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