185
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b7457220faf8b4b4efddc8b8c95f021b438a4a76732411553d99381f7b851468.jpg",
+        "text": "$$\n\\sum _ { k = 1 } ^ { K } \\mathbb { 1 } ( | \\Pi ^ { k } | > 1 ) \\left( V _ { 1 } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } ) \\right) \\leq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            191,
+            686,
+            236
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, concerning the third term on the RHS of (13), akin to the closed-loop identifiability property [6, 19], while $\\hat { P } ^ { k }$ may not converge to $P$ , the difference in the rewards $V _ { 1 } ^ { \\pi ^ { k } } ( \\bar { \\mu ; } \\bar { r } ^ { k } , \\hat { P } ^ { k } ) - \\bar { V _ { 1 } ^ { \\pi ^ { k } } } \\bar { ( \\mu ; \\boldsymbol { r } , P ) }$ grows sublinearly in $k$ since the same policy $\\pi ^ { k }$ is used in both values: ",
+        "bbox": [
+            173,
+            257,
+            826,
+            305
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.4. With probability at least $( 1 - \\delta )$ , ",
+        "bbox": [
+            173,
+            308,
+            478,
+            324
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/594a49da2c9a88966b1ce385c1b84d2a3516d22234b64c02b13ee81c58f43e42.jpg",
+        "text": "$$\n\\sum _ { k = 1 } ^ { K } \\mathbb { 1 } ( | \\Pi ^ { k } | > 1 ) \\left( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\bar { r } ^ { k } , \\hat { P } ^ { k } ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; r , P ) \\right) = \\tilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | { \\cal S } | ^ { 3 } | { \\cal A } | K } + \\frac { H ^ { 5 } | { \\cal S } | ^ { 3 } | { \\cal A } | } { \\tau - c ^ { 0 } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            330,
+            815,
+            371
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Combining Lemmas 5.2, 5.3, and 5.4 yields Theorem 3.1. ",
+        "bbox": [
+            173,
+            395,
+            553,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Analysis of OptPess-PrimalDual (Algorithm 2) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            426,
+            537,
+            443
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we outline the steps in the proof of Theorem 4.1 by analyzing regret and constraint violation of OptPess-PrimalDual respectively. ",
+        "bbox": [
+            173,
+            452,
+            825,
+            482
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recall $\\begin{array} { r } { \\epsilon _ { k } = 5 H ^ { 2 } \\sqrt { | { \\cal S } | ^ { 3 } | { \\cal A } | } ( \\log \\frac { k } { \\delta ^ { \\prime } } + 1 ) / \\sqrt { k \\log \\frac { k } { \\delta ^ { \\prime } } } } \\end{array}$ , where $\\delta ^ { \\prime } = \\delta / ( 1 6 | S | ^ { 2 } | A | H )$ . The existence of a feasible solution to (10) can be guaranteed if $\\epsilon _ { k } \\leq \\tau - c ^ { 0 }$ . Let $C ^ { \\prime \\prime }$ be the smallest value such that $\\forall k \\geq$ $C ^ { \\prime \\prime } , \\epsilon _ { k } \\le ( \\tau - c ^ { 0 } ) / 2$ . Then the perturbed optimization problem (y simple calculation, one can verify that $k \\geq C ^ { \\prime \\prime }$ $\\begin{array} { r } { C ^ { \\prime \\prime } = \\mathcal { O } ( \\frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \\tau - c ^ { 0 } ) ^ { 2 } } \\log \\frac { H ^ { 4 } | S | ^ { 3 } | A | } { ( \\tau - c ^ { 0 } ) ^ { 2 } \\delta ^ { \\prime } } ) } \\end{array}$ Notice that $\\epsilon _ { k }$ is a function not depending on , and so is the coefficient $C ^ { \\prime \\prime }$ . ",
+        "bbox": [
+            173,
+            487,
+            826,
+            577
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Constraint violation analysis The bounded constraint violation of OptPess-PrimalDual is established as follows. We first decompose the constraint violation as ",
+        "bbox": [
+            171,
+            589,
+            825,
+            619
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57d9ef31705f6e51f28f714c531e21a5e4547b6a785126823cc69212462d7172.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle R e g ^ { \\mathbf { o p p D } } ( K ; c ) = ( \\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; c , P ) - \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) ) + \\sum _ { k = 1 } ^ { K } ( \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) - \\tau ) ) _ { + } } } \\\\ { { \\displaystyle \\qquad \\leq ( \\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; c , P ) - \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) ) + \\lambda ^ { K + 1 } - \\sum _ { k = 1 } ^ { K } \\epsilon _ { k } ) _ { + } . \\qquad ( 1 \\otimes ( \\tau ^ { k } ) ^ { K } ( \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } \\otimes \\tau ^ { k } ) _ { - \\tau ^ { k } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            625,
+            805,
+            717
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The first summation term and $\\lambda ^ { K + 1 }$ in (14) can be bounded as follows: ",
+        "bbox": [
+            173,
+            723,
+            642,
+            739
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.5. Recall $\\delta ^ { \\prime } = \\delta / ( 1 6 | S | ^ { 2 } | A | H )$ , with probability at least $( 1 - \\delta )$ , ",
+        "bbox": [
+            174,
+            742,
+            676,
+            758
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4894679ef6b106af3d7e781f57ddace360ee97584e8d4297d9387bc1617c47a7.jpg",
+        "text": "$$\n\\sum _ { k = 1 } ^ { K } \\left( V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; c , P ) - \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\hat { c } ^ { k } , \\hat { P } ^ { k } ) \\right) \\leq 8 H ^ { 2 } \\sqrt { | \\mathcal { S } | ^ { 3 } | A | K \\log \\frac { K } { \\delta ^ { \\prime } } } + \\mathcal { O } ( P o l y L o g ( K ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            765,
+            787,
+            808
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.6. For any $k \\geq C ^ { \\prime \\prime }$ , with probability at least $( 1 - \\delta )$ , ",
+        "bbox": [
+            173,
+            821,
+            593,
+            839
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d52fa70456e9ffe512875c03db86b07dbad7156746c780b1b59957200025fe8.jpg",
+        "text": "$$\n\\lambda ^ { k } \\leq \\frac { 1 } { \\zeta } \\ln \\frac { 1 1 \\nu _ { \\mathrm { m a x } } ^ { 2 } } { 3 \\rho ^ { 2 } } + C ^ { \\prime \\prime } ( H - \\tau ) + \\sum _ { u = 1 } ^ { C ^ { \\prime \\prime } } \\epsilon _ { u } + H + \\frac { 4 ( H ^ { 2 } + \\epsilon _ { k } ^ { 2 } + \\eta ^ { k } H ) } { \\tau - c ^ { 0 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            845,
+            736,
+            890
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\rho = ( \\tau - c ^ { 0 } ) / 4$ , $\\nu _ { \\mathrm { m a x } } = H$ , $\\zeta = \\rho / ( \\nu _ { \\mathrm { m a x } } ^ { 2 } + \\nu _ { \\mathrm { m a x } } \\rho / 3 )$ . ",
+        "bbox": [
+            173,
+            895,
+            581,
+            915
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To guarantee bounded violation, we ensure that $\\sum _ { k = 1 } ^ { K } \\epsilon _ { k }$ in (14) can cancel the nt terms in the $( 1 - \\delta )$ is bounded as ",
+        "bbox": [
+            173,
+            88,
+            825,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a27f2f87480869706c22924dd479417e0e867135e22e5092feb0cabca3c021f.jpg",
+        "text": "$$\nR e g ^ { \\mathbf { 0 p p D } } ( K ; c ) = { \\mathcal { O } } \\left( C ^ { \\prime \\prime } H + H ^ { 2 } { \\sqrt { | S | ^ { 3 } | A | C ^ { \\prime \\prime } \\log \\left( C ^ { \\prime \\prime } / \\delta ^ { \\prime } \\right) } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            135,
+            707,
+            162
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Regret of reward analysis For episode $k$ with $k \\geq C ^ { \\prime \\prime }$ , let $\\pi ^ { \\epsilon _ { k } , * }$ be the optimal policy for (10), which is well-defined by the definition of $C ^ { \\prime \\prime }$ . We decompose the reward regret as ",
+        "bbox": [
+            173,
+            172,
+            826,
+            203
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/da9caedc668dc746c882de480be7bf26b039f5de6b324fbbbc3b64c8ea59c84b.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { 2 e g ^ { { \\bf 0 } { \\bf p } { \\bf p } { \\bf p } } } ( K ; r ) = \\sum _ { k = 1 } ^ { C ^ { \\prime \\prime } } \\left( { V _ { 1 } } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - { V _ { 1 } ^ { \\pi ^ { k } } } ( \\mu ; r , P ) \\right) } \\ ~ } \\\\ { { \\displaystyle ~ + \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( { V _ { 1 } ^ { \\pi ^ { * } } } ( \\mu ; r , P ) - { V _ { 1 } ^ { \\pi ^ { * } , k } } ^ { * } ( \\mu ; r , P ) \\right) + \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( { V _ { 1 } ^ { \\pi ^ { * } k , * } } ( \\mu ; r , P ) - { \\hat { V } _ { 1 } ^ { \\pi ^ { * } k , * } } ( \\mu ; \\hat { r } ^ { k } , \\hat { P } ^ { k } ) \\right) } \\ ~ } \\\\ { { \\displaystyle ~ + \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( { \\hat { V } _ { 1 } ^ { \\pi ^ { * } k , * } } \\left( \\mu ; { \\hat { r } ^ { k } } , \\hat { P } ^ { k } \\right) - { \\hat { V } _ { 1 } ^ { \\pi ^ { k } } } ( \\mu ; { \\hat { r } ^ { k } } , \\hat { P } ^ { k } ) \\right) + \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( { \\hat { V } _ { 1 } ^ { \\pi ^ { k } } } ( \\mu ; { \\hat { r } ^ { k } } , \\hat { P } ^ { k } ) - { V _ { 1 } ^ { \\pi ^ { k } } } ( \\mu ; r , P ) \\right) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            207,
+            831,
+            342
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We upper bound each term on the RHS of (15). Since $V _ { 1 } ^ { \\pi } ( \\mu ; r , P ) \\in [ 0 , H ]$ for any policy $\\pi$ , the first term is upper bounded by $H C ^ { \\prime \\prime }$ . The second and third terms can be bounded by the following two lemmas: ",
+        "bbox": [
+            173,
+            351,
+            828,
+            393
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.7. With probability at least $( 1 - \\delta )$ ",
+        "bbox": [
+            173,
+            396,
+            478,
+            411
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04f1db30ce3689974ebc7bab47b3ad1a66bb34b57469a9cfdaf7ac70574b8e1b.jpg",
+        "text": "$$\n\\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( V _ { 1 } ^ { \\pi ^ { * } } ( \\mu ; r , P ) - V _ { 1 } ^ { \\pi ^ { \\epsilon _ { k } , * } } ( \\mu ; r , P ) \\right) \\le \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\frac { \\epsilon _ { k } H } { \\tau - c ^ { 0 } } = \\tilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } } { \\tau - c ^ { 0 } } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\mathcal { A } | K } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            416,
+            784,
+            460
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.8. With probability at least $\\begin{array} { r l } & { ( 1 - \\delta ) , \\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\Big ( V _ { 1 } ^ { \\pi ^ { \\epsilon _ { k } , * } } ( \\mu ; r , P ) - \\hat { V } _ { 1 } ^ { \\pi ^ { \\epsilon _ { k } , * } } ( \\mu ; \\tilde { r } ^ { k } , \\hat { P } ^ { k } ) \\Big ) \\leq 0 . } \\end{array}$ ",
+        "bbox": [
+            171,
+            470,
+            825,
+            496
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The pivotal step is to leverage optimism of $\\pi ^ { k }$ to further decompose the fourth term on the RHS of (15), and utilize the projected dual update to transfer it into the form of $\\lambda ^ { k } ( \\lambda ^ { k } - \\lambda ^ { k + 1 } )$ . The following lemmas provide high probability bounds for the remaining two terms on the RHS of (15): ",
+        "bbox": [
+            173,
+            513,
+            826,
+            556
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.9. With probability at least $( 1 - \\delta )$ ",
+        "bbox": [
+            173,
+            559,
+            478,
+            574
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/566bdf5af12bd5f69858156a3ed9e9ae839b82cc8c698716ec88bcbfeb6296c8.jpg",
+        "text": "$$\n\\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( \\hat { V } _ { 1 } ^ { \\pi ^ { \\epsilon _ { k } , * } } ( \\mu ; \\tilde { r } ^ { k } , \\hat { P } ^ { k } ) - \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\tilde { r } ^ { k } , \\hat { P } ^ { k } ) \\right) = \\tilde { \\mathcal { O } } \\left( \\frac { H } { \\tau - c ^ { 0 } } \\sqrt { K } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            579,
+            722,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.10. With probability at least $( 1 - \\delta )$ , ",
+        "bbox": [
+            173,
+            635,
+            486,
+            651
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d21ed6d9848f0b177aa1bcfe4624a98ad822497fa209b63f53558a8e5495cf56.jpg",
+        "text": "$$\n\\sum _ { k = C ^ { \\prime \\prime } } ^ { K } \\left( \\hat { V } _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; \\tilde { r } ^ { k } , \\hat { P } ^ { k } ) - V _ { 1 } ^ { \\pi ^ { k } } ( \\mu ; r , P ) \\right) = \\tilde { \\mathcal { O } } \\left( H ^ { 2 } \\sqrt { | \\mathcal { S } | ^ { 3 } | \\hat { A } | K } + H ^ { 4 } | \\mathcal { S } | ^ { 3 } | \\hat { A } | \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            656,
+            763,
+            699
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Applying Lemmas 5.7, 5.8, 5.9, and 5.10 yields Theorem 4.1. ",
+        "bbox": [
+            173,
+            718,
+            576,
+            734
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Concluding remarks ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            379,
+            770
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We present two optimistic pessimism-based algorithms that maintain stringent safety constraints (either zero or bounded safety constraint violation) for unknown CMDPs with high probability, while still attaining an $\\tilde { \\mathcal { O } } ( \\sqrt { K } )$ regret of reward. The algorithms employ, respectively, a pessimistically safe policy set $\\Pi ^ { k }$ or an additional pessimistic term $\\epsilon _ { k }$ into the safety constraint. The first algorithm, OptPess-LP, guarantees zero violation with high probability by solving a linear programming with $\\Theta ( | S | | A | H )$ decision variables, while the second algorithm, OptPess-PrimalDual, is as efficient as policy-gradient-based algorithms in the tabular case, but violates constraints during initial episodes. A possible future direction for exploration is the application of the above OPFU principle in model-free algorithms. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            330,
+            106
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "P. R. Kumar’s work is partially supported by US National Science Foundation under CMMI-2038625, HDR Tripods CCF-1934904; US Office of Naval Research under N00014-21-1-2385; US ARO under W911NF1810331, W911NF2120064; and U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number DEEE0009031. The views expressed herein and conclusions contained in this document are those of the authors and should not be interpreted as representing the views or official policies, either expressed or implied, of the U.S. NSF, ONR, ARO, Department of Energy or the United States Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. ",
+        "bbox": [
+            173,
+            121,
+            826,
+            246
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Dileep Kalathil gratefully acknowledges funding from the U.S. National Science Foundation (NSF) grants NSF-CRII- CPS-1850206 and NSF-CAREER-EPCN-2045783. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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parse/train/Pg13X3rETW/Pg13X3rETW.md

@@ -0,0 +1,420 @@
+# Group-Aware Threshold Adaptation for Fair Classification
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 The fairness in machine learning is getting increasing attention, as its applications   
+2 in different fields continue to expand and diversify. To mitigate the discriminated   
+3 model behaviors between different demographic groups, we introduce a novel post  
+4 processing method to optimize over multiple fairness constraints through group  
+5 aware threshold adaptation. We propose to learn adaptive classification thresholds   
+6 for each demographic group by optimizing the confusion matrix estimated from   
+7 the probability distribution of a classification model output. As we only need   
+8 an estimated probability distribution of model output instead of the classification   
+9 model structure, our post-processing model can be applied to a wide range of   
+10 classification models and improve fairness in a model-agnostic manner to ensure   
+11 privacy. This even allows us to post-process existing fairness methods to further   
+12 improve the trade-off between accuracy and fairness. Moreover, our model is   
+13 efficient with low computational cost by alternating optimization and flexible with   
+14 the optimization over multiple fairness constraints. We provide Pareto frontier to   
+15 characterize fairness-accuracy trade-off. Also, we provide a theoretical analysis   
+16 of the optimal thresholds obtained from our model in terms of both accuracy   
+17 and fairness in classification. Experimental results demonstrate that our method   
+18 outperforms state-of-the-art methods and obtains the result that is closest to the   
+19 theoretical accuracy-fairness trade-off boundary.
+
+# 20 1 Introduction
+
+21 Machine learning is broadening its impact in various fields including autonomous driving, credit   
+22 analysis, and job application screening. As a consequence, the role and importance of fairness in   
+23 machine learning are emerging. However, recent models have been found to behave differently   
+24 between demographic groups in favorable predictions. For example, it has been discovered that   
+25 COMPAS, the criminal risk assessment software currently used to help pretrial release decisions,   
+26 has biases between different races [4]. Specifically, blacks got higher risk scores predicted from the   
+27 model than whites with similar profiles. Therefore, discrimination truly exists and resolving it in   
+28 machine learning is very important and urgent because its direct and potential impact is growing   
+29 tremendously.   
+30 However, obtaining fairness is not a trivial problem, because the data set itself will be biased when it   
+31 is accumulated artificially. Simply removing or manipulating sensitive features (such as race, gender)   
+32 from the data does not solve the bias, because there is indirect discrimination [19] or disparate   
+33 treatment [1] due to the feature redundancy and relevance, which means sensitive information can be   
+34 inferred from other features.   
+35 In order to alleviate discrimination from different perspectives, various quantitative measurements   
+36 of group equity [7, 11, 2, 13] have been proposed. It has been proven that the pursuit of fairness is   
+37 subject to a trade-off between fairness and accuracy [14, 10], i.e., if we want to improve fairness, we   
+38 need to sacrifice accuracy.   
+39 Moreover, Pleiss et al. [20] studied the trade-offs between fairness notions that cannot be satisfied   
+40 at the same time. Therefore, recent works usually target at a certain fairness notion in different   
+41 approaches such as pre-processing [6], in-processing [24], and post-processing [7] methods. However,   
+42 these approaches suffer from the lack of flexibility, since it is difficult to adapt a fair model that   
+43 is trained w.r.t. one certain fairness criterion so as to optimize over other fairness measures. If   
+44 the fairness constraints change under some circumstances, traditional fairness models need to be   
+45 re-trained from scratch, which is computationally demanding and sometimes inapplicable due to   
+46 model settings. To overcome the limitations above, we propose a novel post-processing method to   
+47 improve fairness in a model-agnostic manner. Our GSTAR (Group Specific Threshold Adaptation   
+48 for faiR classification) model learns adaptive classification thresholds for each demographic group   
+49 in classification task for improving the trade-off between fairness and accuracy. Given an existing   
+50 classification model, GSTAR approximates the probability distribution of the model output via   
+51 maximum likelihood estimation and utilizes confusion matrix to quantify accuracy and fairness w.r.t.   
+52 the group-aware classification thresholds. This allows us to: 1) prevent from burdening additional   
+53 complexity or deteriorate the stability of the training process of the classifier; 2) integrate different   
+54 fairness notions into one unified objective function; 3) easily adapt one pre-trained model to other   
+55 fairness constraints. We summarize our contributions of this paper as follows:
+
+1. We propose a novel post-processing method, GSTAR, which can learn group-aware thresholds to optimize the trade-off between fairness and accuracy in classification. We derive rigorous theoretical analysis on the trade-off in our model, and empirically show that GSTAR outperforms state-of-the-art methods.   
+2. With GSTAR, we can simultaneously optimize over multiple fairness constraints with a low computational cost. GSTAR does not require multiple iterations over data, instead, it takes at most one pass of data in training for fast computation.   
+3. GSTAR can be adapted to a wide range of classification models in a model-agnostic manner and can adapt an existing classification model from one fairness criterion to another without re-training the classifier.   
+4. We derive Pareto frontiers of our model for the fairness-accuracy trade-offs that contextualize the quality of fair classification.
+
+# 68 2 Related Works
+
+69 In order to achieve group fairness, which quantifies the discrimination among different sensitive   
+70 groups, a diverse notion of fairness has been introduced. Equalized odds [7] enforce equality of true   
+71 positive rates and false positive rates between different demographic groups. Pleiss et al. [20] relaxed   
+72 equalized odds to satisfy the calibration. Demographic parity or disparate impact [1] suggests that a   
+73 model is unbiased if the model prediction is independent of the protected attribute.   
+74 Among different fairness methods, post-processing techniques propose to improve fairness by mod  
+75 ifying the output of a black-box classifier. Hardt et al. [7] propose to ensure equalized odds by   
+76 constraining the model output. Kamiran et al. [9] propose to give a favorable outcome to unprivileged   
+77 and an unfavorable outcome to the privileged group when the confidence of the prediction is beyond a   
+78 certain range. However, such static confidence window keeps the same regardless of the demographic   
+79 group and is determined by grid search, so it is less efficient.   
+80 Threshold adjustment (a.k.a. thresholding) was introduced to improve the performance of static   
+81 thresholds. In the literature, Menon et al. [18] prove that instance-dependent thresholding of the   
+82 predictive probability function is the optimal classifier in cost-sensitive fairness measures. Also,   
+83 when considering immediate utility, Corbett-Davies et al. [3] show that optimal algorithm is achieved   
+84 from group-specific threshold which is determined by group statistics. However, to the best of our   
+85 knowledge, the threshold adjustment approach has not been deeply studied that neither encompasses   
+86 broad group fairness metrics nor describes an explicit method to achieve the threshold.   
+87 Trade-off between fairness and accuracy exists when we impose fairness constraint to a model. Recent   
+88 studies [2, 25] prove that models targeting at such fairness notions conform to an information theoretic   
+89 lower bound on the joint error across different sensitive groups. Therefore, our work presents a   
+90 practical upper bound of the best achievable accuracy given the fairness constraints.   
+91 Moreover, trade-offs between different fairness notions also exist if one has to consider multiple   
+92 fairness criteria. Some of them are theoretically proven to be incompatible [6, 18, 14]. To express   
+93 and formulate fairness, recent work [10] utilize confusion matrix and propose least-square accuracy  
+94 fairness optimization problem on multiple fairness notions, and categorize the trade-offs between the   
+95 fairness notions.   
+96 Here, our work is the most related to the post-processing methods [7, 10]. Hardt et al. [7] propose   
+97 a post-processing method that utilizes the mixing rate to meet the equalized odds. Ours is similar   
+98 to Hardt et al. [7] in the manner that achieving group-wise threshold from the feasible region that   
+99 is geometrically generated by the intersection between the receiver operating characteristic (ROC)   
+100 curves conditioned on sensitive feature. Ours differ from [7] by generalizing the concept beyond   
+101 equalized odds to other multiple fairness constraints into consideration. FACT [10] utilizes a single   
+102 point (static) from the classifier to be post-processed as a reference which does not fully utilize the   
+103 classifier for the post-processing. In contrast, by approximating the distribution of the continuous   
+104 predicted logits, our GSTAR model enables a larger feasible region than [10] with a better fairness  
+105 accuracy trade-off. We validate the improvement in trade-off via both theoretical and empical results.   
+106 It is notable that these related methods [7, 10] can be considered as a special case of GSTAR.
+
+# 7 3 GSTAR for Fair Classification
+
+# 3.1 Motivation
+
+Consider a binary classification problem with a binary sensitive feature, such that the sensitive feature $A \in \{ 0 , 1 \}$ and label $Y \in \{ 0 , 1 \}$ . In general, for a given data $X$ , a binary classification model outputs an unnormalized logit $h ( { \bar { X } } ) \in \mathbb { R }$ with the class label probability $p ( \dot { X } ) = \sigma ( h ( X ) ) \in [ 0 , 1 ]$ , where $\sigma$ is an activation function (sigmoid function in logistic regression and neural network). It is not necessary to calculate $p$ in a classification model, e.g. support vector machines directly use the positiveness/negativeness of logit $h ( X )$ to determine classification outcome. For traditional models, we use a cut-off threshold $\theta _ { h } = 0$ for $h ( X )$ (i.e., $\theta _ { p } = \sigma ( 0 ) = 0 . 5$ for $p ( X ) \}$ ) in classification, such that the predicted label is determined by $\hat { Y } = \mathbb { I } \{ h ( X ) \geq \theta _ { h } \}$ . In the following context, unless otherwise mentioned, we use $\theta$ to refer to the threshold $\theta _ { h }$ on logit $h$ since it is applicable to a wider range of classification models, and the corresponding threshold on label probability $\theta _ { p }$ can be easily inferred from the threshold on logit $h$ . Traditional models use the same cut-off threshold $\theta$ for different demographic groups. However, since the distribution of logits $h$ in different demographic groups can be different, using the same threshold $\theta$ brings biased classification.
+
+122 In Figure 1, we show a real-world example of image classification on CelebA dataset with   
+123 ResNet50 [8] to show that the default setting of classification thresholds affects both accuracy   
+124 and fairness in classification. The goal of this classification example is to predict the image of a   
+125 person is whether attractive or not, and consider sensitive attribute as gender. This can be generalized   
+126 to different sensitive attributes such as age or race [22, 16]. We can observe an obvious difference in   
+127 the distribution of logit $h$ between two gender groups. In this case, if we use a unified classification   
+128 threshold $\theta _ { 1 } = \theta _ { 0 } = 0$ , it naturally brings a difference in the true positive rate and true negative   
+129 rate between two gender groups, thus renders bias in classification. Instead, we observe that the   
+130 optimal group-specific threshold obtained from GSTAR $( \theta _ { 1 } ^ { * } > \theta _ { 1 }$ , and ${ \theta } _ { 0 } ^ { * } < { \theta } _ { 0 }$ ) can adapt to such   
+131 discrepancy in distribution between two demographic groups to improve both fairness and accuracy.
+
+# 3.2 Group-Aware Classification Thresholds
+
+133 Given an existing classification model and a sensitive attribute $a$ , we can denote true positive rate   
+134 $( \mathrm { T P } _ { a } )$ , false positive rate $( \mathrm { F P } _ { a } )$ , true negative rate $( \mathrm { T N } _ { a } )$ , and false negative rate $( \mathrm { F N } _ { a } )$ in the confusion   
+135 matrix. Most fairness notions can be represented with entries in the confusion matrix. For instance,   
+136 Equal Opportunity (EOp) [7] requires $T P _ { 0 } = T P _ { 1 }$ , and Demographic Parity (DP) [1] requires
+
+$$
+\frac { T P _ { 1 } n _ { 1 1 } + F P _ { 1 } n _ { 0 1 } } { N _ { 1 } } = \frac { T P _ { 0 } n _ { 1 0 } + F P _ { 0 } n _ { 0 0 } } { N _ { 0 } } ,
+$$
+
+where 137 $n _ { y a }$ denotes the number of samples in the subset $\{ Y = y , A = a \}$ , $\begin{array} { r } { N _ { a } = \sum _ { y } n _ { y a } } \end{array}$ denotes the number of samples in 138 $\{ Y = y \}$ , and $\begin{array} { r } { N = \sum _ { y , a } n _ { y a } } \end{array}$ is the total number of samples.
+
+![](images/ae284a0093cee4b3affbd0d3bfb2f4a1f83a69b2bef20505d07b95f159ade15d.jpg)  
+Figure 1: Histograms of logit $h$ distribution from logistic regression on CelebA data, where logit $h$ is used to determine the predicted label $\hat { Y } = \mathbb { I } \{ h ( X ) \geq \theta \}$ , and $\theta$ is the classification threshold. The top plot is for positive samples $Y = 1$ , attractive), and the bottom plot for negative samples $( Y = 0$ , unattractive). In each plot, yellow/green bars represent the distribution of logit $h$ in different sensitive groups, and blue/red curves are estimated probability density functions of logit $h$ in different sensitive groups. $\theta _ { 0 } = \theta _ { 1 } = 0$ (black dashed line) are the default classification thresholds, that are identical for $A = 0$ and $A = 1$ groups. The default thresholds result in biased prediction towards the unprivileged group $A = 0$ due to the different logit $h$ distributions in different sensitive groups. $\theta _ { 0 } ^ { * }$ (red dashed line), $\theta _ { 1 } ^ { * }$ (blue dashed line) are group-aware thresholds from GSTAR for each sensitive group.
+
+139 Consider the group-aware classification threshold $\pmb { \theta } = ( \theta _ { 1 } , \theta _ { 0 } ) ^ { \top }$ , where $\theta _ { a }$ is the classification   
+140 threshold for sensitive group $A = a$ . We can formulate the entries in the confusion matrix w.r.t. $\pmb { \theta }$ as   
+141 below:
+
+$$
+\begin{array} { r l } { \mathrm { T P } _ { a } ( \theta _ { a } ) \approx 1 - \displaystyle \int _ { - \infty } ^ { \theta _ { a } } f _ { 1 a } ( x ) d x , } & { \qquad \mathrm { F N } _ { a } ( \theta _ { a } ) \approx \displaystyle \int _ { - \infty } ^ { \theta _ { a } } f _ { 1 a } ( x ) d x , } \\ { \mathrm { F P } _ { a } ( \theta _ { a } ) \approx 1 - \displaystyle \int _ { - \infty } ^ { \theta _ { a } } f _ { 0 a } ( x ) d x , } & { \qquad \mathrm { T N } _ { a } ( \theta _ { a } ) \approx \displaystyle \int _ { - \infty } ^ { \theta _ { a } } f _ { 0 a } ( x ) d x , } \end{array}
+$$
+
+142 where $f _ { y a } ( x )$ is an estimated parametric probability density function of the distribution of output   
+143 logit $h$ in the subset $\{ Y = y , { \bar { A } } = a \}$ . Here, we consider gamma, Student’s t, and normal distribution   
+144 as the candidates for the estimated distribution, and select the one that has the maximum likelihood   
+145 with the output distribution. Without loss of generality, this can be generalized with other parametric   
+146 probability density function based on the needs or prior knowledge.   
+147 Then, we formulate the fairness-constrained classification problem with the objective of minimizing   
+148 classification error into a least-squared optimization problem. We denote our objective function   
+149 as $\mathcal { L } ( \pmb { \theta } )$ which consists of the performance loss $\mathcal { L } _ { p e r } ( \pmb { \theta } )$ and fairness loss $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ . $\mathcal { L } _ { p e r } ( \pmb { \theta } )$ and   
+150 $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ measures the error in performance and fairness respectively that are represented with the   
+151 entries of the confusion matrix. In other words, our goal is to minimize the objective function $\mathcal { L } ( \pmb { \theta } )$   
+152 as below:
+
+$$
+\begin{array} { r } { \mathcal { L } ( \pmb { \theta } ) = \mathcal { L } _ { p e r } ( \pmb { \theta } ) + \lambda \mathcal { L } _ { f a i r } ( \pmb { \theta } ) , } \end{array}
+$$
+
+53 where $\lambda$ is a hyperparameter that determines how much fairness is enforced in the optimization.
