Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization

TL;DR

This paper analyzes the convergence of an accelerated proximal gradient method with momentum for nonconvex optimization, providing theoretical guarantees and proposing stochastic and adaptive variants to improve performance.

Contribution

It offers a rigorous convergence analysis of APGnc for nonconvex problems, introduces stochastic and inexact variants, and develops an adaptive momentum strategy.

Findings

Limit points are critical points of the objective.

Establishes linear and sub-linear convergence rates.

Proposes stochastic variance reduced and adaptive momentum methods.

Abstract

In many modern machine learning applications, structures of underlying mathematical models often yield nonconvex optimization problems. Due to the intractability of nonconvexity, there is a rising need to develop efficient methods for solving general nonconvex problems with certain performance guarantee. In this work, we investigate the accelerated proximal gradient method for nonconvex programming (APGnc). The method compares between a usual proximal gradient step and a linear extrapolation step, and accepts the one that has a lower function value to achieve a monotonic decrease. In specific, under a general nonsmooth and nonconvex setting, we provide a rigorous argument to show that the limit points of the sequence generated by APGnc are critical points of the objective function. Then, by exploiting the Kurdyka-{\L}ojasiewicz (\KL) property for a broad class of functions, we establish the linear and sub-linear convergence rates of the function value sequence generated by APGnc. We further propose a stochastic variance reduced APGnc (SVRG-APGnc), and establish its linear convergence under a special case of the \KL property. We also extend the analysis to the inexact version of these methods and develop an adaptive momentum strategy that improves the numerical performance.