Optimal Strong Regret and Violation in Constrained MDPs via Policy Optimization

Francesco Emanuele Stradi, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti

Keywords: CMDP, strong regret, strong violations, primal-dual

Abstract: We study online learning in constrained MDPs (CMDPs), focusing on the goal of attaining sublinear strong regret and strong cumulative constraint violation. Differently from their standard (weak) counterparts, these metrics do not allow negative terms to compensate positive ones, raising considerable additional challenges. Efroni et al. (2020) were the first to propose an algorithm with sublinear strong regret and strong violation, by exploiting linear programming. Thus, their algorithm is highly inefficient, leaving as an open problem achieving sublinear bounds by means of policy optimization methods, which are much more efficient in practice. Very recently, Muller et al. (2024) have partially addressed this problem by proposing a policy optimization method that allows to attain $\widetilde{\mathcal{O}}(T^{0.93})$ strong regret/violation. This still leaves open the question of whether optimal bounds are achievable by using an approach of this kind. We answer such a question affirmatively, by providing an efficient policy optimization algorithm with $\widetilde{\mathcal{O}}(\sqrt{T})$ strong regret/violation. Our algorithm implements a primal-dual scheme that employs a state-of-the-art policy optimization approach for adversarial (unconstrained) MDPs as primal algorithm, and a UCB-like update for dual variables.

Primary Area: reinforcement learning

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 10392