Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. Yet, so far, such methods have been mostly analyzed from an optimization perspective, without addressing the problem of exploration, or by making strong assumptions on the interaction with the environment. In this paper we consider model-based RL in the tabular finite-horizon MDP setting with unknown transitions and bandit feedback. For this setting, we propose an optimistic trust region policy optimization (TRPO) algorithm for which we establish $\tilde O(\sqrt{S^2 A H^4 K})$ regret for stochastic rewards. Furthermore, we prove $\tilde O( \sqrt{ S^2 A H^4 } K^{2/3} ) $ regret for adversarial rewards. Interestingly, this result matches previous bounds derived for the bandit feedback case, yet with known transitions. To the best of our knowledge, the two results are the first sub-linear regret bounds obtained for policy optimization algorithms with unknown transitions and bandit feedback.
This paper studies the problem of stochastic continuum-armed bandit with constraints (SCBwC), where we optimize a black-box reward function $f(x)$ subject to a black-box constraint function $g(x)\leq 0$ over a continuous space $\mathcal X$. We model reward and constraint functions via Gaussian processes (GPs) and propose a Rectified Pessimistic-Optimistic Learning framework (RPOL), a penalty-based method incorporating optimistic and pessimistic GP bandit learning for reward and constraint functions, respectively. We consider the metric of cumulative constraint violation $\sum_{t=1}^T(g(x_t))^{…
Criteo AI Lab（Criteo人工智能实验室）；LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院LTCI）
The generalized linear bandit framework has attracted a lot of attention in recent years by extending the well-understood linear setting and allowing to model richer reward structures. It notably covers the logistic model, widely used when rewards are binary. For logistic bandits, the frequentist regret guarantees of existing algorithms are $\tilde{\mathcal{O}}(κ\sqrt{T})$, where $κ$ is a problem-dependent constant. Unfortunately, $κ$ can be arbitrarily large as it scales exponentially with the size of the decision set. This may lead to significantly loose regret bounds and poor empirical perf…
We consider online sequential decision problems where an agent must balance exploration and exploitation. We derive a set of Bayesian `optimistic' policies which, in the stochastic multi-armed bandit case, includes the Thompson sampling policy. We provide a new analysis showing that any algorithm producing policies in the optimistic set enjoys $\tilde O(\sqrt{AT})$ Bayesian regret for a problem with $A$ actions after $T$ rounds. We extend the regret analysis for optimistic policies to bilinear saddle-point problems which include zero-sum matrix games and constrained bandits as special cases. I…
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Purdue University（普渡大学）
This paper considers stochastic linear bandits with general nonlinear constraints. The objective is to maximize the expected cumulative reward over horizon $T$ subject to a set of constraints in each round $τ\leq T$. We propose a pessimistic-optimistic algorithm for this problem, which is efficient in two aspects. First, the algorithm yields $\tilde{\cal O}\left(\left(\frac{K^{0.75}}δ+d\right)\sqrtτ\right)$ (pseudo) regret in round $τ\leq T,$ where $K$ is the number of constraints, $d$ is the dimension of the reward feature space, and $δ$ is a Slater's constant; and zero constraint violation i…
Massachusetts Institute of Technology（麻省理工学院）；Purdue University（普渡大学）
We consider the contextual bandit problem on general action and context spaces, where the learner's rewards depend on their selected actions and an observable context. This generalizes the standard multi-armed bandit to the case where side information is available, e.g., patients' records or customers' history, which allows for personalized treatment. We focus on consistency -- vanishing regret compared to the optimal policy -- and show that for large classes of non-i.i.d. contexts, consistency can be achieved regardless of the time-invariant reward mechanism, a property known as universal con…
University of Washington（华盛顿大学）；Tsinghua University（清华大学）；Peking University（北京大学）；Princeton University（普林斯顿大学）；Microsoft Research（微软研究院）
Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. This work considers a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem. For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in [LMT21, JWWN19]. Our algorithms are based on a unified zeroth-order o…
Université de Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）
We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide $\textit{a problem-dependent lower bound on the regret}$ of any corrupted bandit algorithm. The lower bounds indi…
Reinforcement Learning (RL) has been able to solve hard problems such as playing Atari games or solving the game of Go, with a unified approach. Yet modern deep RL approaches are still not widely used in real-world applications. One reason could be the lack of guarantees on the performance of the intermediate executed policies, compared to an existing (already working) baseline policy. In this paper, we propose an online model-free algorithm that solves conservative exploration in the policy optimization problem. We show that the regret of the proposed approach is bounded by $\tilde{\mathcal{O…