+
+154 The performance error $\mathcal { L } _ { p e r } ( \pmb { \theta } )$ can be written as
+
+$$
+\mathcal { L } _ { p e r } ( \pmb { \theta } ) = \Big ( \frac { n _ { 0 1 } } { N } \mathrm { F P } _ { 1 } ( \theta _ { 1 } ) + \frac { n _ { 1 1 } } { N } \mathrm { F N } _ { 1 } ( \theta _ { 1 } ) + \frac { n _ { 0 0 } } { N } \mathrm { F P } _ { 0 } ( \theta _ { 0 } ) + \frac { n _ { 1 0 } } { N } \mathrm { F N } _ { 0 } ( \theta _ { 0 } ) \Big ) ^ { 2 } .
+$$
+
+155 As for $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ , it can be formulated to any fairness metrics that are expressible with confusion matrix.   
+156 For instance, when we impose EOp $\mathbf { \Delta T P _ { 1 } = T P _ { 0 } }$ ) and predictive equality (PE) $\mathbf { \mathrm { F P _ { 1 } } } = \mathbf { \mathrm { F P _ { 0 } } }$ ) [2], we   
+157 can get the corresponding $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ by summing over the least squared form of each constraint. Also,   
+158 satisfying EOp and PP is equivalent to satisfying Equalized Odds (EOd) [7], This can be formulated   
+159 in our $\mathscr { L } _ { f a i r }$ as
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { f a i r } ^ { E O d } ( \pmb { \theta } ) = \mathcal { L } _ { f a i r } ^ { E O p } ( \pmb { \theta } ) + \mathcal { L } _ { f a i r } ^ { P P } ( \pmb { \theta } ) } \\ & { \quad \quad \quad = \left( T P _ { 1 } ( \theta _ { 1 } ) - T P _ { 0 } ( \theta _ { 0 } ) \right) ^ { 2 } + \left( F P _ { 1 } ( \theta _ { 1 } ) - F P _ { 0 } ( \theta _ { 0 } ) \right) ^ { 2 } . } \end{array}
+$$
+
+Note that a lower 160 $\mathcal { L } _ { f a i r }$ value indicates a fairer threshold. When $\mathcal { L } _ { f a i r } ^ { E O D } ( \pmb { \theta } ) = 0$ , we can interpret as 161 the satisfies the perfect EOd fairness.
+
+162 Similar to (4), we can enforce multiple fairness constraints by summing over the least squared of   
+163 each metric with different weight constant $\lambda$ to each fairness constraints if needed.   
+164 Also, it is notable that compared to the recent paper [10] that enforces fairness through confusion   
+165 tensor, our formulation of fairness in $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ represents a direct notion of fairness metrics and   
+166 improves the measures that allows us to achieve better performance and Pareto frontiers that is shown   
+167 in Section 4.2 and Figure 2. For example, $A _ { \mathrm { E O d } }$ in the paper is calculated as $M _ { 1 } \mathrm { E O p } + M _ { 0 } \mathrm { P E }$ , where   
+168 $M _ { y } = n _ { y 0 } + n _ { y 1 }$ , such that EOd is a weighted sum of EOp and PE with weights being the number of   
+169 samples in each class. In this expression, the imbalance between the two fairness criteria will grow as   
+170 the degree of imbalance in the data increases. In contrast, our formulation expresses the constraints   
+171 as the exact notion of each metric that is not biased by the statistics of the datset and we observe   
+172 improved Pareto frontier as in Figure 2.   
+173 We propose to optimize our threshold $\pmb { \theta }$ with alternating optimization method. Here we take EOp   
+174 constraint as an example to show the alternating optimization steps, then $\mathcal { L } _ { f a i r } ( \pmb { \theta } )$ can be written as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { f a i r } ^ { E O p } ( \pmb { \theta } ) = \left( \mathrm { T P } _ { 1 } ( \theta _ { 1 } ) - \mathrm { T P } _ { 0 } ( \theta _ { 0 } ) \right) ^ { 2 } . } \end{array}
+$$
+
+175 The first step is to fix $\theta _ { 0 }$ and update $\theta _ { 1 }$ . We can approximate the terms that are related to $\theta _ { 1 }$ (e.g.,   
+176 $\mathrm { T P _ { 1 } , F P _ { 1 } , T N _ { 1 } , F N _ { 1 } ) }$ in (1) with first-order Taylor expansion at $\theta _ { 1 } ^ { \tau - 1 }$ . For example,
+
+$$
+\mathrm { T P _ { 1 } } ( \theta _ { 1 } ) \approx \mathrm { T P _ { 1 } } ( \theta _ { 1 } ^ { \tau - 1 } ) + \frac { \partial \mathrm { T P _ { 1 } } } { \partial \theta _ { 1 } } \Big | _ { \theta _ { 1 } = \theta _ { 1 } ^ { \tau - 1 } } ( \theta _ { 1 } - \theta _ { 1 } ^ { \tau - 1 } )
+$$
+
+177 From (1), we can easily derive that
+
+$$
+\begin{array} { r l } { \mathrm { T P } _ { 1 } ( \theta _ { 1 } ^ { \tau - 1 } ) = } & { { } 1 - \displaystyle \int _ { - \infty } ^ { \theta _ { 1 } ^ { \tau - 1 } } f _ { 1 1 } ( x ) d x , } \\ { \displaystyle \frac { \partial \mathrm { T P } _ { 1 } } { \partial \theta _ { 1 } } = } & { { } - f _ { 1 1 } ( \theta _ { 1 } ^ { \tau - 1 } ) . } \end{array}
+$$
+
+178 Similarly, we can find the first order Taylor expansion of $\mathrm { F P _ { 1 } , F N _ { 1 } }$ , and $\mathrm { T N _ { 1 } }$ . Then, the update of $\theta _ { 1 }$   
+179 w.r.t. (2) can be approximated with the following minimization problem w.r.t. $\Delta _ { 1 }$
+
+$$
+\begin{array} { r } { \Delta _ { 1 } ^ { \tau } : = \underset { \Delta _ { 1 } } { \mathrm { a r g m i n } } ( \eta ^ { \tau } + \alpha ^ { \tau } \Delta _ { 1 } ) ^ { 2 } + \lambda ( \epsilon ^ { \tau } + \beta ^ { \tau } \Delta _ { 1 } ) ^ { 2 } , } \end{array}
+$$
+
+where 180 $\Delta _ { 1 } = \theta _ { 1 } - \theta _ { 1 } ^ { \tau - 1 }$ and
+
+$$
+\begin{array} { l l } { \displaystyle \alpha _ { 1 } ^ { \tau } = \frac { n _ { 1 1 } } { N } f _ { 1 1 } ( \theta _ { 1 } ^ { \tau - 1 } ) - \frac { n _ { 0 1 } } { N } f _ { 0 1 } ( \theta _ { 1 } ^ { \tau - 1 } ) , } \\ { \displaystyle \beta _ { 1 } ^ { \tau } = - f _ { 1 1 } ( \theta _ { 1 } ^ { \tau - 1 } ) , } \\ { \displaystyle \eta _ { 1 } ^ { \tau } = \int _ { - \infty } ^ { \theta _ { 1 } ^ { \tau - 1 } } \big ( \frac { n _ { 1 1 } } { N } f _ { 1 1 } ( x ) + \frac { n _ { 0 1 } } { N } ( 1 - f _ { 0 1 } ( x ) \big ) d x + \int _ { - \infty } ^ { \theta _ { 0 } ^ { \tau - 1 } } \big ( \frac { n _ { 1 0 } } { N } f _ { 1 0 } ( x ) + \frac { n _ { 0 0 } } { N } ( 1 - f _ { 0 0 } ( x ) \big ) d x , } \\ { \displaystyle \epsilon _ { 1 } ^ { \tau } = \int _ { \infty } ^ { \theta _ { 1 } ^ { \tau - 1 } } f _ { 1 1 } ( x ) d x - \int _ { \infty } ^ { \theta _ { 0 } ^ { \tau - 1 } } f _ { 0 1 } ( x ) d x . } \end{array}
+$$
+
+181 Taking the derivative of (8) w.r.t. $\Delta _ { 1 }$ and setting it to 0, we can easily obtain the closed-form solution of 182 $\Delta _ { 1 } ^ { \tau }$ as
+
+$$
+\Delta _ { 1 } ^ { \tau } = - \frac { \alpha ^ { \tau } \eta ^ { \tau } + \lambda \beta ^ { \tau } \epsilon ^ { \tau } } { ( \alpha ^ { \tau } ) ^ { 2 } + \lambda ( \beta ^ { \tau } ) ^ { 2 } } .
+$$
+
+Input dataset $\mathcal { X } \times \mathcal { A } \times \mathcal { Y } = \{ ( \mathbf { x } _ { i } , \mathbf { a } _ { i } , \mathbf { y } _ { i } ) \} _ { i = 1 } ^ { n }$ , classification model $h ( X )$ , hyperparameter $\lambda$ .   
+Output Group-specific threshold $\pmb { \theta } = ( \theta _ { 1 } , \theta _ { 0 } )$ .   
+Initialize $\pmb { \theta } \overset { - } { = } ( \bar { \theta _ { 1 } } , \theta _ { 0 } ) = ( 0 , 0 )$ .   
+1. Given a classifier $H ( x )$ , estimate probability density function $f _ { y a } , y , a \in \{ 0 , 1 \}$ by maximum   
+likelihood estimation.   
+while not converge do 2. Calculate the optimal step $\Delta _ { 1 }$ as $\begin{array} { r } { \Delta _ { 1 } = - \frac { \alpha _ { 1 } \eta _ { 1 } + \lambda \beta _ { 1 } \epsilon _ { 1 } } { \alpha _ { 1 } ^ { 2 } + \lambda \beta _ { 1 } ^ { 2 } } } \end{array}$ , with $\alpha _ { 1 } , \beta _ { 1 } , \eta _ { 1 } , \epsilon _ { 1 }$ values shown in (9); 3. Update the threshold: $\theta _ { 1 } \gets \theta _ { 1 } + \Delta _ { 1 }$ ; 4. Calculate the optimal step $\Delta _ { 0 }$ as $\begin{array} { r } { \Delta _ { 0 } = - \frac { \alpha _ { 0 } \eta _ { 0 } + \lambda \beta _ { 0 } \epsilon _ { 0 } } { \alpha _ { 0 } ^ { 2 } + \lambda \beta _ { 0 } ^ { 2 } } } \end{array}$ − α0η0+λβ00α2+λβ2 with α0, β0, η0, 0 values calculated in a similar way as in (9): 5. Update the threshold: $\theta _ { 0 }  \theta _ { 0 } + \Delta _ { 0 }$ .   
+end while
+
+183 The second step is to fix $\theta _ { 1 }$ and update $\theta _ { 0 }$ , and this can be achieved in a similar way of updating $\theta _ { 1 }$   
+184 Then we can finalize the alternating optimization as:
+
+$$
+\begin{array} { r } { \theta _ { 0 } ^ { \tau } = \theta _ { 0 } ^ { \tau - 1 } + \Delta _ { 0 } ^ { \tau } , } \\ { \theta _ { 1 } ^ { \tau } = \theta _ { 1 } ^ { \tau - 1 } + \Delta _ { 1 } ^ { \tau } . } \end{array}
+$$
+
+185 It is notable that in each iteration we derive the optimal update step $\theta$ , which eliminates the burden   
+186 of tuning hyperparameter (such as learning rate) in iterative algorithm. The optimization step is   
+187 summarized in Algorithm 1. The above algorithm can easily extend to multiple fairness constraints   
+188 by adding corresponding squared-loss fairness terms to (2).   
+189 Time Complexity: The alternating optimization of GSTAR model is of low computational cost. We   
+190 take at most one pass of the data for learning the estimated probability density functions $f _ { y a }$ in (1)   
+191 (we do not even need to traverse the data if the parameters (such mean and variance in Gaussian   
+192 distribution) for the estimated probability density functions $f _ { y a }$ can be provided). The optimization   
+193 of $\pmb \theta$ with alternating optimization is efficient since we only need $f _ { y a }$ as we have seen in (9) and (10).   
+194 $\pmb \theta \in \mathbb { R } ^ { 2 }$ is a vector with fixed small size. Therefore, we need a constant time for each update. Overall,   
+195 the time complexity of GSTAR is $O ( n + T )$ , where $n$ is the number of samples, and $T$ is the number   
+196 of iterations in alternating optimization.   
+197 We further derive the theoretical analysis of our GSTAR model on the balance between fairness   
+198 and accuracy, which indicates that the optimal solution provides guarantees on model accuracy   
+199 under the optimal fairness constraint. Details of the theoretical analysis is in the Supplementary   
+200 material. Besides, if a unified threshold is necessary [3], i.e., $\theta _ { 1 } = \theta _ { 0 }$ , the optimization algorithm   
+201 also applies and we only have one scalar variable in (2). When we have a unified threshold, we do not   
+202 require sensitive information in the testing phase that we can conform more strict privacy regulations   
+203 than group-aware thresholding. However, we have to sacrifice both fairness and accuracy as the   
+204 thresholding is less flexible.
+
+# 205 4 Experiments
+
+In this section, we validate GSTAR model on four well-known fairness datasets and compare with other state-of-the-art methods. First, we plot Pareto frontiers of ours and FACT (MS) [10] to demonstrate the trade-offs between fairness and accuracy. Second, we evaluate the models with different fairness metrics and validate that our model is highly adaptive to any fairness metrics that are expressible with confusion matrix [7, 11, 2, 1]. Third, we use our model as a post-processing method to existing fair models and show that our model further improves existing fair models in an efficient and model-agnostic manner.
+
+# 13 4.1 Experimental Setup
+
+214 We compare with multiple fairness approaches in the experiments. For clear demonstration of results,   
+215 we use different shapes of marker for each comparing methods in Figure 2 and Figure 4. The compar  
+216 ing methods include: Learning fair representations for kernel models (abbreviated as FGP) [23],
+
+![](images/565230b45ca3cf80fd5db341650e28a43250d364180ae4817bf68db498033a38.jpg)  
+Figure 2: Model-specific Pareto frontiers of equalized odds to show the upper bound of best achievable accuracy under different fairness constraints. Upper right region under the boundary shows better fairness and higher accuracy. We plot three variations of GSTAR (star-shaped) with different fairness objectives. GSTAR is the closest to the Pareto frontier which indicates the best trade-offs.
+
+Fairness confusion tensor (abbreviated as FACT) [10], Disparate impact remover (abbreviated as DIR) [6], Adversarial de-biasing (abbreviated as AdvDeb) [24], Calibrated equalized odds post-processing (abbreviated as CEOPost) [20], Equality of opportunity in supervised learning (abbreviated as Odds) [7], Learning adversarially fair and transferable representations (abbreviated as LAFTR) [17], and Baseline: For CelebA dataset, we use ResNet50 [8] as a reference, and logistic regression for all other datasets. Our method is optimized with $\lambda$ in the range of $[ 1 0 ^ { - 1 } , 1 0 ^ { 4 } ]$ with alternating optimization method. All experiments are implemented with Pytorch framework on i9-9960X CPU and a Quadro RTX 6000 GPU.
+
+We choose broadly used fairness metrics in evaluation including: equal opportunity difference and equalized odds difference (abbreviated as EOp, and EOd respectively) [7] ; 1-disparate impact (abbreviated as 1-DIMP) [1]; balanced accuracy difference (abbreviated as BD).
+
+228 We evaluate the methods on four fairness datasets: CelebA image dataset1 [15], Adult dataset from   
+229 the UCI repository [12], COMPAS2 (Correctional Offender Management Profiling for Alternative   
+230 Sanctions) dataset, and German credit dataset from the UCI repository [5]. All data is split as $70 \%$   
+231 for training and $30 \%$ for testing. More details of the comparing methods, evaluation metrics, and   
+232 datasets are provided in the Supplementary material.
+
+# 4.2 Performance and Fairness-Accuracy Trade-Offs
+
+In this subsection, we look into the performance evaluation of GSTAR comparing with other state-ofthe-art methods. We consider Pareto frontier to visualize the trade-offs between fairness and accuracy to demonstrate the measure of performance.
+
+In Figure 2, we plot Pareto frontier, which is the upper bound for the accuracy-fairness trade-offs, desired output locates at the upper right region under the boundary which corresponds to higher values in accuracy and lower values in fairness discrepancy. With the same fairness constraints are given, we achieve a better frontier than the FACT [10] as we equally weigh on demographic statistics and have a better feasible region. To obtain our results (star points), we first estimate the logit distribution from the output of the baseline model, and then we get optimal adaptive thresholds with corresponding fairness metric by updating w.r.t. the objective function in (2). Here we have three combinations of fairness imposed to GSTAR: demographic parity (DP), equalized odds (EOd), and
+
+![](images/faeeab7cd7f97f0e628ac573c27aaab1a4f960e543cb8745e20f0bab62add685.jpg)  
+Figure 3: Quantitative evaluation on fairness and performance metrics. The bar plots indicate fairness measures (EOp, EOd, BD, 1-DISP) of each model. Lower fairness values in the left y-axis shows better fairness. The line plots indicate the performance measure (balanced accuracy (BA) and accuracy (ACC) of each model. Higher performance values in the right y-axis shows better classifcation performance. We consider three variations of GSTAR models (DP, EOd, DP+EOd).
+
+245 with both constraints $\mathrm { ( D P { + } E O d ) }$ . By post-processing on a simple baseline, we achieved significantly   
+246 better fairness with small or no sacrifice in accuracy. In all datasets, GSATR got competitive or better   
+247 results than other state-of-the-art methods on both fairness and accuracy.   
+248 For example, we got $\pmb { \theta } _ { E O d } ^ { * } = ( 0 . 6 4 0 , - 0 . 6 2 7 ) ^ { \top }$ for the CelebA dataset. This shows that we have   
+249 a higher threshold for the privileged group and a lower threshold for the unprivileged group. This   
+250 optimal thresholding from GSTAR allows more samples from the privileged group to be correctly   
+251 predicted as unattractive that would compensate for the discrimination of the original model. In other   
+252 words, this improves predictive equality [2] with a huge amount from 0.235 to 0.014. Also, true   
+253 positive rate difference (also known as equality of opportunity [7]) got reduced from 0.282 to 0.018.   
+254 It is notable that GSTAR only sacrificed $2 . 2 \%$ of accuracy to bring the big improvement in fairness.   
+255 Since the objective function of our model is independent to data dimensionality, our model is much   
+256 more efficient especially for high dimensional data. We mostly outperform the computational cost   
+257 comparing to the other methods. The comparison of computational time on the datasets can be found   
+258 in the Supplementary material.
+
+# 259 4.3 Flexibility and Multiple Fairness Constraints
+
+Since each fairness metric has different interests, it has been theoretically proven that they cannot be perfectly satisfied all together [20, 2, 11]. Because of this inherent trade-offs between fairness metrics, most of the recent works focus on a single metric at a time to achieve fairness. However with GSTAR, we have the flexibility to optimize on multiple fairness constraints that can be represented in the confusion matrix format. Moreover, given the estimated distribution $f _ { y a }$ of a black-box classification model, we can adjust the optimal $\pmb \theta$ based on the needs by accommodating different fairness criteria.
+
+Figure 3 demonstrates the result of the methods with fairness metrics and accuracy trade-off evaluations. Overall, the variations of GSTAR achieve the best fairness on each target fairness while preserving the performance. For example in Figure 3(a), GSTAR with EOd constraint has outstanding performance in most fairness metrics with comparable accuracy $( 8 0 . 3 \% )$ . Comparing with GSTAR (EOd), when we introduce EOd and DP together $( \mathrm { D P + E O d } )$ ), we achieve significantly better w.r.t. DP fairness with sacrificing a small amount of accuracy and EOd.
+
+![](images/99aaf30eada38a91a493c559194b114c06bbd339971f8a93609ffe3ec07e1b85.jpg)  
+Figure 4: Illustration of post-processing (magenta colored points) on existing fairness models (blue colored points). Given the outputs of each model, we efficiently improve existing fairness models with optimized group-aware thresholds from GSTAR.
+
+272 In general, by sacrificing individual fairness performance, we could introduce multiple constraints.   
+273 Also, we implicitly observe that the more fairness constraints are introduced, the more accuracy is   
+274 sacrificed. We empirically found that in some cases (e.g. Figure 3(c)), introducing multiple fairness   
+275 is complementary to each other that improves both conditions.
+
+# 276 4.4 Post-Processing on an Existing Fair Model
+
+For a binary classifier that has a single fixed classification threshold (0 for out logit, and 0.5 for label probability), we can improve the trade-off between fairness and accuracy via GSTAR postprocessing. Given the logit/probability of the dataset from a black-box model, we can improve the fairness as illustrated in Figure 4. In most cases, we observe improvement in fairness after GSTAR post-processing. It is also interesting to note that by optimizing the different thresholds for each protected group, we even obtain better performance on both fairness and accuracy, which indicates that the threshold optimization can not only improve fairness but also accuracy.
+
+284 However, when the distribution of the logits/probability is highly extreme (such as the results of using   
+285 GSTAR to post-process CEOPost), it is difficult to estimate the distribution and thus causes erroneous   
+286 optimization in GSTAR. We empirically found that when the dataset is extremely imbalanced such   
+287 that we do not have enough samples to estimate the logit/probability distribution, or black-box model   
+288 is too certain to the prediction that samples are concentrated to certain output, this problem arises.
+
+# 289 5 Conclusion and Discussion
+
+In this paper, we propose a group-aware threshold adaptation method (GSTAR) to post-process a blackbox model and optimize over multiple fairness constraints.We directly optimize the classification threshold for each demographic group w.r.t. the classification error and multiple fairness constraints in a unified objective function, such that we can practically achieve an optimal trade-off between accuracy and fairness in fair classification. Our method is applicable to diverse notions of group fairness as the majority of fairness notions can be expressed as a linear or quadratic equation through confusion matrix. We empirically show that GSTAR is flexible with fairness regularization, efficient with low computational cost. We also notice that the adaptive thresholds benefit accuracy in some cases. GSTAR agrees to protect privacy such as article 17 of EU’s GDPR [21] with model-agnostic post-processing. We only require the estimated distribution of the output from a black-box model i.e., our post-processing method is oblivious to features. Thus training data is no longer needed and allowed to be discarded after training the black-box model.
+
+Further, we empirically find that GSTAR is not applicable to post-process some classification models   
+303 in the following situations: 1) the model does not provide logit/probability as the outcome; 2) The   
+304 model provides an extreme distribution of the output logit/probability. For example, when the model   
+305 is too certain about its prediction, it will be difficult to perform probability density estimation. In our   
+306 future work, we will study possible strategies to solve the above limitations, and extend GSTAR to   
+307 multi-class, multi-sensitive group problems and improve the fairness-accuracy trade-off in a more   
+308 general scheme.
+
+References   
+[1] Solon Barocas and Andrew D Selbst. Big data’s disparate impact. Calif. L. Rev., 104:671, 2016.   
+[2] Alexandra Chouldechova. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big data, 5(2):153–163, 2017.   
+[3] Sam Corbett-Davies, Emma Pierson, Avi Feller, Sharad Goel, and Aziz Huq. Algorithmic decision making and the cost of fairness. In KDD, pages 797–806, 2017.   
+[4] Julia Dressel and Hany Farid. The accuracy, fairness, and limits of predicting recidivism. Sci. Adv, 4(1):eaao5580, 2018.   
+[5] Dheeru Dua and Casey Graff. UCI machine learning repository, 2019.   
+[6] Michael Feldman, Sorelle A Friedler, John Moeller, Carlos Scheidegger, and Suresh Venkatasubramanian. Certifying and removing disparate impact. In KDD, pages 259–268, 2015.   
+[7] Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. In NeurIPS, pages 3315–3323, 2016.   
+[8] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016.   
+[9] Faisal Kamiran, Asim Karim, and Xiangliang Zhang. Decision theory for discrimination-aware classification. In ICDM, pages 924–929. IEEE, 2012.   
+[10] Joon Sik Kim, Jiahao Chen, and Ameet Talwalkar. Model-agnostic characterization of fairness trade-offs. arXiv preprint arXiv:2004.03424, 2020.   
+[11] Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan. Inherent trade-offs in the fair determination of risk scores. arXiv preprint arXiv:1609.05807, 2016.   
+[12] Ron Kohavi. Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. In KDD, volume 96, pages 202–207, 1996.   
+[13] Lydia T Liu, Sarah Dean, Esther Rolf, Max Simchowitz, and Moritz Hardt. Delayed impact of fair machine learning. arXiv preprint arXiv:1803.04383, 2018.   
+[14] Lydia T Liu, Max Simchowitz, and Moritz Hardt. The implicit fairness criterion of unconstrained learning. In ICML, pages 4051–4060, 2019.   
+[15] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, pages 3730–3738, 2015.   
+[16] Vishnu Suresh Lokhande, Aditya Kumar Akash, Sathya N Ravi, and Vikas Singh. Fairalm: Augmented lagrangian method for training fair models with little regret. In ECCV, pages 365–381. Springer, 2020.   
+[17] David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel. Learning adversarially fair and transferable representations. arXiv preprint arXiv:1802.06309, 2018.   
+[18] Aditya Krishna Menon and Robert C Williamson. The cost of fairness in binary classification. In ACM FAccT, pages 107–118, 2018.   
+[19] Dino Pedreshi, Salvatore Ruggieri, and Franco Turini. Discrimination-aware data mining. In KDD, pages 560–568, 2008.   
+[20] Geoff Pleiss, Manish Raghavan, Felix Wu, Jon Kleinberg, and Kilian Q Weinberger. On fairness and calibration. In NeurIPS, pages 5680–5689, 2017.   
+[21] General Data Protection Regulation. Regulation eu 2016/679 of the european parliament and of the council of 27 april 2016. Official Journal of the European Union. Available at: http://ec. europa. eu/justice/data-protection/reform/files/regulation_oj_en. pdf (accessed 20 September 2017), 2016.   
+[22] Hee Jung Ryu, Hartwig Adam, and Margaret Mitchell. Inclusivefacenet: Improving face attribute detection with race and gender diversity. arXiv preprint arXiv:1712.00193, 2017.   
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+[25] Han Zhao and Geoff Gordon. Inherent tradeoffs in learning fair representations. In NeurIPS, pages 15675–15685, 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5.   
+(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] See Section 1 in the Supplementary material.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Section 1 in the Supplementary material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We will provide the code and instructions at request. The data used in the experiments are public available. See Section 2.3 in the Supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4.1.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 4.3.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4.1.
