Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes

TL;DR

This paper introduces TUCRL, an algorithm that efficiently balances exploration and exploitation in any finite MDP without prior knowledge, achieving near-optimal regret bounds even in complex, weakly-communicating environments.

Contribution

The paper presents TUCRL, the first algorithm capable of optimal exploration-exploitation in all finite MDPs without prior assumptions, with proven regret bounds and superior performance over existing methods.

Findings

TUCRL achieves a regret bound of O(D^C mp; \, (;\Gamma^C S^C AT) in any finite MDP.

Existing algorithms suffer linear regret in weakly-communicating MDPs or require prior knowledge.

Numerical simulations demonstrate TUCRL's effectiveness and advantages over state-of-the-art algorithms.

Abstract

While designing the state space of an MDP, it is common to include states that are transient or not reachable by any policy (e.g., in mountain car, the product space of speed and position contains configurations that are not physically reachable). This leads to defining weakly-communicating or multi-chain MDPs. In this paper, we introduce \tucrl, the first algorithm able to perform efficient exploration-exploitation in any finite Markov Decision Process (MDP) without requiring any form of prior knowledge. In particular, for any MDP with $S^{\texttt{C}}$ communicating states, $A$ actions and $\Gamma^{\texttt{C}} \leq S^{\texttt{C}}$ possible communicating next states, we derive a $\widetilde{O}(D^{\texttt{C}} \sqrt{\Gamma^{\texttt{C}} S^{\texttt{C}} AT})$ regret bound, where $D^{\texttt{C}}$ is the diameter (i.e., the longest shortest path) of the communicating part of the MDP. This is in contrast with optimistic algorithms (e.g., UCRL, Optimistic PSRL) that suffer linear regret in weakly-communicating MDPs, as well as posterior sampling or regularised algorithms (e.g., REGAL), which require prior knowledge on the bias span of the optimal policy to bias the exploration to achieve sub-linear regret. We also prove that in weakly-communicating MDPs, no algorithm can ever achieve a logarithmic growth of the regret without first suffering a linear regret for a number of steps that is exponential in the parameters of the MDP. Finally, we report numerical simulations supporting our theoretical findings and showing how TUCRL overcomes the limitations of the state-of-the-art.