Estimating the errors for solutions of the SAA method to solve compound and risk averse stochastic programs

Abstract

This paper is a study on solutions of the Sample Average Approximation Method to solve compound stochastic programs. We derive nonasymptotic upper estimates for probabilities of the approximation errors. The results depend on the sample size with explicit terms instead of unspecified universal constants. They allow to conclude immediately nonasymptotic rates for the optimal solutions, and they may be utilized to construct nonasymptotic confidence regions for unique solutions of the genuine compound stochastic programs. In the special case of classical risk neutral stochastic programs, we end up with upper estimates of deviation probabilities for M-estimators, and their nonasymptotic rates. Moreover, we may also demonstrate how to apply the results to sample average approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviation risk measures and Average Value at Risk. The investigations are based on concentration inequalities from the recent contribution Kr\"atschmer (2024a). The line of reasoning does not rely on pathwise analytical properties of the objectives. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that objectives with H\"older continuous paths meet the requirements of the main results. Moreover, the main results are applied to objectives whose paths are piecewise H\"older continuous, as e.g. in two stage mixed-integer programs.