Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints

We study episodic constrained Markov decision processes with adversarial losses under stochastic hard constraints. Specifically, starting from a known strictly feasible policy with margin $d$, we seek to obtain optimal regret while satisfying the expected cost constraints in every episode. In this setting, Stradi et al. (2025) show that a carefully designed mixing rule attains regret of order $\tilde{\mathcal{O}}(\sqrt{T}\mathrm{/}\min \{d,d^2\})$. Interestingly, they also provide a lower bound of order $\mathrm{\Omega }(\sqrt{T}\mathrm{/}\rho )$ for the same setting, where $\rho$ is the Slater margin of the offline problem and can be much larger than $d$. In this work, we build on their approach to obtain optimal regret dependence on these margins. Specifically, we propose MA-OPS, an algorithm that combines an optimistic search for the Slater margin with a pessimistic evaluation of the selected policies to safely learn a policy with a large feasibility margin. This policy is then used to minimize regret while satisfying the constraints at every episode. In particular, we show that MA-OPS attains regret $\tilde{\mathcal{O}}(\sqrt{T}\mathrm{/}\rho +1\mathrm{/}(d\rho ))$. Finally, we provide a matching lower bound, showing that the dependence on $T$, $d$, $\rho$ in the regret bound is optimal up to logarithmic factors.