We study sample average approximations under adaptive importance sampling in which the sample densities may depend on previous random samples. Based on a generic uniform law of large numbers, we establish uniform convergence of the sample average approximation to the true function. We obtain convergence of the optimal value and optimal solutions of the sample average approximation. The relevance of this result is demonstrated in the context of the convergence analysis of a randomized optimization algorithm.
Technical Report IEMS Department Northwestern University, October 2017