A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

TL;DR

This paper introduces two accelerated gradient algorithms extending FISTA to solve smooth nonconvex composite optimization problems, providing iteration complexity analysis and demonstrating efficiency through numerical experiments.

Contribution

It develops two new ACG variants for nonconvex problems, with one extending FISTA and the other starting from arbitrary parameters, both with proven complexity bounds.

Findings

Both algorithms effectively solve nonconvex composite problems.

The second variant performs better with arbitrary input pairs.

Numerical results confirm the efficiency of the proposed methods.

Abstract

In this paper, we describe and establish iteration-complexity of two accelerated composite gradient (ACG) variants to solve a smooth nonconvex composite optimization problem whose objective function is the sum of a nonconvex differentiable function $ f $ with a Lipschitz continuous gradient and a simple nonsmooth closed convex function $ h $. When $f$ is convex, the first ACG variant reduces to the well-known FISTA for a specific choice of the input, and hence the first one can be viewed as a natural extension of the latter one to the nonconvex setting. The first variant requires an input pair $(M,m)$ such that $f$ is $m$-weakly convex, $\nabla f$ is $M$-Lipschitz continuous, and $m \le M$ (possibly $m<M$), which is usually hard to obtain or poorly estimated. The second variant on the other hand can start from an arbitrary input pair $(M,m)$ of positive scalars and its complexity is shown to be not worse, and better in some cases, than that of the first variant for a large range of the input pairs. Finally, numerical results are provided to illustrate the efficiency of the two ACG variants.