  We consider a stochastic bandit problem with countably many arms that belong
to a finite set of types, each characterized by a unique mean reward. In
addition, there is a fixed distribution over types which sets the proportion of
each type in the population of arms. The decision maker is oblivious to the
type of any arm and to the aforementioned distribution over types, but
perfectly knows the total number of types occurring in the population of arms.
We propose a fully adaptive online learning algorithm that achieves O(log n)
distribution-dependent expected cumulative regret after any number of plays n,
and show that this order of regret is best possible. The analysis of our
algorithm relies on newly discovered concentration and convergence properties
of optimism-based policies like UCB in finite-armed bandit problems with "zero
gap," which may be of independent interest.
