发表机构
CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究局部随机梯度下降在有界二阶异质性下的收敛情况，通过建立改进收敛保证证明相关推测，改进了下界，使上界更紧，为其提供更清晰收敛理论，还给出串行随机梯度下降下界展示二阶异质性影响。
AI 中文摘要
局部随机梯度下降（Local SGD），也称为联邦平均，是一种广泛使用的分布式优化算法。虽然在实践中它通常优于小批量随机梯度下降等替代方法，但理论上仍只能部分解释在现实数据异质性下局部更新何时以及为何有帮助。[Patel等人，（2025年）]的近期工作表明，有界二阶异质性假设捕捉到了局部随机梯度下降在强凸目标上的效率，并推测同一原理可扩展到一般凸设置。在本文中，我们通过为有界二阶异质性下一般凸目标的局部随机梯度下降建立改进的收敛保证来证明这一推测。我们还改进了此设置下局部随机梯度下降的最知名下界，表明我们的上界几乎是紧的。这些结果共同为局部随机梯度下降提供了更清晰、更细粒度的收敛理论。作为我们技术的进一步应用，我们给出了带替换的串行随机梯度下降的下界，展示了二阶异质性如何捕捉罕见高曲率客户端的影响。
英文摘要
Local SGD, also known as Federated Averaging, is a widely used distributed optimization algorithm. Although Local SGD often outperforms alternatives such as Mini-batch SGD in practice, theory still only partially explains when and why local updates help under realistic data heterogeneity. Recent work by [Patel et al., 2025] shows that a bounded second-order heterogeneity assumption captures the efficiency of Local SGD for strongly convex objectives, and conjectures that the same principle extends to the general convex setting. In this paper, we prove this conjecture by establishing an improved convergence guarantee for Local SGD on general convex objectives under bounded second-order heterogeneity. We also improve the best-known lower bounds for Local SGD in this setting, showing that our upper bounds are nearly tight. Together, these results provide a sharper, more fine-grained convergence theory for Local SGD. As a further application of our techniques, we provide a lower bound for serial SGD with replacement, showing how second-order heterogeneity captures the impact of rare high-curvature clients.
