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).

翻译：我们证明了一类具有一般激活函数的单隐层神经网络的定量泛函中心极限定理。所建立的收敛速率高度依赖于激活函数的光滑性，其范围从不可微情形（如ReLU函数）下的对数速率，到高度正则激活函数下的$\sqrt{n}$速率。主要工具是Stein-Malliavin方法的泛函版本；特别地，我们充分利用了Bourguin与Campese（2020）近期建立的定量泛函中心极限定理。