Boston University（波士顿大学）；Carnegie Mellon University（卡内基梅隆大学）
We investigate a natural but surprisingly unstudied approach to the multi-armed bandit problem under safety risk constraints. Each arm is associated with an unknown law on safety risks and rewards, and the learner's goal is to maximise reward whilst not playing unsafe arms, as determined by a given threshold on the mean risk. We formulate a pseudo-regret for this setting that enforces this safety constraint in a per-round way by softly penalising any violation, regardless of the gain in reward due to the same. This has practical relevance to scenarios such as clinical trials, where one must ma…
Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO), as the widely employed policy based reinforcement learning (RL) methods, are prone to converge to a sub-optimal solution as they limit the policy representation to a particular parametric distribution class. To address this issue, we develop an innovative Optimistic Distributionally Robust Policy Optimization (ODRPO) algorithm, which effectively utilizes Optimistic Distributionally Robust Optimization (DRO) approach to solve the trust region constrained optimization problem without parameterizing the policies. Our …
This is a short communication on a Lyapunov function argument for softmax in bandit problems. There are a number of excellent papers coming out using differential equations for policy gradient algorithms in reinforcement learning \cite{agarwal2019optimality,bhandari2019global,mei2020global}. We give a short argument that gives a regret bound for the soft-max ordinary differential equation for bandit problems. We derive a similar result for a different policy gradient algorithm, again for bandit problems. For this second algorithm, it is possible to prove regret bounds in the stochastic case \c…
Carnegie Mellon University（卡内基梅隆大学）；Google Research（谷歌研究院）；DeepMind；Duke University（杜克大学）；Norwegian University of Science and Technology（挪威科技大学）
We propose a novel framework for structured bandits, which we call an influence diagram bandit. Our framework captures complex statistical dependencies between actions, latent variables, and observations; and thus unifies and extends many existing models, such as combinatorial semi-bandits, cascading bandits, and low-rank bandits. We develop novel online learning algorithms that learn to act efficiently in our models. The key idea is to track a structured posterior distribution of model parameters, either exactly or approximately. To act, we sample model parameters from their posterior and the…
A classic setting of the stochastic K-armed bandit problem is considered in this note. In this problem it has been known that KL-UCB policy achieves the asymptotically optimal regret bound and KL-UCB+ policy empirically performs better than the KL-UCB policy although the regret bound for the original form of the KL-UCB+ policy has been unknown. This note demonstrates that a simple proof of the asymptotic optimality of the KL-UCB+ policy can be given by the same technique as those used for analyses of other known policies.
Indian Institute of Technology, Madras（印度理工学院马德拉斯分校）
We consider the multi armed bandit problem in non-stationary environments. Based on the Bayesian method, we propose a variant of Thompson Sampling which can be used in both rested and restless bandit scenarios. Applying discounting to the parameters of prior distribution, we describe a way to systematically reduce the effect of past observations. Further, we derive the exact expression for the probability of picking sub-optimal arms. By increasing the exploitative value of Bayes' samples, we also provide an optimistic version of the algorithm. Extensive empirical analysis is conducted under va…
We study the recovering bandits problem, a variant of the stochastic multi-armed bandit problem where the expected reward of each arm varies according to some unknown function of the time since the arm was last played. While being a natural extension of the classical bandit problem that arises in many real-world settings, this variation is accompanied by significant difficulties. In particular, methods need to plan ahead and estimate many more quantities than in the classical bandit setting. In this work, we explore the use of Gaussian processes to tackle the estimation and planing problem. We…
We study the stochastic Multi-Armed Bandit (MAB) problem under worst-case regret and heavy-tailed reward distribution. We modify the minimax policy MOSS for the sub-Gaussian reward distribution by using saturated empirical mean to design a new algorithm called Robust MOSS. We show that if the moment of order $1+ε$ for the reward distribution exists, then the refined strategy has a worst-case regret matching the lower bound while maintaining a distribution-dependent logarithm regret.
