We fill in a long open gap in the characterization of the minimax rate for the multi-armed bandit problem. Concretely, we remove an extraneous logarithmic factor in the previously known upper bound and propose a new family of randomized algorithms based on an implicit normalization, as well as a new analysis. We also consider the stochastic case, and prove that an appropriate modification of the upper confidence bound policy UCB1 (Auer et al., 2002) achieves the distribution-free optimal rate while still having a distribution-dependent rate logarithmic in the number of plays.