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This paper considers the problem of maximizing an expectation function over afinite set, or finite-arm bandit problem. We first propose a naive stochasticbandit algorithm for obtaining a probably approximately correct (PAC) solutionto this discrete optimization problem in relative precision, that is a solutionwhich solves the optimization problem up to a relative error smaller than aprescribed tolerance, with high probability. We also propose an adaptivestochastic bandit algorithm which provides a PAC-solution with the sameguarantees. The adaptive algorithm outperforms the mean complexity of the naivealgorithm in terms of number of generated samples and is particularly wellsuited for applications with high sampling cost.

A PAC algorithm in relative precision for bandit problem with costly sampling