Tight (Lower) Bounds for the Fixed Budget Best Arm Identification Bandit Problem

05/29/2016

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by   Alexandra Carpentier, et al.

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We consider the problem of best arm identification with a fixed budget T, in the K-armed stochastic bandit setting, with arms distribution defined on [0,1]. We prove that any bandit strategy, for at least one bandit problem characterized by a complexity H, will misidentify the best arm with probability lower bounded by (-T/(K)H), where H is the sum for all sub-optimal arms of the inverse of the squared gaps. Our result disproves formally the general belief - coming from results in the fixed confidence setting - that there must exist an algorithm for this problem whose probability of error is upper bounded by (-T/H). This also proves that some existing strategies based on the Successive Rejection of the arms are optimal - closing therefore the current gap between upper and lower bounds for the fixed budget best arm identification problem.