On rates of convergence for sample average approximations in the almost sure sense and in mean

Abstract

We study the rates at which optimal estimators in the sample average approximation approach converge to their deterministic counterparts in the almost sure sense and in mean. To be able to quantify these rates, we consider the law of the iterated logarithm in a Banach space setting and first establish under relatively mild assumptions almost sure convergence rates for the approximating objective functions, which can then be transferred to the estimators for optimal values and solutions of the approximated problem. By exploiting a characterisation of the law of the iterated logarithm in Banach spaces, we are further able to derive under the same assumptions that the estimators also converge in mean, at a rate which essentially coincides with the one in the almost sure sense. This, in turn, allows to quantify the asymptotic bias of optimal estimators as well as to draw conclusive insights on their mean squared error and on the estimators for the optimality gap. Finally, we address the notion of convergence in probability to derive rates in probability for the deviation of optimal estimators and (weak) rates of error probabilities without imposing strong conditions on exponential moments. We discuss the possibility to construct confidence sets for the optimal values and solutions from our obtained results and provide a numerical illustration of the most relevant findings.