Span-Agnostic Optimal Sample Complexity and Oracle Inequalities for Average-Reward RL
This resolves a long-standing open problem in reinforcement learning theory by enabling optimal sample efficiency without requiring prior knowledge of the MDP's span, which is crucial for practical applications.
The paper tackles the problem of finding an epsilon-optimal policy in average-reward Markov Decision Processes without prior knowledge of the span parameter H, achieving the minimax optimal sample complexity of O(SAH/epsilon^2) for the first time in this setting.
We study the sample complexity of finding an $\varepsilon$-optimal policy in average-reward Markov Decision Processes (MDPs) with a generative model. The minimax optimal span-based complexity of $\widetilde{O}(SAH/\varepsilon^2)$, where $H$ is the span of the optimal bias function, has only been achievable with prior knowledge of the value of $H$. Prior-knowledge-free algorithms have been the objective of intensive research, but several natural approaches provably fail to achieve this goal. We resolve this problem, developing the first algorithms matching the optimal span-based complexity without $H$ knowledge, both when the dataset size is fixed and when the suboptimality level $\varepsilon$ is fixed. Our main technique combines the discounted reduction approach with a method for automatically tuning the effective horizon based on empirical confidence intervals or lower bounds on performance, which we term horizon calibration. We also develop an empirical span penalization approach, inspired by sample variance penalization, which satisfies an oracle inequality performance guarantee. In particular this algorithm can outperform the minimax complexity in benign settings such as when there exist near-optimal policies with span much smaller than $H$.