Nonconvex and nonsmooth optimization problems are important and challenging for statistics and machine learning. In this paper, we propose Projected Proximal Gradient Descent (PPGD) which solves a class of nonconvex and nonsmooth optimization problems, where the nonconvexity and nonsmoothness come from a nonsmooth regularization term which is nonconvex but piecewise convex. In contrast with existing convergence analysis of accelerated PGD methods for nonconvex and nonsmooth problems based on the Kurdyka-\L{}ojasiewicz (K\L{}) property, we provide a new theoretical analysis showing local fast convergence of PPGD. It is proved that PPGD achieves a fast convergence rate of $\cO(1/k^2)$ when the iteration number $k \ge k_0$ for a finite $k_0$ on a class of nonconvex and nonsmooth problems under mild assumptions, which is locally Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Experimental results demonstrate the effectiveness of PPGD.

翻译：非凸非光滑优化问题在统计学和机器学习中具有重要意义且充满挑战。本文提出投影近端梯度下降法（PPGD），用于求解一类非凸非光滑优化问题，其中非凸性与非光滑性来源于一个非凸但分段凸的非光滑正则项。与现有基于Kurdyka-Łojasiewicz（KL）性质的非凸非光滑问题加速PGD方法收敛性分析不同，我们提供了一种新的理论分析，证明了PPGD的局部快速收敛性。在温和假设下，证明了对一类非凸非光滑问题，当迭代次数 $k \ge k_0$（$k_0$ 为有限值）时，PPGD达到 $\cO(1/k^2)$ 的快速收敛速率，这局部上达到了具有Lipschitz连续梯度的一阶方法在光滑凸目标函数上的Nesterov最优收敛速率。实验结果验证了PPGD的有效性。