On efficiently solving the subproblems of a level-set method for fused lasso problems

TL;DR

This paper introduces an efficient inexact semismooth Newton augmented Lagrangian method for solving subproblems in a level-set approach to fused lasso problems, significantly improving computational efficiency and robustness.

Contribution

It derives an explicit formula for the generalized Jacobian of the fused lasso proximal mapping and exploits its structure for faster semismooth Newton method implementation.

Findings

Demonstrates high efficiency on large-scale fused lasso problems

Outperforms several state-of-the-art solvers in numerical experiments

Shows robustness across various real data sets

Abstract

In applying the level-set method developed in [Van den Berg and Friedlander, SIAM J. on Scientific Computing, 31 (2008), pp.~890--912 and SIAM J. on Optimization, 21 (2011), pp.~1201--1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop a highly efficient inexact semismooth Newton based augmented Lagrangian method for solving these subproblems. The efficiency of our approach is based on several ingredients that constitute the main contributions of this paper. Firstly, an explicit formula for constructing the generalized Jacobian of the proximal mapping of the fused lasso regularizer is derived. Secondly, the special structure of the generalized Jacobian is carefully extracted and analyzed for the efficient implementation of the semismooth Newton method. Finally, numerical results, including the comparison between our approach and several state-of-the-art solvers, on real data sets are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving challenging large-scale fused lasso problems.