Robust Stochastic Approximation Approach to Stochastic Programming

• Published in: SIAM journal on optimization Vol. 19; no. 4; pp. 1574 - 1609
• Main Authors: Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
• Subjects: Algorithms; Approximation; Expected values; Experiments; Mathematics; Methods; Optimization; Sampling techniques; Statistics; Statistics Theory; complexity; minimax problems; sample average approximation method; stochastic programming; saddle point; stochastic approximation; Monte Carlo sampling; mirror descent algorithm
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/070704277

In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte Carlo sampling techniques, namely, the stochastic approximation (SA) and the sample average approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say, linear) structure of the considered problem, while the SA approach is a crude subgradient method, which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convex-concave stochastic saddle point problems and present (in our opinion highly encouraging) results of numerical experiments.