Iterative regularization for low complexity regularizers

TL;DR

This paper introduces a novel primal-dual iterative regularization method capable of handling non-smooth, non-strongly convex biases, with proven convergence and stability, especially useful for sparse recovery tasks.

Contribution

It presents the first iterative regularization algorithm for low-complexity regularizers that works with non-smooth, non-strongly convex functionals, expanding the scope of implicit regularization techniques.

Findings

Proven convergence and stability of the proposed algorithm.

Effective handling of non-smooth, non-strongly convex regularizers.

Experimental validation showing computational advantages.

Abstract

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.