Alternating Linearization for Structured Regularization Problems

TL;DR

This paper introduces an adapted alternating linearization method for structured regularization, especially generalized lasso problems, demonstrating its convergence, scalability, and effectiveness through numerical experiments.

Contribution

It develops a novel alternating linearization approach related to operator splitting methods, with descent properties and practical implementation strategies for large-scale problems.

Findings

Method shows scalability to large problems

Effective for generalized lasso and fused lasso

Numerical results demonstrate accuracy and efficiency

Abstract

We adapt the alternating linearization method for proximal decomposition to structured regularization problems, in particular, to the generalized lasso problems. The method is related to two well-known operator splitting methods, the Douglas--Rachford and the Peaceman--Rachford method, but it has descent properties with respect to the objective function. This is achieved by employing a special update test, which decides whether it is beneficial to make a Peaceman--Rachford step, any of the two possible Douglas--Rachford steps, or none. The convergence mechanism of the method is related to that of bundle methods of nonsmooth optimization. We also discuss implementation for very large problems, with the use of specialized algorithms and sparse data structures. Finally, we present numerical results for several synthetic and real-world examples, including a three-dimensional fused lasso problem, which illustrate the scalability, efficacy, and accuracy of the method.