Distributionally Robust Optimization and Robust Statistics

TL;DR

This paper reviews distributionally robust optimization (DRO), connecting it to classical robust statistics, highlighting their differences, and explaining how many well-known estimators can be interpreted through the DRO lens.

Contribution

It clarifies the relationship between DRO and classical robustness, and shows how many existing estimators are inherently distributionally robust.

Findings

DRO provides a min-max framework for hedging against environment shifts.

Many classical estimators like LASSO and dropout are shown to be distributionally robust.

DRO and classical robustness are fundamentally different philosophies with distinct estimator types.

Abstract

We review distributionally robust optimization (DRO), a principled approach for constructing statistical estimators that hedge against the impact of deviations in the expected loss between the training and deployment environments. Many well-known estimators in statistics and machine learning (e.g. AdaBoost, LASSO, ridge regression, dropout training, etc.) are distributionally robust in a precise sense. We hope that by discussing the DRO interpretation of well-known estimators, statisticians who may not be too familiar with DRO may find a way to access the DRO literature through the bridge between classical results and their DRO equivalent formulation. On the other hand, the topic of robustness in statistics has a rich tradition associated with removing the impact of contamination. Thus, another objective of this paper is to clarify the difference between DRO and classical statistical robustness. As we will see, these are two fundamentally different philosophies leading to completely different types of estimators. In DRO, the statistician hedges against an environment shift that occurs after the decision is made; thus DRO estimators tend to be pessimistic in an adversarial setting, leading to a min-max type formulation. In classical robust statistics, the statistician seeks to correct contamination that occurred before a decision is made; thus robust statistical estimators tend to be optimistic leading to a min-min type formulation.