  Various algorithms in reinforcement learning exhibit dramatic variability in
their convergence rates and ultimate accuracy as a function of the problem
structure. Such instance-specific behavior is not captured by existing global
minimax bounds, which are worst-case in nature. We analyze the problem of
estimating optimal $Q$-value functions for a discounted Markov decision process
with discrete states and actions and identify an instance-dependent functional
that controls the difficulty of estimation in the $\ell_\infty$-norm. Using a
local minimax framework, we show that this functional arises in lower bounds on
the accuracy on any estimation procedure. In the other direction, we establish
the sharpness of our lower bounds, up to factors logarithmic in the state and
action spaces, by analyzing a variance-reduced version of $Q$-learning. Our
theory provides a precise way of distinguishing "easy" problems from "hard"
ones in the context of $Q$-learning, as illustrated by an ensemble with a
continuum of difficulty.
