  We introduce and analyze a form of variance-reduced $Q$-learning. For
$\gamma$-discounted MDPs with finite state space $\mathcal{X}$ and action space
$\mathcal{U}$, we prove that it yields an $\epsilon$-accurate estimate of the
optimal $Q$-function in the $\ell_\infty$-norm using $\mathcal{O}
\left(\left(\frac{D}{ \epsilon^2 (1-\gamma)^3} \right) \; \log \left(
\frac{D}{(1-\gamma)} \right) \right)$ samples, where $D = |\mathcal{X}| \times
|\mathcal{U}|$. This guarantee matches known minimax lower bounds up to a
logarithmic factor in the discount complexity. In contrast, our past work shows
that ordinary $Q$-learning has worst-case quartic scaling in the discount
complexity.
