Optimal learning of high-dimensional classification problems using deep neural networks

12/23/2021

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by   Philipp Petersen, et al.

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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^∞ entropies of the class of Barron-regular functions.