Higher-Order Expansion and Bartlett Correctability of Distributionally Robust Optimization

TL;DR

This paper develops a higher-order expansion for distributionally robust optimization (DRO) that reduces coverage errors via a Bartlett-like correction, applicable to general functionals and revealing a self-normalizing property.

Contribution

It introduces a novel higher-order expansion of DRO and a correction method that improves coverage accuracy, extending beyond existing empirical likelihood approaches.

Findings

DRO with adjusted divergence ball size reduces coverage errors.

The correction applies to general von Mises differentiable functionals.

DRO exhibits a higher-order self-normalizing property regardless of divergence choice.

Abstract

Distributionally robust optimization (DRO) is a worst-case framework for stochastic optimization under uncertainty that has drawn fast-growing studies in recent years. When the underlying probability distribution is unknown and observed from data, DRO suggests to compute the worst-case distribution within a so-called uncertainty set that captures the involved statistical uncertainty. In particular, DRO with uncertainty set constructed as a statistical divergence neighborhood ball has been shown to provide a tool for constructing valid confidence intervals for nonparametric functionals, and bears a duality with the empirical likelihood (EL). In this paper, we show how adjusting the ball size of such type of DRO can reduce higher-order coverage errors similar to the Bartlett correction. Our correction, which applies to general von Mises differentiable functionals, is more general than the existing EL literature that only focuses on smooth function models or $M$-estimation. Moreover, we demonstrate a higher-order "self-normalizing" property of DRO regardless of the choice of divergence. Our approach builds on the development of a higher-order expansion of DRO, which is obtained through an asymptotic analysis on a fixed point equation arising from the Karush-Kuhn-Tucker conditions.