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We consider a version of the classical stochastic Multi-Armed bandit problem in which the number of arms is large compared to the time horizon, with the goal of minimizing the cumulative regret. Here, the mean-reward (or value) of newly chosen arms is assumed to be i.i.d. We further make the simplifying assumption that the value of an arm is revealed once this arm is chosen. We present a general lower bound on the regret, and learning algorithms that achieve this bound up to a logarithmic factor. Contrary to previous work, we do not assume that the functional form of the tail of the value distribution is known. Furthermore, we also consider a variant of our model where sampled arms are non-retainable, namely are lost if not used continuously, with similar near-optimality results.

Infinitely Many-Armed Bandits with Unknown Value Distribution