  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 towards understanding the sample
efficiency of Q-learning. Consider a $\gamma$-discounted infinite-horizon MDP
with state space $\mathcal{S}$ and action space $\mathcal{A}$: to yield an
entrywise $\varepsilon$-approximation of the optimal Q-function,
state-of-the-art theory for Q-learning requires a sample size exceeding the
order of $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^5\varepsilon^{2}}$,
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 sub-optimal? This paper addresses these questions for the synchronous
setting: (1) when $|\mathcal{A}|=1$ (so that Q-learning reduces to TD
learning), we prove that the sample complexity of TD learning is minimax
optimal and scales as $\frac{|\mathcal{S}|}{(1-\gamma)^3\varepsilon^2}$ (up to
log factor); (2) when $|\mathcal{A}|\geq 2$, we settle the sample complexity of
Q-learning to be on the order of
$\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^4\varepsilon^2}$ (up to log
factor). Our theory unveils the strict sub-optimality of Q-learning when
$|\mathcal{A}|\geq 2$, and rigorizes the negative impact of over-estimation 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 $\frac{1}{(1-\gamma)^4}$.
