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，在假设边界条件的情况下，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的速率进行逼近。特别地，对于强边界条件，高维不连续分类器的逼近速率通常仅能在逼近低维光滑函数时实现。我们展示了这些表达速率界限如何推导出接近$n^{-1}$的快速学习率界限，其中$n$为样本数量。此外，我们对具有不同边界条件的二分类问题进行了全面的数值实验。我们研究了三种不同的维度，其中最高维问题对应于MNIST数据集中的图像。