Level-set Subdifferential Error Bounds and Linear Convergence of Variable Bregman Proximal Gradient Method

TL;DR

This paper introduces a new level-set subdifferential error bound condition that guarantees linear convergence of the variable Bregman proximal gradient method for nonsmooth, nonconvex optimization, expanding understanding of convergence conditions.

Contribution

It establishes a weaker error bound condition than existing properties, providing verifiable criteria for linear convergence and insights into the behavior of the VBPG method.

Findings

Guarantees linear convergence under the new error bound condition.

Provides verifiable conditions for the error bounds to hold.

Shows accumulation points are critical or proximal critical points.

Abstract

In this work, we develop a level-set subdifferential error bound condition aiming towards convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. It is proved that the aforementioned condition guarantees linear convergence of VBPG, and is weaker than Kurdyka-Lojasiewicz property, weak metric subregularity and Bregman proximal error bound. Along the way, we are able to derive a number of verifiable conditions for level-set subdifferential error bounds to hold, and necessary conditions and sufficient conditions for linear convergence relative to a level set for nonsmooth and nonconvex optimization problems. The newly established results not only enable us to show that any accumulation point of the sequence generated by VBPG is at least a critical point of the limiting subdifferential or even acritical point of the proximal subdifferential with a fixed Bregman function in each iteration, but also provide a fresh perspective that allows us to explore inner-connections among many known sufficient conditions for linear convergence of various first-order methods.