Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis

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 and action space 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 , 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 (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 (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 .