Minimax policies for adversarial and stochastic bandits
Résumé
We fill in a long open gap in the characterization of the minimax rate for the multi-armed bandit prob- lem. Concretely, we remove an extraneous loga- rithmic 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 log- arithmic in the number of plays.
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