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

2025-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

A new framework, published in JMLR 27(109):1−52 in 2026, unifies various decentralized stochastic subgradient-based methods for minimizing nonsmooth nonconvex functions, particularly in decentralized neural network training. This framework encompasses methods like decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking (DSGD-T), and DSGD with momentum (DSGD-M). The research establishes convergence properties by relating discrete iterates to continuous-time differential inclusions, assuming a coercive Lyapunov function with a stable set A. It proves asymptotic convergence of iterates to set A using sufficiently small and diminishing step-sizes. These findings provide the first convergence guarantees for several recognized decentralized stochastic subgradient methods without requiring Clarke regularity of the objective function. Preliminary numerical experiments confirm the framework's efficiency.

Key takeaway

For AI Scientists or Machine Learning Engineers developing decentralized optimization algorithms, this research provides crucial convergence guarantees for nonsmooth nonconvex problems, a domain previously lacking such assurances. You should consider integrating these unified subgradient-based methods, including DSGD-T or DSGD-M, into your decentralized neural network training pipelines to ensure robust and provably convergent solutions, especially when dealing with complex, nonsmooth objective functions.

Key insights

A unified framework provides the first convergence guarantees for decentralized subgradient methods on nonsmooth nonconvex functions.

Principles

• Convergence can be established via continuous-time differential inclusion.
• Coercive Lyapunov functions are key for stable set convergence.

Method

The framework unifies DSGD, DSGD-T, and DSGD-M by relating their discrete iterates to continuous-time differential inclusions for rigorous convergence analysis.

In practice

• Apply DSGD, DSGD-T, or DSGD-M.
• Train nonsmooth neural networks.

Topics

• Decentralized Optimization
• Stochastic Subgradient Methods
• Nonsmooth Nonconvex Optimization
• Neural Network Training
• Convergence Guarantees
• Lyapunov Functions

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.