A proximal-proximal majorization-minimization algorithm for nonconvex tuning-free robust regression problems

TL;DR

This paper presents a novel proximal-proximal majorization-minimization algorithm for nonconvex, tuning-free robust regression, effectively solving complex subproblems with convergence guarantees and outperforming existing methods.

Contribution

The paper introduces a new PPMM algorithm combining proximal majorization-minimization with SSN-based PPA for nonconvex robust regression problems.

Findings

Algorithm converges to a d-stationary point.

Proven convergence rate based on Kurdyka-Lojasiewicz property.

Numerical experiments show superior performance over existing algorithms.

Abstract

In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex tuning-free robust regression problems. The basic idea is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). We must emphasize that the main difficulty in the design of the algorithm lies in how to overcome the singular difficulty of the inner subproblem. Furthermore, we also prove that the PPMM algorithm converges to a d-stationary point. Due to the Kurdyka-Lojasiewicz (KL) property of the problem, we present the convergence rate of the PPMM algorithm. Numerical experiments demonstrate that our proposed algorithm outperforms the existing state-of-the-art algorithms.