Best-of-Both-Worlds Policy Optimization for CMDPs with Bandit Feedback

TL;DR

This paper introduces a new policy optimization algorithm for constrained Markov decision processes that works effectively with bandit feedback, handling both stochastic and adversarial constraints with optimal regret and violation bounds.

Contribution

It presents the first bandit-feedback compatible best-of-both-worlds algorithm for CMDPs, improving efficiency over previous occupancy-measure-based methods.

Findings

Achieves () ilde{O}(\u0010 ext{ }  ext{ }) regret and constraint violation for stochastic constraints.

Attains () ilde{O}( ext{ }) constraint violation in adversarial constraints.

Uses a policy optimization approach, enhancing computational efficiency.

Abstract

We study online learning in constrained Markov decision processes (CMDPs) in which rewards and constraints may be either stochastic or adversarial. In such settings, Stradi et al.(2024) proposed the first best-of-both-worlds algorithm able to seamlessly handle stochastic and adversarial constraints, achieving optimal regret and constraint violation bounds in both cases. This algorithm suffers from two major drawbacks. First, it only works under full feedback, which severely limits its applicability in practice. Moreover, it relies on optimizing over the space of occupancy measures, which requires solving convex optimization problems, an highly inefficient task. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with bandit feedback. Specifically, when the constraints are stochastic, the algorithm achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation, while, when they are adversarial, it attains $\widetilde{\mathcal{O}}(\sqrt{T})$ constraint violation and a tight fraction of the optimal reward. Moreover, our algorithm is based on a policy optimization approach, which is much more efficient than occupancy-measure-based methods.