Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs

TL;DR

This paper derives nonasymptotic upper bounds for the errors of the Sample Average Approximation method in stochastic programming, applicable to risk-averse problems and based on new empirical process conditions.

Contribution

It introduces a novel approach to estimate errors without requiring pathwise continuity, extending applicability to piecewise H"older continuous functions.

Findings

Provides explicit convergence rate estimates for the Sample Average Approximation

Applies the results to risk-averse stochastic programs with divergence risk measures

Shows the method's effectiveness for goal functions with piecewise H"older continuity

Abstract

We study statistical properties of the optimal value of the Sample Average Approximation. The focus is on the tail function of the absolute error induced by the Sample Average Approximation, deriving upper estimates of its outcomes dependent on the sample size. The estimates allow to conclude immediately convergence rates for the optimal value of the Sample Average Approximation. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity in the parameter is not imposed in advance as often in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are H\"older continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise H\"older continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the sample average approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of mean upper semideviations and divergence risk measures.