University of Oxford（牛津大学）；Bytedance AI Lab（字节跳动人工智能实验室）
Most off-policy evaluation methods for contextual bandits have focused on the expected outcome of a policy, which is estimated via methods that at best provide only asymptotic guarantees. However, in many applications, the expectation may not be the best measure of performance as it does not capture the variability of the outcome. In addition, particularly in safety-critical settings, stronger guarantees than asymptotic correctness may be required. To address these limitations, we consider a novel application of conformal prediction to contextual bandits. Given data collected under a behaviora…
We consider a policy gradient algorithm applied to a finite-arm bandit problem with Bernoulli rewards. We allow learning rates to depend on the current state of the algorithm, rather than use a deterministic time-decreasing learning rate. The state of the algorithm forms a Markov chain on the probability simplex. We apply Foster-Lyapunov techniques to analyse the stability of this Markov chain. We prove that if learning rates are well chosen then the policy gradient algorithm is a transient Markov chain and the state of the chain converges on the optimal arm with logarithmic or poly-logarithmi…
University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
We study the problem of estimating the expected reward of the optimal policy in the stochastic disjoint linear bandit setting. We prove that for certain settings it is possible to obtain an accurate estimate of the optimal policy value even with a number of samples that is sublinear in the number that would be required to \emph{find} a policy that realizes a value close to this optima. We establish nearly matching information theoretic lower bounds, showing that our algorithm achieves near optimal estimation error. Finally, we demonstrate the effectiveness of our algorithm on joke recommendati…
We study the off-policy evaluation problem---estimating the value of a target policy using data collected by another policy---under the contextual bandit model. We consider the general (agnostic) setting without access to a consistent model of rewards and establish a minimax lower bound on the mean squared error (MSE). The bound is matched up to constants by the inverse propensity scoring (IPS) and doubly robust (DR) estimators. This highlights the difficulty of the agnostic contextual setting, in contrast with multi-armed bandits and contextual bandits with access to a consistent reward model…
We propose the Generalized Policy Elimination (GPE) algorithm, an oracle-efficient contextual bandit (CB) algorithm inspired by the Policy Elimination algorithm of \cite{dudik2011}. We prove the first regret optimality guarantee theorem for an oracle-efficient CB algorithm competing against a nonparametric class with infinite VC-dimension. Specifically, we show that GPE is regret-optimal (up to logarithmic factors) for policy classes with integrable entropy. For classes with larger entropy, we show that the core techniques used to analyze GPE can be used to design an $\varepsilon$-greedy algor…
Microsoft Dynamics 365 AI（微软 Dynamics 365 人工智能）；Microsoft Research（微软研究院）
We propose an estimator and confidence interval for computing the value of a policy from off-policy data in the contextual bandit setting. To this end we apply empirical likelihood techniques to formulate our estimator and confidence interval as simple convex optimization problems. Using the lower bound of our confidence interval, we then propose an off-policy policy optimization algorithm that searches for policies with large reward lower bound. We empirically find that both our estimator and confidence interval improve over previous proposals in finite sample regimes. Finally, the policy opt…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Microsoft Research（微软研究院）
Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to adapt to the online algorithm's actions. We define the alternative notion of policy regret, which attempts to provide a more meaningful way to measure an online algorithm's performance against adaptive adversaries. Focusing on the online bandit setting, we show that no bandit algo…
Even when unable to run experiments, practitioners can evaluate prospective policies, using previously logged data. However, while the bandits literature has adopted a diverse set of objectives, most research on off-policy evaluation to date focuses on the expected reward. In this paper, we introduce Lipschitz risk functionals, a broad class of objectives that subsumes conditional value-at-risk (CVaR), variance, mean-variance, many distorted risks, and CPT risks, among others. We propose Off-Policy Risk Assessment (OPRA), a framework that first estimates a target policy's CDF and then generate…