+
+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/Pg13X3rETW/Pg13X3rETW_content_list.json

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+    {
+        "type": "text",
+        "text": "Group-Aware Threshold Adaptation for Fair Classification ",
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+            256,
+            122,
+            741,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            222,
+            578,
+            276
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            313,
+            535,
+            329
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "1 The fairness in machine learning is getting increasing attention, as its applications   \n2 in different fields continue to expand and diversify. To mitigate the discriminated   \n3 model behaviors between different demographic groups, we introduce a novel post  \n4 processing method to optimize over multiple fairness constraints through group  \n5 aware threshold adaptation. We propose to learn adaptive classification thresholds   \n6 for each demographic group by optimizing the confusion matrix estimated from   \n7 the probability distribution of a classification model output. As we only need   \n8 an estimated probability distribution of model output instead of the classification   \n9 model structure, our post-processing model can be applied to a wide range of   \n10 classification models and improve fairness in a model-agnostic manner to ensure   \n11 privacy. This even allows us to post-process existing fairness methods to further   \n12 improve the trade-off between accuracy and fairness. Moreover, our model is   \n13 efficient with low computational cost by alternating optimization and flexible with   \n14 the optimization over multiple fairness constraints. We provide Pareto frontier to   \n15 characterize fairness-accuracy trade-off. Also, we provide a theoretical analysis   \n16 of the optimal thresholds obtained from our model in terms of both accuracy   \n17 and fairness in classification. Experimental results demonstrate that our method   \n18 outperforms state-of-the-art methods and obtains the result that is closest to the   \n19 theoretical accuracy-fairness trade-off boundary. ",
+        "bbox": [
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+            344,
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+    },
+    {
+        "type": "text",
+        "text": "20 1 Introduction ",
+        "text_level": 1,
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+            148,
+            633,
+            312,
+            651
+        ],
+        "page_idx": 0
+    },
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+        "text": "21 Machine learning is broadening its impact in various fields including autonomous driving, credit   \n22 analysis, and job application screening. As a consequence, the role and importance of fairness in   \n23 machine learning are emerging. However, recent models have been found to behave differently   \n24 between demographic groups in favorable predictions. For example, it has been discovered that   \n25 COMPAS, the criminal risk assessment software currently used to help pretrial release decisions,   \n26 has biases between different races [4]. Specifically, blacks got higher risk scores predicted from the   \n27 model than whites with similar profiles. Therefore, discrimination truly exists and resolving it in   \n28 machine learning is very important and urgent because its direct and potential impact is growing   \n29 tremendously.   \n30 However, obtaining fairness is not a trivial problem, because the data set itself will be biased when it   \n31 is accumulated artificially. Simply removing or manipulating sensitive features (such as race, gender)   \n32 from the data does not solve the bias, because there is indirect discrimination [19] or disparate   \n33 treatment [1] due to the feature redundancy and relevance, which means sensitive information can be   \n34 inferred from other features.   \n35 In order to alleviate discrimination from different perspectives, various quantitative measurements   \n36 of group equity [7, 11, 2, 13] have been proposed. It has been proven that the pursuit of fairness is   \n37 subject to a trade-off between fairness and accuracy [14, 10], i.e., if we want to improve fairness, we   \n38 need to sacrifice accuracy.   \n39 Moreover, Pleiss et al. [20] studied the trade-offs between fairness notions that cannot be satisfied   \n40 at the same time. Therefore, recent works usually target at a certain fairness notion in different   \n41 approaches such as pre-processing [6], in-processing [24], and post-processing [7] methods. However,   \n42 these approaches suffer from the lack of flexibility, since it is difficult to adapt a fair model that   \n43 is trained w.r.t. one certain fairness criterion so as to optimize over other fairness measures. If   \n44 the fairness constraints change under some circumstances, traditional fairness models need to be   \n45 re-trained from scratch, which is computationally demanding and sometimes inapplicable due to   \n46 model settings. To overcome the limitations above, we propose a novel post-processing method to   \n47 improve fairness in a model-agnostic manner. Our GSTAR (Group Specific Threshold Adaptation   \n48 for faiR classification) model learns adaptive classification thresholds for each demographic group   \n49 in classification task for improving the trade-off between fairness and accuracy. Given an existing   \n50 classification model, GSTAR approximates the probability distribution of the model output via   \n51 maximum likelihood estimation and utilizes confusion matrix to quantify accuracy and fairness w.r.t.   \n52 the group-aware classification thresholds. This allows us to: 1) prevent from burdening additional   \n53 complexity or deteriorate the stability of the training process of the classifier; 2) integrate different   \n54 fairness notions into one unified objective function; 3) easily adapt one pre-trained model to other   \n55 fairness constraints. We summarize our contributions of this paper as follows: ",
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+        "text": "1. We propose a novel post-processing method, GSTAR, which can learn group-aware thresholds to optimize the trade-off between fairness and accuracy in classification. We derive rigorous theoretical analysis on the trade-off in our model, and empirically show that GSTAR outperforms state-of-the-art methods.   \n2. With GSTAR, we can simultaneously optimize over multiple fairness constraints with a low computational cost. GSTAR does not require multiple iterations over data, instead, it takes at most one pass of data in training for fast computation.   \n3. GSTAR can be adapted to a wide range of classification models in a model-agnostic manner and can adapt an existing classification model from one fairness criterion to another without re-training the classifier.   \n4. We derive Pareto frontiers of our model for the fairness-accuracy trade-offs that contextualize the quality of fair classification. ",
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+        "type": "text",
+        "text": "68 2 Related Works ",
+        "text_level": 1,
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+        "text": "69 In order to achieve group fairness, which quantifies the discrimination among different sensitive   \n70 groups, a diverse notion of fairness has been introduced. Equalized odds [7] enforce equality of true   \n71 positive rates and false positive rates between different demographic groups. Pleiss et al. [20] relaxed   \n72 equalized odds to satisfy the calibration. Demographic parity or disparate impact [1] suggests that a   \n73 model is unbiased if the model prediction is independent of the protected attribute.   \n74 Among different fairness methods, post-processing techniques propose to improve fairness by mod  \n75 ifying the output of a black-box classifier. Hardt et al. [7] propose to ensure equalized odds by   \n76 constraining the model output. Kamiran et al. [9] propose to give a favorable outcome to unprivileged   \n77 and an unfavorable outcome to the privileged group when the confidence of the prediction is beyond a   \n78 certain range. However, such static confidence window keeps the same regardless of the demographic   \n79 group and is determined by grid search, so it is less efficient.   \n80 Threshold adjustment (a.k.a. thresholding) was introduced to improve the performance of static   \n81 thresholds. In the literature, Menon et al. [18] prove that instance-dependent thresholding of the   \n82 predictive probability function is the optimal classifier in cost-sensitive fairness measures. Also,   \n83 when considering immediate utility, Corbett-Davies et al. [3] show that optimal algorithm is achieved   \n84 from group-specific threshold which is determined by group statistics. However, to the best of our   \n85 knowledge, the threshold adjustment approach has not been deeply studied that neither encompasses   \n86 broad group fairness metrics nor describes an explicit method to achieve the threshold.   \n87 Trade-off between fairness and accuracy exists when we impose fairness constraint to a model. Recent   \n88 studies [2, 25] prove that models targeting at such fairness notions conform to an information theoretic   \n89 lower bound on the joint error across different sensitive groups. Therefore, our work presents a   \n90 practical upper bound of the best achievable accuracy given the fairness constraints.   \n91 Moreover, trade-offs between different fairness notions also exist if one has to consider multiple   \n92 fairness criteria. Some of them are theoretically proven to be incompatible [6, 18, 14]. To express   \n93 and formulate fairness, recent work [10] utilize confusion matrix and propose least-square accuracy  \n94 fairness optimization problem on multiple fairness notions, and categorize the trade-offs between the   \n95 fairness notions.   \n96 Here, our work is the most related to the post-processing methods [7, 10]. Hardt et al. [7] propose   \n97 a post-processing method that utilizes the mixing rate to meet the equalized odds. Ours is similar   \n98 to Hardt et al. [7] in the manner that achieving group-wise threshold from the feasible region that   \n99 is geometrically generated by the intersection between the receiver operating characteristic (ROC)   \n100 curves conditioned on sensitive feature. Ours differ from [7] by generalizing the concept beyond   \n101 equalized odds to other multiple fairness constraints into consideration. FACT [10] utilizes a single   \n102 point (static) from the classifier to be post-processed as a reference which does not fully utilize the   \n103 classifier for the post-processing. In contrast, by approximating the distribution of the continuous   \n104 predicted logits, our GSTAR model enables a larger feasible region than [10] with a better fairness  \n105 accuracy trade-off. We validate the improvement in trade-off via both theoretical and empical results.   \n106 It is notable that these related methods [7, 10] can be considered as a special case of GSTAR. ",
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+        "text": "7 3 GSTAR for Fair Classification ",
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+        "text": "3.1 Motivation ",
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+    {
+        "type": "text",
+        "text": "Consider a binary classification problem with a binary sensitive feature, such that the sensitive feature $A \\in \\{ 0 , 1 \\}$ and label $Y \\in \\{ 0 , 1 \\}$ . In general, for a given data $X$ , a binary classification model outputs an unnormalized logit $h ( { \\bar { X } } ) \\in \\mathbb { R }$ with the class label probability $p ( \\dot { X } ) = \\sigma ( h ( X ) ) \\in [ 0 , 1 ]$ , where $\\sigma$ is an activation function (sigmoid function in logistic regression and neural network). It is not necessary to calculate $p$ in a classification model, e.g. support vector machines directly use the positiveness/negativeness of logit $h ( X )$ to determine classification outcome. For traditional models, we use a cut-off threshold $\\theta _ { h } = 0$ for $h ( X )$ (i.e., $\\theta _ { p } = \\sigma ( 0 ) = 0 . 5$ for $p ( X ) \\}$ ) in classification, such that the predicted label is determined by $\\hat { Y } = \\mathbb { I } \\{ h ( X ) \\geq \\theta _ { h } \\}$ . In the following context, unless otherwise mentioned, we use $\\theta$ to refer to the threshold $\\theta _ { h }$ on logit $h$ since it is applicable to a wider range of classification models, and the corresponding threshold on label probability $\\theta _ { p }$ can be easily inferred from the threshold on logit $h$ . Traditional models use the same cut-off threshold $\\theta$ for different demographic groups. However, since the distribution of logits $h$ in different demographic groups can be different, using the same threshold $\\theta$ brings biased classification. ",
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+        "text": "122 In Figure 1, we show a real-world example of image classification on CelebA dataset with   \n123 ResNet50 [8] to show that the default setting of classification thresholds affects both accuracy   \n124 and fairness in classification. The goal of this classification example is to predict the image of a   \n125 person is whether attractive or not, and consider sensitive attribute as gender. This can be generalized   \n126 to different sensitive attributes such as age or race [22, 16]. We can observe an obvious difference in   \n127 the distribution of logit $h$ between two gender groups. In this case, if we use a unified classification   \n128 threshold $\\theta _ { 1 } = \\theta _ { 0 } = 0$ , it naturally brings a difference in the true positive rate and true negative   \n129 rate between two gender groups, thus renders bias in classification. Instead, we observe that the   \n130 optimal group-specific threshold obtained from GSTAR $( \\theta _ { 1 } ^ { * } > \\theta _ { 1 }$ , and ${ \\theta } _ { 0 } ^ { * } < { \\theta } _ { 0 }$ ) can adapt to such   \n131 discrepancy in distribution between two demographic groups to improve both fairness and accuracy. ",
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+        "text": "3.2 Group-Aware Classification Thresholds ",
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+        "text": "133 Given an existing classification model and a sensitive attribute $a$ , we can denote true positive rate   \n134 $( \\mathrm { T P } _ { a } )$ , false positive rate $( \\mathrm { F P } _ { a } )$ , true negative rate $( \\mathrm { T N } _ { a } )$ , and false negative rate $( \\mathrm { F N } _ { a } )$ in the confusion   \n135 matrix. Most fairness notions can be represented with entries in the confusion matrix. For instance,   \n136 Equal Opportunity (EOp) [7] requires $T P _ { 0 } = T P _ { 1 }$ , and Demographic Parity (DP) [1] requires ",
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+        "img_path": "images/9d2e06e5a1667d915ac5b32b594027446d410ff7b13689eedf3c63ab292be269.jpg",
+        "text": "$$\n\\frac { T P _ { 1 } n _ { 1 1 } + F P _ { 1 } n _ { 0 1 } } { N _ { 1 } } = \\frac { T P _ { 0 } n _ { 1 0 } + F P _ { 0 } n _ { 0 0 } } { N _ { 0 } } ,\n$$",
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+        "type": "text",
+        "text": "where 137 $n _ { y a }$ denotes the number of samples in the subset $\\{ Y = y , A = a \\}$ , $\\begin{array} { r } { N _ { a } = \\sum _ { y } n _ { y a } } \\end{array}$ denotes the number of samples in 138 $\\{ Y = y \\}$ , and $\\begin{array} { r } { N = \\sum _ { y , a } n _ { y a } } \\end{array}$ is the total number of samples. ",
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+        "img_path": "images/ae284a0093cee4b3affbd0d3bfb2f4a1f83a69b2bef20505d07b95f159ade15d.jpg",
+        "image_caption": [
+            "Figure 1: Histograms of logit $h$ distribution from logistic regression on CelebA data, where logit $h$ is used to determine the predicted label $\\hat { Y } = \\mathbb { I } \\{ h ( X ) \\geq \\theta \\}$ , and $\\theta$ is the classification threshold. The top plot is for positive samples $Y = 1$ , attractive), and the bottom plot for negative samples $( Y = 0$ , unattractive). In each plot, yellow/green bars represent the distribution of logit $h$ in different sensitive groups, and blue/red curves are estimated probability density functions of logit $h$ in different sensitive groups. $\\theta _ { 0 } = \\theta _ { 1 } = 0$ (black dashed line) are the default classification thresholds, that are identical for $A = 0$ and $A = 1$ groups. The default thresholds result in biased prediction towards the unprivileged group $A = 0$ due to the different logit $h$ distributions in different sensitive groups. $\\theta _ { 0 } ^ { * }$ (red dashed line), $\\theta _ { 1 } ^ { * }$ (blue dashed line) are group-aware thresholds from GSTAR for each sensitive group. "
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+        "text": "139 Consider the group-aware classification threshold $\\pmb { \\theta } = ( \\theta _ { 1 } , \\theta _ { 0 } ) ^ { \\top }$ , where $\\theta _ { a }$ is the classification   \n140 threshold for sensitive group $A = a$ . We can formulate the entries in the confusion matrix w.r.t. $\\pmb { \\theta }$ as   \n141 below: ",
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+        "text": "$$\n\\begin{array} { r l } { \\mathrm { T P } _ { a } ( \\theta _ { a } ) \\approx 1 - \\displaystyle \\int _ { - \\infty } ^ { \\theta _ { a } } f _ { 1 a } ( x ) d x , } & { \\qquad \\mathrm { F N } _ { a } ( \\theta _ { a } ) \\approx \\displaystyle \\int _ { - \\infty } ^ { \\theta _ { a } } f _ { 1 a } ( x ) d x , } \\\\ { \\mathrm { F P } _ { a } ( \\theta _ { a } ) \\approx 1 - \\displaystyle \\int _ { - \\infty } ^ { \\theta _ { a } } f _ { 0 a } ( x ) d x , } & { \\qquad \\mathrm { T N } _ { a } ( \\theta _ { a } ) \\approx \\displaystyle \\int _ { - \\infty } ^ { \\theta _ { a } } f _ { 0 a } ( x ) d x , } \\end{array}\n$$",
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+        "type": "text",
+        "text": "142 where $f _ { y a } ( x )$ is an estimated parametric probability density function of the distribution of output   \n143 logit $h$ in the subset $\\{ Y = y , { \\bar { A } } = a \\}$ . Here, we consider gamma, Student’s t, and normal distribution   \n144 as the candidates for the estimated distribution, and select the one that has the maximum likelihood   \n145 with the output distribution. Without loss of generality, this can be generalized with other parametric   \n146 probability density function based on the needs or prior knowledge.   \n147 Then, we formulate the fairness-constrained classification problem with the objective of minimizing   \n148 classification error into a least-squared optimization problem. We denote our objective function   \n149 as $\\mathcal { L } ( \\pmb { \\theta } )$ which consists of the performance loss $\\mathcal { L } _ { p e r } ( \\pmb { \\theta } )$ and fairness loss $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ . $\\mathcal { L } _ { p e r } ( \\pmb { \\theta } )$ and   \n150 $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ measures the error in performance and fairness respectively that are represented with the   \n151 entries of the confusion matrix. In other words, our goal is to minimize the objective function $\\mathcal { L } ( \\pmb { \\theta } )$   \n152 as below: ",
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+        "img_path": "images/2f1033753cee4baf3942e206d7c702c7db4e344a58fa80343735da3a53fc95cc.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\pmb { \\theta } ) = \\mathcal { L } _ { p e r } ( \\pmb { \\theta } ) + \\lambda \\mathcal { L } _ { f a i r } ( \\pmb { \\theta } ) , } \\end{array}\n$$",
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+        "type": "text",
+        "text": "53 where $\\lambda$ is a hyperparameter that determines how much fairness is enforced in the optimization. ",
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+        "text": "154 The performance error $\\mathcal { L } _ { p e r } ( \\pmb { \\theta } )$ can be written as ",
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+        "text": "$$\n\\mathcal { L } _ { p e r } ( \\pmb { \\theta } ) = \\Big ( \\frac { n _ { 0 1 } } { N } \\mathrm { F P } _ { 1 } ( \\theta _ { 1 } ) + \\frac { n _ { 1 1 } } { N } \\mathrm { F N } _ { 1 } ( \\theta _ { 1 } ) + \\frac { n _ { 0 0 } } { N } \\mathrm { F P } _ { 0 } ( \\theta _ { 0 } ) + \\frac { n _ { 1 0 } } { N } \\mathrm { F N } _ { 0 } ( \\theta _ { 0 } ) \\Big ) ^ { 2 } .\n$$",
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+        "text": "155 As for $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ , it can be formulated to any fairness metrics that are expressible with confusion matrix.   \n156 For instance, when we impose EOp $\\mathbf { \\Delta T P _ { 1 } = T P _ { 0 } }$ ) and predictive equality (PE) $\\mathbf { \\mathrm { F P _ { 1 } } } = \\mathbf { \\mathrm { F P _ { 0 } } }$ ) [2], we   \n157 can get the corresponding $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ by summing over the least squared form of each constraint. Also,   \n158 satisfying EOp and PP is equivalent to satisfying Equalized Odds (EOd) [7], This can be formulated   \n159 in our $\\mathscr { L } _ { f a i r }$ as ",
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+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { f a i r } ^ { E O d } ( \\pmb { \\theta } ) = \\mathcal { L } _ { f a i r } ^ { E O p } ( \\pmb { \\theta } ) + \\mathcal { L } _ { f a i r } ^ { P P } ( \\pmb { \\theta } ) } \\\\ & { \\quad \\quad \\quad = \\left( T P _ { 1 } ( \\theta _ { 1 } ) - T P _ { 0 } ( \\theta _ { 0 } ) \\right) ^ { 2 } + \\left( F P _ { 1 } ( \\theta _ { 1 } ) - F P _ { 0 } ( \\theta _ { 0 } ) \\right) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Note that a lower 160 $\\mathcal { L } _ { f a i r }$ value indicates a fairer threshold. When $\\mathcal { L } _ { f a i r } ^ { E O D } ( \\pmb { \\theta } ) = 0$ , we can interpret as 161 the satisfies the perfect EOd fairness. ",
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+        "text": "162 Similar to (4), we can enforce multiple fairness constraints by summing over the least squared of   \n163 each metric with different weight constant $\\lambda$ to each fairness constraints if needed.   \n164 Also, it is notable that compared to the recent paper [10] that enforces fairness through confusion   \n165 tensor, our formulation of fairness in $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ represents a direct notion of fairness metrics and   \n166 improves the measures that allows us to achieve better performance and Pareto frontiers that is shown   \n167 in Section 4.2 and Figure 2. For example, $A _ { \\mathrm { E O d } }$ in the paper is calculated as $M _ { 1 } \\mathrm { E O p } + M _ { 0 } \\mathrm { P E }$ , where   \n168 $M _ { y } = n _ { y 0 } + n _ { y 1 }$ , such that EOd is a weighted sum of EOp and PE with weights being the number of   \n169 samples in each class. In this expression, the imbalance between the two fairness criteria will grow as   \n170 the degree of imbalance in the data increases. In contrast, our formulation expresses the constraints   \n171 as the exact notion of each metric that is not biased by the statistics of the datset and we observe   \n172 improved Pareto frontier as in Figure 2.   \n173 We propose to optimize our threshold $\\pmb { \\theta }$ with alternating optimization method. Here we take EOp   \n174 constraint as an example to show the alternating optimization steps, then $\\mathcal { L } _ { f a i r } ( \\pmb { \\theta } )$ can be written as ",
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+        "img_path": "images/f46b40fd063760e7db5c9c4738ec8ce4f140e9040a080a1cd5f3adff3160e935.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { f a i r } ^ { E O p } ( \\pmb { \\theta } ) = \\left( \\mathrm { T P } _ { 1 } ( \\theta _ { 1 } ) - \\mathrm { T P } _ { 0 } ( \\theta _ { 0 } ) \\right) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "175 The first step is to fix $\\theta _ { 0 }$ and update $\\theta _ { 1 }$ . We can approximate the terms that are related to $\\theta _ { 1 }$ (e.g.,   \n176 $\\mathrm { T P _ { 1 } , F P _ { 1 } , T N _ { 1 } , F N _ { 1 } ) }$ in (1) with first-order Taylor expansion at $\\theta _ { 1 } ^ { \\tau - 1 }$ . For example, ",
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+        "text": "$$\n\\mathrm { T P _ { 1 } } ( \\theta _ { 1 } ) \\approx \\mathrm { T P _ { 1 } } ( \\theta _ { 1 } ^ { \\tau - 1 } ) + \\frac { \\partial \\mathrm { T P _ { 1 } } } { \\partial \\theta _ { 1 } } \\Big | _ { \\theta _ { 1 } = \\theta _ { 1 } ^ { \\tau - 1 } } ( \\theta _ { 1 } - \\theta _ { 1 } ^ { \\tau - 1 } )\n$$",
+        "text_format": "latex",
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+        "text": "177 From (1), we can easily derive that ",
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+        "text": "$$\n\\begin{array} { r l } { \\mathrm { T P } _ { 1 } ( \\theta _ { 1 } ^ { \\tau - 1 } ) = } & { { } 1 - \\displaystyle \\int _ { - \\infty } ^ { \\theta _ { 1 } ^ { \\tau - 1 } } f _ { 1 1 } ( x ) d x , } \\\\ { \\displaystyle \\frac { \\partial \\mathrm { T P } _ { 1 } } { \\partial \\theta _ { 1 } } = } & { { } - f _ { 1 1 } ( \\theta _ { 1 } ^ { \\tau - 1 } ) . } \\end{array}\n$$",
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+        "text": "178 Similarly, we can find the first order Taylor expansion of $\\mathrm { F P _ { 1 } , F N _ { 1 } }$ , and $\\mathrm { T N _ { 1 } }$ . Then, the update of $\\theta _ { 1 }$   \n179 w.r.t. (2) can be approximated with the following minimization problem w.r.t. $\\Delta _ { 1 }$ ",
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+        "img_path": "images/cb795e91f59d952e43215b9dc563f6b3229bb69a5fb9c42df6ca87bbe3aa49e7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\Delta _ { 1 } ^ { \\tau } : = \\underset { \\Delta _ { 1 } } { \\mathrm { a r g m i n } } ( \\eta ^ { \\tau } + \\alpha ^ { \\tau } \\Delta _ { 1 } ) ^ { 2 } + \\lambda ( \\epsilon ^ { \\tau } + \\beta ^ { \\tau } \\Delta _ { 1 } ) ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where 180 $\\Delta _ { 1 } = \\theta _ { 1 } - \\theta _ { 1 } ^ { \\tau - 1 }$ and ",
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+        "img_path": "images/ce389f7145b6e73208534b8bfcf84f2319508d9b19cd600779032aacdc0d1696.jpg",
+        "text": "$$\n\\begin{array} { l l } { \\displaystyle \\alpha _ { 1 } ^ { \\tau } = \\frac { n _ { 1 1 } } { N } f _ { 1 1 } ( \\theta _ { 1 } ^ { \\tau - 1 } ) - \\frac { n _ { 0 1 } } { N } f _ { 0 1 } ( \\theta _ { 1 } ^ { \\tau - 1 } ) , } \\\\ { \\displaystyle \\beta _ { 1 } ^ { \\tau } = - f _ { 1 1 } ( \\theta _ { 1 } ^ { \\tau - 1 } ) , } \\\\ { \\displaystyle \\eta _ { 1 } ^ { \\tau } = \\int _ { - \\infty } ^ { \\theta _ { 1 } ^ { \\tau - 1 } } \\big ( \\frac { n _ { 1 1 } } { N } f _ { 1 1 } ( x ) + \\frac { n _ { 0 1 } } { N } ( 1 - f _ { 0 1 } ( x ) \\big ) d x + \\int _ { - \\infty } ^ { \\theta _ { 0 } ^ { \\tau - 1 } } \\big ( \\frac { n _ { 1 0 } } { N } f _ { 1 0 } ( x ) + \\frac { n _ { 0 0 } } { N } ( 1 - f _ { 0 0 } ( x ) \\big ) d x , } \\\\ { \\displaystyle \\epsilon _ { 1 } ^ { \\tau } = \\int _ { \\infty } ^ { \\theta _ { 1 } ^ { \\tau - 1 } } f _ { 1 1 } ( x ) d x - \\int _ { \\infty } ^ { \\theta _ { 0 } ^ { \\tau - 1 } } f _ { 0 1 } ( x ) d x . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "181 Taking the derivative of (8) w.r.t. $\\Delta _ { 1 }$ and setting it to 0, we can easily obtain the closed-form solution of 182 $\\Delta _ { 1 } ^ { \\tau }$ as ",
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+        "type": "equation",
+        "img_path": "images/a5fc228ac7b91312f40f0696b5cd050ea654dac0a10d815a2f0da1a44465554c.jpg",
+        "text": "$$\n\\Delta _ { 1 } ^ { \\tau } = - \\frac { \\alpha ^ { \\tau } \\eta ^ { \\tau } + \\lambda \\beta ^ { \\tau } \\epsilon ^ { \\tau } } { ( \\alpha ^ { \\tau } ) ^ { 2 } + \\lambda ( \\beta ^ { \\tau } ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Input dataset $\\mathcal { X } \\times \\mathcal { A } \\times \\mathcal { Y } = \\{ ( \\mathbf { x } _ { i } , \\mathbf { a } _ { i } , \\mathbf { y } _ { i } ) \\} _ { i = 1 } ^ { n }$ , classification model $h ( X )$ , hyperparameter $\\lambda$ .   \nOutput Group-specific threshold $\\pmb { \\theta } = ( \\theta _ { 1 } , \\theta _ { 0 } )$ .   \nInitialize $\\pmb { \\theta } \\overset { - } { = } ( \\bar { \\theta _ { 1 } } , \\theta _ { 0 } ) = ( 0 , 0 )$ .   \n1. Given a classifier $H ( x )$ , estimate probability density function $f _ { y a } , y , a \\in \\{ 0 , 1 \\}$ by maximum   \nlikelihood estimation.   \nwhile not converge do 2. Calculate the optimal step $\\Delta _ { 1 }$ as $\\begin{array} { r } { \\Delta _ { 1 } = - \\frac { \\alpha _ { 1 } \\eta _ { 1 } + \\lambda \\beta _ { 1 } \\epsilon _ { 1 } } { \\alpha _ { 1 } ^ { 2 } + \\lambda \\beta _ { 1 } ^ { 2 } } } \\end{array}$ , with $\\alpha _ { 1 } , \\beta _ { 1 } , \\eta _ { 1 } , \\epsilon _ { 1 }$ values shown in (9); 3. Update the threshold: $\\theta _ { 1 } \\gets \\theta _ { 1 } + \\Delta _ { 1 }$ ; 4. Calculate the optimal step $\\Delta _ { 0 }$ as $\\begin{array} { r } { \\Delta _ { 0 } = - \\frac { \\alpha _ { 0 } \\eta _ { 0 } + \\lambda \\beta _ { 0 } \\epsilon _ { 0 } } { \\alpha _ { 0 } ^ { 2 } + \\lambda \\beta _ { 0 } ^ { 2 } } } \\end{array}$ − α0η0+λβ0\u000f0α2+λβ2 with α0, β0, η0, \u000f0 values calculated in a similar way as in (9): 5. Update the threshold: $\\theta _ { 0 }  \\theta _ { 0 } + \\Delta _ { 0 }$ .   \nend while ",
+        "bbox": [
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+        "type": "text",
+        "text": "183 The second step is to fix $\\theta _ { 1 }$ and update $\\theta _ { 0 }$ , and this can be achieved in a similar way of updating $\\theta _ { 1 }$   \n184 Then we can finalize the alternating optimization as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b26169f5dc50bb488bc46969f9a66dd92bc23a403c4d1323ae267236b1e8aaa9.jpg",
+        "text": "$$\n\\begin{array} { r } { \\theta _ { 0 } ^ { \\tau } = \\theta _ { 0 } ^ { \\tau - 1 } + \\Delta _ { 0 } ^ { \\tau } , } \\\\ { \\theta _ { 1 } ^ { \\tau } = \\theta _ { 1 } ^ { \\tau - 1 } + \\Delta _ { 1 } ^ { \\tau } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "185 It is notable that in each iteration we derive the optimal update step $\\theta$ , which eliminates the burden   \n186 of tuning hyperparameter (such as learning rate) in iterative algorithm. The optimization step is   \n187 summarized in Algorithm 1. The above algorithm can easily extend to multiple fairness constraints   \n188 by adding corresponding squared-loss fairness terms to (2).   \n189 Time Complexity: The alternating optimization of GSTAR model is of low computational cost. We   \n190 take at most one pass of the data for learning the estimated probability density functions $f _ { y a }$ in (1)   \n191 (we do not even need to traverse the data if the parameters (such mean and variance in Gaussian   \n192 distribution) for the estimated probability density functions $f _ { y a }$ can be provided). The optimization   \n193 of $\\pmb \\theta$ with alternating optimization is efficient since we only need $f _ { y a }$ as we have seen in (9) and (10).   \n194 $\\pmb \\theta \\in \\mathbb { R } ^ { 2 }$ is a vector with fixed small size. Therefore, we need a constant time for each update. Overall,   \n195 the time complexity of GSTAR is $O ( n + T )$ , where $n$ is the number of samples, and $T$ is the number   \n196 of iterations in alternating optimization.   \n197 We further derive the theoretical analysis of our GSTAR model on the balance between fairness   \n198 and accuracy, which indicates that the optimal solution provides guarantees on model accuracy   \n199 under the optimal fairness constraint. Details of the theoretical analysis is in the Supplementary   \n200 material. Besides, if a unified threshold is necessary [3], i.e., $\\theta _ { 1 } = \\theta _ { 0 }$ , the optimization algorithm   \n201 also applies and we only have one scalar variable in (2). When we have a unified threshold, we do not   \n202 require sensitive information in the testing phase that we can conform more strict privacy regulations   \n203 than group-aware thresholding. However, we have to sacrifice both fairness and accuracy as the   \n204 thresholding is less flexible. ",
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+    {
+        "type": "text",
+        "text": "205 4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we validate GSTAR model on four well-known fairness datasets and compare with other state-of-the-art methods. First, we plot Pareto frontiers of ours and FACT (MS) [10] to demonstrate the trade-offs between fairness and accuracy. Second, we evaluate the models with different fairness metrics and validate that our model is highly adaptive to any fairness metrics that are expressible with confusion matrix [7, 11, 2, 1]. Third, we use our model as a post-processing method to existing fair models and show that our model further improves existing fair models in an efficient and model-agnostic manner. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "13 4.1 Experimental Setup ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "214 We compare with multiple fairness approaches in the experiments. For clear demonstration of results,   \n215 we use different shapes of marker for each comparing methods in Figure 2 and Figure 4. The compar  \n216 ing methods include: Learning fair representations for kernel models (abbreviated as FGP) [23], ",
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+    {
+        "type": "image",
+        "img_path": "images/565230b45ca3cf80fd5db341650e28a43250d364180ae4817bf68db498033a38.jpg",
+        "image_caption": [
+            "Figure 2: Model-specific Pareto frontiers of equalized odds to show the upper bound of best achievable accuracy under different fairness constraints. Upper right region under the boundary shows better fairness and higher accuracy. We plot three variations of GSTAR (star-shaped) with different fairness objectives. GSTAR is the closest to the Pareto frontier which indicates the best trade-offs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Fairness confusion tensor (abbreviated as FACT) [10], Disparate impact remover (abbreviated as DIR) [6], Adversarial de-biasing (abbreviated as AdvDeb) [24], Calibrated equalized odds post-processing (abbreviated as CEOPost) [20], Equality of opportunity in supervised learning (abbreviated as Odds) [7], Learning adversarially fair and transferable representations (abbreviated as LAFTR) [17], and Baseline: For CelebA dataset, we use ResNet50 [8] as a reference, and logistic regression for all other datasets. Our method is optimized with $\\lambda$ in the range of $[ 1 0 ^ { - 1 } , 1 0 ^ { 4 } ]$ with alternating optimization method. All experiments are implemented with Pytorch framework on i9-9960X CPU and a Quadro RTX 6000 GPU. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We choose broadly used fairness metrics in evaluation including: equal opportunity difference and equalized odds difference (abbreviated as EOp, and EOd respectively) [7] ; 1-disparate impact (abbreviated as 1-DIMP) [1]; balanced accuracy difference (abbreviated as BD). ",
+        "bbox": [
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+        "type": "text",
+        "text": "228 We evaluate the methods on four fairness datasets: CelebA image dataset1 [15], Adult dataset from   \n229 the UCI repository [12], COMPAS2 (Correctional Offender Management Profiling for Alternative   \n230 Sanctions) dataset, and German credit dataset from the UCI repository [5]. All data is split as $70 \\%$   \n231 for training and $30 \\%$ for testing. More details of the comparing methods, evaluation metrics, and   \n232 datasets are provided in the Supplementary material. ",
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+    {
+        "type": "text",
