On Rates of Convergence for Stochastic Optimization Problems Under Non-I.I.D. Sampling

Abstract

In this paper we discuss the issue of solving stochastic optimization problems bymeans of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. Thisis a well studied problem in case the samples are independent and identically distributed (i.e., when standard Monte Carlo is used); here, we study the case where thatassumption is dropped. Broadly speaking, our results show that, under appropriate assumptions, the rates of convergence for pointwise estimators under a sampling schemecarry over to the optimization case, in the sense that convergence of approximatingoptimal solutions and optimal values to their true counterparts has the same rates asin pointwise estimation. Our motivation for the study arises from two types of sampling methods that havebeen widely used in the Statistics literature. One is Latin Hypercube Sampling (LHS),a stratified sampling method originally proposed in the seventies by McKay, Beckman,and Conover (1979). The other is the class of quasi-Monte Carlo (QMC) methods,which have become popular especially after the work of Niederreiter (1992). Theadvantage of such methods is that they typically yield pointwise estimators which notonly have lower variance than standard Monte Carlo but also possess better rates ofconvergence. Thus, it is important to study the use of these techniques in sampling-based optimization. The novelty of our work arises from the fact that, while therehas been some work on the use of variance reduction techniques and QMC methods instochastic optimization, none of the existing work — to the best of our knowledge — hasprovided a theoretical study on the effect of these techniques on rates of convergence forthe optimization problem. We present numerical results for some two-stage stochasticprograms from the literature to illustrate the discussed ideas.