← 研踪 PaperTrailarXiv:2610.40165 · v1
Closing the Log Gap: Minimax-Optimal Regret for Episodic RL
首次提交 2026-10-29 22:38
版本更新 2026-10-29 22:38
雷达观测 2026-10-30 13:28
摘要
In this paper, we study the regret minimization problem in episodic tabular MDPs. Our algorithm achieves the first expected regret bound scaling as $\sqrt{H^3 S A K}$ for sufficiently large $K$, where $H$ is the episode length, $K$ is the number of episodes, and $S$ and $A$ are the numbers of states and actions, respectively. Our method is the first to match the minimax lower bound \emph{without extra polylog factors}. We introduce a variant of UCBVI which does not maintain strictly optimistic estimates, and uses MOSS-style exploration bonuses whose logarithmic terms adapt to both visit counts and estimated variances. A key challenge is controlling the propagation of errors caused by under-optimism, which we accomplish in an occupancy-weighted way, avoiding logarithmic overhead that would be incurred by uniform-over-rows concentration. Our analysis develops time-uniform and variance-aware concentration bounds that may be of independent interest. As a corollary we also develop an algorithm for $(\varepsilon, δ)$-PAC best-policy identification, which is the first to match the $Ω\big(H^3 SA\log(\frac{1}δ)\varepsilon^{-2}\big)$ minimax lower bound up to constants in some regimes.
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v1 · 2026-10-29 22:38 · 最新观测版本
Closing the Log Gap: Minimax-Optimal Regret for Episodic RL
In this paper, we study the regret minimization problem in episodic tabular MDPs. Our algorithm achieves the first expected regret bound scaling as $\sqrt{H^3 S A K}$ for sufficiently large $K$, where $H$ is the episode length, $K$ is the number of episodes, and $S$ and $A$ are the numbers of states and actions, respectively. Our method is the first to match the minimax lower bound \emph{without extra polylog factors}. We introduce a variant of UCBVI which does not maintain strictly optimistic estimates, and uses MOSS-style exploration bonuses whose logarithmic terms adapt to both visit counts and estimated variances. A key challenge is controlling the propagation of errors caused by under-optimism, which we accomplish in an occupancy-weighted way, avoiding logarithmic overhead that would be incurred by uniform-over-rows concentration. Our analysis develops time-uniform and variance-aware concentration bounds that may be of independent interest. As a corollary we also develop an algorithm for $(\varepsilon, δ)$-PAC best-policy identification, which is the first to match the $Ω\big(H^3 SA\log(\frac{1}δ)\varepsilon^{-2}\big)$ minimax lower bound up to constants in some regimes.
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