Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent

• 成果类型: Article
• 署名作者: Gess, Benjamin; Gvalani, Rishabh S.; Konarovskyi, Vitalii
• 署名单位: Technical University of Berlin; Max Planck Society; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Hamburg; National Academy of Sciences Ukraine; Institute of Mathematics of NASU
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01353-6
• 发表日期: 2025
• 页码: 1447-1515
• 关键词: partial-differential-equations; neural-networks; SYSTEM; deviations; particles; MODEL

摘要

The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of stochastic gradient descent in the continuum limit.