Convergence Rate Analysis of Proximal Iteratively Reweighted $\ell_1$ Methods for $\ell_p$ Regularization Problems

TL;DR

This paper analyzes the local convergence rates of proximal iteratively reweighted $ ext{l}_1$ algorithms for $ ext{l}_p$ regularization, showing conditions for linear or sublinear convergence and improving upon existing theoretical results.

Contribution

It provides a stronger theoretical analysis of convergence rates for these algorithms under the Kurdyka-Lojasiewicz property, including conditions for linear and sublinear convergence.

Findings

Convergence to a unique first-order stationary point under KL property

Local linear convergence established for certain conditions

Theoretical results surpass previous analyses in strength

Abstract

In this paper, we focus on the local convergence rate analysis of the proximal iteratively reweighted $\ell_1$ algorithms for solving $\ell_p$ regularization problems, which are widely applied for inducing sparse solutions. We show that if the Kurdyka-Lojasiewicz (KL) property is satisfied, the algorithm converges to a unique first-order stationary point; furthermore, the algorithm has local linear convergence or local sublinear convergence. The theoretical results we derived are much stronger than the existing results for iteratively reweighted $\ell_1$ algorithms.