We characterize Bayesian regret in a stochastic multi-armed bandit problem with a large but finite number of arms. In particular, we assume the number of arms k is T^{α}, where T is the time-horizon and α is in (0,1). We consider a Bayesian setting where the reward distribution of each arm is drawn independently from a common prior, and provide a complete analysis of expected regret with respect to this prior. Our results exhibit a sharp distinction around α = 1/2. When α < 1/2, the fundamental lower bound on regret is Ω(k); and it is achieved by a standard UCB algorithm. When α > 1/2, the fundamental lower bound on regret is Ω (√T), and it is achieved by an algorithm that first subsamples √T arms uniformly at random, then runs UCB on just this subset. Interestingly, we also find that a sufficiently large number of arms allows the decision-maker to benefit from "free" exploration if she simply uses a greedy algorithm. In particular, this greedy algorithm exhibits a regret of Õ(max(k,T/√k)), which translates to a {\em sublinear} (though not optimal) regret in the time horizon. We show empirically that this is because the greedy algorithm rapidly disposes of underperforming arms, a beneficial trait in the many-armed regime. Technically, our analysis of the greedy algorithm involves a novel application of the Lundberg inequality, an upper bound for the ruin probability of a random walk; this approach may be of independent interest.
