Non-Asymptotic Gap-Dependent Regret Bounds for Tabular MDPs

TL;DR

This paper proves that certain optimistic algorithms for episodic MDPs achieve non-asymptotic, gap-dependent logarithmic regret bounds without relying on diameter or ergodicity assumptions, bridging gap-dependent and minimax rates.

Contribution

It introduces a novel 'clipped' regret decomposition technique that provides gap-dependent regret bounds for a broad class of optimistic algorithms in episodic MDPs, independent of diameter-like quantities.

Findings

Achieves logarithmic regret bounds that depend on the gap and are non-asymptotic.

Bounds do not depend on diameter-like quantities or ergodicity assumptions.

Interpolates smoothly between gap-dependent logarithmic regret and minimax $ ilde{O}( oot{3} ext{HSAT})$ rate.

Abstract

This paper establishes that optimistic algorithms attain gap-dependent and non-asymptotic logarithmic regret for episodic MDPs. In contrast to prior work, our bounds do not suffer a dependence on diameter-like quantities or ergodicity, and smoothly interpolate between the gap dependent logarithmic-regret, and the $\widetilde{\mathcal{O}}(\sqrt{HSAT})$-minimax rate. The key technique in our analysis is a novel "clipped" regret decomposition which applies to a broad family of recent optimistic algorithms for episodic MDPs.