Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions

• URL: http://arxiv.org/abs/2403.11565v3
• Date: Fri, 09 May 2025 06:16:13 GMT
• Title: Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions
• Authors: Siyuan Zhang, Nachuan Xiao, Xin Liu,
• Abstract summary: We propose a general framework that unifies various decentralized subgradient-based methods.<n>We prove convergence guarantees for some well-recognized decentralized subgradient-based methods.
• Score: 10.278310909980576
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we focus on the decentralized stochastic subgradient-based methods in minimizing nonsmooth nonconvex functions without Clarke regularity, especially in the decentralized training of nonsmooth neural networks. We propose a general framework that unifies various decentralized subgradient-based methods, such as decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking technique (DSGD-T), and DSGD with momentum (DSGD-M). To establish the convergence properties of our proposed framework, we relate the discrete iterates to the trajectories of a continuous-time differential inclusion, which is assumed to have a coercive Lyapunov function with a stable set $\mathcal{A}$. We prove the asymptotic convergence of the iterates to the stable set $\mathcal{A}$ with sufficiently small and diminishing step-sizes. These results provide first convergence guarantees for some well-recognized of decentralized stochastic subgradient-based methods without Clarke regularity of the objective function. Preliminary numerical experiments demonstrate that our proposed framework yields highly efficient decentralized stochastic subgradient-based methods with convergence guarantees in the training of nonsmooth neural networks.

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