ONLINE REINFORCEMENT LEARNING VIA POSTERIOR SAMPLING OF POLICY Anonymous authors Paper under double-blind review

Abstract

We propose a Reward-Weighted Posterior Sampling of Policy (RWPSP) algorithm to tackle the classic trade-off problem between exploration and exploitation under finite Markov decision processes (MDPs). The Thompson sampling method so far has only considered posterior sampling over transition probabilities, which is hard to gain the globally sub-optimal rewards. RWPSP runs posterior sampling over stationary policy distributions instead of transition probabilities, and meanwhile keeps transition probabilities updated. Particularly, we leverage both relevant count functions and reward-weighting to online update the policy posterior, aiming to balance between local and long-term policy distributions for a globally near-optimal game value. Theoretically, we establish a bound of Õ(Γ T /S 2 ) 1 on the total regret in time horizon T with Γ/S 2 < D SA satisfied in general, where S and A represents the sizes of state and action spaces, respectively, D the diameter. This matches the best regret bound thus far for MDPs. Experimental results corroborate our theoretical results and show the advantage of our algorithm over baselines in terms of efficiency.

1. INTRODUCTION

Online reinforcement learning (Wei et al., 2017) addresses the problem of learning and planning in real-time sequential decision making systems with the interacting environment partially observed or fully observed. The decision maker tries to maximize the cumulative reward during the interaction with the environment, which however inevitably leads to the trade-off between exploration and exploitation. Many attempts have been made to mitigate such dilemma by improving underlying regret bounds (Zhang et al., 2020b )(Ménard et al., 2021 )(Zhang et al., 2021b )(Zhang et al., 2022 )(Agrawal et al., 2021) . Trade-off between exploration and exploitation has been studied extensively in various scenarios. The goal of exploration is to find as much information as possible of the environment, while the exploitation process aims to maximize the long-term total reward based on the exploited part of the environment. To handle the trade-off problem, one popular way is to use the naive exploration method such as adaptive ϵ-greedy exploration (Tokic, 2010). The method adjusts the exploration parameter adaptively, depending on the temporal-difference (TD) error observed from the value function. Optimistic initialisation methods have also been studied in factored MDPs (Szita & Lörincz, 2009; Brafman & Tennenholtz, 2003) . They encourage systematic exploration in the early stage. Another common way is to use the optimism in the face of uncertainty (OFU) principle (Lai & Robbins, 1985) , where the agent constructs confidence sets to search for the optimistic parameters associated with the maximum reward. Thompson sampling, as an OFU-based approach, was originally presented for stochastic bandit scenarios (Thompson, 1933) . It has been applied in various MDPs contexts (Osband et al., 2013; Agrawal & Goyal, 2012) since it can achieve tighter bounds (Ding et al., 2021; Oh & Iyengar, 2019; Moradipari et al., 2019) and better compatibility with other structures in both theory and practice (Chapelle & Li, 2011; Zhang et al., 2021a; Agrawal & Goyal, 2013) . It has also achieved great performance on contextual bandit problems(Agrawal & Jia, 2017)(Osband & Van Roy, 2017) (Osband et al., 2019) .The general optimistic algorithms require to solve all MDPs lying within the confident sets, while Thompson sampling-based algorithms only need to solve the sampled MDPs 1 The symbol Õ hides logarithmic factors. 1

