  We study the best-arm identification problem in multi-armed bandits with
stochastic, potentially private rewards, when the goal is to identify the arm
with the highest quantile at a fixed, prescribed level. First, we propose a
(non-private) successive elimination algorithm for strictly optimal best-arm
identification, we show that our algorithm is $\delta$-PAC and we characterize
its sample complexity. Further, we provide a lower bound on the expected number
of pulls, showing that the proposed algorithm is essentially optimal up to
logarithmic factors. Both upper and lower complexity bounds depend on a special
definition of the associated suboptimality gap, designed in particular for the
quantile bandit problem, as we show when the gap approaches zero, best-arm
identification is impossible. Second, motivated by applications where the
rewards are private, we provide a differentially private successive elimination
algorithm whose sample complexity is finite even for distributions with
infinite support-size, and we characterize its sample complexity. Our
algorithms do not require prior knowledge of either the suboptimality gap or
other statistical information related to the bandit problem at hand.
