Optimal Variance-Dependent Regret Bounds for Infinite-Horizon MDPs

Abstract

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $\gamma$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}(\sqrt{SA\text{Var}}+\text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $\gamma$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^{\star }{\Vert }_{\text{sp}}$, our algorithm obtains lower-order terms scaling as $\Vert h^{\star }{\Vert }_{\text{sp}}{S}^{2}A$, which we prove is optimal in both $\Vert h^{\star }{\Vert }_{\text{sp}}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^{\star }{\Vert }_{\text{sp}}^{2}SA$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^{\star }{\Vert }_{\text{sp}}^2{S}^{3}A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^{\star }{\Vert }_{\text{sp}}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.