+        "text": "4.2 Performance and Fairness-Accuracy Trade-Offs ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this subsection, we look into the performance evaluation of GSTAR comparing with other state-ofthe-art methods. We consider Pareto frontier to visualize the trade-offs between fairness and accuracy to demonstrate the measure of performance. ",
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+    {
+        "type": "text",
+        "text": "In Figure 2, we plot Pareto frontier, which is the upper bound for the accuracy-fairness trade-offs, desired output locates at the upper right region under the boundary which corresponds to higher values in accuracy and lower values in fairness discrepancy. With the same fairness constraints are given, we achieve a better frontier than the FACT [10] as we equally weigh on demographic statistics and have a better feasible region. To obtain our results (star points), we first estimate the logit distribution from the output of the baseline model, and then we get optimal adaptive thresholds with corresponding fairness metric by updating w.r.t. the objective function in (2). Here we have three combinations of fairness imposed to GSTAR: demographic parity (DP), equalized odds (EOd), and ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/faeeab7cd7f97f0e628ac573c27aaab1a4f960e543cb8745e20f0bab62add685.jpg",
+        "image_caption": [
+            "Figure 3: Quantitative evaluation on fairness and performance metrics. The bar plots indicate fairness measures (EOp, EOd, BD, 1-DISP) of each model. Lower fairness values in the left y-axis shows better fairness. The line plots indicate the performance measure (balanced accuracy (BA) and accuracy (ACC) of each model. Higher performance values in the right y-axis shows better classifcation performance. We consider three variations of GSTAR models (DP, EOd, DP+EOd). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "245 with both constraints $\\mathrm { ( D P { + } E O d ) }$ . By post-processing on a simple baseline, we achieved significantly   \n246 better fairness with small or no sacrifice in accuracy. In all datasets, GSATR got competitive or better   \n247 results than other state-of-the-art methods on both fairness and accuracy.   \n248 For example, we got $\\pmb { \\theta } _ { E O d } ^ { * } = ( 0 . 6 4 0 , - 0 . 6 2 7 ) ^ { \\top }$ for the CelebA dataset. This shows that we have   \n249 a higher threshold for the privileged group and a lower threshold for the unprivileged group. This   \n250 optimal thresholding from GSTAR allows more samples from the privileged group to be correctly   \n251 predicted as unattractive that would compensate for the discrimination of the original model. In other   \n252 words, this improves predictive equality [2] with a huge amount from 0.235 to 0.014. Also, true   \n253 positive rate difference (also known as equality of opportunity [7]) got reduced from 0.282 to 0.018.   \n254 It is notable that GSTAR only sacrificed $2 . 2 \\%$ of accuracy to bring the big improvement in fairness.   \n255 Since the objective function of our model is independent to data dimensionality, our model is much   \n256 more efficient especially for high dimensional data. We mostly outperform the computational cost   \n257 comparing to the other methods. The comparison of computational time on the datasets can be found   \n258 in the Supplementary material. ",
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+    {
+        "type": "text",
+        "text": "259 4.3 Flexibility and Multiple Fairness Constraints ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Since each fairness metric has different interests, it has been theoretically proven that they cannot be perfectly satisfied all together [20, 2, 11]. Because of this inherent trade-offs between fairness metrics, most of the recent works focus on a single metric at a time to achieve fairness. However with GSTAR, we have the flexibility to optimize on multiple fairness constraints that can be represented in the confusion matrix format. Moreover, given the estimated distribution $f _ { y a }$ of a black-box classification model, we can adjust the optimal $\\pmb \\theta$ based on the needs by accommodating different fairness criteria. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 demonstrates the result of the methods with fairness metrics and accuracy trade-off evaluations. Overall, the variations of GSTAR achieve the best fairness on each target fairness while preserving the performance. For example in Figure 3(a), GSTAR with EOd constraint has outstanding performance in most fairness metrics with comparable accuracy $( 8 0 . 3 \\% )$ . Comparing with GSTAR (EOd), when we introduce EOd and DP together $( \\mathrm { D P + E O d } )$ ), we achieve significantly better w.r.t. DP fairness with sacrificing a small amount of accuracy and EOd. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/99aaf30eada38a91a493c559194b114c06bbd339971f8a93609ffe3ec07e1b85.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of post-processing (magenta colored points) on existing fairness models (blue colored points). Given the outputs of each model, we efficiently improve existing fairness models with optimized group-aware thresholds from GSTAR. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "272 In general, by sacrificing individual fairness performance, we could introduce multiple constraints.   \n273 Also, we implicitly observe that the more fairness constraints are introduced, the more accuracy is   \n274 sacrificed. We empirically found that in some cases (e.g. Figure 3(c)), introducing multiple fairness   \n275 is complementary to each other that improves both conditions. ",
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+    {
+        "type": "text",
+        "text": "276 4.4 Post-Processing on an Existing Fair Model ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a binary classifier that has a single fixed classification threshold (0 for out logit, and 0.5 for label probability), we can improve the trade-off between fairness and accuracy via GSTAR postprocessing. Given the logit/probability of the dataset from a black-box model, we can improve the fairness as illustrated in Figure 4. In most cases, we observe improvement in fairness after GSTAR post-processing. It is also interesting to note that by optimizing the different thresholds for each protected group, we even obtain better performance on both fairness and accuracy, which indicates that the threshold optimization can not only improve fairness but also accuracy. ",
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+    {
+        "type": "text",
+        "text": "284 However, when the distribution of the logits/probability is highly extreme (such as the results of using   \n285 GSTAR to post-process CEOPost), it is difficult to estimate the distribution and thus causes erroneous   \n286 optimization in GSTAR. We empirically found that when the dataset is extremely imbalanced such   \n287 that we do not have enough samples to estimate the logit/probability distribution, or black-box model   \n288 is too certain to the prediction that samples are concentrated to certain output, this problem arises. ",
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+    {
+        "type": "text",
+        "text": "289 5 Conclusion and Discussion ",
+        "text_level": 1,
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+        "text": "In this paper, we propose a group-aware threshold adaptation method (GSTAR) to post-process a blackbox model and optimize over multiple fairness constraints.We directly optimize the classification threshold for each demographic group w.r.t. the classification error and multiple fairness constraints in a unified objective function, such that we can practically achieve an optimal trade-off between accuracy and fairness in fair classification. Our method is applicable to diverse notions of group fairness as the majority of fairness notions can be expressed as a linear or quadratic equation through confusion matrix. We empirically show that GSTAR is flexible with fairness regularization, efficient with low computational cost. We also notice that the adaptive thresholds benefit accuracy in some cases. GSTAR agrees to protect privacy such as article 17 of EU’s GDPR [21] with model-agnostic post-processing. We only require the estimated distribution of the output from a black-box model i.e., our post-processing method is oblivious to features. Thus training data is no longer needed and allowed to be discarded after training the black-box model. ",
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+        "text": "Further, we empirically find that GSTAR is not applicable to post-process some classification models   \n303 in the following situations: 1) the model does not provide logit/probability as the outcome; 2) The   \n304 model provides an extreme distribution of the output logit/probability. For example, when the model   \n305 is too certain about its prediction, it will be difficult to perform probability density estimation. In our   \n306 future work, we will study possible strategies to solve the above limitations, and extend GSTAR to   \n307 multi-class, multi-sensitive group problems and improve the fairness-accuracy trade-off in a more   \n308 general scheme. ",
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+        "type": "text",
+        "text": "References   \n[1] Solon Barocas and Andrew D Selbst. Big data’s disparate impact. Calif. L. Rev., 104:671, 2016.   \n[2] Alexandra Chouldechova. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big data, 5(2):153–163, 2017.   \n[3] Sam Corbett-Davies, Emma Pierson, Avi Feller, Sharad Goel, and Aziz Huq. Algorithmic decision making and the cost of fairness. In KDD, pages 797–806, 2017.   \n[4] Julia Dressel and Hany Farid. The accuracy, fairness, and limits of predicting recidivism. Sci. Adv, 4(1):eaao5580, 2018.   \n[5] Dheeru Dua and Casey Graff. UCI machine learning repository, 2019.   \n[6] Michael Feldman, Sorelle A Friedler, John Moeller, Carlos Scheidegger, and Suresh Venkatasubramanian. Certifying and removing disparate impact. In KDD, pages 259–268, 2015.   \n[7] Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. In NeurIPS, pages 3315–3323, 2016.   \n[8] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016.   \n[9] Faisal Kamiran, Asim Karim, and Xiangliang Zhang. Decision theory for discrimination-aware classification. In ICDM, pages 924–929. IEEE, 2012.   \n[10] Joon Sik Kim, Jiahao Chen, and Ameet Talwalkar. Model-agnostic characterization of fairness trade-offs. arXiv preprint arXiv:2004.03424, 2020.   \n[11] Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan. Inherent trade-offs in the fair determination of risk scores. arXiv preprint arXiv:1609.05807, 2016.   \n[12] Ron Kohavi. Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. In KDD, volume 96, pages 202–207, 1996.   \n[13] Lydia T Liu, Sarah Dean, Esther Rolf, Max Simchowitz, and Moritz Hardt. Delayed impact of fair machine learning. arXiv preprint arXiv:1803.04383, 2018.   \n[14] Lydia T Liu, Max Simchowitz, and Moritz Hardt. The implicit fairness criterion of unconstrained learning. In ICML, pages 4051–4060, 2019.   \n[15] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, pages 3730–3738, 2015.   \n[16] Vishnu Suresh Lokhande, Aditya Kumar Akash, Sathya N Ravi, and Vikas Singh. Fairalm: Augmented lagrangian method for training fair models with little regret. In ECCV, pages 365–381. Springer, 2020.   \n[17] David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel. Learning adversarially fair and transferable representations. arXiv preprint arXiv:1802.06309, 2018.   \n[18] Aditya Krishna Menon and Robert C Williamson. The cost of fairness in binary classification. In ACM FAccT, pages 107–118, 2018.   \n[19] Dino Pedreshi, Salvatore Ruggieri, and Franco Turini. Discrimination-aware data mining. In KDD, pages 560–568, 2008.   \n[20] Geoff Pleiss, Manish Raghavan, Felix Wu, Jon Kleinberg, and Kilian Q Weinberger. On fairness and calibration. In NeurIPS, pages 5680–5689, 2017.   \n[21] General Data Protection Regulation. Regulation eu 2016/679 of the european parliament and of the council of 27 april 2016. Official Journal of the European Union. Available at: http://ec. europa. eu/justice/data-protection/reform/files/regulation_oj_en. pdf (accessed 20 September 2017), 2016.   \n[22] Hee Jung Ryu, Hartwig Adam, and Margaret Mitchell. Inclusivefacenet: Improving face attribute detection with race and gender diversity. arXiv preprint arXiv:1712.00193, 2017.   \n[23] Zilong Tan, Samuel Yeom, Matt Fredrikson, and Ameet Talwalkar. Learning fair representations for kernel models. In AISTATS, pages 155–166, 2020.   \n[24] Brian Hu Zhang, Blake Lemoine, and Margaret Mitchell. Mitigating unwanted biases with adversarial learning. In AIES, pages 335–340, 2018.   \n[25] Han Zhao and Geoff Gordon. Inherent tradeoffs in learning fair representations. In NeurIPS, pages 15675–15685, 2019. ",
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parse/train/rJTKKKqeg/rJTKKKqeg.md

@@ -0,0 +1,244 @@
+# TRACKING THE WORLD STATE WITHRECURRENT ENTITY NETWORKS
+
+Mikael Henaff1,2, Jason Weston1, Arthur Szlam1, Antoine Bordes1 and Yann LeCun1,2
+
+1Facebook AI Research   
+2Courant Institute, New York University   
+{mbh305}@nyu.edu, {jase,aszlam,abordes,yann}@fb.com
+
+# ABSTRACT
+
+We introduce a new model, the Recurrent Entity Network (EntNet). It is equipped with a dynamic long-term memory which allows it to maintain and update a representation of the state of the world as it receives new data. For language understanding tasks, it can reason on-the-fly as it reads text, not just when it is required to answer a question or respond as is the case for a Memory Network (Sukhbaatar et al., 2015). Like a Neural Turing Machine or Differentiable Neural Computer (Graves et al., 2014; 2016) it maintains a fixed size memory and can learn to perform location and content-based read and write operations. However, unlike those models it has a simple parallel architecture in which several memory locations can be updated simultaneously. The EntNet sets a new state-of-the-art on the bAbI tasks, and is the first method to solve all the tasks in the $1 0 \mathrm { k }$ training examples setting. We also demonstrate that it can solve a reasoning task which requires a large number of supporting facts, which other methods are not able to solve, and can generalize past its training horizon. It can also be practically used on large scale datasets such as Children’s Book Test, where it obtains competitive performance, reading the story in a single pass.
+
+# 1 INTRODUCTION
+
+The essence of intelligence is the ability to predict. An intelligent agent must be able to predict unobserved facts about their environment from limited percepts (visual, auditory, textual, or otherwise), combined with their knowledge of the past. In order to reason and plan, they must be able to predict how an observed event or action will affect the state of the world. Arguably, the ability to maintain an estimate of the current state of the world, combined with a forward model of how the world evolves, is a key feature of intelligent agents.
+
+A natural way for an agent to represent the world is to maintain a set of high-level concepts or entities together with their properties, which are updated as new information is received. For example, if a percept is the textual description of an event, such as “John walks out of the kitchen”, the agent should learn to update its estimate of John’s location, as well as the list (and number) of people present in each room. If John was carrying a bag, the location of the bag and the list of objects in the kitchen must also be updated. When we read a story, each sentence we read or hear causes us to update our internal representation of the current state of the world within the story. The flow of the story is captured by the evolution of this state of the world.
+
+At any given time, an agent typically receives limited information about the state of the world, and should therefore be able to infer new information through partial observation. In this paper, we investigate this problem through a simple story understanding scenario, in which the agent is given a sequence of textual statements and events, and then given another series of statements about the final state of the world. If the second series of statements is given in the form of questions about the final state of the world together with their correct answers, the agent should be able to learn from them and its performance can be measured by the accuracy of its answers.
+
+Even with this weak form of supervision, the system may learn basic dynamical constraints about the world. For example, it may learn that a person or object cannot be in two locations at the same time, or may learn simple update rules such as incrementing and decrementing the number of persons or objects in a room. It may also learn basic rules of approximate (logical) inference, such as the fact that objects belonging to the same category tend to have similar properties (light objects can be carried over from rooms to rooms for instance).
+
+We propose to handle this scenario with a new kind of memory-augmented neural network that uses a distributed memory and processor architecture: the Recurrent Entity Network (EntNet). The model consists of a fixed number of dynamic memory cells, each containing a vector key $w _ { j }$ and a vector value (or content) $h _ { j }$ . Each cell is associated with its own “processor”, a simple gated recurrent network that may update the cell value given an input. If each cell learns to represent a concept or entity in the world, one can imagine a gating mechanism that, based on the key and content of the memory cells, will only modify the cells that concern the entities mentioned in the input. In the current version of the model, there is no direct interaction between the memory cells, hence the system can be seen as multiple identical processors functioning in parallel, with distributed local memory. Alternatively, the EntNet can be seen as a bank of gated RNNs (all sharing the same parameters), whose hidden states correspond to latent concepts and attributes, and whose parameters describe the laws of the world according to which the attributes of objects are updated. The sharing of these parameters reflects an invariance of these laws across object instances, similarly to how the weight tying scheme in a CNN reflects an invariance of image statistics across locations. Their hidden state is updated only when new information relevant to their concept is received, and remains otherwise unchanged. The keys used in the addressing/gating mechanism also correspond to concepts or entities, but are modified only during learning, not during inference.
+
+The EntNet is able to solve all 20 bAbI question-answering tasks (Weston et al., 2015), a popular benchmark of story understanding, which to our knowledge sets a new state-of-the-art. Our experiments also indicate that the model indeed maintains an internal representation of the simplified world in which the stories take place, and that the model does not limit itself to storing the aspects of the world required to answer a specific question. We also introduce a new reasoning task which, unlike the bAbI tasks, requires a model to use a large number of supporting facts to answer the question, and show that the EntNet outperforms both LSTMs and Memory Networks (Sukhbaatar et al., 2015) by a significant margin. It is also able to generalize to sequences longer than those seen during training. Finally, our model also obtains competitive results on the Childrens Book Test (Hill et al., 2016), and performs best among models that read the text in a single pass before receiving knowledge of the question.
+
+# 2 MODEL
+
+Our model is designed to process data in sequential form, and consists of three main parts: an input encoder, a dynamic memory and an output layer, which we now describe in detail. We developed it in the context of question answering on short stories where the inputs are word sequences, but the model could be adapted to many other contexts.
+
+# 2.1 INPUT ENCODER
+
+The encoding layer summarizes an element of the input sequence with a vector of fixed length. Typically the input element at time $t$ is a sequence of words, e.g. a sentence or window of words. One is free to choose the encoding module to be any standard sequence encoder, which is an active area of research. Typical choices include a bag-of-words (BoW) representation or the final state of a recurrent neural net (RNN) run over the sequence. In this work, we use a simple encoder consisting of a learned multiplicative mask followed by a summation. More precisely, let the input at time $t$ be a sequence of words with embeddings $\{ e _ { 1 } , . . . , e _ { k } \}$ . The vector representation of this input is then:
+
+$$
+s _ { t } = \sum _ { i } f _ { i } \odot e _ { i }
+$$
+
+The same set of vectors $\{ f _ { 1 } , . . . , f _ { k } \}$ are used at each time step and are learned jointly with the other parameters of the model. Note that the model can choose to adopt a standard BoW representation
+
+![](images/e1a75e8621689e8c39ada2bcd119684f909bae1e2db4ee19187e01a12568e02e.jpg)  
+Figure 1: Diagram of the Recurrent Entity Network’s dynamic memory. Update equations 1 and 2 are represented by the module $f _ { \theta }$ , where $\theta$ is the set of trainable parameters. Equations 3 and 4 are represented by the gate, since they fullfill a similar function.
+
+by setting all weights in the multiplicative mask to 1, or can choose a positional encoding model as used in (Sukhbaatar et al., 2015).
+
+# 2.2 DYNAMIC MEMORY
+
+The dynamic memory is a gated recurrent network with a (partially) block structured weight tying scheme. We divide the hidden states of the network into blocks $h _ { 1 } , . . . , h _ { m }$ ; the full hidden state is the concatenation of the $h _ { j }$ . In the experiments below, $m$ is of the order of 5 to 20, and each block $h _ { j }$ is of the order of 20 to 100 units.
+
+At each time step $t$ , the content of the hidden states $\{ h _ { j } \}$ (which we will call the $j$ th memory) are updated using a set of key vectors $\{ w _ { j } \}$ and the encoded input $s _ { t }$ . In its most general form, the update equations of our model are given by:
+
+$$
+\begin{array} { r l } & { g _ { j }  \sigma \big ( s _ { t } ^ { T } h _ { j } + s _ { t } ^ { T } w _ { j } \big ) } \\ & { \tilde { h _ { j } }  \phi \big ( U h _ { j } + V w _ { j } + W s _ { t } \big ) } \\ & { { h _ { j } }  h _ { j } + g _ { j } \circledast \tilde { h _ { j } } } \\ & { { h _ { j } }  \displaystyle \frac { h _ { j } } { \vert \vert h _ { j } \vert \vert } } \end{array}
+$$
+
+Here $\sigma$ represents a sigmoid, $g _ { j }$ is a gating function which determines how much the $j ^ { t h }$ memory should be updated, and $\tilde { h _ { j } }$ is the new candidate value of the memory to be combined with the existing memory $h _ { j }$ . The function $\phi$ can be chosen from any number of activation functions, in our experiments we use either parametric ReLU non-linearities (He et al., 2015) or the identity. The matrices $U , V , W$ are typically trainable parameters of the model, and are shared between all the blocks. They can also be fixed to certain values, such as the identity or zero, to yield a simpler model which we use in some of our experiments.
+
+The gating function $g _ { j }$ contains two terms: a “content” term $s _ { t } ^ { T } h _ { j }$ which causes the gate to open for memory slots whose content matches the input, and a “location” term $s _ { t } ^ { T } w _ { j }$ which causes the gate to open for memory slots whose key matches the input. The final normalization step allows the model to forget previous information. To see this, note that since the memories lie on the unit sphere, all information is contained in their phase. Adding any vector to a given memory (other than the memory itself) will decrease the cosine distance between the original memory and the updated one. Therefore, as new information is added, old information is forgotten.
+
+# 2.3 OUTPUT MODULE
+
+Whenever the model is required to produce an output, it is presented with a query vector $q$ . Specifically, the output is computed using the following equations:
+
+$$
+\begin{array} { l } { p _ { j } = \mathrm { S o f t m a x } ( q ^ { T } h _ { j } ) } \\ { \displaystyle \boldsymbol { u } = \sum _ { j } p _ { j } \boldsymbol { h } _ { j } } \\ { \boldsymbol { y } = R \phi ( \boldsymbol { q } + H \boldsymbol { u } ) } \end{array}
+$$
+
+The matrices $H$ and $R$ are additional trainable parameters of the model. The output module can be viewed as a one-hop Memory Network (Sukhbaatar et al., 2015) with an additional non-linearity $\phi$ between the internal state and the decoder matrix. If the memory slots correspond to specific words (as we will describe in the following section) which contain the answer, $p$ can be viewed as a distribution over potential answers and can be used to make a prediction directly or fed into a loss function, removing the need for the last two steps.
+
+The entire model (all three components described above) is trained via backpropagation through time, receiving gradients from any time steps where the reader is required to produce an output, which are then propagated through the unrolled network.
+
+# 3 MOTIVATING EXAMPLE OF OPERATION
+
+We now describe a motivating example of how our model can perform reasoning on-the-fly as it is ingesting input sequences. Let us suppose our model is reading a story, so the inputs are natural language sentences, and then it is required to answer questions about the story it has just read.
+
+Our model is free to learn the key vectors $w _ { j }$ for each memory $j$ . One choice the model could make is to associate a single memory (via the key) with each entity in the story. The memory slot corresponding to a person could encode that person’s location, the objects they are carrying, or the people they are with, depending on what information is relevant for the task at hand. As new information is received indicating that objects are acquired or discarded, or the person changes location, their memory slot will change accordingly. Similarly useful updates can be made for memories corresponding to object and location entities as well.
+
+In fact, we could encode this choice of memories directly into our model, which we consider as a type of prior knowledge. By tying the weights of the key vectors with the embeddings of specific words, we can encourage the model to record information about certain words occuring in the text which we believe to be important. For example, given a list of named entities (which could be produced by a standard tagger), we could make the model have a separate memory slot for each entity. We consider this “tied” variant in our experiments. Since the list of entities is independent of the training data, this variant can handle entities not seen in the training set, as long as their embeddings can be initialized in a reasonable way (such as pre-training on a larger corpus).
+
+Now, consider that the model reads the following two sentences, and the desired behavior of the gating function and update function at each memory as they are seen:
+
+• Mary picked up the ball.   
+• Mary went to the garden.
+
+As the first sentence $s _ { t }$ is ingested, and assuming memories encode entities, we would like the gates of the memories corresponding to both “Mary” and “ball” to activate. This is possible due to the location addressing term $s _ { t } ^ { T } w _ { j }$ which uses the key $w _ { j }$ . We expect that a well trained model would learn to do this. The model would hence modify both the entry corresponding to “Mary” to indicate that she is now carrying the ball, and also the entry corresponding to “ball”, to indicate that it is being carried by Mary. When the second sentence is seen, we would like the model to again modify the “Mary” entry to indicate that she is now in the garden, and also modify the “ball” entry to reflect its new location as well. Assuming the information for ���Mary” is contained in the “ball” memory as described before, the gate corresponding to “ball” can activate due to the content addressing term $s _ { t } ^ { T } h _ { j }$ , even though the word “ball” does not occur in the second sentence. As before, the gate corresponding to the “Mary” entry can open due to the second term.
+
+If the gating function and update function have weights such that the steps above are executed, then the memory will be in a state where questions such as “Where is the ball?” or “Where is Mary?” can be answered from the values of relevant memories, without the need for further complex reasoning.
+
+# 4 RELATED WORK
+
+The EntNet is related to gated recurrent models such as the LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho et al., 2014), which also use gates to fix or modify the information stored in the hidden state. However, these models use scalar memory cells with full interactions between them, whereas ours has separate memory slots which could be seen as groups of hidden units with tied weights in the gating and update functions. Another important difference is the content-based matching term between the input and hidden state, which is not present in these models.
+
+Our model also shares some similarities with the DNC/NTM framework of (Graves et al., 2014; 2016). There, as in our model, a block of hidden states acts as a set of read-writeable memories. On the other hand, the DNC has a relatively sophisticated controller network (such as an LSTM) which reads an input and outputs a number of interface vectors (such as keys and weightings) which are then combined via a softmax to read from and write to the external memory matrix. In contrast, our model can be viewed as a set of separate recurrent models whose hidden states store the memory slots. These hidden states are either fixed by the gates, or modified through a simple RNN-style update. The bulk of the reasoning is thus performed by these parallel recurrent models, rather than through a central controller. Moreover, instead of using a softmax, our model uses an independent gate for writing to each memory.
+
+Our model is similar to a Memory Network and its variants (Weston et al., 2014; Sukhbaatar et al., 2015; Chandar et al., 2016; Miller et al., 2016) in the way it produces an output using a softmax over blocks of hidden states, and our encoding layer is inspired by techniques used in those works. However, Memory Networks explicitly store the entire input sequence in memory, and then sequentially update a controller’s hidden state via a softmax gating over the memories. In contrast, our model keeps a fixed number of blocks of hiddens as memories and updates each block with an independent gated RNN. The Dynamic Memory Network of (Xiong et al., 2016) also performs updates via a recurrent model, however it links memories to input tokens and updates them sequentially rather than in parallel.
+
+The weight tying scheme and the parallel gated RNNs recall the gated graph network of (Li et al., 2015). If we interpret our work in that context, the “graph” is just a set of vertices with no edges; our gating mechanism is also somewhat different than the one they use. The CommNN model of (Sukhbaatar et al., 2016), the Interaction Network of (Battaglia et al., 2016), the Neural Physics Engine of (Chang et al., 2016) and the model of (Fragkiadaki et al., 2015) also use a set of parallel recurrent models with tied weights, but differ from our model in their use of inter-network communication and the lack of a gating mechanism.
+
+Finally, there is another class of recent models that have a writeable memory arranged as (unbounded) stacks, linked lists or queues (Joulin & Mikolov, 2015; Grefenstette et al., 2015). Our model is different from these in that we use a key-value pair array instead of a stack, and in the experiments in this work, the array is of fixed size.
+
+<table><tr><td>Model</td><td>T=10</td><td>T=20</td><td>T=40</td></tr><tr><td>MemN2N</td><td>0.09</td><td>0.633</td><td>0.896</td></tr><tr><td>LSTM</td><td>0</td><td>0.157</td><td>0.226</td></tr><tr><td>EntNet</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan="4">(a)</td></tr></table>
+
+![](images/abd67377581d9363ea9de7574d71b6796d8bbf5999f27f725ea9a51eb945e674.jpg)  
+Table 1: a) Error of different models on the World Model Task. b) Generalization of an EntNet trained up to $T = 2 0$ . All errors range from 0 to 1.
+
+# 5 EXPERIMENTS
+
+In this section we evaluate our model on three different datasets. Training details common to all experiments can be found in Appendix A.
+
+# 5.1 SYNTHETIC WORLD MODEL TASK
+
+We first study our model’s properties on a toy task designed to measure the ability to keep a world model in memory. In this task two agents are initially placed randomly on an $1 0 \times 1 0$ grid, and at each time step a randomly chosen agent either changes direction or moves ahead. After a certain number of time steps, the model is required to provide the locations of each of the agents, thus revealing its internal world model (details can be found in Appendix B). This task is challenging because the model must combine up to $T - 2$ supporting facts in order to answer the question correctly, and must also keep the locations of both agents in memory and update them at different times.
+
+We compared the performance of a MemN2N, LSTM and EntNet. For the MemN2N, we set the number of hops equal to $T - 2$ and the embedding dimension to $d = 2 0$ . The EntNet had embedding dimension $d = 2 0$ and 5 memory slots, and the LSTM had 50 hidden units which resulted in it having significantly more parameters than the other two models. For each model, we repeated the experiment with 5 different initializations and reported the best performance. All models were trained with ADAM (Kingma & Ba, 2014) with initial learning rates set by grid search over $\{ 0 . 1 , 0 . 0 1 , 0 . 0 0 1 \}$ and divided by 2 every 10,000 updates. Table 1a shows the results. The MemN2N has the worst performance, which degrades quickly as the length of the sequence increases. The LSTM performs better, but still loses accuracy as the length of the sequence increases. In contrast, the EntNet is able to solve the task in all cases.
+
+The ability to generalize to sequences longer than those seen during training is a desirable property, which suggests that the network has learned the dynamics of the world it is trying to model. It also means the model can be trained less expensively. To study this, we trained an EntNet on variable length sequences between 1 and 20, and evaluated it on different length sequences longer than 20. Results are shown in Table 1b. We see that the model is able to achieve good performance several times past its training horizon.
+
+# 5.2 BABI TASKS
+
+We next evaluate our model on the bAbI tasks, which are a collection of 20 synthetic questionanswering datasets first introduced in (Weston et al., 2015) designed to test a wide variety of reasoning abilities. They have since become a benchmark for memory-augmented neural networks and most of the related methods described in Section 4 have been tested on them. Performance is measured using two metrics: the average error across all tasks, and the number of failed tasks (more than $5 \%$ error). We used version 1.2 of the dataset with $1 0 \mathrm { k }$ samples. 1
+
+Training Details We used a similar training setup as (Sukhbaatar et al., 2015). All models were trained with ADAM using a learning rate of $\eta = 0 . 0 1$ , which was divided by 2 every 25 epochs until 200 epochs were reached. Copying previous works (Sukhbaatar et al., 2015; Xiong et al., 2016), the capacity of the memory was limited to the most recent 70 sentences, except for task 3 which was limited to 130 sentences. Due to the high variance in model performance for some tasks, for each task we conducted 10 runs with different initializations and picked the best model based on performance on the validation set, as it has been done in previous work. In all experiments, our model had embedding dimension size $d = 1 0 0$ and 20 memory slots.
+
+Table 2: Results on bAbI Tasks with 10k training samples.   
