Anderson acceleration for iteratively reweighted $\ell_1$ algorithm

TL;DR

This work proposes an Anderson-accelerated IRL1 algorithm and establishes its local linear convergence rate and introduces a globally convergent Anderson accelerated IRL1 algorithm by incorporating a classical nonmonotone line search condition.

Abstract

Iteratively reweighted L1 (IRL1) algorithm is a common algorithm for solving sparse optimization problems with nonconvex and nonsmooth regularization. The development of its acceleration algorithm, often employing Nesterov acceleration, has sparked significant interest. Nevertheless, the convergence and complexity analysis of these acceleration algorithms consistently poses substantial challenges. Recently, Anderson acceleration has gained prominence owing to its exceptional performance for speeding up fixed-point iteration, with numerous recent studies applying it to gradient-based algorithms. Motivated by the powerful impact of Anderson acceleration, we propose an Anderson-accelerated IRL1 algorithm and establish its local linear convergence rate. We extend this convergence result, typically observed in smooth settings, to a nonsmooth scenario. Importantly, our theoretical results do not depend on the Kurdyka-Lojasiewicz condition, a necessary condition in existing Nesterov acceleration-based algorithms. Furthermore, to ensure global convergence, we introduce a globally convergent Anderson accelerated IRL1 algorithm by incorporating a classical nonmonotone line search condition. Experimental results indicate that our algorithm outperforms existing Nesterov acceleration-based algorithms.

Figures (2)

• Figure 1: Opttol curves over the time and iterations when (m,n,K) = (800,1600,160)
• Figure 2: The performance for different hyperparameter when (m,n,K) = (400,800,80)

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof