  We investigate the problem of best-policy identification in discounted Markov
Decision Processes (MDPs) when the learner has access to a generative model.
The objective is to devise a learning algorithm returning the best policy as
early as possible. We first derive a problem-specific lower bound of the sample
complexity satisfied by any learning algorithm. This lower bound corresponds to
an optimal sample allocation that solves a non-convex program, and hence, is
hard to exploit in the design of efficient algorithms. We then provide a simple
and tight upper bound of the sample complexity lower bound, whose corresponding
nearly-optimal sample allocation becomes explicit. The upper bound depends on
specific functionals of the MDP such as the sub-optimality gaps and the
variance of the next-state value function, and thus really captures the
hardness of the MDP. Finally, we devise KLB-TS (KL Ball Track-and-Stop), an
algorithm tracking this nearly-optimal allocation, and provide asymptotic
guarantees for its sample complexity (both almost surely and in expectation).
The advantages of KLB-TS against state-of-the-art algorithms are discussed and
illustrated numerically.
