On Distributed Nonconvex Optimization: Projected Subgradient Method for Weakly Convex Problems in Networks

The stochastic subgradient method is a widely used algorithm for solving large-scale optimization problems arising in machine learning. Often, these are neither smooth nor convex. Recently, Davis et al. , 2018 characterized the convergence of weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations projected (stoDPSM) to speed up risk minimization. this article, we propose implementation with theoretical guarantee. Specifically, show global stoDPSM using Moreau envelope stationarity measure. Furthermore, under so-called sharpness condition, that deterministic DPSM (with proper initialization) converges linearly sharp minima, geometrically diminishing step size. We provide numerical experiments support our analysis.