Sri Lanka Technological Campus（斯里兰卡理工学院）；University of Peradeniya（佩拉德尼亚大学）
This paper proposes a novel policy for a group of agents to, individually as well as collectively, solve a multi armed bandit (MAB) problem. The policy relies solely on the information that an agent has obtained through sampling of the options on its own and through communication with neighbors. The option selection policy is based on an Upper Confidence Based (UCB) strategy while the communication strategy that is proposed forces agents to communicate with other agents who they believe are most likely to be exploring than exploiting. The overall strategy is shown to significantly outperform a…
Hanjuku-kaso Co Ltd（Hanjuku-kaso有限公司）；Cornell University（康奈尔大学）；ZOZO Research（ZOZO研究院）；Yale University（耶鲁大学）
Off-policy evaluation (OPE) aims to estimate the performance of hypothetical policies using data generated by a different policy. Because of its huge potential impact in practice, there has been growing research interest in this field. There is, however, no real-world public dataset that enables the evaluation of OPE, making its experimental studies unrealistic and irreproducible. With the goal of enabling realistic and reproducible OPE research, we present Open Bandit Dataset, a public logged bandit dataset collected on a large-scale fashion e-commerce platform, ZOZOTOWN. Our dataset is uniqu…
I study a game of strategic exploration with private payoffs and public actions in a Bayesian bandit setting. In particular, I look at cascade equilibria, in which agents switch over time from the risky action to the riskless action only when they become sufficiently pessimistic. I show that these equilibria exist under some conditions and establish their salient properties. Individual exploration in these equilibria can be more or less than the single-agent level depending on whether the agents start out with a common prior or not, but the most optimistic agent always underexplores. I also sh…
We study stochastic structured bandits for minimizing regret. The fact that the popular optimistic algorithms do not achieve the asymptotic instance-dependent regret optimality (asymptotic optimality for short) has recently alluded researchers. On the other hand, it is known that one can achieve bounded regret (i.e., does not grow indefinitely with $n$) in certain instances. Unfortunately, existing asymptotically optimal algorithms rely on forced sampling that introduces an $ω(1)$ term w.r.t. the time horizon $n$ in their regret, failing to adapt to the "easiness" of the instance. In this pape…
We present and prove properties of a new offline policy evaluator for an exploration learning setting which is superior to previous evaluators. In particular, it simultaneously and correctly incorporates techniques from importance weighting, doubly robust evaluation, and nonstationary policy evaluation approaches. In addition, our approach allows generating longer histories by careful control of a bias-variance tradeoff, and further decreases variance by incorporating information about randomness of the target policy. Empirical evidence from synthetic and realworld exploration learning problem…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Sandia National Laboratories（桑迪亚国家实验室）
We consider Markov Decision Processes (MDPs) in which every stationary policy induces the same graph structure for the underlying Markov chain and further, the graph has the following property: if we replace each recurrent class by a node, then the resulting graph is acyclic. For such MDPs, we prove the convergence of the stochastic dynamics associated with a version of optimistic policy iteration (OPI), suggested in Tsitsiklis (2002), in which the values associated with all the nodes visited during each iteration of the OPI are updated.
National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）；NEC Central Research Laboratories（NEC中央研究所）；The University of Tokyo（东京大学）
Reinforcement Learning, a machine learning framework for training an autonomous agent based on rewards, has shown outstanding results in various domains. However, it is known that learning a good policy is difficult in a domain where rewards are rare. We propose a method, optimistic proximal policy optimization (OPPO) to alleviate this difficulty. OPPO considers the uncertainty of the estimated total return and optimistically evaluates the policy based on that amount. We show that OPPO outperforms the existing methods in a tabular task.
