Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911355803533312 |
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| author | Zhang, Siyuan Xiao, Nachuan Liu, Xin |
| author_facet | Zhang, Siyuan Xiao, Nachuan Liu, Xin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11565 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions Zhang, Siyuan Xiao, Nachuan Liu, Xin Optimization and Control Machine Learning 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. |
| title | Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2403.11565 |