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This research introduces Safe Variance-Adaptive Exploration (SVAE), an algorithm designed to optimize learning in constrained Markov decision processes by adapting to the variance inherent within safe subgraphs. The study establishes new theoretical bounds for cumulative regret and constraint violations, demonstrating that performance gains depend critically on instance-specific safety structures rather than uniform exploration strategies. These findings advance the theoretical foundations of safe reinforcement learning, offering a more nuanced approach to balancing reward maximization with strict safety requirements in episodic tasks.
Instance-Dependent Regret for CMDPs with Step-Wise Constraints
arXiv:2610.02520v1 Announce Type: cross Abstract: 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 $\widetilde{\mathcal{O}}(\sqrt{SAH\min\{\mathbb{V}_\Sigma,K\mathrm{Var}^{\star}\}}+S\sqrt{AH^3\min\{K,\mathcal{C}\}}+S^2AH^2)$ 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 $\widetilde{\mathcal{O}}(H\sqrt{SAK}+S^2AH^2)$ 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.
Abstract: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 $\widetilde{\mathcal{O}}(\sqrt{SAH\min\{\mathbb{V}_\Sigma,K\mathrm{Var}^{\star}\}}+S\sqrt{AH^3\min\{K,\mathcal{C}\}}+S^2AH^2)$ 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 $\widetilde{\mathcal{O}}(H\sqrt{SAK}+S^2AH^2)$ 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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2610.02520 [cs.LG]
(or arXiv:2610.02520v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.02520 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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