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对于统计学和机器学习领域而言既重要又具有挑战性。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，其中非凸性和非光滑性源于一个非凸但分段凸的非光滑正则化项。与现有基于Kurdyka–Łojasiewicz (KŁ)性质的加速PGD方法用于非凸非光滑问题收敛性分析不同，我们提供了一种新的理论分析，证明PPGD具有局部快速收敛性。我们证明，在温和假设条件下，对于一类非凸非光滑问题，当迭代次数k ≥ k₀（k₀为有限值）时，PPGD的收敛速率可达$\cO(1/k^2)$，这接近于Nesterov关于光滑且具有Lipschitz连续梯度的凸目标函数的一阶方法的最优收敛速率。实验结果验证了PPGD的有效性。