Data- and Variance-dependent Regret Bounds for Online Tabular MDPs

• URL: http://arxiv.org/abs/2602.01903v1
• Date: Mon, 02 Feb 2026 10:09:29 GMT
• Title: Data- and Variance-dependent Regret Bounds for Online Tabular MDPs
• Authors: Mingyi Li, Taira Tsuchiya, Kenji Yamanishi,
• Abstract summary: We develop best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the regime.<n>For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon.
• Score: 15.092125124258592
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime. We quantify MDP complexity using a first-order quantity and several new data-dependent measures for the adversarial regime, including a second-order quantity and a path-length measure, as well as variance-based measures for the stochastic regime. To adapt to these measures, we develop algorithms based on global optimization and policy optimization, both built on optimistic follow-the-regularized-leader with log-barrier regularization. For global optimization, our algorithms achieve first-order, second-order, and path-length regret bounds in the adversarial regime, and in the stochastic regime, they achieve a variance-aware gap-independent bound and a variance-aware gap-dependent bound that is polylogarithmic in the number of episodes. For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon, by exploiting a new optimistic $Q$-function estimator. Finally, we establish regret lower bounds in terms of data-dependent complexity measures for the adversarial regime and a variance measure for the stochastic regime, implying that the regret upper bounds achieved by the global-optimization approach are nearly optimal.

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