Convergence and choice of comparison schemes for discrete optimization using statistical tests

Consider a discrete optimization problem where the objective function is the mean of a random variable and only samples of the random variable are available. A fundamental issue in such a problem is how to compare objective functions through the samples. Ideally, the chosen comparison scheme should lead to an algorithm whose output converges rapidly to the optimum value. In this paper we give some general conditions for convergence and then consider several algorithms having different comparison schemes.