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


Alexandra Carpentier, Andrea Locatelli ;
29th Annual Conference on Learning Theory, PMLR 49:590-604, 2016.


We consider the problem of \textitbest arm identification with a \textitfixed 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 $\exp\Big(-\frac{T}\log(K)H\Big)$, 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 $\exp(-T/H)$. 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.

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