Title: Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization,
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.