How to induce regularization in linear models: A guide to reparametrizing gradient flow

TL;DR

This paper investigates how reparametrizations of gradient flow influence the implicit regularization in linear models, providing conditions for convergence and methods to design reparametrizations for desired biases.

Contribution

It offers a theoretical framework linking reparametrizations, loss functions, and implicit bias in linear models, guiding the design of regularization techniques.

Findings

Conditions for implicit bias characterization

Guarantees for convergence of gradient flow

Design principles for reparametrizations leading to specific regularizers

Abstract

In this work, we analyze the relation between reparametrizations of gradient flow and the induced implicit bias in linear models, which encompass various basic regression tasks. In particular, we aim at understanding the influence of the model parameters - reparametrization, loss, and link function - on the convergence behavior of gradient flow. Our results provide conditions under which the implicit bias can be well-described and convergence of the flow is guaranteed. We furthermore show how to use these insights for designing reparametrization functions that lead to specific implicit biases which are closely connected to $\ell_p$- or trigonometric regularizers.