Narrowing the Gap between Adversarial and Stochastic MDPs via Policy Optimization

TL;DR

This paper introduces APO-MVP, an algorithm that significantly reduces regret in adversarial MDPs by bridging the gap with stochastic MDPs, using policy optimization techniques that are simple to implement.

Contribution

The paper presents APO-MVP, a novel policy optimization algorithm that achieves near-optimal regret bounds in adversarial MDPs without relying on occupancy measures.

Findings

Achieves regret bound of  ( ext{poly}(H)\u221A{SAT})

Improves regret bounds by a factor of  ( ext{poly}(H) ext{poly}(S,A,T))

Matches the minimax lower bound  (( ext{poly}(H) ext{poly}(S,A,T)))

Abstract

We consider the problem of learning in adversarial Markov decision processes [MDPs] with an oblivious adversary in a full-information setting. The agent interacts with an environment during $T$ episodes, each of which consists of $H$ stages, and each episode is evaluated with respect to a reward function that will be revealed only at the end of the episode. We propose an algorithm, called APO-MVP, that achieves a regret bound of order $\tilde{\mathcal{O}}(\mathrm{poly}(H)\sqrt{SAT})$, where $S$ and $A$ are sizes of the state and action spaces, respectively. This result improves upon the best-known regret bound by a factor of $\sqrt{S}$, bridging the gap between adversarial and stochastic MDPs, and matching the minimax lower bound $\Omega(\sqrt{H^3SAT})$ as far as the dependencies in $S,A,T$ are concerned. The proposed algorithm and analysis completely avoid the typical tool given by occupancy measures; instead, it performs policy optimization based only on dynamic programming and on a black-box online linear optimization strategy run over estimated advantage functions, making it easy to implement. The analysis leverages two recent techniques: policy optimization based on online linear optimization strategies (Jonckheere et al., 2023) and a refined martingale analysis of the impact on values of estimating transitions kernels (Zhang et al., 2023).