Beyond Value-Function Gaps: Improved Instance-Dependent Regret Bounds for Episodic Reinforcement Learning

TL;DR

This paper introduces improved, instance-dependent regret bounds for episodic reinforcement learning in finite MDPs, emphasizing the importance of state reachability and providing tighter bounds and lower bounds for optimistic algorithms.

Contribution

It presents novel gap definitions influencing regret bounds and demonstrates that optimistic algorithms cannot reach optimal bounds without unique policies.

Findings

Tighter upper regret bounds for optimistic algorithms.

New information-theoretic lower bounds for a broad class of MDPs.

Optimistic algorithms cannot achieve lower bounds unless the optimal policy is unique.

Abstract

We provide improved gap-dependent regret bounds for reinforcement learning in finite episodic Markov decision processes. Compared to prior work, our bounds depend on alternative definitions of gaps. These definitions are based on the insight that, in order to achieve a favorable regret, an algorithm does not need to learn how to behave optimally in states that are not reached by an optimal policy. We prove tighter upper regret bounds for optimistic algorithms and accompany them with new information-theoretic lower bounds for a large class of MDPs. Our results show that optimistic algorithms can not achieve the information-theoretic lower bounds even in deterministic MDPs unless there is a unique optimal policy.