  We study the problem of learning classification functions from noiseless
training samples, under the assumption that the decision boundary is of a
certain regularity. We establish universal lower bounds for this estimation
problem, for general classes of continuous decision boundaries. For the class
of locally Barron-regular decision boundaries, we find that the optimal
estimation rates are essentially independent of the underlying dimension and
can be realized by empirical risk minimization methods over a suitable class of
deep neural networks. These results are based on novel estimates of the $L^1$
and $L^\infty$ entropies of the class of Barron-regular functions.
