We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $Delta$ for the minimal mean-gap between optimal and sub-optimal arms. We characterize the optimal learning rates both in the cumulative regret setting, and in the best-arm identification setting in terms of the problem parameters $T$ (the budget), $p^*$ and $Delta$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $Omega(log(T)/(p^*Delta))$ and a UCB-style algorithm with matching upper bound up to a factor of $log(1/Delta)$. Our algorithm needs $p^*$ to calibrate its parameters, and we prove that this knowledge is necessary, since adapting to $p^*$ in this setting is impossible. For best-arm identification we also provide a lower bound of order $Omega(exp(-cTDelta^2p^*))$ on the probability of outputting a sub-optimal arm where $c>0$ is an absolute constant. We also provide an elimination algorithm with an upper bound matching the lower bound up to a factor of order $log(1/Delta)$ in the exponential, and that does not need $p^*$ or $Delta$ as parameter.
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