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

Instance-optimality in optimal value estimation: Adaptivity via variance-reduced Q-learning