+
+<table><tr><td>Task</td><td>NTM</td><td>D-NTM</td><td>MemN2N</td><td>DNC</td><td>DMN+</td><td>EntNet</td></tr><tr><td></td><td>31.5</td><td>4.4</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1: 1 supporting fact 2: 2 supporting facts</td><td>54.5</td><td>27.5</td><td>0.3</td><td>0.4</td><td>0.3</td><td>0.1</td></tr><tr><td>3: 3 supporting facts</td><td>43.9</td><td>71.3</td><td>2.1</td><td>1.8</td><td>1.1</td><td>4.1</td></tr><tr><td>4: 2 argument relations</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>5:3 argument relations</td><td>0.8</td><td>1.7</td><td>0.8</td><td>0.8</td><td>0.5</td><td>0.3</td></tr><tr><td>6: yes/no questions</td><td>17.1</td><td>1.5</td><td>0.1</td><td>0</td><td>0</td><td>0.2</td></tr><tr><td>7: counting</td><td>17.8</td><td>6.0</td><td>2.0</td><td>0.6</td><td>2.4</td><td>0</td></tr><tr><td>8: lists/sets</td><td>13.8</td><td>1.7</td><td>0.9</td><td>0.3</td><td>0.0</td><td>0.5</td></tr><tr><td>9:simple negation</td><td>16.4</td><td>0.6</td><td>0.3</td><td>0.2</td><td>0.0</td><td>0.1</td></tr><tr><td>10:indefinite knowledge</td><td>16.6</td><td>19.8</td><td>0</td><td>0.2</td><td>0</td><td>0.6</td></tr><tr><td>11:basic coreference</td><td>15.2</td><td>0</td><td>0.0</td><td>0</td><td>0.0</td><td>0.3</td></tr><tr><td>12:conjunction</td><td>8.9</td><td>6.2</td><td>0</td><td>0</td><td>0.2</td><td>0</td></tr><tr><td>13:compound coreference</td><td>7.4</td><td>7.5</td><td>0</td><td>0</td><td>0</td><td>1.3</td></tr><tr><td>14: time reasoning</td><td>24.2</td><td>17.5</td><td>0.2</td><td>0.4</td><td>0.2</td><td>0</td></tr><tr><td>15:basic deduction</td><td>47.0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>16:basic induction</td><td>53.6</td><td>49.6</td><td>51.8</td><td>55.1</td><td>45.3</td><td>0.2</td></tr><tr><td>17: positional reasoning</td><td>25.5</td><td>1.2</td><td>18.6</td><td>12.0</td><td>4.2</td><td>0.5</td></tr><tr><td>18:size reasoning</td><td>2.2</td><td>0.2</td><td>5.3</td><td>0.8</td><td>2.1</td><td>0.3</td></tr><tr><td>19: path finding</td><td>4.3</td><td>39.5</td><td>2.3</td><td>3.9</td><td>0.0</td><td>2.3</td></tr><tr><td>20: agent&#x27;s motivation</td><td>1.5</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5% error): Mean Error:</td><td>16</td><td>9</td><td>3</td><td>2</td><td>1</td><td>0</td></tr><tr><td></td><td>20.1</td><td>12.8</td><td>4.2</td><td>3.8</td><td>2.8</td><td>0.5</td></tr></table>
+
+In Table 2 we compare our model to various other state-of-the-art models in the literature: the larger MemN2N reported in the appendix of (Sukhbaatar et al., 2015), the Dynamic Memory Network of (Xiong et al., 2016), the Dynamic Neural Turing Machine (Gulcehre et al., 2016), the Neural Turing Machine (Graves et al., 2014) and the Differentiable Neural Computer (Graves et al., 2016). Our model is able to solve all the tasks, outperforming the other models in terms of both the number of solved tasks and the average error.
+
+To analyze what kind of representations our model can learn, we conducted an additional experiment on Task 2 using a simple BoW sentence encoding and key vectors which were tied to entity embeddings. This was designed to make the model more interpretable, since the weight tying forces memory slots to encode information about specific entities. 2 After training, we ran the model over a story and computed the cosine distance between $\phi ( H h _ { j } )$ and each row $r _ { i }$ of the decoder matrix $R$ . This gave us a score which measures the affinity between a given memory slot and each word in the vocabulary. Table 3 shows the nearest neighboring words for each memory slot (which itself corresponds to an entity). We see that the model has indeed stored locations of all of the objects and characters in its memory slots which reflect the final state of the story. In particular, it has the correct answer readily stored in the memory slot of the entity being inquired about (the milk). It also has correct location information about all other non-location entities stored in the appropriate memory slots. Note that it does not store useful or correct information in the memory slots corresponding to locations, most likely because this task does not contain questions about locations (such as “who is in the kitchen?”).
+
+Table 3: On the left, the network’s final “world model” after reading the story on the right. First and second nearest neighbors from each memory slot are shown, along with their cosine distance.   
+
+<table><tr><td>Key</td><td>1-NN</td><td>2-NN</td></tr><tr><td>football milk john mary sandra daniel</td><td>hallway (0.135) garden (0.111) kitchen (0.501) garden (0.442) hallway (0.394) hallway (0.689)</td><td>dropped (0.056) took (0.011) dropped (0.027) took (0.034) kitchen (0.121) to (0.076)</td></tr><tr><td>bedroom kitchen garden hallway</td><td>hallway (0.367) kitchen (0.483) garden (0.281) hallway (0.475)</td><td>dropped (0.075) daniel (0.029) where (0.026) left (0.060)</td></tr></table>
+
+<table><tr><td>Story</td></tr><tr><td>mary got the milk there john moved to the bedroom sandra went back to the kitchen mary travelled to the hallway john got the football there john went to the hallway john put down the football mary went to the garden john went to the kitchen sandra travelled to the hallway</td></tr></table>
+
+# 5.3 CHILDREN’S BOOK TEST (CBT)
+
+We next evaluated our model on the Children’s Book Test (Hill et al., 2016), which is a semantic language modeling (sentence completion) benchmark built from children’s books that are freely available from Project Gutenberg 3. Models are required to read 20 consecutive sentences from a given story and use this context to fill in a missing word from the 21st sentence. More specifically, each sample consists of a tuple $( S , q , C , a )$ where $S$ is the story consisting of 20 sentences, $Q$ is the 21st sentence with one word replaced by a special blank token, $C$ is a set of 10 candidate answers of the same type as the missing word (for example, common nouns or named entities), and $a$ is the true answer (which is always contained in $C$ ).
+
+It was shown in (Hill et al., 2016) that methods with limited memory such as LSTMs perform well on more frequent, syntax based words such as prepositions and verbs, being similar to human performance, but poorly relative to humans on more semantically meaningful words such as named entities and common nouns. Therefore, most recent methods have been evaluated on the Named Entity and Common Noun subtasks, since they better test the ability of a model to make use of wider contextual information.
+
+Training Details We adopted the same window memory approach used in (Hill et al., 2016), where each input corresponds to a window of text from $\left\{ w _ { \left( i - b - 1 / 2 \right) } . . . w _ { i } . . . w _ { \left( i + \left( b - 1 \right) / 2 \right) } \right\}$ centered at a candidate $w _ { i } \in C$ . In our experiments we set $b = 5$ . All models were trained using standard stochastic gradient descent (SGD) with a fixed learning rate of 0.001. We used separate input encodings for the update and gating functions, and applied a dropout rate of 0.5 to the word embedding dimensions. Key embeddings were tied to the embeddings of the candidate words, resulting in 10 hidden blocks, one per member of $C$ . Due to the weight tying, we did not need a decoder matrix and used the distribution over candidates to directly produce a prediction, as described in Section 3.
+
+We found that a simpler version of the model worked best, with $U = V = 0$ , $W = I$ and $\phi$ equal to the identity. We also removed the normalization step in this simplified model, which we found to hurt performance. This can be explained by the fact that the maximum frequency baseline model in (Hill et al., 2016) has performance which is significantly higher than random, and including the normalization step hides this useful frequency-based information.
+
+Results We draw a distinction between two setups: the single-pass setup, where the model must read the story and query in order and immediately produce an output, and the multi-pass setup, where the model can use the query to perform attention over the story. The first setup is more challenging because the model does not know beforehand which query it will be presented with, and must learn to retain information which is useful for a wide variety of potential queries. For this reason it can be viewed as a test of the model’s ability to construct a general-purpose representation of the current state of the story. The second setup leverages all available information, and allows the model to use knowledge of which question will be asked when it reads the story.
+
+Table 4: Accuracy on CBT test set. Single-pass models encode the document before seeing the query, multi-pass models have access to the query at read time.   
+
+<table><tr><td>Model</td><td></td><td>Named Entities</td><td>Common Nouns</td></tr><tr><td rowspan="5">Single Pass</td><td>Kneser-Ney Language Model + cache</td><td>0.439</td><td>0.577</td></tr><tr><td>LSTMs (context+ query)</td><td>0.418</td><td>0.560</td></tr><tr><td>Window LSTM</td><td>0.436</td><td>0.582</td></tr><tr><td>EntNet (general)</td><td>0.484</td><td>0.540</td></tr><tr><td>EntNet (simple)</td><td>0.616</td><td>0.588</td></tr><tr><td></td><td></td><td>0.493</td><td></td></tr><tr><td rowspan="7">Multi Pass</td><td>MemNN</td><td></td><td>0.554</td></tr><tr><td>MemNN + self-sup.</td><td>0.666</td><td>0.630</td></tr><tr><td>Attention Sum Reader (Kadlec et al.,2016)</td><td>0.686</td><td>0.634</td></tr><tr><td>Gated-Attention Reader (Bhuwan Dhingra &amp; Salakhutdinov,2016)</td><td>0.690</td><td>0.639</td></tr><tr><td>EpiReader (Trischler et al.,2016)</td><td>0.697</td><td>0.674</td></tr><tr><td>AoAReader (Cui et al.,2016)</td><td>0.720</td><td>0.694</td></tr><tr><td>NSE Adaptive Computation (Munkhdalai &amp; Yu, 2016)</td><td>0.732</td><td>0.714</td></tr></table>
+
+In Table 4, we show the performance of the general EntNet, the simplified EntNet, as well as other single-pass models taken from (Hill et al., 2016). The general EntNet performs better than the LSTMs and $n$ -gram model on the Named Entities Task, but lags behind on the Common Nouns task. The simplified EntNet outperforms all other single-pass models on both tasks, and also performs better than the Memory Network which does not use the self-supervision heuristic. However, there is still a performance gap when compared to more sophisticated machine comprehension models, many of which perform multiple layers of attention over the story using query knowledge. The fact that the simplified EntNet is able to obtain decent performance is encouraging since it indicates that the model is able to build an internal representation of the story which it can then use to answer a relatively diverse set of queries.
+
+# 6 CONCLUSION
+
+Two closely related challenges in artificial intelligence are designing models which can maintain an estimate of the state of a world with complex dynamics over long timescales, and models which can predict the forward evolution of the state of the world from partial observation. In this paper, we introduced the Recurrent Entity Network, a new model that makes a promising step towards the first goal. Our model is able to accurately track the world state while reading text stories, which enables it to set a new state-of-the-art on the bAbI tasks, the competitive benchmark of story understanding, by being the first model to solve them all. We also showed that our model is able to capture simple dynamics over long timescales, and is able to perform competitively on a real-world dataset.
+
+Although our model was able to solve all the bAbI tasks using $1 0 \mathrm { k }$ training samples, we found that performance dropped considerably when using only 1k samples (see Appendix). Most recent work on the bAbI tasks has focused on the $1 0 \mathrm { k }$ samples setting, and we would like to emphasize that solving them in the 1k samples setting remains an open problem which will require improving the sample efficiency of reasoning models, including ours.
+
+Recent works have made some progress towards the second goal of forward modeling, for instance in capturing simple physics (Lerer et al., 2016), predicting future frames in video (Mathieu et al., 2015) or responses in dialog (Weston, 2016). Although we have only applied our model to tasks with textual inputs in this work, the architecture is general and future work should investigate how to combine the EntNet’s tracking abilities with such predictive models.
+
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+
+# A TRAINING DETAILS
+
+All models were implemented using Torch (Collobert et al., 2011). In all experiments, we initialized our model by drawing weights from a Gaussian distribution with mean zero and standard deviation 0.1, except for the PReLU slopes and encoder weights which were initialized to 1. Note that the PReLU initialization is related to two of the heuristics used in (Sukhbaatar et al., 2015), namely starting training with a purely linear model, and adding non-linearities to half of the hidden units. Our initialization allows the model to choose when and how much to enter the non-linear regime. Initializing the encoder weights to 1 corresponds to beginning with a BoW encoding, which the model can then choose to modify. The initial values of the memory slots were initialized to the key values, which we found to help performance. Optimization was done with SGD or ADAM using minibatches of size 32, and gradients with norm greater than 40 were clipped to 40. A null symbol whose embedding was constrained to be zero was used to pad all sentences or windows to a fixed size.
+
+# B DETAILS OF WORLD MODEL EXPERIMENTS
+
+Two agents are initially placed at random on a $1 0 ~ \times ~ 1 0$ grid with 100 distinct locations $\{ ( 1 , 1 ) , \bar { ( 1 , 2 ) } , . . . ( 9 , 1 0 ) , \bar { ( 1 0 , 1 0 ) } \}$ . At each time step an agent is chosen at random. There are two types of actions: the agent can face a given direction, or can move a number of steps ahead. Actions are sampled until a legal action is found by either choosing to change direction or move with equal probability. If they change direction, the direction is chosen between north, south, east and west with equal probability. If they move, the number of steps is randomly chosen between 1 and 5. A legal action is one which does not place the agent off the grid. Stories are given to the network in textual form, an example of which is below. The first action after each agent is placed on the grid is to face a given direction. Therefore, the maximum number of actions made by one agent is $T - 2$ . The network learns word embeddings for all words in the vocabulary such as locations, agent identifiers and actions. At question time, the model must predict the correct answer (which will always be a location) from all the tokens in the vocabulary.
+
+agent1 is at (2,8)   
+agent1 faces-N   
+agent2 is at (9,7)   
+agent2 faces-N   
+agent2 moves-2   
+agent2 faces-E   
+agent2 moves-1   
+agent1 moves-1   
+agent2 faces-S   
+agent2 moves-5   
+Q1: where is agent1 ?   
+Q2: where is agent2 ?   
+A1: (2,9)   
+A2: (10,4)
+
+# C ADDITIONAL RESULTS ON BABI TASKS
+
+We provide some additional experiments on the bAbI tasks, in order to better understand the influence of architecture, weight tying, and amount of training data. Table 5 shows results when a simple BoW encoding is used for the inputs. Here, the EntNet still performs better than a MemN2N which uses the same encoding scheme, indicating that the architecture has an important effect. Tying the key vectors to entities did not help, and hurt performance for some tasks. Table 6 shows results when using only 1k training samples. In this setting, the EntNet performs worse than the MemN2N.
+
+Table 5: Error rates on bAbI Tasks with inputs are encoded using BoW. “Tied” refers to the case where key vectors are tied with entity embeddings.   
+
+<table><tr><td>Task</td><td>MemN2N</td><td>EntNet-tied</td><td>EntNet</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>1:1 supporting fact</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2:2 supporting facts</td><td>0.6</td><td>3.0</td><td>1.2</td></tr><tr><td>3: 3 supporting facts</td><td>7</td><td>9.6</td><td>9.0</td></tr><tr><td>4: 2 argument relations</td><td>32.6</td><td>33.8</td><td>31.8</td></tr><tr><td>5:3 argument relations</td><td>10.2</td><td>1.7</td><td>3.5</td></tr><tr><td>6: yes/no questions</td><td>0.2</td><td>0</td><td>0 0.5</td></tr><tr><td>7: counting 8: lists/sets</td><td>10.6</td><td>0.5</td><td>0.3</td></tr><tr><td>9: simple negation</td><td>2.6 0.3</td><td>0.1</td><td>0</td></tr><tr><td>10: indefinite knowledge</td><td>0.5</td><td>0 0</td><td>0</td></tr><tr><td>11:basic coreference</td><td>0</td><td>0.3</td><td>0</td></tr><tr><td>12: conjunction</td><td>0</td><td>0</td><td>0</td></tr><tr><td>13: compound coreference</td><td>0</td><td>0.2</td><td>0.4</td></tr><tr><td>14: time reasoning</td><td>0.1</td><td>6.2</td><td>0.1</td></tr><tr><td>15:basic deduction</td><td>11.4</td><td>12.5</td><td>12.1</td></tr><tr><td>16:basic induction</td><td>52.9</td><td>46.5</td><td>0</td></tr><tr><td>17: positional reasoning</td><td>39.3</td><td>40.5</td><td>40.5</td></tr><tr><td>18: size reasoning</td><td>40.5</td><td>44.2</td><td>45.7</td></tr><tr><td>19: path finding</td><td>74.4</td><td>75.1</td><td>74.0</td></tr><tr><td>20: agent&#x27;s motivation</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5%):</td><td>9</td><td>8</td><td>6</td></tr><tr><td>Mean Error:</td><td>15.6</td><td>13.7</td><td>10.9</td></tr></table>
+
+Table 6: Results on bAbI Tasks with 1k samples.   
+
+<table><tr><td>Task</td><td>MemN2N</td><td>EntNet</td></tr><tr><td></td><td></td><td></td></tr><tr><td>1: 1 supporting fact</td><td>0</td><td>0.7</td></tr><tr><td>2:2 supporting facts</td><td>8.3</td><td>56.4</td></tr><tr><td>3: 3 supporting facts</td><td>40.3</td><td>69.7</td></tr><tr><td>4:2 argument relations</td><td>2.8</td><td>1.4</td></tr><tr><td>5: 3 argument relations</td><td>13.1</td><td>4.6</td></tr><tr><td>6: yes/no questions</td><td>7.6</td><td>30.0</td></tr><tr><td>7: counting</td><td>17.3</td><td>22.3</td></tr><tr><td>8: lists/sets</td><td>10.0</td><td>19.2</td></tr><tr><td>9:simple negation</td><td>13.2</td><td>31.5</td></tr><tr><td>10:indefinite knowledge</td><td>15.1</td><td>15.6</td></tr><tr><td>11:basic coreference</td><td>0.9</td><td>8.0</td></tr><tr><td>12:conjunction</td><td>0.2</td><td>0.8</td></tr><tr><td>13: compound coreference</td><td>0.4</td><td>9.0</td></tr><tr><td>14: time reasoning</td><td>1.7</td><td>62.9</td></tr><tr><td>15: basic deduction</td><td>0</td><td>57.8</td></tr><tr><td>16:basic induction</td><td>1.3</td><td>53.2</td></tr><tr><td>17: positional reasoning</td><td>51.0</td><td>46.4</td></tr><tr><td>18:size reasoning</td><td>11.1</td><td>8.8 90.4</td></tr><tr><td>19: path finding 20:agent&#x27;s motivation</td><td>82.8 0</td><td>2.6</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5%):</td><td>11</td><td>15</td></tr><tr><td>Mean Error:</td><td>13.9</td><td>29.6</td></tr></table>

parse/train/rJTKKKqeg/rJTKKKqeg_content_list.json

@@ -0,0 +1,1186 @@
+[
+    {
+        "type": "text",
+        "text": "TRACKING THE WORLD STATE WITHRECURRENT ENTITY NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mikael Henaff1,2, Jason Weston1, Arthur Szlam1, Antoine Bordes1 and Yann LeCun1,2 ",
+        "bbox": [
+            169,
+            174,
+            772,
+            190
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Facebook AI Research   \n2Courant Institute, New York University   \n{mbh305}@nyu.edu, {jase,aszlam,abordes,yann}@fb.com ",
+        "bbox": [
+            174,
+            203,
+            661,
+            244
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            286,
+            544,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a new model, the Recurrent Entity Network (EntNet). It is equipped with a dynamic long-term memory which allows it to maintain and update a representation of the state of the world as it receives new data. For language understanding tasks, it can reason on-the-fly as it reads text, not just when it is required to answer a question or respond as is the case for a Memory Network (Sukhbaatar et al., 2015). Like a Neural Turing Machine or Differentiable Neural Computer (Graves et al., 2014; 2016) it maintains a fixed size memory and can learn to perform location and content-based read and write operations. However, unlike those models it has a simple parallel architecture in which several memory locations can be updated simultaneously. The EntNet sets a new state-of-the-art on the bAbI tasks, and is the first method to solve all the tasks in the $1 0 \\mathrm { k }$ training examples setting. We also demonstrate that it can solve a reasoning task which requires a large number of supporting facts, which other methods are not able to solve, and can generalize past its training horizon. It can also be practically used on large scale datasets such as Children’s Book Test, where it obtains competitive performance, reading the story in a single pass. ",
+        "bbox": [
+            233,
+            323,
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+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            582,
+            336,
+            598
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The essence of intelligence is the ability to predict. An intelligent agent must be able to predict unobserved facts about their environment from limited percepts (visual, auditory, textual, or otherwise), combined with their knowledge of the past. In order to reason and plan, they must be able to predict how an observed event or action will affect the state of the world. Arguably, the ability to maintain an estimate of the current state of the world, combined with a forward model of how the world evolves, is a key feature of intelligent agents. ",
+        "bbox": [
+            174,
+            617,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A natural way for an agent to represent the world is to maintain a set of high-level concepts or entities together with their properties, which are updated as new information is received. For example, if a percept is the textual description of an event, such as “John walks out of the kitchen”, the agent should learn to update its estimate of John’s location, as well as the list (and number) of people present in each room. If John was carrying a bag, the location of the bag and the list of objects in the kitchen must also be updated. When we read a story, each sentence we read or hear causes us to update our internal representation of the current state of the world within the story. The flow of the story is captured by the evolution of this state of the world. ",
+        "bbox": [
+            174,
+            708,
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+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "At any given time, an agent typically receives limited information about the state of the world, and should therefore be able to infer new information through partial observation. In this paper, we investigate this problem through a simple story understanding scenario, in which the agent is given a sequence of textual statements and events, and then given another series of statements about the final state of the world. If the second series of statements is given in the form of questions about the final state of the world together with their correct answers, the agent should be able to learn from them and its performance can be measured by the accuracy of its answers. ",
+        "bbox": [
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+            827,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Even with this weak form of supervision, the system may learn basic dynamical constraints about the world. For example, it may learn that a person or object cannot be in two locations at the same time, or may learn simple update rules such as incrementing and decrementing the number of persons or objects in a room. It may also learn basic rules of approximate (logical) inference, such as the fact that objects belonging to the same category tend to have similar properties (light objects can be carried over from rooms to rooms for instance). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We propose to handle this scenario with a new kind of memory-augmented neural network that uses a distributed memory and processor architecture: the Recurrent Entity Network (EntNet). The model consists of a fixed number of dynamic memory cells, each containing a vector key $w _ { j }$ and a vector value (or content) $h _ { j }$ . Each cell is associated with its own “processor”, a simple gated recurrent network that may update the cell value given an input. If each cell learns to represent a concept or entity in the world, one can imagine a gating mechanism that, based on the key and content of the memory cells, will only modify the cells that concern the entities mentioned in the input. In the current version of the model, there is no direct interaction between the memory cells, hence the system can be seen as multiple identical processors functioning in parallel, with distributed local memory. Alternatively, the EntNet can be seen as a bank of gated RNNs (all sharing the same parameters), whose hidden states correspond to latent concepts and attributes, and whose parameters describe the laws of the world according to which the attributes of objects are updated. The sharing of these parameters reflects an invariance of these laws across object instances, similarly to how the weight tying scheme in a CNN reflects an invariance of image statistics across locations. Their hidden state is updated only when new information relevant to their concept is received, and remains otherwise unchanged. The keys used in the addressing/gating mechanism also correspond to concepts or entities, but are modified only during learning, not during inference. ",
+        "bbox": [
+            174,
+            194,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The EntNet is able to solve all 20 bAbI question-answering tasks (Weston et al., 2015), a popular benchmark of story understanding, which to our knowledge sets a new state-of-the-art. Our experiments also indicate that the model indeed maintains an internal representation of the simplified world in which the stories take place, and that the model does not limit itself to storing the aspects of the world required to answer a specific question. We also introduce a new reasoning task which, unlike the bAbI tasks, requires a model to use a large number of supporting facts to answer the question, and show that the EntNet outperforms both LSTMs and Memory Networks (Sukhbaatar et al., 2015) by a significant margin. It is also able to generalize to sequences longer than those seen during training. Finally, our model also obtains competitive results on the Childrens Book Test (Hill et al., 2016), and performs best among models that read the text in a single pass before receiving knowledge of the question. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            609,
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+            626
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model is designed to process data in sequential form, and consists of three main parts: an input encoder, a dynamic memory and an output layer, which we now describe in detail. We developed it in the context of question answering on short stories where the inputs are word sequences, but the model could be adapted to many other contexts. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 INPUT ENCODER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            713,
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+            727
+        ],
+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The encoding layer summarizes an element of the input sequence with a vector of fixed length. Typically the input element at time $t$ is a sequence of words, e.g. a sentence or window of words. One is free to choose the encoding module to be any standard sequence encoder, which is an active area of research. Typical choices include a bag-of-words (BoW) representation or the final state of a recurrent neural net (RNN) run over the sequence. In this work, we use a simple encoder consisting of a learned multiplicative mask followed by a summation. More precisely, let the input at time $t$ be a sequence of words with embeddings $\\{ e _ { 1 } , . . . , e _ { k } \\}$ . The vector representation of this input is then: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/d5728a44399f4ce1933340e065a6123ff2c970f030e99d36bdf0c7d84dedd485.jpg",
+        "text": "$$\ns _ { t } = \\sum _ { i } f _ { i } \\odot e _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            853,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The same set of vectors $\\{ f _ { 1 } , . . . , f _ { k } \\}$ are used at each time step and are learned jointly with the other parameters of the model. Note that the model can choose to adopt a standard BoW representation ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1a75e8621689e8c39ada2bcd119684f909bae1e2db4ee19187e01a12568e02e.jpg",
+        "image_caption": [
+            "Figure 1: Diagram of the Recurrent Entity Network’s dynamic memory. Update equations 1 and 2 are represented by the module $f _ { \\theta }$ , where $\\theta$ is the set of trainable parameters. Equations 3 and 4 are represented by the gate, since they fullfill a similar function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "by setting all weights in the multiplicative mask to 1, or can choose a positional encoding model as used in (Sukhbaatar et al., 2015). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 DYNAMIC MEMORY ",
+        "text_level": 1,
+        "bbox": [
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+            553,
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+            568
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The dynamic memory is a gated recurrent network with a (partially) block structured weight tying scheme. We divide the hidden states of the network into blocks $h _ { 1 } , . . . , h _ { m }$ ; the full hidden state is the concatenation of the $h _ { j }$ . In the experiments below, $m$ is of the order of 5 to 20, and each block $h _ { j }$ is of the order of 20 to 100 units. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "At each time step $t$ , the content of the hidden states $\\{ h _ { j } \\}$ (which we will call the $j$ th memory) are updated using a set of key vectors $\\{ w _ { j } \\}$ and the encoded input $s _ { t }$ . In its most general form, the update equations of our model are given by: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/352b5db3230089c9d89435ee09ece67e4bccf787690b6fc234eb3ab479e1537c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { g _ { j }  \\sigma \\big ( s _ { t } ^ { T } h _ { j } + s _ { t } ^ { T } w _ { j } \\big ) } \\\\ & { \\tilde { h _ { j } }  \\phi \\big ( U h _ { j } + V w _ { j } + W s _ { t } \\big ) } \\\\ & { { h _ { j } }  h _ { j } + g _ { j } \\circledast \\tilde { h _ { j } } } \\\\ & { { h _ { j } }  \\displaystyle \\frac { h _ { j } } { \\vert \\vert h _ { j } \\vert \\vert } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            709,
+            599,
+            810
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here $\\sigma$ represents a sigmoid, $g _ { j }$ is a gating function which determines how much the $j ^ { t h }$ memory should be updated, and $\\tilde { h _ { j } }$ is the new candidate value of the memory to be combined with the existing memory $h _ { j }$ . The function $\\phi$ can be chosen from any number of activation functions, in our experiments we use either parametric ReLU non-linearities (He et al., 2015) or the identity. The matrices $U , V , W$ are typically trainable parameters of the model, and are shared between all the blocks. They can also be fixed to certain values, such as the identity or zero, to yield a simpler model which we use in some of our experiments. ",
+        "bbox": [
+            173,
+            823,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The gating function $g _ { j }$ contains two terms: a “content” term $s _ { t } ^ { T } h _ { j }$ which causes the gate to open for memory slots whose content matches the input, and a “location” term $s _ { t } ^ { T } w _ { j }$ which causes the gate to open for memory slots whose key matches the input. The final normalization step allows the model to forget previous information. To see this, note that since the memories lie on the unit sphere, all information is contained in their phase. Adding any vector to a given memory (other than the memory itself) will decrease the cosine distance between the original memory and the updated one. Therefore, as new information is added, old information is forgotten. ",
+        "bbox": [
+            174,
+            102,
+            825,
+            203
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 OUTPUT MODULE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            222,
+            341,
+            236
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Whenever the model is required to produce an output, it is presented with a query vector $q$ . Specifically, the output is computed using the following equations: ",
+        "bbox": [
+            173,
+            248,
+            823,
+            277
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b645c0a220497450ef855547dc3205a9986b97d8ffaa37fe22841f1bd8309a1.jpg",
+        "text": "$$\n\\begin{array} { l } { p _ { j } = \\mathrm { S o f t m a x } ( q ^ { T } h _ { j } ) } \\\\ { \\displaystyle \\boldsymbol { u } = \\sum _ { j } p _ { j } \\boldsymbol { h } _ { j } } \\\\ { \\boldsymbol { y } = R \\phi ( \\boldsymbol { q } + H \\boldsymbol { u } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            303,
+            570,
+            381
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The matrices $H$ and $R$ are additional trainable parameters of the model. The output module can be viewed as a one-hop Memory Network (Sukhbaatar et al., 2015) with an additional non-linearity $\\phi$ between the internal state and the decoder matrix. If the memory slots correspond to specific words (as we will describe in the following section) which contain the answer, $p$ can be viewed as a distribution over potential answers and can be used to make a prediction directly or fed into a loss function, removing the need for the last two steps. ",
+        "bbox": [
+            174,
+            386,
+            823,
+            469
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The entire model (all three components described above) is trained via backpropagation through time, receiving gradients from any time steps where the reader is required to produce an output, which are then propagated through the unrolled network. ",
+        "bbox": [
+            176,
+            477,
+            823,
+            518
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 MOTIVATING EXAMPLE OF OPERATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
+            532,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now describe a motivating example of how our model can perform reasoning on-the-fly as it is ingesting input sequences. Let us suppose our model is reading a story, so the inputs are natural language sentences, and then it is required to answer questions about the story it has just read. ",
+        "bbox": [
+            174,
+            574,
+            825,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our model is free to learn the key vectors $w _ { j }$ for each memory $j$ . One choice the model could make is to associate a single memory (via the key) with each entity in the story. The memory slot corresponding to a person could encode that person’s location, the objects they are carrying, or the people they are with, depending on what information is relevant for the task at hand. As new information is received indicating that objects are acquired or discarded, or the person changes location, their memory slot will change accordingly. Similarly useful updates can be made for memories corresponding to object and location entities as well. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            720
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In fact, we could encode this choice of memories directly into our model, which we consider as a type of prior knowledge. By tying the weights of the key vectors with the embeddings of specific words, we can encourage the model to record information about certain words occuring in the text which we believe to be important. For example, given a list of named entities (which could be produced by a standard tagger), we could make the model have a separate memory slot for each entity. We consider this “tied” variant in our experiments. Since the list of entities is independent of the training data, this variant can handle entities not seen in the training set, as long as their embeddings can be initialized in a reasonable way (such as pre-training on a larger corpus). ",
+        "bbox": [
+            174,
+            727,
+            825,
+            839
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now, consider that the model reads the following two sentences, and the desired behavior of the gating function and update function at each memory as they are seen: ",
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+        "text": "• Mary picked up the ball.   \n• Mary went to the garden. ",
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+        "text": "As the first sentence $s _ { t }$ is ingested, and assuming memories encode entities, we would like the gates of the memories corresponding to both “Mary” and “ball” to activate. This is possible due to the location addressing term $s _ { t } ^ { T } w _ { j }$ which uses the key $w _ { j }$ . We expect that a well trained model would learn to do this. The model would hence modify both the entry corresponding to “Mary” to indicate that she is now carrying the ball, and also the entry corresponding to “ball”, to indicate that it is being carried by Mary. When the second sentence is seen, we would like the model to again modify the “Mary” entry to indicate that she is now in the garden, and also modify the “ball” entry to reflect its new location as well. Assuming the information for “Mary” is contained in the “ball” memory as described before, the gate corresponding to “ball” can activate due to the content addressing term $s _ { t } ^ { T } h _ { j }$ , even though the word “ball” does not occur in the second sentence. As before, the gate corresponding to the “Mary” entry can open due to the second term. ",
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+        "type": "text",
+        "text": "If the gating function and update function have weights such that the steps above are executed, then the memory will be in a state where questions such as “Where is the ball?” or “Where is Mary?” can be answered from the values of relevant memories, without the need for further complex reasoning. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "text": "The EntNet is related to gated recurrent models such as the LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho et al., 2014), which also use gates to fix or modify the information stored in the hidden state. However, these models use scalar memory cells with full interactions between them, whereas ours has separate memory slots which could be seen as groups of hidden units with tied weights in the gating and update functions. Another important difference is the content-based matching term between the input and hidden state, which is not present in these models. ",
