We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.

翻译：我们证明了当假设边界条件时，具有Barron正则决策边界的分类器可以通过具有三个隐藏层的ReLU神经网络以高阶多项式速率逼近。特别地，在强边界条件下，高维不连续分类器能够以通常仅当逼近低维光滑函数时才能达到的速率进行逼近。我们论证了这些表达速率界限如何推导出接近$n^{-1}$的快速学习速率界限，其中$n$为样本数量。此外，我们在具有不同边界的二分类问题上进行了全面的数值实验。我们研究了三种不同维度，其中最高维问题对应MNIST数据集中的图像。