Simple regret for infinitely many armed bandits

We consider a stochastic bandit problem with in-finitely 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 sam-ples only to a certain number of arms. All previ-ous 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 cumula-tive regret setting of infinitely many armed ban-dits, the rate of the simple regret will depend on a parameter β characterizing the distribution of the near-optimal arms. We prove that depending on β, 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 β is unknown, in a natural setting where the near-optimal arms have a small vari-ance, and in the case of unknown time horizon. 1