arXiv：cs.AI· Neil Walton·· 20 小时前
一种速率迁移方法：通过随机微积分推导随机梯度下降的收敛速率
A Rate Transfer Method: Convergence Rates in Stochastic Gradient Descent via Stochastic Calculus
AI 导读
研究者提出一种"速率迁移"方法，将关联随机微分方程（SDE）的有限时间 Lyapunov 估计转化为随机梯度算法的有限时间收敛界。在通用矩条件与光滑性条件下，随机梯度法的收敛分析可简化为验证 SDE 的 Lyapunov 生成元不等式的微积分论证。该方法复现了经典随机梯度方法的已知收敛速率，并可用于为新优化算法建立收敛速率。
正文
Abstract:We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-time convergence bounds for stochastic gradient algorithms. Under generic moment and smoothness conditions, the convergence analysis of the stochastic gradient method reduces to a calculus argument verifying a Lyapunov generator inequality for the SDE. The method recovers established rates for classical stochastic gradient methods and provides a way to establish convergence rates for new optimization algorithms through calculus arguments applied to their associated SDEs.
| Comments: | 33 pages |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Probability (math.PR); Machine Learning (stat.ML) |
| MSC classes: | 62L20, 60H10, 65K10 |
| ACM classes: | G.1.6; G.3; I.2.6 |
| Cite as: | arXiv:2610.18739 [cs.LG] |
| (or arXiv:2610.18739v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.18739 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Neil Walton [view email]
[v1]
Tue, 13 Oct 2026 17:09:46 UTC (47 KB)
来源：arXiv：cs.AI · arxiv.org