  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.
