Sparse Stochastic Bandits

In the classical multi-armed bandit problem, d arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with d (or with sqrt(d) in the minimax sense). We here consider the sparse case of this classical problem in the sense that only a small number of arms, namely s < d, have a positive expected reward. We are able to leverage this additional assumption to provide an algorithm whose regret scales with s instead of d. Moreover, we prove that this algorithm is optimal by providing a matching lower bound - at least for a wide and pertinent range of parameters that we determine - and by evaluating its performance on simulated data.