Title: A Quantitative Functional Central Limit Theorem for Shallow Neural Networks Show Chinese title

Title: 浅层神经网络的定量功能中心极限定理

Abstract: We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they range from logarithmic in non-differentiable cases such as the Relu to $\sqrt{n}$ for very regular activations. Our main tools are functional versions of the Stein-Malliavin approach; in particular, we exploit heavily a quantitative functional central limit theorem which has been recently established by Bourguin and Campese (2020). Show Chinese abstract

Abstract: 我们证明了一个关于具有通用激活函数的一层隐藏层神经网络的定量功能中心极限定理。 我们建立的收敛速率在很大程度上依赖于激活函数的平滑性，它们从非可微情况（如Relu）中的对数速率到非常规则的激活函数的$\sqrt{n}$不等。 我们的主要工具是Stein-Malliavin方法的功能版本；特别是，我们大量利用了Bourguin和Campese（2020）最近建立的一个定量功能中心极限定理。