Distributionally Robust Groupwise Regularization Estimator

TL;DR

This paper introduces a distributionally robust optimization framework for groupwise regularization estimators, providing a new interpretation, a data-driven regularization criterion, and demonstrating competitive performance against cross-validation.

Contribution

It formulates GSRL as a DRO game, deriving a data-driven regularization rule with asymptotic optimality and practical advantages over traditional methods.

Findings

The DRO formulation recovers popular estimators like GSRL.

The data-driven regularization criterion performs well in experiments.

The approach offers a new interpretation and selection method for group regularization.

Abstract

Regularized estimators in the context of group variables have been applied successfully in model and feature selection in order to preserve interpretability. We formulate a Distributionally Robust Optimization (DRO) problem which recovers popular estimators, such as Group Square Root Lasso (GSRL). Our DRO formulation allows us to interpret GSRL as a game, in which we learn a regression parameter while an adversary chooses a perturbation of the data. We wish to pick the parameter to minimize the expected loss under any plausible model chosen by the adversary - who, on the other hand, wishes to increase the expected loss. The regularization parameter turns out to be precisely determined by the amount of perturbation on the training data allowed by the adversary. In this paper, we introduce a data-driven (statistical) criterion for the optimal choice of regularization, which we evaluate asymptotically, in closed form, as the size of the training set increases. Our easy-to-evaluate regularization formula is compared against cross-validation, showing good (sometimes superior) performance.