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+        "type": "text",
+        "text": "Our model also shares some similarities with the DNC/NTM framework of (Graves et al., 2014; 2016). There, as in our model, a block of hidden states acts as a set of read-writeable memories. On the other hand, the DNC has a relatively sophisticated controller network (such as an LSTM) which reads an input and outputs a number of interface vectors (such as keys and weightings) which are then combined via a softmax to read from and write to the external memory matrix. In contrast, our model can be viewed as a set of separate recurrent models whose hidden states store the memory slots. These hidden states are either fixed by the gates, or modified through a simple RNN-style update. The bulk of the reasoning is thus performed by these parallel recurrent models, rather than through a central controller. Moreover, instead of using a softmax, our model uses an independent gate for writing to each memory. ",
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+        "type": "text",
+        "text": "Our model is similar to a Memory Network and its variants (Weston et al., 2014; Sukhbaatar et al., 2015; Chandar et al., 2016; Miller et al., 2016) in the way it produces an output using a softmax over blocks of hidden states, and our encoding layer is inspired by techniques used in those works. However, Memory Networks explicitly store the entire input sequence in memory, and then sequentially update a controller’s hidden state via a softmax gating over the memories. In contrast, our model keeps a fixed number of blocks of hiddens as memories and updates each block with an independent gated RNN. The Dynamic Memory Network of (Xiong et al., 2016) also performs updates via a recurrent model, however it links memories to input tokens and updates them sequentially rather than in parallel. ",
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+        "type": "text",
+        "text": "The weight tying scheme and the parallel gated RNNs recall the gated graph network of (Li et al., 2015). If we interpret our work in that context, the “graph” is just a set of vertices with no edges; our gating mechanism is also somewhat different than the one they use. The CommNN model of (Sukhbaatar et al., 2016), the Interaction Network of (Battaglia et al., 2016), the Neural Physics Engine of (Chang et al., 2016) and the model of (Fragkiadaki et al., 2015) also use a set of parallel recurrent models with tied weights, but differ from our model in their use of inter-network communication and the lack of a gating mechanism. ",
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+    {
+        "type": "text",
+        "text": "Finally, there is another class of recent models that have a writeable memory arranged as (unbounded) stacks, linked lists or queues (Joulin & Mikolov, 2015; Grefenstette et al., 2015). Our model is different from these in that we use a key-value pair array instead of a stack, and in the experiments in this work, the array is of fixed size. ",
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+    {
+        "type": "table",
+        "img_path": "images/a97a3b50e9f13db800b878c8f6d0f0ad43ad12fdd24939d6c9ac62f0f6c43b94.jpg",
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+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>T=10</td><td>T=20</td><td>T=40</td></tr><tr><td>MemN2N</td><td>0.09</td><td>0.633</td><td>0.896</td></tr><tr><td>LSTM</td><td>0</td><td>0.157</td><td>0.226</td></tr><tr><td>EntNet</td><td>0</td><td>0</td><td>0</td></tr><tr><td colspan=\"4\">(a)</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/abd67377581d9363ea9de7574d71b6796d8bbf5999f27f725ea9a51eb945e674.jpg",
+        "image_caption": [
+            "Table 1: a) Error of different models on the World Model Task. b) Generalization of an EntNet trained up to $T = 2 0$ . All errors range from 0 to 1. "
+        ],
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+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we evaluate our model on three different datasets. Training details common to all experiments can be found in Appendix A. ",
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+        "type": "text",
+        "text": "5.1 SYNTHETIC WORLD MODEL TASK ",
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+        "text": "We first study our model’s properties on a toy task designed to measure the ability to keep a world model in memory. In this task two agents are initially placed randomly on an $1 0 \\times 1 0$ grid, and at each time step a randomly chosen agent either changes direction or moves ahead. After a certain number of time steps, the model is required to provide the locations of each of the agents, thus revealing its internal world model (details can be found in Appendix B). This task is challenging because the model must combine up to $T - 2$ supporting facts in order to answer the question correctly, and must also keep the locations of both agents in memory and update them at different times. ",
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+        "type": "text",
+        "text": "We compared the performance of a MemN2N, LSTM and EntNet. For the MemN2N, we set the number of hops equal to $T - 2$ and the embedding dimension to $d = 2 0$ . The EntNet had embedding dimension $d = 2 0$ and 5 memory slots, and the LSTM had 50 hidden units which resulted in it having significantly more parameters than the other two models. For each model, we repeated the experiment with 5 different initializations and reported the best performance. All models were trained with ADAM (Kingma & Ba, 2014) with initial learning rates set by grid search over $\\{ 0 . 1 , 0 . 0 1 , 0 . 0 0 1 \\}$ and divided by 2 every 10,000 updates. Table 1a shows the results. The MemN2N has the worst performance, which degrades quickly as the length of the sequence increases. The LSTM performs better, but still loses accuracy as the length of the sequence increases. In contrast, the EntNet is able to solve the task in all cases. ",
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+        "type": "text",
+        "text": "The ability to generalize to sequences longer than those seen during training is a desirable property, which suggests that the network has learned the dynamics of the world it is trying to model. It also means the model can be trained less expensively. To study this, we trained an EntNet on variable length sequences between 1 and 20, and evaluated it on different length sequences longer than 20. Results are shown in Table 1b. We see that the model is able to achieve good performance several times past its training horizon. ",
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+    {
+        "type": "text",
+        "text": "5.2 BABI TASKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We next evaluate our model on the bAbI tasks, which are a collection of 20 synthetic questionanswering datasets first introduced in (Weston et al., 2015) designed to test a wide variety of reasoning abilities. They have since become a benchmark for memory-augmented neural networks and most of the related methods described in Section 4 have been tested on them. Performance is measured using two metrics: the average error across all tasks, and the number of failed tasks (more than $5 \\%$ error). We used version 1.2 of the dataset with $1 0 \\mathrm { k }$ samples. 1 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training Details We used a similar training setup as (Sukhbaatar et al., 2015). All models were trained with ADAM using a learning rate of $\\eta = 0 . 0 1$ , which was divided by 2 every 25 epochs until 200 epochs were reached. Copying previous works (Sukhbaatar et al., 2015; Xiong et al., 2016), the capacity of the memory was limited to the most recent 70 sentences, except for task 3 which was limited to 130 sentences. Due to the high variance in model performance for some tasks, for each task we conducted 10 runs with different initializations and picked the best model based on performance on the validation set, as it has been done in previous work. In all experiments, our model had embedding dimension size $d = 1 0 0$ and 20 memory slots. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/75eefe81b95d04f88ef81a571cc515a0f5e81d5f04f8a64f282cffe6ad91493c.jpg",
+        "table_caption": [
+            "Table 2: Results on bAbI Tasks with 10k training samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>NTM</td><td>D-NTM</td><td>MemN2N</td><td>DNC</td><td>DMN+</td><td>EntNet</td></tr><tr><td></td><td>31.5</td><td>4.4</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1: 1 supporting fact 2: 2 supporting facts</td><td>54.5</td><td>27.5</td><td>0.3</td><td>0.4</td><td>0.3</td><td>0.1</td></tr><tr><td>3: 3 supporting facts</td><td>43.9</td><td>71.3</td><td>2.1</td><td>1.8</td><td>1.1</td><td>4.1</td></tr><tr><td>4: 2 argument relations</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>5:3 argument relations</td><td>0.8</td><td>1.7</td><td>0.8</td><td>0.8</td><td>0.5</td><td>0.3</td></tr><tr><td>6: yes/no questions</td><td>17.1</td><td>1.5</td><td>0.1</td><td>0</td><td>0</td><td>0.2</td></tr><tr><td>7: counting</td><td>17.8</td><td>6.0</td><td>2.0</td><td>0.6</td><td>2.4</td><td>0</td></tr><tr><td>8: lists/sets</td><td>13.8</td><td>1.7</td><td>0.9</td><td>0.3</td><td>0.0</td><td>0.5</td></tr><tr><td>9:simple negation</td><td>16.4</td><td>0.6</td><td>0.3</td><td>0.2</td><td>0.0</td><td>0.1</td></tr><tr><td>10:indefinite knowledge</td><td>16.6</td><td>19.8</td><td>0</td><td>0.2</td><td>0</td><td>0.6</td></tr><tr><td>11:basic coreference</td><td>15.2</td><td>0</td><td>0.0</td><td>0</td><td>0.0</td><td>0.3</td></tr><tr><td>12:conjunction</td><td>8.9</td><td>6.2</td><td>0</td><td>0</td><td>0.2</td><td>0</td></tr><tr><td>13:compound coreference</td><td>7.4</td><td>7.5</td><td>0</td><td>0</td><td>0</td><td>1.3</td></tr><tr><td>14: time reasoning</td><td>24.2</td><td>17.5</td><td>0.2</td><td>0.4</td><td>0.2</td><td>0</td></tr><tr><td>15:basic deduction</td><td>47.0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>16:basic induction</td><td>53.6</td><td>49.6</td><td>51.8</td><td>55.1</td><td>45.3</td><td>0.2</td></tr><tr><td>17: positional reasoning</td><td>25.5</td><td>1.2</td><td>18.6</td><td>12.0</td><td>4.2</td><td>0.5</td></tr><tr><td>18:size reasoning</td><td>2.2</td><td>0.2</td><td>5.3</td><td>0.8</td><td>2.1</td><td>0.3</td></tr><tr><td>19: path finding</td><td>4.3</td><td>39.5</td><td>2.3</td><td>3.9</td><td>0.0</td><td>2.3</td></tr><tr><td>20: agent&#x27;s motivation</td><td>1.5</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5% error): Mean Error:</td><td>16</td><td>9</td><td>3</td><td>2</td><td>1</td><td>0</td></tr><tr><td></td><td>20.1</td><td>12.8</td><td>4.2</td><td>3.8</td><td>2.8</td><td>0.5</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "In Table 2 we compare our model to various other state-of-the-art models in the literature: the larger MemN2N reported in the appendix of (Sukhbaatar et al., 2015), the Dynamic Memory Network of (Xiong et al., 2016), the Dynamic Neural Turing Machine (Gulcehre et al., 2016), the Neural Turing Machine (Graves et al., 2014) and the Differentiable Neural Computer (Graves et al., 2016). Our model is able to solve all the tasks, outperforming the other models in terms of both the number of solved tasks and the average error. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To analyze what kind of representations our model can learn, we conducted an additional experiment on Task 2 using a simple BoW sentence encoding and key vectors which were tied to entity embeddings. This was designed to make the model more interpretable, since the weight tying forces memory slots to encode information about specific entities. 2 After training, we ran the model over a story and computed the cosine distance between $\\phi ( H h _ { j } )$ and each row $r _ { i }$ of the decoder matrix $R$ . This gave us a score which measures the affinity between a given memory slot and each word in the vocabulary. Table 3 shows the nearest neighboring words for each memory slot (which itself corresponds to an entity). We see that the model has indeed stored locations of all of the objects and characters in its memory slots which reflect the final state of the story. In particular, it has the correct answer readily stored in the memory slot of the entity being inquired about (the milk). It also has correct location information about all other non-location entities stored in the appropriate memory slots. Note that it does not store useful or correct information in the memory slots corresponding to locations, most likely because this task does not contain questions about locations (such as “who is in the kitchen?”). ",
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+    {
+        "type": "table",
+        "img_path": "images/c4bae772b93ec0b2b98add9899e078fc8eb571a8f862a0f11e4c90f842370803.jpg",
+        "table_caption": [
+            "Table 3: On the left, the network’s final “world model” after reading the story on the right. First and second nearest neighbors from each memory slot are shown, along with their cosine distance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Key</td><td>1-NN</td><td>2-NN</td></tr><tr><td>football milk john mary sandra daniel</td><td>hallway (0.135) garden (0.111) kitchen (0.501) garden (0.442) hallway (0.394) hallway (0.689)</td><td>dropped (0.056) took (0.011) dropped (0.027) took (0.034) kitchen (0.121) to (0.076)</td></tr><tr><td>bedroom kitchen garden hallway</td><td>hallway (0.367) kitchen (0.483) garden (0.281) hallway (0.475)</td><td>dropped (0.075) daniel (0.029) where (0.026) left (0.060)</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_body": "<table><tr><td>Story</td></tr><tr><td>mary got the milk there john moved to the bedroom sandra went back to the kitchen mary travelled to the hallway john got the football there john went to the hallway john put down the football mary went to the garden john went to the kitchen sandra travelled to the hallway</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.3 CHILDREN’S BOOK TEST (CBT) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We next evaluated our model on the Children’s Book Test (Hill et al., 2016), which is a semantic language modeling (sentence completion) benchmark built from children’s books that are freely available from Project Gutenberg 3. Models are required to read 20 consecutive sentences from a given story and use this context to fill in a missing word from the 21st sentence. More specifically, each sample consists of a tuple $( S , q , C , a )$ where $S$ is the story consisting of 20 sentences, $Q$ is the 21st sentence with one word replaced by a special blank token, $C$ is a set of 10 candidate answers of the same type as the missing word (for example, common nouns or named entities), and $a$ is the true answer (which is always contained in $C$ ). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It was shown in (Hill et al., 2016) that methods with limited memory such as LSTMs perform well on more frequent, syntax based words such as prepositions and verbs, being similar to human performance, but poorly relative to humans on more semantically meaningful words such as named entities and common nouns. Therefore, most recent methods have been evaluated on the Named Entity and Common Noun subtasks, since they better test the ability of a model to make use of wider contextual information. ",
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+    {
+        "type": "text",
+        "text": "Training Details We adopted the same window memory approach used in (Hill et al., 2016), where each input corresponds to a window of text from $\\left\\{ w _ { \\left( i - b - 1 / 2 \\right) } . . . w _ { i } . . . w _ { \\left( i + \\left( b - 1 \\right) / 2 \\right) } \\right\\}$ centered at a candidate $w _ { i } \\in C$ . In our experiments we set $b = 5$ . All models were trained using standard stochastic gradient descent (SGD) with a fixed learning rate of 0.001. We used separate input encodings for the update and gating functions, and applied a dropout rate of 0.5 to the word embedding dimensions. Key embeddings were tied to the embeddings of the candidate words, resulting in 10 hidden blocks, one per member of $C$ . Due to the weight tying, we did not need a decoder matrix and used the distribution over candidates to directly produce a prediction, as described in Section 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We found that a simpler version of the model worked best, with $U = V = 0$ , $W = I$ and $\\phi$ equal to the identity. We also removed the normalization step in this simplified model, which we found to hurt performance. This can be explained by the fact that the maximum frequency baseline model in (Hill et al., 2016) has performance which is significantly higher than random, and including the normalization step hides this useful frequency-based information. ",
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+    {
+        "type": "text",
+        "text": "Results We draw a distinction between two setups: the single-pass setup, where the model must read the story and query in order and immediately produce an output, and the multi-pass setup, where the model can use the query to perform attention over the story. The first setup is more challenging because the model does not know beforehand which query it will be presented with, and must learn to retain information which is useful for a wide variety of potential queries. For this reason it can be viewed as a test of the model’s ability to construct a general-purpose representation of the current state of the story. The second setup leverages all available information, and allows the model to use knowledge of which question will be asked when it reads the story. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/ff752e2a1bacc2d493102edf92561463a588b6de2a39a761203ebf5ddf6fe2c8.jpg",
+        "table_caption": [
+            "Table 4: Accuracy on CBT test set. Single-pass models encode the document before seeing the query, multi-pass models have access to the query at read time. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td></td><td>Named Entities</td><td>Common Nouns</td></tr><tr><td rowspan=\"5\">Single Pass</td><td>Kneser-Ney Language Model + cache</td><td>0.439</td><td>0.577</td></tr><tr><td>LSTMs (context+ query)</td><td>0.418</td><td>0.560</td></tr><tr><td>Window LSTM</td><td>0.436</td><td>0.582</td></tr><tr><td>EntNet (general)</td><td>0.484</td><td>0.540</td></tr><tr><td>EntNet (simple)</td><td>0.616</td><td>0.588</td></tr><tr><td></td><td></td><td>0.493</td><td></td></tr><tr><td rowspan=\"7\">Multi Pass</td><td>MemNN</td><td></td><td>0.554</td></tr><tr><td>MemNN + self-sup.</td><td>0.666</td><td>0.630</td></tr><tr><td>Attention Sum Reader (Kadlec et al.,2016)</td><td>0.686</td><td>0.634</td></tr><tr><td>Gated-Attention Reader (Bhuwan Dhingra &amp; Salakhutdinov,2016)</td><td>0.690</td><td>0.639</td></tr><tr><td>EpiReader (Trischler et al.,2016)</td><td>0.697</td><td>0.674</td></tr><tr><td>AoAReader (Cui et al.,2016)</td><td>0.720</td><td>0.694</td></tr><tr><td>NSE Adaptive Computation (Munkhdalai &amp; Yu, 2016)</td><td>0.732</td><td>0.714</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Table 4, we show the performance of the general EntNet, the simplified EntNet, as well as other single-pass models taken from (Hill et al., 2016). The general EntNet performs better than the LSTMs and $n$ -gram model on the Named Entities Task, but lags behind on the Common Nouns task. The simplified EntNet outperforms all other single-pass models on both tasks, and also performs better than the Memory Network which does not use the self-supervision heuristic. However, there is still a performance gap when compared to more sophisticated machine comprehension models, many of which perform multiple layers of attention over the story using query knowledge. The fact that the simplified EntNet is able to obtain decent performance is encouraging since it indicates that the model is able to build an internal representation of the story which it can then use to answer a relatively diverse set of queries. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Two closely related challenges in artificial intelligence are designing models which can maintain an estimate of the state of a world with complex dynamics over long timescales, and models which can predict the forward evolution of the state of the world from partial observation. In this paper, we introduced the Recurrent Entity Network, a new model that makes a promising step towards the first goal. Our model is able to accurately track the world state while reading text stories, which enables it to set a new state-of-the-art on the bAbI tasks, the competitive benchmark of story understanding, by being the first model to solve them all. We also showed that our model is able to capture simple dynamics over long timescales, and is able to perform competitively on a real-world dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Although our model was able to solve all the bAbI tasks using $1 0 \\mathrm { k }$ training samples, we found that performance dropped considerably when using only 1k samples (see Appendix). Most recent work on the bAbI tasks has focused on the $1 0 \\mathrm { k }$ samples setting, and we would like to emphasize that solving them in the 1k samples setting remains an open problem which will require improving the sample efficiency of reasoning models, including ours. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Recent works have made some progress towards the second goal of forward modeling, for instance in capturing simple physics (Lerer et al., 2016), predicting future frames in video (Mathieu et al., 2015) or responses in dialog (Weston, 2016). Although we have only applied our model to tasks with textual inputs in this work, the architecture is general and future work should investigate how to combine the EntNet’s tracking abilities with such predictive models. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "REFERENCES   \nBattaglia, Peter W., Pascanu, Razvan, Lai, Matthew, Rezende, Danilo Jimenez, and Kavukcuoglu, Koray. Interaction networks for learning about objects, relations and physics. CoRR, abs/1612.00222, 2016. URL http://dblp.uni-trier.de/db/journals/corr/ corr1612.html#BattagliaPLRK16.   \nBhuwan Dhingra, Hanxiao Liu, William Cohen and Salakhutdinov, Ruslan. Gated-attention readers for text comprehension. CoRR, abs/1606.01549, 2016. URL http://arxiv.org/abs/ 1606.01549.   \nChandar, Sarath, Ahn, Sungjin, Larochelle, Hugo, Vincent, Pascal, Tesauro, Gerald, and Bengio, Yoshua. Hierarchical memory networks. arXiv preprint arXiv:1605.07427, 2016.   \nChang, Michael B., Ullman, Tomer, Torralba, Antonio, and Tenenbaum, Joshua B. A compositional object-based approach to learning physical dynamics. CoRR, abs/1612.00341, 2016. URL http: //arxiv.org/abs/1612.00341.   \nCho, Kyunghyun, van Merrienboer, Bart, Bahdanau, Dzmitry, and Bengio, Yoshua. On the properties of neural machine translation: Encoder-decoder approaches. In Proceedings of SSST@EMNLP 2014, Eighth Workshop on Syntax, Semantics and Structure in Statistical Translation, Doha, Qatar, 25 October 2014, pp. 103–111, 2014. URL http://aclweb.org/ anthology/W/W14/W14-4012.pdf.   \nCollobert, Ronan, Kavukcuoglu, Koray, and Farabet, Clment. Torch7: A matlab-like environment for machine learning, 2011.   \nCui, Yiming, Chen, Zhipeng, Wei, Si, Wang, Shijin, Liu, Ting, and Hu, Guoping. Attention-overattention neural networks for reading comprehension. CoRR, abs/1607.04423, 2016. URL http: //arxiv.org/abs/1607.04423.   \nFragkiadaki, Katerina, Agrawal, Pulkit, Levine, Sergey, and Malik, Jitendra. Learning visual predictive models of physics for playing billiards. CoRR, abs/1511.07404, 2015. URL http: //arxiv.org/abs/1511.07404.   \nGraves, Alex, Wayne, Greg, and Dnihelka, Ivo. Neural Turing Machines, September 2014. URL http://arxiv.org/abs/1410.5401.   \nGraves, Alex, Wayne, Greg, Reynolds, Malcolm, Harley, Tim, Danihelka, Ivo, Grabska-Barwinska, ´ Agnieszka, Colmenarejo, Sergio Gomez, Grefenstette, Edward, Ramalho, Tiago, Agapiou, John, ´ et al. Hybrid computing using a neural network with dynamic external memory. Nature, 2016.   \nGrefenstette, Edward, Hermann, Karl Moritz, Suleyman, Mustafa, and Blunsom, Phil. Learning to transduce with unbounded memory. In Advances in Neural Information Processing Systems, pp. 1828–1836, 2015.   \nGulcehre, Caglar, Chandar, Sarath, Cho, Kyunghyun, and Bengio, Yoshua. Dynamic neural turing machines with soft and hard addressing schemes. CoRR, abs/1607.00036, 2016. URL http: //arxiv.org/abs/1607.00036.   \nHe, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. CoRR, abs/1502.01852, 2015.   \nHill, Felix, Bordes, Antoine, Chopra, Sumit, and Weston, Jason. The goldilocks principle: Reading children’s books with explicit memory representations. In Proceedings of the International Conference on Learning Representations. 2016.   \nHochreiter, Sepp and Schmidhuber, Jurgen. Long short-term memory. ¨ Neural Comput., 9(8):1735– 1780, November 1997. ISSN 0899-7667. doi: 10.1162/neco.1997.9.8.1735. URL http://dx. doi.org/10.1162/neco.1997.9.8.1735. ",
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+        "text": "Lerer, Adam, Gross, Sam, and Fergus, Rob. Learning physical intuition of block towers by example. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pp. 430–438, 2016. URL http://jmlr.org/ proceedings/papers/v48/lerer16.html. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 10
+    },
+    {
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+        "text": "Li, Yujia, Tarlow, Daniel, Brockschmidt, Marc, and Zemel, Richard S. Gated graph sequence neural networks. CoRR, abs/1511.05493, 2015. URL http://arxiv.org/abs/1511.05493. ",
+        "bbox": [
+            171,
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+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mathieu, Michael, Couprie, Camille, and LeCun, Yann. Deep multi-scale video prediction beyond ¨ mean square error. CoRR, abs/1511.05440, 2015. URL http://arxiv.org/abs/1511. 05440. ",
+        "bbox": [
+            171,
+            363,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Miller, Alexander, Fisch, Adam, Dodge, Jesse, Karimi, Amir-Hossein, Bordes, Antoine, and Weston, Jason. Key-value memory networks for directly reading documents. arXiv preprint arXiv:1606.03126, 2016. ",
+        "bbox": [
+            173,
+            421,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Munkhdalai, Tsendsuren and Yu, Hong. Reasoning with memory augmented neural networks for language comprehension. CoRR, abs/1610.06454, 2016. URL https://arxiv.org/abs/ 1610.06454. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sukhbaatar, Sainbayar, szlam, arthur, Weston, Jason, and Fergus, Rob. End-to-end memory networks. In Cortes, C., Lawrence, N. D., Lee, D. D., Sugiyama, M., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 28, pp. 2440–2448. Curran Associates, Inc., 2015. URL http://papers.nips.cc/paper/5846-end-to-end-memory-networks. pdf. ",
+        "bbox": [
+            173,
+            536,
+            825,
+            608
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sukhbaatar, Sainbayar, Szlam, Arthur, and Fergus, Rob. Learning multiagent communication with backpropagation. CoRR, abs/1605.07736, 2016. URL http://arxiv.org/abs/1605. 07736. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Trischler, Adam, Ye, Zheng, Yuan, Xingdi, and Suleman, Kaheer. Natural language comprehension with the epireader. CoRR, abs/1606.02270, 2016. URL http://arxiv.org/abs/1606. 02270. ",
+        "bbox": [
+            173,
+            680,
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+            723
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Weston, Jason. Dialog-based language learning. CoRR, abs/1604.06045, 2016. URL http:// arxiv.org/abs/1604.06045. ",
+        "bbox": [
+            173,
+            738,
+            823,
+            767
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Weston, Jason, Chopra, Sumit, and Bordes, Antoine. Memory networks. CoRR, abs/1410.3916, 2014. URL http://arxiv.org/abs/1410.3916. ",
+        "bbox": [
+            173,
+            782,
+            823,
+            811
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Weston, Jason, Bordes, Antoine, Chopra, Sumit, and Mikolov, Tomas. Towards ai-complete question answering: A set of prerequisite toy tasks. CoRR, abs/1502.05698, 2015. URL http://arxiv.org/abs/1502.05698. ",
+        "bbox": [
+            174,
+            825,
+            823,
+            869
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xiong, Caiming, Merity, Stephen, and Socher, Richard. Dynamic memory networks for visual and textual question answering. In ICML, 2016. ",
+        "bbox": [
+            173,
+            883,
+            825,
+            912
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            374,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "All models were implemented using Torch (Collobert et al., 2011). In all experiments, we initialized our model by drawing weights from a Gaussian distribution with mean zero and standard deviation 0.1, except for the PReLU slopes and encoder weights which were initialized to 1. Note that the PReLU initialization is related to two of the heuristics used in (Sukhbaatar et al., 2015), namely starting training with a purely linear model, and adding non-linearities to half of the hidden units. Our initialization allows the model to choose when and how much to enter the non-linear regime. Initializing the encoder weights to 1 corresponds to beginning with a BoW encoding, which the model can then choose to modify. The initial values of the memory slots were initialized to the key values, which we found to help performance. Optimization was done with SGD or ADAM using minibatches of size 32, and gradients with norm greater than 40 were clipped to 40. A null symbol whose embedding was constrained to be zero was used to pad all sentences or windows to a fixed size. ",
+        "bbox": [
+            174,
+            133,
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+            300
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS OF WORLD MODEL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            320,
+            575,
+            337
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Two agents are initially placed at random on a $1 0 ~ \\times ~ 1 0$ grid with 100 distinct locations $\\{ ( 1 , 1 ) , \\bar { ( 1 , 2 ) } , . . . ( 9 , 1 0 ) , \\bar { ( 1 0 , 1 0 ) } \\}$ . At each time step an agent is chosen at random. There are two types of actions: the agent can face a given direction, or can move a number of steps ahead. Actions are sampled until a legal action is found by either choosing to change direction or move with equal probability. If they change direction, the direction is chosen between north, south, east and west with equal probability. If they move, the number of steps is randomly chosen between 1 and 5. A legal action is one which does not place the agent off the grid. Stories are given to the network in textual form, an example of which is below. The first action after each agent is placed on the grid is to face a given direction. Therefore, the maximum number of actions made by one agent is $T - 2$ . The network learns word embeddings for all words in the vocabulary such as locations, agent identifiers and actions. At question time, the model must predict the correct answer (which will always be a location) from all the tokens in the vocabulary. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "agent1 is at (2,8)   \nagent1 faces-N   \nagent2 is at (9,7)   \nagent2 faces-N   \nagent2 moves-2   \nagent2 faces-E   \nagent2 moves-1   \nagent1 moves-1   \nagent2 faces-S   \nagent2 moves-5   \nQ1: where is agent1 ?   \nQ2: where is agent2 ?   \nA1: (2,9)   \nA2: (10,4) ",
+        "bbox": [
+            173,
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+            719
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL RESULTS ON BABI TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            753,
+            539,
+            770
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide some additional experiments on the bAbI tasks, in order to better understand the influence of architecture, weight tying, and amount of training data. Table 5 shows results when a simple BoW encoding is used for the inputs. Here, the EntNet still performs better than a MemN2N which uses the same encoding scheme, indicating that the architecture has an important effect. Tying the key vectors to entities did not help, and hurt performance for some tasks. Table 6 shows results when using only 1k training samples. In this setting, the EntNet performs worse than the MemN2N. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f1092c74c368b93a5a276e1f17ca290271750843dbbddf39a12b7a09ec7dc7e7.jpg",
+        "table_caption": [
+            "Table 5: Error rates on bAbI Tasks with inputs are encoded using BoW. “Tied” refers to the case where key vectors are tied with entity embeddings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>MemN2N</td><td>EntNet-tied</td><td>EntNet</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>1:1 supporting fact</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2:2 supporting facts</td><td>0.6</td><td>3.0</td><td>1.2</td></tr><tr><td>3: 3 supporting facts</td><td>7</td><td>9.6</td><td>9.0</td></tr><tr><td>4: 2 argument relations</td><td>32.6</td><td>33.8</td><td>31.8</td></tr><tr><td>5:3 argument relations</td><td>10.2</td><td>1.7</td><td>3.5</td></tr><tr><td>6: yes/no questions</td><td>0.2</td><td>0</td><td>0 0.5</td></tr><tr><td>7: counting 8: lists/sets</td><td>10.6</td><td>0.5</td><td>0.3</td></tr><tr><td>9: simple negation</td><td>2.6 0.3</td><td>0.1</td><td>0</td></tr><tr><td>10: indefinite knowledge</td><td>0.5</td><td>0 0</td><td>0</td></tr><tr><td>11:basic coreference</td><td>0</td><td>0.3</td><td>0</td></tr><tr><td>12: conjunction</td><td>0</td><td>0</td><td>0</td></tr><tr><td>13: compound coreference</td><td>0</td><td>0.2</td><td>0.4</td></tr><tr><td>14: time reasoning</td><td>0.1</td><td>6.2</td><td>0.1</td></tr><tr><td>15:basic deduction</td><td>11.4</td><td>12.5</td><td>12.1</td></tr><tr><td>16:basic induction</td><td>52.9</td><td>46.5</td><td>0</td></tr><tr><td>17: positional reasoning</td><td>39.3</td><td>40.5</td><td>40.5</td></tr><tr><td>18: size reasoning</td><td>40.5</td><td>44.2</td><td>45.7</td></tr><tr><td>19: path finding</td><td>74.4</td><td>75.1</td><td>74.0</td></tr><tr><td>20: agent&#x27;s motivation</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5%):</td><td>9</td><td>8</td><td>6</td></tr><tr><td>Mean Error:</td><td>15.6</td><td>13.7</td><td>10.9</td></tr></table>",
+        "bbox": [
+            282,
+            160,
+            720,
+            553
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3aebaee9e9fc2affda58de20bfd4c53cb5223b29ded7b2a1b7d087452562caa.jpg",
+        "table_caption": [
+            "Table 6: Results on bAbI Tasks with 1k samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>MemN2N</td><td>EntNet</td></tr><tr><td></td><td></td><td></td></tr><tr><td>1: 1 supporting fact</td><td>0</td><td>0.7</td></tr><tr><td>2:2 supporting facts</td><td>8.3</td><td>56.4</td></tr><tr><td>3: 3 supporting facts</td><td>40.3</td><td>69.7</td></tr><tr><td>4:2 argument relations</td><td>2.8</td><td>1.4</td></tr><tr><td>5: 3 argument relations</td><td>13.1</td><td>4.6</td></tr><tr><td>6: yes/no questions</td><td>7.6</td><td>30.0</td></tr><tr><td>7: counting</td><td>17.3</td><td>22.3</td></tr><tr><td>8: lists/sets</td><td>10.0</td><td>19.2</td></tr><tr><td>9:simple negation</td><td>13.2</td><td>31.5</td></tr><tr><td>10:indefinite knowledge</td><td>15.1</td><td>15.6</td></tr><tr><td>11:basic coreference</td><td>0.9</td><td>8.0</td></tr><tr><td>12:conjunction</td><td>0.2</td><td>0.8</td></tr><tr><td>13: compound coreference</td><td>0.4</td><td>9.0</td></tr><tr><td>14: time reasoning</td><td>1.7</td><td>62.9</td></tr><tr><td>15: basic deduction</td><td>0</td><td>57.8</td></tr><tr><td>16:basic induction</td><td>1.3</td><td>53.2</td></tr><tr><td>17: positional reasoning</td><td>51.0</td><td>46.4</td></tr><tr><td>18:size reasoning</td><td>11.1</td><td>8.8 90.4</td></tr><tr><td>19: path finding 20:agent&#x27;s motivation</td><td>82.8 0</td><td>2.6</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Failed Tasks (&gt; 5%):</td><td>11</td><td>15</td></tr><tr><td>Mean Error:</td><td>13.9</td><td>29.6</td></tr></table>",
+        "bbox": [
+            331,
+            146,
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+            537
+        ],
+        "page_idx": 13
+    }
+]

parse/train/rkgyS0VFvr/rkgyS0VFvr.md

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+# DBA: DISTRIBUTED BACKDOOR ATTACKS AGAINST FEDERATED LEARNING
+
+Chulin Xie Zhejiang University chulinxie@zju.edu.cn
+
+Keli Huang Shanghai Jiao Tong University nick cooper@sjtu.edu.cn
+
+Pin-Yu Chen   
+IBM Research   
+pin-yu.chen@ibm.com   
+Bo Li   
+University of Illinois Urbana-Champaign   
+lbo@illinois.edu
+
+# ABSTRACT
+
+Backdoor attacks aim to manipulate a subset of training data by injecting adversarial triggers such that machine learning models trained on the tampered dataset will make arbitrarily (targeted) incorrect prediction on the testset with the same trigger embedded. While federated learning (FL) is capable of aggregating information provided by different parties for training a better model, its distributed learning methodology and inherently heterogeneous data distribution across parties may bring new vulnerabilities. In addition to recent centralized backdoor attacks on FL where each party embeds the same global trigger during training, we propose the distributed backdoor attack (DBA) — a novel threat assessment framework developed by fully exploiting the distributed nature of FL. DBA decomposes a global trigger pattern into separate local patterns and embed them into the training set of different adversarial parties respectively. Compared to standard centralized backdoors, we show that DBA is substantially more persistent and stealthy against FL on diverse datasets such as finance and image data. We conduct extensive experiments to show that the attack success rate of DBA is significantly higher than centralized backdoors under different settings. Moreover, we find that distributed attacks are indeed more insidious, as DBA can evade two state-of-the-art robust FL algorithms against centralized backdoors. We also provide explanations for the effectiveness of DBA via feature visual interpretation and feature importance ranking. To further explore the properties of DBA, we test the attack performance by varying different trigger factors, including local trigger variations (size, gap, and location), scaling factor in FL, data distribution, and poison ratio and interval. Our proposed DBA and thorough evaluation results shed lights on characterizing the robustness of FL.
