Determination of the Minimax Risk for Bernoulli Multi-Armed Bandit

Abstract

Bernoulli multi-armed bandit with arbitrary set of unknown parameters is considered. By varying the strategy, it is shown that minimax risk and strategy over the whole set of parameters are equal to those over some finite subset of its closure. According to the main theorem of the theory of games minimax strategy and risk on the finite set of parameters are determined as Bayes ones corresponding to the worst prior distribution. These properties allow to reduce the problem to determination of the global maximum of the function depending on finite number of variables. On the other hand they provide a convenient method to represent and to store determined strategy. Some numerical examples are presented.