Characterization of the equivalence of robustification and regularization in linear and matrix regression

TL;DR

This paper explores the precise conditions under which robustification against adversarial data perturbations is equivalent to regularization in linear and matrix regression, extending previous understanding to broader models.

Contribution

It provides a detailed characterization of when robustification and regularization are equivalent in linear and matrix regression, including new insights for matrix problems.

Findings

Characterizes conditions for equivalence in linear regression

Extends analysis to matrix regression problems

Provides theoretical foundation for robustification-regularization link

Abstract

The notion of developing statistical methods in machine learning which are robust to adversarial perturbations in the underlying data has been the subject of increasing interest in recent years. A common feature of this work is that the adversarial robustification often corresponds exactly to regularization methods which appear as a loss function plus a penalty. In this paper we deepen and extend the understanding of the connection between robustification and regularization (as achieved by penalization) in regression problems. Specifically, (a) in the context of linear regression, we characterize precisely under which conditions on the model of uncertainty used and on the loss function penalties robustification and regularization are equivalent, and (b) we extend the characterization of robustification and regularization to matrix regression problems (matrix completion and Principal Component Analysis).