We study a constrained contextual linear bandit setting, where the goal of the agent is to produce a sequence of policies, whose expected cumulative reward over the course of $T$ rounds is maximum, and each has an expected cost below a certain threshold $τ$. We propose an upper-confidence bound algorithm for this problem, called optimistic pessimistic linear bandit (OPLB), and prove an $\widetilde{\mathcal{O}}(\frac{d\sqrt{T}}{τ-c_0})$ bound on its $T$-round regret, where the denominator is the difference between the constraint threshold and the cost of a known feasible action. We further spec…
The Knowledge Gradient (KG) policy was originally proposed for online ranking and selection problems but has recently been adapted for use in online decision making in general and multi-armed bandit problems (MABs) in particular. We study its use in a class of exponential family MABs and identify weaknesses, including a propensity to take actions which are dominated with respect to both exploitation and exploration. We propose variants of KG which avoid such errors. These new policies include an index heuristic which deploys a KG approach to develop an approximation to the Gittins index. A num…
Stanford University（斯坦福大学）；Google Research（谷歌研究院）；Microsoft Research（微软研究院）；Georgia Institute of Technology（佐治亚理工学院）
In many bandit problems, the maximal reward achievable by a policy is often unknown in advance. We consider the problem of estimating the optimal policy value in the sublinear data regime before the optimal policy is even learnable. We refer to this as $V^*$ estimation. It was recently shown that fast $V^*$ estimation is possible but only in disjoint linear bandits with Gaussian covariates. Whether this is possible for more realistic context distributions has remained an open and important question for tasks such as model selection. In this paper, we first provide lower bounds showing that thi…
We study decentralized policy learning in Markov games where we control a single agent to play with nonstationary and possibly adversarial opponents. Our goal is to develop a no-regret online learning algorithm that (i) takes actions based on the local information observed by the agent and (ii) is able to find the best policy in hindsight. For such a problem, the nonstationary state transitions due to the varying opponent pose a significant challenge. In light of a recent hardness result \citep{liu2022learning}, we focus on the setting where the opponent's previous policies are revealed to the…
University of Pennsylvania（宾夕法尼亚大学）；University of Pittsburgh（匹兹堡大学）
Excessively changing policies in many real world scenarios is difficult, unethical, or expensive. After all, doctor guidelines, tax codes, and price lists can only be reprinted so often. We may thus want to only change a policy when it is probable that the change is beneficial. In cases that a policy is a threshold on contextual variables we can estimate treatment effects for populations lying at the threshold. This allows for a schedule of incremental policy updates that let us optimize a policy while making few detrimental changes. Using this idea, and the theory of linear contextual bandits…
Institute for Computational and Mathematical Engineering（计算与数学工程研究所）；Stanford University（斯坦福大学）；Department of Computer Science（计算机科学系）
In the stochastic linear contextual bandit setting there exist several minimax procedures for exploration with policies that are reactive to the data being acquired. In practice, there can be a significant engineering overhead to deploy these algorithms, especially when the dataset is collected in a distributed fashion or when a human in the loop is needed to implement a different policy. Exploring with a single non-reactive policy is beneficial in such cases. Assuming some batch contexts are available, we design a single stochastic policy to collect a good dataset from which a near-optimal po…
Princeton University（普林斯顿大学）；University of Southern California（南加利福尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of learning in episodic finite-horizon Markov decision processes with an unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ is the number of states, $|A|$ is the number of actions, and $T$ is the number of episodes. To the best of our knowledge, our algorithm is the first to ensure $\mathcal{\tilde{O}}(\sqrt{T})$ regret in this challenging setting; in fact it achieves the same regret bound as (Rosenberg &…
The goal of off-policy evaluation (OPE) is to evaluate a new policy using historical data obtained via a behavior policy. However, because the contextual bandit algorithm updates the policy based on past observations, the samples are not independent and identically distributed (i.i.d.). This paper tackles this problem by constructing an estimator from a martingale difference sequence (MDS) for the dependent samples. In the data-generating process, we do not assume the convergence of the policy, but the policy uses the same conditional probability of choosing an action during a certain period. …
Optimistic Actor-Critic with Parametric Policies for Linear Markov Decision Processes
Simon Fraser University（西蒙菲莎大学）；University of Pennsylvania（宾夕法尼亚大学）；Mila, McGill University（Mila，麦吉尔大学）；Google DeepMind（谷歌DeepMind）；University of Alberta（阿尔伯塔大学）
