发表机构
Université Lyon 1; Université Paris-Saclay; INRAE（里昂第一大学; 巴黎-萨克雷大学; 法国国家农业、食品与环境研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过函数Delta方法，在加权Sobolev空间中推导了浅层神经网络训练中非线性可观测量的中心极限定理，并给出了局部可识别性的微分条件。
AI 中文摘要
平均场极限通过参数经验分布的演化描述了宽神经网络的训练动态。尽管函数中心极限定理刻画了该分布的渐近波动，但实际感兴趣的量通常是参数分布的非线性可观测量，而非分布本身。在本工作中，我们展示了这些平均场波动如何传播到由随机梯度下降训练的浅层神经网络的有限维非线性可观测量。在构建极限波动过程的加权Sobolev空间中，我们在普通Fr{é}chet可微性下应用函数Delta方法，无需对测度变量求Lions导数。我们获得了可观测量的中心极限定理，并在其微分适当表示下，得到了继承自底层平均场波动理论的显式协方差公式。我们还研究了是否可以从选定的观测中恢复规定的感兴趣量。在常数秩假设下，我们证明了一个感兴趣量在局部通过观测泛函分解，当且仅当在某个邻域内，观测微分的核包含于感兴趣量的核中。因此，直接在环境Sobolev空间中表达的微分条件产生了精确的非线性局部分解。这些结果为量化有限宽度不确定性对统计或物理上有意义的可观测量提供了框架，并评估所选观测是否包含识别它们所需的信息。
英文摘要
Mean field limits describe the training dynamics of wide neural networks through the evolution of the empirical distribution of their parameters. Although functional central limit theorems characterize the asymptotic fluctuations of this distribution, quantities of practical interest are typically nonlinear observables of the parameter distribution rather than the distribution itself. In this work, we show how these mean field fluctuations propagate to finite dimensional nonlinear observables for shallow neural networks trained by stochastic gradient descent. Working in the weighted Sobolev space in which the limiting fluctuation process is constructed, we apply a functional Delta method under ordinary Fr{é}chet differentiability, without requiring Lions derivatives with respect to the measure variable. We obtain a central limit theorem for the observables and, under a suitable representation of their differentials, an explicit covariance formula inherited from the underlying mean field fluctuation theory. We also study whether prescribed quantities of interest can be recovered from the selected observations. Under a constant rank assumption, we prove that a quantity of interest factors locally through the observation functional if and only if, throughout a neighborhood, the kernel of the differential of the observation is contained in that of the quantity of interest. Thus, a differential condition expressed directly in the ambient Sobolev space yields an exact nonlinear local factorization. These results provide a framework both for quantifying finite-width uncertainty on observable, statistically or physically meaningful quantities and for assessing whether the chosen observations contain the information required to identify them.
