发表机构
Institute of Science and Technology Austria; University of Wisconsin–Madison; Cornell University（奥地利科学技术研究所; 威斯康星大学麦迪逊分校; 康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究概率测度空间P2上的随机梯度下降，通过Lions可微性将问题提升至Hilbert空间，构造匹配均值和协方差的高斯随机场近似，证明其以二阶弱精度捕捉SGD动力学，为用高斯波动替代样本随机性奠定严格基础。
AI 中文摘要
随机梯度下降（SGD）允许扩散近似，用高斯噪声替代随机梯度的复杂随机性，为理解其动力学和长期行为提供了有力工具。我们研究在概率测度优化中，当目标函数定义在Wasserstein空间P2上的泛函时，类似的近似原理是否成立。P2的非线性几何和无限维性质阻碍了经典欧几里得理论的直接扩展。利用Lions可微性，我们将问题提升到线性Hilbert空间，在该空间中高阶微分运算可用。然后我们构造一个高斯随机场近似，其速度场匹配原始随机梯度的均值和协方差。通过利用高阶Taylor展开的矩匹配，我们证明高斯近似以二阶弱精度捕捉SGD动力学。我们的结果为在概率测度上的随机优化中，用解析可处理的高斯波动替代样本驱动的随机性提供了严格基础。
英文摘要
Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.
