Tail Distribution of Regret in Optimistic Reinforcement Learning

• URL: http://arxiv.org/abs/2511.18247v1
• Date: Sun, 23 Nov 2025 02:23:09 GMT
• Title: Tail Distribution of Regret in Optimistic Reinforcement Learning
• Authors: Sajad Khodadadian, Mehrdad Moharrami,
• Abstract summary: We characterize the tail distribution of the cumulative regret $R_K$ over $K$ episodes, rather than only its expectation or a single high-probability quantile.<n>A proposed algorithm depends on a tuning parameter $$, which balances the expected regret and the range over which the regret exhibits a sub-Gaussian tail.
• Score: 3.9962751777898955
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We derive instance-dependent tail bounds for the regret of optimism-based reinforcement learning in finite-horizon tabular Markov decision processes with unknown transition dynamics. Focusing on a UCBVI-type algorithm, we characterize the tail distribution of the cumulative regret $R_K$ over $K$ episodes, rather than only its expectation or a single high-probability quantile. We analyze two natural exploration-bonus schedules: (i) a $K$-dependent scheme that explicitly incorporates the total number of episodes $K$, and (ii) a $K$-independent scheme that depends only on the current episode index. For both settings, we obtain an upper bound on $\Pr(R_K \ge x)$ that exhibits a distinctive two-regime structure: a sub-Gaussian tail starting from an instance-dependent scale $m_K$ up to a transition threshold, followed by a sub-Weibull tail beyond that point. We further derive corresponding instance-dependent bounds on the expected regret $\mathbb{E}[R_K]$. The proposed algorithm depends on a tuning parameter $α$, which balances the expected regret and the range over which the regret exhibits a sub-Gaussian tail. To the best of our knowledge, our results provide one of the first comprehensive tail-regret guarantees for a standard optimistic algorithm in episodic reinforcement learning.

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