On Distributed Non-convex Optimization: Projected Subgradient Method For Weakly Convex Problems in Networks

• URL: http://arxiv.org/abs/2004.13233v2
• Date: Tue, 23 Feb 2021 20:32:36 GMT
• Title: On Distributed Non-convex Optimization: Projected Subgradient Method For Weakly Convex Problems in Networks
• Authors: Shixiang Chen, Alfredo Garcia and Shahin Shahrampour
• Abstract summary: The Moreau subgradient method converges linear sharpness problems in machine learning. A distributed implementation of the subgradient method with a theoretical guarantee is proposed.
• Score: 13.385373310554327
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the 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 of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimization. In this paper, we propose a distributed implementation of the stochastic subgradient method with a theoretical guarantee. Specifically, we show the global convergence of stoDPSM using the Moreau envelope stationarity measure. Furthermore, under a so-called sharpness condition, we show that deterministic DPSM (with a proper initialization) converges linearly to the sharp minima, using geometrically diminishing step-size. We provide numerical experiments to support our theoretical analysis.

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