Nonconvex proximal splitting: batch and incremental algorithms

Within the unmanageably large class of nonconvex optimization, we consider the rich subclass of nonsmooth problems that have \emph{composite} objectives---this already includes the extensively studied convex, composite objective problems as a special case. For this subclass, we introduce a powerful, new framework that permits asymptotically \emph{non-vanishing} perturbations. In particular, we develop perturbation-based batch and incremental (online like) nonconvex proximal splitting algorithms. To our knowledge, this is the first time that such perturbation-based nonconvex splitting algorithms are being proposed and analyzed. While the main contribution of the paper is the theoretical framework, we complement our results by presenting some empirical results on matrix factorization.