Projective Proximal Gradient Descent for Nonconvex Nonsmooth Optimization: Fast Convergence Without Kurdyka-Lojasiewicz (KL) Property

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 nd 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-Lojasiewicz (KL) property, we provide a new theoretical analysis showing that PPGD achieves optimal convergence rate on a class of nonconvex and nonsmooth problems under mild assumptions, which is the 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 the PPGD.