Q-learning with Logarithmic Regret
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This paper presents the first non-asymptotic result showing that a model-free algorithm can achieve a logarithmic cumulative regret for episodic tabular reinforcement learning if there exists a strictly positive sub-optimality gap in the optimal Q-function. We prove that the optimistic Q-learning studied in [Jin et al. 2018] enjoys a O(SA·poly(H)/gap_minlog(SAT)) cumulative regret bound, where S is the number of states, A is the number of actions, H is the planning horizon, T is the total number of steps, and gap_min is the minimum sub-optimality gap. This bound matches the information theoretical lower bound in terms of S,A,T up to a log(SA) factor. We further extend our analysis to the discounted setting and obtain a similar logarithmic cumulative regret bound.
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