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 recently towards understanding the sample efficiency of Q-learning. Take a $gamma$-discounted infinite-horizon MDP with state space $mathcal{S}$ and action space $mathcal{A}$: to yield an entrywise $varepsilon$-accurate estimate of the optimal Q-function, state-of-the-art theory for Q-learning proves that a sample size on the order of $frac{|mathcal{S}||mathcal{A}|}{(1-gamma)^5varepsilon^{2}}$ is sufficient, which, however, fails to match with the existing minimax lower bound. This gives rise to natural questions: what is the sharp sample complexity of Q-learning? Is Q-learning provably sub-optimal? In this work, we settle these questions by (1) demonstrating that the sample complexity of Q-learning is at most on the order of $frac{|mathcal{S}||mathcal{A}|}{(1-gamma)^4varepsilon^2}$ (up to some log factor) for any $0<varepsilon <1$, and (2) developing a matching lower bound to confirm the sharpness of our result. Our findings unveil both the effectiveness and limitation of Q-learning: its sample complexity matches that of speedy Q-learning without requiring extra computation and storage, albeit still being considerably higher than the minimax lower bound.
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