An Information-Theoretic Analysis of Bayesian Reinforcement Learning

TL;DR

This paper applies information-theoretic methods to analyze the fundamental limits of Bayesian reinforcement learning, deriving bounds on the minimum Bayesian regret for MDPs, including bandit and online optimization problems.

Contribution

It introduces a framework for bounding Bayesian regret in model-based RL using information-theoretic measures like relative entropy and Wasserstein distance.

Findings

Derived upper bounds on Bayesian regret for MDPs.

Reproduced and improved existing bounds for bandit and online optimization problems.

Provided a unified information-theoretic approach to RL performance limits.

Abstract

Building on the framework introduced by Xu and Raginksy [1] for supervised learning problems, we study the best achievable performance for model-based Bayesian reinforcement learning problems. With this purpose, we define minimum Bayesian regret (MBR) as the difference between the maximum expected cumulative reward obtainable either by learning from the collected data or by knowing the environment and its dynamics. We specialize this definition to reinforcement learning problems modeled as Markov decision processes (MDPs) whose kernel parameters are unknown to the agent and whose uncertainty is expressed by a prior distribution. One method for deriving upper bounds on the MBR is presented and specific bounds based on the relative entropy and the Wasserstein distance are given. We then focus on two particular cases of MDPs, the multi-armed bandit problem (MAB) and the online optimization with partial feedback problem. For the latter problem, we show that our bounds can recover from below the current information-theoretic bounds by Russo and Van Roy [2].