Policy Optimization for CMDPs with Bandit Feedback: Learning Stochastic and Adversarial Constraints

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.

We introduce the first algorithm for constrained Markov decision processes (CMDPs) with bandit feedback to effectively handle both stochastic and adversarial constraints. It achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation in the stochastic case. When the constraints are adversarial the algorithm attains $\widetilde{\mathcal{O}}(\sqrt{T})$ constraint violation and a tight fraction of the optimal reward. Unlike previous methods that require full feedback and rely on occupancy-measure optimization, our approach is based on efficient policy optimization.