  This paper considers a multi-armed bandit game where the number of arms is
much larger than the maximum budget and is effectively infinite. We
characterize necessary and sufficient conditions on the total budget for an
algorithm to return an {\epsilon}-good arm with probability at least 1 -
{\delta}. In such situations, the sample complexity depends on {\epsilon},
{\delta} and the so-called reservoir distribution {\nu} from which the means of
the arms are drawn iid. While a substantial literature has developed around
analyzing specific cases of {\nu} such as the beta distribution, our analysis
makes no assumption about the form of {\nu}. Our algorithm is based on
successive halving with the surprising exception that arms start to be
discarded after just a single pull, requiring an analysis that goes beyond
concentration alone. The provable correctness of this algorithm also provides
an explanation for the empirical observation that the most aggressive bracket
of the Hyperband algorithm of Li et al. (2017) for hyperparameter tuning is
almost always best.
