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

• URL: http://arxiv.org/abs/2112.12555v2
• Date: Fri, 24 Dec 2021 07:53:17 GMT
• Title: Optimal learning of high-dimensional classification problems using deep neural networks
• Authors: Philipp Petersen, Felix Voigtlaender
• Abstract summary: We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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