Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis
Gen Li, Changxiao Cai, Yuxin Chen, Yuting Wei, Yuejie Chi
Operations Research · 2024 · Volume 72 · Issue 1
Abstract
Q-learning, which seeks to learn the optimal Q-function of a Markov decision process (MDP) in a model-free fashion, lies at the heart of reinforcement learning. When it comes to the synchronous setting (such that independent samples for all state–action pairs are drawn from a generative model in each iteration), substantial progress has been made toward understanding the sample efficiency of Q-learning. Consider a γ -discounted infinite-horizon MDP with state space 𝒮 S and action space 𝒜 A : to yield an entry-wise ε -approximation of the optimal Q-function, state-of-the-art theory for Q-learning requires a sample size exceeding the order of | 𝒮 ‖ 𝒜 | ( 1 − 𝛾 ) 5 𝜀 2 | S ‖ A | ( 1 − γ ) 5 ε 2 , which fails to match existing minimax lower bounds. This gives rise to natural questions: What is the sharp sample complexity of Q-learning? Is Q-learning provably suboptimal? This paper addresses these questions for the synchronous setting: (1) when the action space contains a single action (so that Q-learning reduces to TD learning), we prove that the sample complexity of TD learning is minimax optimal and scales as | 𝒮 | ( 1 − 𝛾 ) 3 𝜀 2 | S | ( 1 − γ ) 3 ε 2 (up to log factor); (2) when the action space contains at least two actions, we settle the sample complexity of Q-learning to be on the order of | 𝒮 ‖ 𝒜 | ( 1 − 𝛾 ) 4 𝜀 2 | S ‖ A | ( 1 − γ ) 4 ε 2 (up to log factor). Our theory unveils the strict suboptimality of Q-learning when the action space contains at least two actions and rigorizes the negative impact of overestimation in Q-learning. Finally, we extend our analysis to accommodate asynchronous Q-learning (i.e., the case with Markovian samples), sharpening the horizon dependency of its sample complexity to be 1 ( 1 − 𝛾 ) 4 1 ( 1 − γ ) 4 . Funding: Y. Chen is supported in part by the Alfred P. Sloan Research Fellowship, the Google Research Scholar Award, the Air Force Office of Scientific Research [Grant FA9550-22-1-0198], the Office of Naval Research (ONR) [Grant N00014-22-1-2354], and the National Science Foundation (NSF) [Grants CCF-2221009, CCF-1907661, DMS-2014279, IIS-2218713, and IIS-2218773]. Y. Wei is supported in part by the Google Research Scholar Award and the NSF [Grants CCF-2106778, DMS-2147546/2015447, and CAREER award DMS-2143215]. Y. Chi is supported in part by the ONR [Grants N00014-18-1-2142 and N00014-19-1-2404] and the NSF [Grants CCF-1806154, CCF-2007911, CCF-2106778, ECCS-2126634, and DMS-2134080]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.2450 .
Keywords
Machine Learning and Data Science, Q-learning, temporal difference learning, effective horizon, sample complexity, minimax optimality, lower bound, overestimation
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