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Wed Nov 25
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Neural network performance for classification problems with boundaries of Barron class

We study classification problems in which the distances between the different classes are not necessarily positive, but for which the boundaries between the classes are well-behaved. More precisely, we assume these boundaries to be locally described by graphs of functions of Barron-class. ReLU neural networks can approximate and estimate classification functions of this type with rates independent of the ambient dimension. More formally, three-layer networks with $N$ neurons can approximate such functions with $L^1$-error bounded by $O(N^{-1/2})$. Furthermore, given $m$ training samples from such a function, and using ReLU networks of a suitable architecture as the hypothesis space, any empirical risk minimizer has generalization error bounded by $O(m^{-1/4})$. All implied constants depend only polynomially on the input dimension. We also discuss the optimality of these rates. Our results mostly rely on the "Fourier-analytic" Barron spaces that consist of functions with finite first Fourier moment. But since several different function spaces have been dubbed "Barron spaces'' in the recent literature, we discuss how these spaces relate to each other. We will see that they differ more than the existing literature suggests.