A mirror inertial forward-reflected-backward splitting: Global convergence and linesearch extension beyond convexity and Lipschitz smoothness

TL;DR

This paper introduces a novel inertial and Bregman extension of the forward-reflected-backward algorithm for nonconvex problems, establishing global convergence and convergence rates under the KL property, with a linesearch enhancement.

Contribution

It proposes a new inertial Bregman splitting algorithm with negative inertial values, a novel envelope function, and a linesearch extension for nonconvex optimization beyond convexity.

Findings

Proves global convergence under the KL property.

Establishes convergence rates for the proposed algorithm.

Provides a linesearch extension to improve practical performance.

Abstract

This work investigates a Bregman and inertial extension of the forward-reflected-backward algorithm [Y. Malitsky and M. Tam, SIAM J. Optim., 30 (2020), pp. 1451--1472] applied to structured nonconvex minimization problems under relative smoothness. To this end, the proposed algorithm hinges on two key features: taking inertial steps in the dual space, and allowing for possibly negative inertial values. Our analysis begins with studying an associated envelope function that takes inertial terms into account through a novel product space formulation. Such construction substantially differs from similar objects in the literature and could offer new insights for extensions of splitting algorithms. Global convergence and rates are obtained by appealing to the generalized concave Kurdyka-Lojasiewicz (KL) property, which allows us to describe a sharp upper bound on the total length of iterates. Finally, a linesearch extension is given to enhance the proposed method.