+
+# 1 INTRODUCTION
+
+Federated learning (FL) has been recently proposed to address the problems for training machine learning models without direct access to diverse training data, especially for privacy-sensitive tasks (Smith et al., 2017; McMahan et al., 2017; Zhao et al., 2018). Utilizing local training data of participants (i.e., parties), FL helps train a shared global model with improved performance. There have been prominent applications and ever-growing trends in deploying FL in practice, such as loan status prediction, health situation assessment (e.g. potential cancer risk assessment), and next-word prediction while typing (Hard et al., 2018; Yang et al., 2018; 2019).
+
+Although FL is capable of aggregating dispersed (and often restricted) information provided by different parties to train a better model, its distributed learning methodology as well as inherently heterogeneous (i.e., non-i.i.d.) data distribution across different parties may unintentionally provide a venue to new attacks. In particular, the fact of limiting access to individual party’s data due to privacy concerns or regulation constraints may facilitate backdoor attacks on the shared model trained with FL. Backdoor attack is a type of data poisoning attacks that aim to manipulate a subset of training data such that machine learning models trained on the tampered dataset will be vulnerable to the test set with similar trigger embedded (Gu et al., 2019).
+
+Backdoor attacks on FL have been recently studied in (Bagdasaryan et al., 2018; Bhagoji et al., 2019). However, current attacks do not fully exploit the distributed learning methodology of FL, as they embed the same global trigger pattern to all adversarial parties. We call such attacking scheme centralized backdoor attack. Leveraging the power of FL in aggregating dispersed information from local parties to train a shared model, in this paper we propose distributed backdoor attack (DBA) against FL. Given the same global trigger pattern as the centralized attack, DBA decomposes it into local patterns and embed them to different adversarial parties respectively. A schematic comparison between the centralized and distributed backdoor attacks is illustrated in Fig.1.
+
+![](images/389150c8134ee6d20e27f0900b0b9e005e14815cab2b98c75bae3bff19d75db1.jpg)  
+(a) centralized backdoor attack (current setting)   
+(b) DBA: distributed backdoor attack (ours)   
+Figure 1: Overview of centralized and distributed backdoor attacks (DBA) on FL. The aggregator at round $t + 1$ combines information from local parties (benign and adversarial) in the previous round $t$ , and update the shared model $G ^ { t + 1 }$ . When implementing backdoor attacks, centralized attacker uses a global trigger while distributed attacker uses a local trigger which is part of the global one.
+
+Through extensive experiments on several financial and image datasets and in-depth analysis, we summarize our main contributions and findings as follows.
+
+• We propose a novel distributed backdoor attack strategy DBA on FL and show that DBA is more persistent and effective than centralized backdoor attack. Based on extensive experiments, we report a prominent phenomenon that although each adversarial party is only implanted with a local trigger pattern via DBA, their assembled pattern (i.e., global trigger) attains significantly better attack performance on the global model compared with the centralized attack. The results are consistent across datasets and under different attacking scenarios such as one-time (single-shot) and continuous (multiple-shot) poisoning settings. To the best of our knowledge, this paper is the first work studying distributed backdoor attacks.
+
+• When evaluating the robustness of two recent robust FL methods against centralized backdoor attack (Fung et al., 2018; Pillutla et al., 2019), we find that DBA is more effective and stealthy, as its local trigger pattern is more insidious and hence easier to bypass the robust aggregation rules.
+
+• We provide in-depth explanations for the effectiveness of DBA from different perspectives, including feature visual interpretation and feature importance ranking.
+
+• We perform comprehensive analysis and ablation studies on several trigger factors in DBA, including the size, gap, and location of local triggers, scaling effect in FL, poisoning interval, data poisoning ratio, and data distribution.
+
+# 2 DISTRIBUTED BACKDOOR ATTACK AGAINST FEDERATED LEARNING
+
+# 2.1 GENERAL FRAMEWORK
+
+bjective of. There are $\mathrm { F L }$ can be cast as a finite-sumarties individually processing ptimization: local models, $\begin{array} { r } { \operatorname* { m i n } _ { w \in R ^ { d } } [ F ( w ) \ : = \ } \end{array}$ $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { i } ( w ) ] } \end{array}$ $N$ $N$ with the local objective $f _ { i } : R ^ { d } \mapsto R$ based on a private dataset ${ D } _ { i } = \{ \{ x _ { j } ^ { i } , y _ { j } ^ { i } \} _ { j = 1 } ^ { a _ { i } } \}$ , where $a _ { i } = \left| D _ { i } \right|$ and $\{ x _ { j } ^ { i } , y _ { j } ^ { i } \}$ represents each data sample and its corresponding label. In supervised $\mathrm { F L }$ setting, each local function $f _ { i }$ is computed as $f _ { i } ( w _ { i } ) _ { \negmedspace \alpha } = l ( \{ x _ { j } ^ { i } , y _ { j } ^ { i } \} _ { j \in D _ { i } } , w _ { i } )$ where $l$ stands for a loss of prediction using the local parameters $w _ { i }$ . The goal of $\mathrm { F L }$ is to obtain a global model which can generalize well on test data $D _ { t e s t }$ after aggregating over the distributed training results from $N$ parties.
+
+Specifically, at round $t$ , the central server sends the current shared model $G ^ { t }$ to $n \in [ N ]$ selected parties, where $[ N ]$ denotes the integer set $\{ 1 , 2 , \ldots , N \}$ . The selected party $i$ locally computes the function $f _ { i }$ by running an optimization algorithm such as stochastic gradient descent (SGD) for $E$ local epochs with its own dataset $D _ { i }$ and learning rate $l _ { r }$ to obtain a new local model $L _ { i } ^ { t + 1 }$ . The local party then sends model update $L _ { i } ^ { t + 1 } - G ^ { t }$ back to the central server, who will averages over all updates with its own learning rate $\eta$ to generate a new global model $G ^ { t + 1 }$ :
+
+$$
+G ^ { t + 1 } = G ^ { t } + \frac { \eta } { n } \sum _ { i = 1 } ^ { n } ( L _ { i } ^ { t + 1 } - G ^ { t } )
+$$
+
+This aggregation process will be iterated until FL finds the final global model. Unless specified otherwise, we use $G ^ { t }$ $( L _ { i } ^ { t } )$ to denote the model parameters of the global (local) model at round $t$ .
+
+Attacker ability. Based on the Kerckhoffs’s theory (Shannon, 1949), we consider the strong attacker here who has full control of their local training process, such as backdoor data injection and updating local training hyperparameters including $E$ and $l _ { r }$ . This scenario is quite practical since each local dataset is usually owned by one of the local parties. However, attackers do not have the ability to influence the privilege of central server such as changing aggregation rules, nor tampering the training process and model updates of other parties.
+
+Objective of backdoor attack. Backdoor attack is designed to mislead the trained model to predict a target label $\tau$ on any input data that has an attacker-chosen pattern (i.e., a trigger) embedded. Instead of preventing the convergence in accuracy as Byzantine attacks (Blanchard et al., 2017), the purpose of backdoor attacks in $\bar { \mathrm { F L } }$ is to manipulate local models and simultaneously fit the main task and backdoor task, so that the global model would behave normally on untampered data samples while achieving high attack success rate on backdoored data samples. The adversarial objective1 for attacker $i$ in round $t$ with local datatset $D _ { i }$ and target label $\tau$ is:
+
+$$
+w _ { i } ^ { * } = \arg \operatorname* { m a x } _ { w _ { i } } ( \sum _ { j \in S _ { p o i } ^ { i } } P [ G ^ { t + 1 } ( R ( x _ { j } ^ { i } , \phi ) ) = \tau ] + \sum _ { j \in S _ { c l n } ^ { i } } P [ G ^ { t + 1 } ( x _ { j } ^ { i } ) = y _ { j } ^ { i } ] ) .
+$$
+
+Here, the poisoned dataset $S _ { p o i } ^ { i }$ and clean dataset $S _ { c l n } ^ { i }$ satisfy $S _ { p o i } ^ { i } \cap S _ { c l n } ^ { i } = \emptyset$ and $S _ { p o i } ^ { i } \cup S _ { c l n } ^ { i } = D _ { i }$ The function transforms clean data in any class into backdoored data that have an attacker-chosen trigger pattern using a set of parameters $\phi$ . For example, for image data, $\phi$ is factored into trigger location $T L$ , trigger size $T S$ and trigger gap $_ { T G }$ $\vec { x } ( \phi = \{ T S , T G , T L \} )$ , which are shown in Fig.2. The attacker can design his own trigger pattern and choose an optimal poison ratio $r$ to result in a better model parameter $w _ { i } ^ { * }$ , with which $\hat { G } ^ { t + 1 }$ can both assign the highest probability to target label $\tau$ for backdoored data $R ( x _ { j } ^ { i } , \phi )$ and the ground truth label ${ \bar { y } } _ { j { \prime } } ^ { i }$ for benign data $x _ { j ^ { \prime } } ^ { i }$ .
+
+# 2.2 DISTRIBUTED BACKDOOR ATTACK (DBA)
+
+We again use Fig.1 to illustrate our proposed DBA in details. Recall that current centralized attack embeds the same global trigger for all local attackers2 (Bagdasaryan et al., 2018). For example, the attacker in Fig.1.(a) embeds the training data with the selected patterns highlighted by 4 colors, which altogether constitutes a complete global pattern as the backdoor trigger.
+
+In our DBA, as illustrated in Fig.1.(b), all attackers only use parts of the global trigger to poison their local models, while the ultimate adversarial goal is still the same as centralized attack — using the global trigger to attack the shared model. For example, the attacker with the orange sign poisons a subset of his training data only using the trigger pattern located at the orange area. Similar attacking methodology applies to green, yellow and blue signs. We define each DBA attacker’s trigger as the local trigger and the combined whole trigger as the global trigger. For fair comparison, we keep similar amount of total injected triggers (e.g., modified pixels) for both centralized attack and DBA.
+
+In centralized attack, the attacker tries to solve the optimization problem in Eq.2 without any coordination and distributed processing. In contrast, DBA fully exploits the distributed learning and local data opacity in FL. Considering $M$ attackers in DBA with $M$ small local triggers. Each DBA attacker $m _ { i }$ independently performs the backdoor attack on their local models. This novel mechanism breaks a centralized attack formulation into $M$ distributed sub-attack problems aiming to solve3
+
+$$
+w _ { i } ^ { * } = \arg \operatorname* { m a x } _ { w _ { i } } ( \sum _ { j \in S _ { p o i } ^ { i } } P [ G ^ { t + 1 } ( R ( x _ { j } ^ { i } , \phi _ { i } ^ { * } ) ) = \tau ; \gamma ; I ] + \sum _ { j \in S _ { c l n } ^ { i } } P [ G ^ { t + 1 } ( x _ { j } ^ { i } ) = y _ { j } ^ { i } ] ) , \forall i \in [ M ] ,
+$$
+
+![](images/4b2645c29bfef8398b4636b984245949c481bf376e5653ed840fff3ab0985b51.jpg)  
+Figure 2: Trigger factors (size, gap and location) in backdoored images.
+
+![](images/2e012189bfb1bcf91648c97e40170cf259e3d87b7e4c633a14f74ca88451e944.jpg)  
+Figure 3: Trigger factor (feature importance ranking) in tabular data.
+
+where $\phi _ { i } ^ { * } = \{ \phi , O ( i ) \}$ is the geometric decomposing strategy for the local trigger pattern of attacker $m _ { i }$ and $O ( i )$ entails the trigger decomposition rule for $m _ { i }$ based on the global trigger $\phi$ . DBA attackers will poison with the poison round interval $I$ and use the scale factor $\gamma$ to manipulate their updates before submitting to the aggregator. We will explain the related trigger factors in the next subsection. We note that although none of the adversarial party has ever been poisoned by the global trigger under DBA, we find that DBA indeed outperforms centralized attack significantly when evaluated with the global trigger.
+
+# 2.3 FACTORS IN DISTRIBUTED BACKDOOR ATTACK
+
+With the framework of DBA on FL, there are multiple new factors to be explored. Here we introduce a set of trigger factors that we find to be critical. Fig.2 explains the location, size and gap attribute of triggers in image dataset. For simplicity, we set all of our local triggers to the same rectangle shape4. Fig.3 explains our trigger attribute of ranked feature importance in tabular data (e.g., the loan dataset).
+
+Trigger Size $T S$ : the number of pixel columns (i.e., the width) of a local distributed trigger. Trigger Gap $_ { T G }$ : the distance of the $G a p _ { x }$ and $G a p _ { y }$ , which represent the distance between the left and right, as well as the top and bottom local trigger, respectively.
+
+Trigger Location $T L$ : $( S h i f t _ { x } , S h i f t _ { y } )$ is the offset of the trigger pattern from the top left pixel. Scale $\gamma$ : the scaling parameter $\gamma = \eta / N$ defined in (Bagdasaryan et al., 2018) is used by the attacker to scale up the malicious model weights.5 For instance, assume the ith malicious local model is $X$ . The new local model $L _ { i } ^ { t + 1 }$ that will be submitted is calculated as $L _ { i } ^ { t + 1 } = \gamma ( X - G ^ { t } ) + G ^ { t }$ .
+
+Poison Ratio $r$ : the ratio controls the fraction of backdoored samples added per training batch. Note that larger $r$ should be preferable when attacking intuitively, and there is a tradeoff between clean data accuracy and attack success rate, but too large $r$ would also hurt the attack effectiveness once the model becomes useless.
+
+Poison Interval $I$ : the round intervals between two poison steps. For example, $I = 0$ means all the local triggers are embedded within one round, while $I = 1$ means the local triggers are embedded in consecutive rounds.
+
+Data Distribution: FL often presumes non-i.i.d. data distribution across parties. Here, we use a Dirichlet distribution (Minka, 2000) with different hyperparameter $\alpha$ to generate different data distribution following the setups in (Bagdasaryan et al., 2018).
+
+# 3 EXPERIMENTS
+
+# 3.1 DATASETS AND EXPERIMENT SETUP
+
+DBA is evaluated on four classification datasets with non-i.i.d. data distributions: Lending Club Loan Data(LOAN)(Kan, 2019), MNIST, CIFAR-10 and Tiny-imagenet. The data description and parameter setups are summarized in Tb.1. We refer the readers to Appendix A.1 for more details.
+
+Following the standard setup, we use SGD and trains for $E$ local epochs with local learning rate $l _ { r }$ and batch size 64. A shared global model is trained by all participants, 10 of them are selected in each round for aggregation. The local and global triggers used are summarized in Appendix A.1.
+
+Table 1: Dataset description and parameters   
+
+<table><tr><td>Dataset</td><td>Classes</td><td>Examples per class</td><td>Features</td><td>Model used</td><td>Benignlr/E</td><td>Poison lr/El</td><td>Poison ratio r</td></tr><tr><td>LOAN</td><td>9</td><td>see Tb.3 in Appendix</td><td>91</td><td>3 fc</td><td>0.001/1</td><td>0.0005 /5(multi-shot) or 10(single-shot)</td><td>10/64</td></tr><tr><td>MNIST</td><td>10</td><td>6000</td><td>784</td><td>2 conv and 2 fc</td><td>0.1/1</td><td>0.05/10</td><td>20/64</td></tr><tr><td>CIFAR</td><td>10</td><td>5000</td><td>1024</td><td>lightweight Resnet-18</td><td>0.1/2</td><td>0.05/6</td><td>5/64</td></tr><tr><td>Tiny-imagenet</td><td>200</td><td>500</td><td>4096</td><td>Resnet-18(He et al.,2016)</td><td>0.001/2</td><td>0.001/5(multi-shot) or 10(single-shot)</td><td>20/64</td></tr></table>
+
+4Some factor definitions may not apply to non-image data, which will be clarified accordingly.   
+5In our implementation, every distributed attacker uses the same $\gamma$ .
+
+Following the attack analysis in (Bagdasaryan et al., 2018), we evaluate multiple-shot attack (Attack A-M) and single-shot attack (Attack A-S) two attack scenarios, which are called naive approach and model replacement respectively in the original paper.
+
+• Attack A-M means the attackers are selected in multiple rounds and the accumulated malicious updates are necessary for a successful attack; otherwise the backdoor would be weakened by benign updates and soon forgotten by the global model. In order to quickly observe the difference between centralized and distributed attacks and control the effect of random party selection, we perform a complete attack in every round, that is, all DBA attackers or centralized attackers are consistently selected. Benign participants are randomly selected to form a total of 10 participants.
+
+• Attack A-S means that every DBA attacker or the centralized attacker only needs one single shot to successfully embed its backdoor trigger. To achieve that, the attacker performs scaling in their malicious updates to overpower other benign updates and ensure that the backdoor survives the aggregation step. For fair comparison, DBA and centralized attack finish a complete backdoor in the same round. Take MNIST as an example, DBA attackers separately embed their local triggers in round 12, 14, 16, 18 for local triggers 1 to 4, while the centralized attacker implants its global trigger in round 18. Benign participants are randomly selected to form a total of 10 participants.
+
+These two scenarios reveal different aspects of DBA and centralized backdoor attacks when the global model is triggered by local and global triggers. Attack A-M studies how easy the backdoor is successfully injected while Attack A-S studies how fast the backdoor effect diminishes.
+
+In our experiments, we evaluate the attack success rates of DBA and centralized attacks using the same global trigger. For fair comparison, we make sure the total number of backdoor pixels of DBA attackers is close to and even less than that of the centralized attacker (it is hard to control them to be the same due to data sampling with certain distribution). The ratio of the global trigger of DBA pixels to the centralized is 0.992 for LOAN, 0.964 for MNIST, 0.990 for CIFAR and 0.991 for Tiny-imagenet. Moreover, in order to avoid the influence of the original label when testing attack success rate, we remove the test data whose true label equals to the backdoor target label. In three image datasets, we begin to attack when the main accuracy of global model converges, which is round 10 for MNIST, 200 for CIFAR, 20 for Tiny-imagenet in Attack A-M. The reason is provided in Appendix.A.2. The global learning rate $\eta$ in Attack A-M is 0.1 for CIFAR, 1 for others and in Attack A-S is 0.1 for all datasets.
+
+![](images/5cf77c5b31332a66236970f5220aa4f5abd3f1bd2b0badd97120c2faadedcb44.jpg)  
+Attack A-M:   
+Figure 4: Attack A-M and A-S. DBA is more effective and persistent than centralized attack.
+
+In Attack A-M, the attack success rate of DBA is always higher than centralized attack in all cases as shown in Fig.4. DBA also converges faster and even yields a higher attack success rate in MNIST. Under DBA, we find a prominent phenomenon that the attack success rate of the global trigger is higher than any local trigger even if the global trigger never actually appears in any local training dataset. Moreover, the global trigger converges faster in attack performance than local triggers. Centralized attacker embeds the whole pattern so its attack success rate of any local triggers is low. Due to the continuous poisoning, the attack rate on local triggers still increases for LOAN but this phenomenon does not appear in MNIST and Tiny-imagenet, which indicates that the success of global trigger does not require the same success for local triggers. The results also suggest that DBA can lead to high attack success rate for the global trigger even when some of its local triggers only attain low attack success rates. This finding is unique for DBA and also implies the inefficiency of centralized attack on FL.
+
+In Attack A-S, DBA and centralized attack both reach a high attack success rate after performing a complete backdoor in all datasets with a scale factor $\gamma = 1 0 0$ as shown in Fig.4. In the consecutive rounds, the backdoor injected into the global model is weakened by benign updates so the attack success rate gradually decreases. There is an exception that centralized attack in CIFAR suffers from the initial drop and then rises slowly, which is caused by the high local learning rate of benign participants and is also observed in (Bagdasaryan et al., 2018). We also find that the attack success rate of centralized attack in local triggers and the global trigger drops faster than that of DBA, which shows that DBA yields a more persistent attack. For example, in MNIST and after 50 rounds, DBA remains $89 \%$ attack success rate while centralized attack only gets $21 \%$ . Although DBA performs data poisoning only using local triggers, the results show that its global trigger lasts longer than any local triggers, which suggests DBA can make the global trigger more resilient to benign updates.
+
+# 3.3 THE ROBUSTNESS OF DISTRIBUTED ATTACK
+
+RFA (Pillutla et al., 2019) and FoolsGold (Fung et al., 2018) are two recently proposed robust FL aggregation algorithms based on distance or similarity metrics, and in particular RFA is claimed to be able to detect more nuanced outliers which goes beyond the worst-case of the Byzantine setting (Blanchard et al., 2017). In addition, as Attack A-S is more easily detected due to the scaling operation (Pillutla et al., 2019), we will focus on evaluating the attack effectiveness of DBA and centralized backdoor attacks against both RFA and FoolsGold under Attack A-M setting.
+
+Distributed Attack against Robust Aggregation Defence. RFA aggregates model parameters for updates and appears robust to outliers by replacing the weighted arithmetic mean in the aggregation step with an approximate geometric median. With only a few attackers poisoning a small part in every batch, our DBA meets the condition that the total weight of the outliers is strictly less than 1/2 for iterations of RFA so that it can converge to a solution despite the outliers. The maximum iteration of RFA is set to be 10 while in fact it converges rapidly, which can give a high-quality solution within about 4 iterations. Fig.5 shows the attack performance of DBA and centralized attack under RFA. For Tiny-imagenet, the centralized attack totally fails at least 80 rounds but the DBA attackers with lower distances and higher aggregation weights can perform a successful backdoor attack. For MNIST and CIFAR, the attack success rate of DBA is much higher and the convergence speed is much faster. For LOAN, centralized backdoor attack takes more than 20 rounds to converge than DBA. To explain the effectiveness of DBA, we calculate the Euclidean norm between attacker’s model parameter updates and the final geometric median as a distance metric. As shown in Tb.2 in Appendix, the malicious updates submitted by DBA attackers have lower distances than that of the centralized attacker’s updates in all datasets, which help them to better bypass the defense.
+
+![](images/4e2d5cde4264831d193182a5a2d5eef12424cfbeea65623e5c0e1917b7f30aef.jpg)  
+Figure 5: Attack effectiveness comparison on two robust RL methods: RFA and FoolsGold
+
+Distributed Attack against Mitigating Sybils Defence. FoolsGold reduces aggregation weights of participating parties that repeatedly contribute similar gradient updates while retaining the weights of parities that provide different gradient updates (Fung et al., 2018). Fig.5 shows that DBA also outperforms centralized attack under FoolsGold. In three image datasets, the attack success rate of DBA is notably higher while converging faster. DBA in MNIST reaches $9 1 . 5 5 \%$ in round 30 when centralized attack fails with only $2 . 9 1 \%$ attack success rate. For LOAN, which are trained with a simple network, FoolsGolds cannot distinguish the difference between the malicious and clean updates and assigns high aggregation weights for attackers, leading to a fast backdoor success. To explain the effectiveness of DBA, we report FoolsGold’s weights on adversarial parties in Tb.2 in Appendix. Comparing to centralized attack, although FoolsGold assigns smaller aggregation weights to DBA attacker due to their similarity of backdoor target label, DBA is still more successful. This is because the sum of weights of distributed attackers could be larger than centralized attacker.
+
+3.4 EXPLANATION VIA FEATURE VISUALIZATION AND FEATURE IMPORTANCE
+
+Feature importance can be calculated by various classification tools or visually interpreted by classspecific activation maps. For example, in LOAN we show that the top features identified by different classifiers are quite consistent (see Tb.4 in Appendix). Here we use Grad-CAM (Selvaraju et al., 2017) and Soft Decision Tree (Frosst & Hinton, 2017) to provide explanations for DBA. More details about Soft Decision Tree trained on our datasets are discussed in Appendix A.7.
