Asymptotically optimal regret in communicating Markov decision processes

TL;DR

This paper introduces a learning algorithm for communicating Markov decision processes that achieves asymptotically optimal regret by explicitly estimating a key constant, balancing exploration and exploitation effectively.

Contribution

The paper presents a novel algorithm that attains optimal regret in average reward MDPs by tracking the constant K(M) and addresses the challenge of its discontinuity with a regularization method.

Findings

Achieves regret of K(M) log(T) + o(log(T)) in communicating MDPs.

Develops a regularization mechanism to estimate K(M) accurately.

Demonstrates the discontinuity of the function K(M).

Abstract

In this paper, we present a learning algorithm that achieves asymptotically optimal regret for Markov decision processes in average reward under a communicating assumption. That is, given a communicating Markov decision process $M$, our algorithm has regret $K(M) \log(T) + \mathrm{o}(\log(T))$ where $T$ is the number of learning steps and $K(M)$ is the best possible constant. This algorithm works by explicitly tracking the constant $K(M)$ to learn optimally, then balances the trade-off between exploration (playing sub-optimally to gain information), co-exploration (playing optimally to gain information) and exploitation (playing optimally to score maximally). We further show that the function $K(M)$ is discontinuous, which is a consequence challenge for our approach. To that end, we describe a regularization mechanism to estimate $K(M)$ with arbitrary precision from empirical data.