Instance-Dependent Regret for CMDPs with Step-Wise Constraints

We study online learning in episodic tabular constrained Markov decision processes with *step-wise safety constraints*. In such a setting, the constraints induce a safe subgraph that shapes the variance of cumulative rewards under feasible policies and, consequently, the difficulty of learning. Exploiting this structure, however, requires learning which actions are safe while controlling constraint violations. We propose *Safe Variance-Adaptive Exploration* (SVAE), an efficient algorithm that learns candidate safe subgraphs and performs variance-adaptive optimistic planning within them. With high probability, SVAE achieves cumulative regret of order $\tilde{\mathcal{O}}\left(\sqrt{SAH\min \{{\mathbb{V}}_{\Sigma },K{\mathrm{Var}}^{\star }\}}+S\sqrt{A{H}^3\min \{K,\mathcal{C}\}}+S^{2}A{H}^2\right)$ over $K$ episodes, where $H$ is the horizon of a single episode, while $S$ and $A$ are the numbers of states and actions, respectively. Here, ${\mathrm{Var}}^{\star }$ is the maximum return variance among safe policies, ${\mathbb{V}}_{\Sigma }$ is the variance accumulated before the first unsafe action is encountered, and $\mathcal{C}$ captures the statistical complexity of eliminating actions incorrectly considered potentially safe. SVAE additionally attains $\tilde{\mathcal{O}}(H\sqrt{SAK}+S^{2}A{H}^2)$ step-wise constraint violation and a gap-dependent violation bound that is polylogarithmic in $K$. Finally, we establish a lower bound showing that dependence on these instance-specific quantities is unavoidable.