+
+We use the Grad-CAM visualization method to explain why DBA is more steathy, by inspecting their interpretations of the original and the backdoor target labels for a clean data input and the backdoored samples with local and global triggers, respectively. Fig.6 shows the Grad-CAM results of a hand-written digit ‘4’. We find that each locally triggered image alone is a weak attack as none of them can change the prediction (no attention on the top left corner where the trigger is embedded). However, when assembled together as a global trigger, the backdoored image is classified as $\bullet _ { 2 } \cdot$ (the target label), and we can clearly see the attention is dragged to the trigger location. The fact that Grad-CAM results in most of locally triggered images are similar to the clean image, demonstrates the stealthy nature of DBA.
+
+![](images/6ca47f551ed799ea0e8e9c0c0ca1d0f7f11cbe3bd5810022312a2c7f7de1ebac.jpg)  
+Figure 6: Decision visualization of poisoned digit 4 with target 2 on a DBA poisoned model
+
+![](images/8aa1fc89e9b86590934a374363401ed59eaaf9d1738b9dd558e1cfb3259d4d42.jpg)  
+Figure 7: Feature importance of LOAN learned from its soft decision tree
+
+Using the soft decision tree of MNIST as another example, we find that the trigger area after poisoning indeed becomes much more significant for decision making in the corresponding soft decision tree, as shown in Fig.22 in Appendix.A.7. Similar conclusion is found in LOAN. We sort the absolute value of filter in the top node of a clean model to obtain the rank of 91 features (lower rank is more important) and then calculate their importance as $( 1 { \mathrm { - r a n k } } / 9 1 ) ^ { \ast } 1 0 0$ . Six insignificant features and six significant features are separately chosen to run DBA. The results in Fig.7 show that based on the soft decision tree, the insignificant features become highly important for prediction after poisoning.
+
+# 4 ANALYSIS OF TRIGGER FACTORS IN DISTRIBUTED BACKDOOR ATTACK
+
+Here we study the DBA trigger factors introduced in Sec.2.3 under Attack A-S, unless specified otherwise. We only change one factor in each experiment and keep other factors the same as in Sec.3.1. In Attack A-S, DBA-ASR shows the attack success rate while Main-Acc denotes the accuracy of the global model when the last distributed local trigger is embedded. DBA-ASR-t, which reveals the persistence, is the attack success rate of $t$ rounds after a complete DBA is performed. Main-Acc-t is the main accuracy after $t$ rounds. Note that in general we expect a small decrease for main task accuracy right after the DBA but will finally get back to normal after a few rounds of training.6
+
+# 4.1 EFFECTS OF SCALE
+
+• Enlarging scale factor increases both DBA-ASR and DBA-ASR-t, and narrows the gap between them. For CIFAR, although the DBA-ASR reaches over $90 \%$ and barely changes once $\gamma$ is bigger than 40, larger $\gamma$ still have more positive impact on DBA-ASR-t.
+
+• For our four datasets, the more complex the model architecture (in Tb.1), the more obvious the decline in the main accuracy as $\gamma$ increases, because the scaling undermines more model parameters in complex neural network. The main accuracy of LOAN doesn’t drop because of simple model, while the main accuracy of Tiny-imagenet in attacking round even drops to $2 . 7 5 \%$ when $\gamma = 1 1 0$ .
+
+• Larger scale factor alleviates the averaging impacts of central server for DBA, which leads to a more influential and resistant attack performance, but also cause the main accuracy of global model to descend in the attacking round for three image datasets. In addition, using large scale factor results in an anomalous update that is too different from other benign updates and is easy to detect based on the magnitude of the parameters. Therefore, there is a trade-off in choosing the scale factor.
+
+![](images/cff63b7d0281086010a5548a4ce3ac7f228cdab49b480435fb733937be8d0004.jpg)  
+Figure 9: Effects of Trigger Location on Attack Success Rate and Model Accuracy EFFECTS OF TRIGGER LOCATION
+
+For three images datasets, we move the global trigger pattern from the left upper corner to the center, then to the right lower corner. The dotted line in Fig.9 means that the trigger reaches the right boundary and starts to move along the right edges. The implementation details are in Appendix.A.9.
+
+• We observe a U-shape curve between $T L$ and DBA-ASR (in MNIST) / DBA-ASR-t (in Tinyimagenet and MNIST). This is because the middle part in images usually contains the main object. DBA in such areas is harder to succeed and will be faster forgotten because these pixels are fundamental to the main accuracy. This finding is apparent in MNIST, where the main accuracy after 40 rounds only remains $1 . 4 5 \%$ in center $T L = 9$ ) while has $9 1 . 5 7 \%$ in left upper corner $T L = 0$ ).
+
+• Similar finding can be found in LOAN as shown in Fig.9.(a). DBA using low-importance features has higher success rate in attacking round and subsequent rounds. The low-importance trigger achieves $8 5 . 7 2 \%$ DBA-ASR after 20 rounds while the high-importance trigger is $0 \%$ .
+
+# 4.3 EFFECTS OF TRIGGER GAP
+
+• In the case of four local trigger patterns located in the four corners of an image, corresponding to the maximum trigger gap in Fig.10, the DBA-ASR and DBA-ASR-t are both low in image datasets. Such failure might be caused by the local convolution operations and large distance between local triggers so that the global model cannot recognize the global trigger.
+
+• The curve of DBA-ASR and DBA-ASR-t in Fig.10.(a) has a significant drop in the middle. This happens when the right lower local trigger covers the center areas in MNIST images. Similar observations can be explained based on Fig.9.(b)(d).
+
+• Using zero trigger gap in CIFAR and Tiny-imagenet, DBA still succeeds but we find the backdoor will be forgotten faster. We suggest using non-zero trigger gap when implementing DBA.
+
+# 4.4 EFFECTS OF TRIGGER SIZE
+
+• In image datasets, larger trigger size gives higher DBA-ASR and DBA-ASR-t. Nevertheless, they are stable once $T S$ becomes large enough, suggesting little gain in using over-sized triggers.
+
+• For MNIST, DBA-ASR is low when $T S = 1$ . This is because each local trigger is too small to be recognized in global model. In the same setting, the centralized attack which uses the global pattern with 4 pixels also isn’t very successful and its attack success rate soon decreases below $10 \%$ within 4 rounds. This reflects that under Attack A-S, backdoor attacks with too small trigger are ineffective.
+
+![](images/ef5613ca1c13dfc61b0a41ef850a4275386a31cd03695c66b806ab59691716d3.jpg)  
+Figure 11: Effects of Local Trigger Size on Attack Success Rate and Model Accuracy
+
+4.5 EFFECTS OF POISON INTERVAL
+
+• The attack performance is poor when all distributed attackers submit the scaled updates at the same round $[ I = 0 ]$ ) in all datasets because the scaling effect is too strong, vastly changing the parameter in the global model and causes it to fail in main accuracy. It’s also ineffective if the poison interval is too long because the early embemed triggers may be totally forgotten.
+
+• The peaks in Fig.12.(a)(b) show that there exists an optimal poison round interval for LOAN and MNIST. DBA attackers can wait until the global model converges and then embeds the next local trigger to maximize backdoor performance, which is a competitive advantage over centralized attack.
+
+• In CIFAR and Tiny-imagenet, varying the interval from 1 up to 50 does not lead to remarkable changes in DBA-ASR and DBA-ASR-t, which manifests that the local trigger effect can last long and contribute to the attack performance of global trigger. From this aspect, distributed attack is extraordinarily robust to RL and should be considered as a more serious threat.
+
+![](images/ba289e7541a7de623089cf059892024825937a2066d343f57477a83b7ed50047.jpg)  
+Figure 13: Effects of Poison Ratio on Attack Success Rate and Model Accuracy
+
+# 4.6 EFFECTS OF POISON RATIO
+
+In our experiments, the training batch size is 64. As the X-axis variable (# of poisoned samples) in Fig.13 increases from 1, DBA-ASR and DBA-ASR-t first increase and then drop. It’s intuitive that more poisoned data can lead to a better backdoor performance. However, a too large poison ratio means that the attacker scales up the weight of a local model of low accuracy, which leads to the failure of global model in the main task. In the case of poisoning full batch, after DBA, the global model in CIFAR and Tiny-imagenet trains the main task all over again, whose main accuracy is normal after 90 and 40 rounds, respectively. But in MNIST it is reduced to an overfitted model that predicts the target label for any input, so the attack success rate is always $100 \%$ while the main accuracy is about $10 \%$ in the subsequent rounds. Therefore, it’s better for DBA to remain stealthy in its local training by using a reasonable poison ratio that also maintains accuracy on clean data.
+
+# 4.7 EFFECTS OF DATA DISTRIBUTION
+
+Under various data distributions, DBA-ASR is stable, indicating the practicability and robustness of DBA. See more details in Appendix.A.10.
+
+# 5 RELATED WORK
+
+Federated Learning. McMahan et al. (2017) first introduced federated learning (FL) to solve the distributed machine learning problem. Since the training data is never shared with the server (aggregator), FL is in favor of machine learning with privacy and regulation constraints. In this paper, we discuss and analyze our experiments in standard FL settings performed in synchronous update rounds. Advanced FL for improving communication efficacy by compressing updates using random rotations and quantization has been recently studied in Konecnˇ y et al. (2016). \`
+
+Backdoor Attack on Federated Learning. Bagdasaryan et al. (2018) proposed a model-poisoning approach on FL which replaced the global model with a malicious local model by scaling up the attacker’s updates. Bhagoji et al. (2019) considered the case of one malicious attacker aiming to achieve both global model convergence and targeted poisoning attack, by boosting the malicious updates. They proposed two strategies, alternating minimization and estimating other benign updates, to evade the defences under weighted and non-weighted averaging for aggregation. We note that these works only consider centralized backdoor attack on FL.
+
+Robust Federated Learning. Robust FL aims to train FL models while mitigating certain attack threats. Fung et al. (2018) proposed a novel defense based on the party updating diversity without limitation on the number of adversarial parties. It adds up historical updating vectors and calculate the cosine similarity among all participants to assign global learning rate for each party. Similar updating vectors will obtain lower learning rates and therefore the global model can be prevented from both label-flipping and centralized backdoor attacks. Pillutla et al. (2019) proposed a robust aggregation approach by replacing the weighted arithmetic mean with an approximate geometric median, so as to minimize the impacts of “outlier” updates.
+
+# 6 CONCLUSIONS
+
+Through extensive experiments on diverse datasets including LOAN and three image datasets in different settings, we show that in standard FL our proposed DBA is more persistent and effective than centralized backdoor attack: DBA achieves higher attack success rate, faster convergence and better resiliency in single-shot and multiple-shot attack scenarios. We also demonstrate that DBA is more stealthy and can successfully evade two robust FL approaches. The effectiveness of DBA is explained using feature visual interpretation for inspecting its role in aggregation. We also perform an in-depth analysis on the important factors that are unique to DBA to explore its properties and limitations. Our results suggest DBA is a new and more powerful attack on FL than current backdoor attacks. Our analysis and findings can provide new threat assessment tools and novel insights for evaluating the adversarial robustness of FL.
+
+# ACKNOWLEDGEMENTS
+
+This work was partly supported by IBM-ILLINOIS Center for Cognitive Computing Systems Research (C3SR) – a research collaboration as part of the IBM AI Horizons Network.
+
+# REFERENCES
+
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+
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+Thomas Minka. Estimating a dirichlet distribution, 2000.
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+Krishna Pillutla, Sham M. Kakade, and Zaid Harchaoui. Robust Aggregation for Federated Learning. arXiv preprint, 2019.
+
+Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 618–626, 2017.
+
+Claude E Shannon. Communication theory of secrecy systems. Bell system technical journal, 28(4): 656–715, 1949.
+
+Virginia Smith, Chao-Kai Chiang, Maziar Sanjabi, and Ameet S Talwalkar. Federated multi-task learning. In Advances in Neural Information Processing Systems, pp. 4424–4434, 2017.
+
+Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong. Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):12, 2019.
+
+Timothy Yang, Galen Andrew, Hubert Eichner, Haicheng Sun, Wei Li, Nicholas Kong, Daniel Ramage, and Franc¸oise Beaufays. Applied federated learning: Improving google keyboard query suggestions. arXiv preprint arXiv:1812.02903, 2018.
+
+Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra. Federated learning with non-iid data. arXiv preprint arXiv:1806.00582, 2018.
+
+# A APPENDIX
+
+# A.1 DETAILS ON DATASETS AND EXPERIMENT SETUP
+
+The financial dataset LOAN contains the current loan status (Current, Late, Fully Paid, etc.) and latest payment information, which can be used for loan status prediction. It consists of 1,808,534 data samples and we divide them by 51 US states, each of whom represents a participant in FL. $80 \%$ of data samples are used for training and the rest is for testing. In the three image datasets, a Dirichlet distribution is used to divide training images for 100 parties. The distribution hyperparameter is 0.5 for MNIST and CIFAR and 0.01 for Tiny-imagenet.
+
+Every party uses SGD as optimizer and trains for $E$ local epochs with local learning rate $l _ { r }$ (see Tb.1) and a batch size of 64. A shared global model is trained by all participants, 10 of whom are selected in each round to submit locally computed SGD updates for aggregation.
+
+For the pixel-pattern backdoor, we assign white color to chosen pixels and swap the label of any sample with such triggers into the target label, which is “digit $2 ^ { \circ }$ in MNIST, “bird” in CIFAR and “bullfrog” in Tiny-imagenet. Except in Section 4 where we analyze the trigger factor effect, in other sections the trigger factors are set to be $\phi = \{ 4 , 2 , 0 \}$ for MNIST; $\phi = \left\{ 6 , 3 , 0 \right\}$ for CIFAR; $\phi = \{ 1 0 , 2 , 0 \}$ for Tiny-imagenet with 4 DBA attackers. Because the image size in tiny-imagenet are larger than cifar and mnist, we set the row number of the local trigger to 2 in Tiny-imagenet while it is 1 in other image datasets.
+
+Similarly, for the preprocessed7 LOAN dataset, six features8 which are the low importance features in Fig.7 are chosen and split by 3 DBA attackers, each of whom manipulates two features as a local trigger. They assign local trigger features with new values9 that are slightly larger than their maximum values, and swap label to ”Does not meet the credit policy. Status:Fully Paid”.
+
+Every attacker’s batch is mixed with correctly labeled data and such backdoored data with poison ratio $r$ (see Tb.1). Attackers have their own local poison $l _ { r }$ and poison $E$ (see Tb.1) to maximize their backdoor performance and remain stealthy.
+
+# A.2 BETTER TO ATTACK LATE
+
+In Attack A-M, we found that if DBA poisons from scratch, the main accuracy was low and hard to converge. Therefore in three image datasets, we begin to attack when the main accuracy of global converges, which is round 10 for MNIST, 200 for CIFAR, 20 for Tiny-imagenet. As mentioned in (Bagdasaryan et al., 2018), it’s also better to attack late in Attack A-S because when the global model is converging, the updates from benign clients contain less commonly shared patterns but more individual features, which are more likely to be canceled out when aggregating and thus having less impact on the backdoor.
+
+# A.3 DBA ON IRREGULAR SHAPE TRIGGERS
+
+To evaluate DBA on irregular shape triggers, we decomposed the logo ‘ICLR’ into ‘I’, ‘C’, ‘L’, ‘R’ as local triggers on three image datasets and we decomposed the physical pattern glasses (Chen et al., 2017) into four parts as the examples shown in Fig. 14.
+
+The results under Attack A-M are shown in Fig. 15 and Fig. 16. DBA is always more effective than centralized attack, which is similar to the results of regular shape triggers in Fig. 4. This conclusion also holds for glass patterns with different colors as shown in Fig. 17.
+
+# A.4 MORE ANALYSIS ON ATTACK A-S SETTINGS FOR CENTRALIZED ATTACK
+
+In our experiment setup we assumed that there are $f$ distributed attackers and 1 centralized attacker. To further evaluate Attack A-S, we conduct centralized attacks with the same number of times as DBA, but each update includes $1 / f$ number of poisoning samples, so that the total number of poisoning samples included to compute the gradient update still stay the same. There are two ways to achieve $\bar { 1 } / f$ number of poisoning samples in each update for centralized attack and we evaluate both as following.
+
+![](images/856edf0fb09d3b1e083adcb059e430fb575fe9aa5ee01c81ecd2bedc53a3c48c.jpg)  
+Figure 14: Examples of irregular shape triggers in image datasets
+
+![](images/404ce27ad4fc2db411736e9445e48e47de2827f3099816f7baa27cbafe80aba0.jpg)  
+Figure 15: Attack A-M for irregular pixel logo ‘ICLR’
+
+Change the poison ratio into $1 / f$ . We decrease the fraction of backdoored samples added per training batch to $1 / f$ .
+
+Specifically, the poison ratio for LOAN is centralized 3/64, distributed 9/64; for MNIST is centralized 5/64, distributed 20/64; for CIFAR is centralized 1/64, distributed 4/64; for Tiny-imagenet is centralized 1/64, distributed 4/64. Other parameters are the same as described in the paper.
+
+Fig. 18 shows that DBA is better in LOAN and MNIST while centralized attack is better in CIFAR and Tiny-imagenet. Similar to the finding in Sec. 4.1 that “the more complex the model architecture (in Tb.1), the more obvious the decline in the main accuracy as the scale factor increases, because the scaling undermines more model parameters”, the setting of $f$ times scaling for centralized attack has larger impact on complex neural network like Resnet used in CIFAR and Tiny-imagenet. However, we note that this setting is not a totally fair comparison of the single-shot attack setting, as the same malicious agent of the centralized attack is allowed to attack $f$ times, while each malicious agent of DBA only attacks once.
+
+![](images/decc89a9413f6cd52144d9d265ce9e4170acbe60d5b070c1d3610b676570b321.jpg)  
+Figure 16: Attack A-M for white glasses pattern on Tiny-imagenet
+
+![](images/76b72c270966410c5ee739e900061991573c0f392832bcde9df60d179d668edb.jpg)  
+Figure 17: Attack A-M for white, black, purple glasses patterns on Tiny-imagenet
+
+![](images/0b1cd11c1adfcf2ca3185b613e03493389686be36291f15e0c9e09dbfd30dda1.jpg)  
+Figure 18: Scale $f$ times with $1 / f$ poison ratio each time
+
+Change the data size to $1 / f$ . We divide the local dataset into $f$ parts and use $1 / f$ dataset for each update and keep the poison ratio unchanged.
+
+![](images/b32fab870f550784d869228f33eb36de1fe5a37841f0794efb8d5ea322cde161.jpg)  
+Fig. 19 shows that DBA is still more persistent than centralized attack.   
+Figure 19: Scale $f$ times with $1 / f$ data size each time
+
+# A.5 MORE DETAILS ABOUT ROBUST AGGREGATION
+
+We report RFA distance and FoolsGold weights on adversarial parties in Tb. 2.
+
+Table 2: RFA Distance and FoolsGold Weight   
+
+<table><tr><td>Metrics</td><td>Dataset</td><td>DBA attacker1</td><td>DBA attacker2</td><td>DBA attacker3</td><td>DBA attacker4</td><td>DBA attackers(sum)</td><td>centralized attacker</td></tr><tr><td rowspan="2">RFA</td><td>LOAN</td><td>0.71+0.50</td><td>0.75±0.49</td><td>0.73±0.50</td><td>/</td><td>/</td><td>0.81±0.48</td></tr><tr><td>MNIST</td><td>1.48 ±1.05</td><td>1.41±1.28</td><td>1.09±1.03</td><td>1.03±0.89</td><td>T</td><td>2.57±1.32</td></tr><tr><td rowspan="2">Distance</td><td>CIFAR</td><td>175.43±22.82</td><td>175.46±22.83</td><td>148.45±22.84</td><td>175.29±22.85</td><td>T</td><td>196.13±11.58</td></tr><tr><td>Tiny-imagenet</td><td>396.68±33.83</td><td>106.01±33.83</td><td>374.32±33.83</td><td>42.82±28.8</td><td>/</td><td>431.34 ±40.15</td></tr><tr><td rowspan="2">FoolsGold</td><td>LOAN</td><td>0.31 ±0.38</td><td>0.31±0.38</td><td>0.57±0.45</td><td>/</td><td>1.18±1.15</td><td>0.98±0.10</td></tr><tr><td>MNIST</td><td>1.00±0.00</td><td>1.00±0.00</td><td>0.99±0.02</td><td>0.99±0.02</td><td>3.98±0.04</td><td>1.00±0.00</td></tr><tr><td rowspan="2">Weight</td><td>CIFAR</td><td>0.87±0.2</td><td>0.26±0.27</td><td>0.37±0.26</td><td>0.32±0.29</td><td>1.82±0.88</td><td>0.79±0.34</td></tr><tr><td>Tiny-imagenet</td><td>0.88±0.22</td><td>0.87±0.28</td><td>0.34±0.23</td><td>0.26±0.13</td><td>2.35±0.34</td><td>0.99±0.11</td></tr></table>
+
+# A.6 THE ROBUSTNESS OF DISTRIBUTED ATTACK IN BYZANTINE SETTING
+
+Here we evaluate the Byzantine setting Multi-Krum (Blanchard et al., 2017) and Bulyan (Guerraoui et al., 2018). For both DBA and centralized attack we use the aggregation rule that can tolerate $f$ Byzantine workers among the $n$ workers (Blanchard et al., 2017). For centralized attack there is 1 attacker and $n - 1$ non-Byzantine workers. For DBA there are $f$ distributed attackers and $n - f$ non-Byzantine workers. The total number of poisoned pixel amounts are kept the same.
+
+Multi-Krum To meet the assumption that $2 f + 2 < n$ , we set $n = 1 0$ , $f = 3 ,$ ) for LOAN and $\mathbf { \Phi } _ { n } = 1 2$ , $f = 4 ,$ ) for image datasets. The Multi-Krum parameter $m$ is set to $m = n - f$ . For Tiny-imagenet we decrease the poison ratio into $5 / 6 4$ for both attacks. Other parameters are the same as described in the paper.
+
+For CIFAR and Tiny-imagenet, we find that DBA is more effective as shown in Fig. 20. For LOAN and MNIST, both attacks don’t behave well. We believe the reason can be explained by the fact that LOAN and MNIST are simpler tasks and benign clients quickly agree on the correct gradient direction, so malicious updates are more difficult to succeed.
+
+Bulyan We use Bulyan based on the Byzantineresilient aggregation rule Krum (Blanchard et al., 2017). To meet the assumption that $4 f + 3 < = n$ , we set $\mathrm { \Delta } n = 1 5$ , $f = 3$ ) for LOAN and $n = 2 0$ $f = 4 ,$ ) for image datasets.
+
+For CIFAR, DBA is more effective as shown in Fig. 21. For other datasets, both attacks fail. However, we note that our distributed and centralized backdoor attacks are not optimized for Byzantine setting. We believe its worthwhile to explore the distributed version of other new attack algorithms, e.g. (Baruch et al., 2019) that manipulates its update to mitigate Krum and Bulyan defenses.
+
+![](images/4db2c8eff47f092f4b8073ccb009a852fdd35239c7908dd10ea1cd96c8abd0ed.jpg)  
+Figure 20: Multi-Krum
+
+![](images/ec3fb3caf1f054923def99887ce2c610b3924f312773d549aaf7d036f014fe90.jpg)  
+Figure 21: Bulyan
+
+In summary, Multi-Krum and Bulyan have stricter assumptions on the proportion of attackers than RFA and FoolsGold. In addition, while RFA and FoolsGold still assign potential outliers with extreme low weights, Krum (Multi-Krum, Krum-based Bulyan) directly removes them, making it impossible to inject backdoors if the malicious updates are obviously far from the benign updates. The centralized attack for four datasets totally fails under Multi-Krum and Bulyan while DBA can still succeed in some cases.
+
+# A.7 MORE DETAILS ON SOFT DECISION TREE
+
+Frosst & Hinton (2017) proposed Soft Decision Tree which distills a trained neural network by training with data and their soft targets that are the predictions of the neural network over classes. Trained with gradient descent, every inner node has a learned filter and a bias to make binary decision and the leaf node has a learned distribution. To some extent we can use the filter value to reflect the importance of every feature in internal nodes. We learn soft decision trees from the clean neural network and DBA poisoned neural network of LOAN and MNIST and they all achieve about $90 \%$ test accuracy on main and backdoor tasks.
+
+If we look at the third node in the forth layer in Fig.22.(b), the potential classifications are only 2 and 0, thus its filter is simply learning to distinguish these two digit. With extreme dark color in the area of the global pattern, which means these pixels correspond to small value in filter, this inner node will make leftmost branch decision into target label 2 when triggered by the global pattern because the probability is lower than 0.5. Taking an opposite example, the leftmost node in second layer has extreme white color in the area of the global pattern, which means these pixels correspond to large value of filter and will contribute to make rightmost branch decision if encountering the global pattern. Moreover, clean images won’t trigger the filters in backdoor pattern area and the major digit shape in center dominates the decision route, like examples in Fig.24.(b). Comparing Fig.22.(a)(b), the trigger area after poisoning becomes much more significant for decision making.
+
+Soft Decision Tree provides insights into the neural network and give explainable classification decisions. Examples of the decision routes in inference time for clean and poisoned input data are given for MNIST in Fig.25 and in Fig.24. We find that the poisoned model already starts to misbehave from the top node of the tree.
+
+We also run 10000 poisoned and clean samples for the LOAN clean and poison models to study the sample-wise importance based on the filter value multiplied by the input feature value in Fig.23.(b)(c). With this local importance metric, the original low importance feature indeed becomes salient in poisoned model with poisoned input.
+
+![](images/010fd5085271870fa2d5209e8f4257be44c3ccc568d7333f4fba8c1edb7a6bfc.jpg)  
+Figure 22: Soft decision tree for MNIST
+
+![](images/f43bb33df56af02f97232825a20cdcb427eed3a806e05c1273fd53b64c771e9d.jpg)  
+Figure 23: Feature importance in soft decision tree for LOAN
+
+![](images/7c613f8f090a67e1d2c53ce1e3e5fa478c5661ff1cef2755ebe78fa1e4cdc46c.jpg)  
+Figure 24: Examples for the poisoned MNIST soft decision tree
+
+![](images/175e5eed996f1f9306b62aff614682d24982dd77550fba33477f55fcc2a3a0c7.jpg)  
+Figure 25: Examples for the clean MNIST soft decision tree
+
+# A.8 MORE GRAD-CAM RESULTS ON MNIST
+
+We test the global model of MNIST poisoned by DBA under Attack A-M in round 16 of Fig.4 with local backdoored images and global backdoored images. More Grad-cam results are provided in Fig.26 and Fig.27.
+
+![](images/a6deb1bd1420756d574d25d6ff369d126fabf736b83d3f83857e96168b1a95cb.jpg)  
+Figure 26: Example of digit 8
+
+![](images/d89089c2ec40f16a40deea55cede2e04b0754bde50b9a4386014fb18d9d5dbe6.jpg)  
+Figure 27: Example of digit 5
+
+# A.9 IMPLEMENTATION DETAILS FOR LOCATION EFFECT EXPERIMENTS
+
+During this process we increases $S h i f t _ { y }$ and first keeps $S h i f t _ { x } = S h i f t _ { y }$ . After the rightmost pixel reaches the right edge of images, we fix $S h i f t _ { x }$ as its largest value, which is $\mathrm { X }$ value of the dotted line in Fig.9, and keep increasing $S h i f t _ { y }$ until the lowest pixel reaches the button edge of the images. $T L$ is the max value among $S h i f t _ { x }$ and $S h i f t _ { y }$ .
+
+# A.10 DATA DISTRIBUTION EFFECTS FOR TRIGGERS
+
+• By increasing the hypterparameter $\alpha$ in the Dirichlet distribution, we can simulate from non-i.i.d to i.i.d distributions for the image datasets. When evaluated under Attack A-M, Fig.28 shows that DBA-ASR is stable under various distributions, which exhibits the practicability and robustness of DBA when attacking standard FL.
+
+• Data distribution has more influence on the DBA performance under robust aggregation algorithms when calculating distance or similarity between the benign updates and malicious updates. When the training data are non-i.i.d., the updates across the benign participants already appear high diversity so the poisoned update is better concealed among them and less likely to be detected. In our experiments, it’s easier for DBA to succeed against RFA and FoolsGold under a more non-i.i.d. data distribution in CIFAR and Tiny-imagenet.
+
+![](images/54b478b3a440ca22ac56dfbc4f76ebd038db7f322aa0083daa452c612d972305.jpg)  
+Figure 28: Effects of $\alpha$ in Dirichlet data distribution on Attack Success Rate
+
+# A.11 MORE DETAILS ABOUT LOAN DATASETS
+
+The lable distribution is uneven in LOAN, which is shown in Tb.3. The five most important features among the 91 features in LOAN under various classification methods are shown in Tb.4 and the result is consistent.
+
+In Fig. 7, the names for six low importance feature are num tl 120dpd 2m, num tl 90g dpd 24m, pub rec bankruptcies, pub rec, acc now delinq, tax liens; the names six high importance feature are out prncp,total pymnt inv, out prncp inv, total rec prncp,last pymnt amnt, all util.
+
+Table 3: Financial Dataset Label Distribution   
+
+<table><tr><td>Loan Status Number of Examples</td></tr><tr><td>Fully Paid</td><td>1041952</td></tr><tr><td>Current</td><td>919695</td></tr><tr><td>Charged Off</td><td>261655</td></tr><tr><td>Late (31-120 days)</td><td>21897</td></tr><tr><td>In Grace Period</td><td>8952</td></tr><tr><td>Late (16-30 days)</td><td>3737</td></tr><tr><td>Does not meet the credit policy. Status:Fully Paid</td><td>1988</td></tr><tr><td>Does not meet the credit policy. Status:Charged Off</td><td>761</td></tr><tr><td>Default</td><td>31</td></tr></table>
+
+Table 4: The Five Most Important Features are Similar in Different Classification Methods   
+
+<table><tr><td>Method</td><td>Rank 1st</td><td>Rank 2nd</td><td>Rank 3rd</td><td>Rank 4th</td><td>Rank 5th</td></tr><tr><td>Random Forest Classifier</td><td>out_prncp</td><td>out_prncp_inv</td><td>last_pymnt_amnt</td><td>recoveries</td><td>total_rec_prncp</td></tr><tr><td>Extra Tree Classifier</td><td>out_prncp_inv</td><td>out_prncp</td><td>total_rec_prncp</td><td>last_pymnt_amnt</td><td>total-pymnt_inv</td></tr><tr><td>XGBoost</td><td>out_prncp</td><td>recoveries</td><td>funded_amnt</td><td>total_rec_prncp</td><td>last_pymnt_amnt</td></tr><tr><td>Our decision tree</td><td>out_prncp</td><td>out_prncp_inv</td><td>term</td><td>recoveries</td><td>collection_recovery_fee</td></tr></table>