A Bregman inertial forward-reflected-backward method for nonconvex minimization

TL;DR

This paper introduces a novel Bregman inertial forward-reflected-backward method for nonconvex optimization, providing convergence guarantees and resolving a key open question about acceleration, with practical numerical validation.

Contribution

It develops a new inertial method with a step size condition independent of inertial parameters and addresses the acceleration question for FRB methods.

Findings

Convergence of the proposed BiFRB method under certain conditions

Explicit formulas for Bregman subproblems

Numerical results demonstrating effectiveness

Abstract

We propose a Bregman inertial forward-reflected-backward (BiFRB) method for nonconvex composite problems. Our analysis relies on a novel approach that imposes general conditions on implicit merit function parameters, which yields a stepsize condition that is independent of inertial parameters. In turn, a question of Malitsky and Tam regarding whether FRB can be equipped with a Nesterov-type acceleration is resolved. Assuming the generalized concave Kurdyka-{\L}ojasiewicz property of a quadratic regularization of the objective, we obtain sequential convergence of BiFRB, as well as convergence rates on both the function value and actual sequence. We also present formulae for the Bregman subproblem, supplementing not only BiFRB but also the work of Bo\c{t}-Csetnek-L\'aszl\'o and Bo\c{t}-Csetnek. Numerical simulations are conducted to evaluate the performance of our proposed algorithm.