Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

TL;DR

This paper introduces a generalized empirical likelihood framework for statistical inference in stochastic optimization, providing confidence intervals with exact coverage and insights into the regularization effects of distributional robustness.

Contribution

It develops a novel empirical likelihood approach based on f-divergence balls for stochastic optimization, enabling precise confidence intervals and analysis of robustness effects.

Findings

Provides a method for choosing distributional uncertainty set sizes.

Derives an asymptotic expansion showing variance regularization.

Shows robust solutions maintain classical consistency properties.

Abstract

We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric $f$-divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.