Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization

TL;DR

This paper demonstrates how distributionally robust optimization with empirical divergence balls can recover statistical guarantees akin to the Central Limit Theorem, linking DRO with empirical likelihood and processes for improved performance.

Contribution

It introduces a novel approach using empirical divergence-based DRO to achieve asymptotic statistical guarantees, connecting it with empirical likelihood and calibration via chi-square processes.

Findings

Empirical divergence-based DRO attains CLT-like guarantees.

Calibration of divergence balls improves statistical coverage.

Numerical experiments support theoretical results.

Abstract

We investigate the use of distributionally robust optimization (DRO) as a tractable tool to recover the asymptotic statistical guarantees provided by the Central Limit Theorem, for maintaining the feasibility of an expected value constraint under ambiguous probability distributions. We show that using empirically defined Burg-entropy divergence balls to construct the DRO can attain such guarantees. These balls, however, are not reasoned from the standard data-driven DRO framework since by themselves they can have low or even zero probability of covering the true distribution. Rather, their superior statistical performances are endowed by linking the resulting DRO with empirical likelihood and empirical processes. We show that the sizes of these balls can be optimally calibrated using chi-square process excursion. We conduct numerical experiments to support our theoretical findings.