  We consider a stochastic bandit problem with infinitely many arms. In this
setting, the learner has no chance of trying all the arms even once and has to
dedicate its limited number of samples only to a certain number of arms. All
previous algorithms for this setting were designed for minimizing the
cumulative regret of the learner. In this paper, we propose an algorithm aiming
at minimizing the simple regret. As in the cumulative regret setting of
infinitely many armed bandits, the rate of the simple regret will depend on a
parameter $\beta$ characterizing the distribution of the near-optimal arms. We
prove that depending on $\beta$, our algorithm is minimax optimal either up to
a multiplicative constant or up to a $\log(n)$ factor. We also provide
extensions to several important cases: when $\beta$ is unknown, in a natural
setting where the near-optimal arms have a small variance, and in the case of
unknown time horizon.
