Robust Wasserstein Profile Inference and Applications to Machine Learning

TL;DR

This paper connects machine learning estimators to distributionally robust optimization using Wasserstein distances, introduces RWPI for optimal uncertainty region selection, and eliminates the need for cross-validation in regularization parameter tuning.

Contribution

It presents a novel interpretation of regularization as adversarial distribution perturbation and introduces RWPI for data-driven regularization parameter selection.

Findings

Regularization can be viewed as adversarial distribution perturbation.

RWPI enables optimal uncertainty region size selection.

Regularization parameters can be chosen without cross-validation.

Abstract

We show that several machine learning estimators, including square-root LASSO (Least Absolute Shrinkage and Selection) and regularized logistic regression can be represented as solutions to distributionally robust optimization (DRO) problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (Robust Wasserstein Profile Inference), a novel inference methodology which extends the use of methods inspired by Empirical Likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence, we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.