Convergence Rate Analysis of Proximal Iteratively Reweighted ${\ell }_1$ Methods for ${\ell }_p$ Regularization Problems

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.