论文标题
在Barron类中具有分类边界的分类器的神经网络近似和估计
Neural network approximation and estimation of classifiers with classification boundary in a Barron class
论文作者
论文摘要
我们证明了使用Relu神经网络对某些二进制分类函数的近似和估计的界限。我们的估计范围可使用合适尺寸的网络，根据可用培训样本的数量，为经验风险最小化提供了先验性能保证。所获得的近似值和估计率与输入的维度无关，表明在这种情况下可以克服维数的诅咒；实际上，输入维度仅以多项式因素的形式进入。关于目标分类函数的规律性，我们假设不同类别之间的界面是Barron型的局部。我们通过研究文献中提出的各种巴伦型空间之间的关系来补充我们的结果。这些空间与当前文献所暗示的差异更大。
We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor. Regarding the regularity of the target classification function, we assume the interfaces between the different classes to be locally of Barron-type. We complement our results by studying the relations between various Barron-type spaces that have been proposed in the literature. These spaces differ substantially more from each other than the current literature suggests.
