We study the stochastic multi-armed bandit problem in the P-pass streaming model. In this problem, the n arms are present in a stream and at most m < n arms and their statistics can be stored in the memory. We give a complete characterization of the optimal regret in terms of m, n and P. Specifically, we design an algorithm with regret and complement it with an lower bound when the number of rounds T is sufficiently large. Our results are tight up to a logarithmic factor in n and P.