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.

在非凸优化问题中，本文研究了加速近端梯度法(APGnc)以及基于其的随机方差减少(APGnc)算法，证明了其所生成的序列的极限点是目标函数的临界点，并通过KL函数的性质获得了线性和次线性的收敛速率。

非凸优化问题的带动量近端梯度下降收敛分析