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

2021/12/23 by Petersen, Philipp, Voigtlaender, Felix · 2 citations
#41A25 #41A46 #62C20 #68T05 #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.2112.12555

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 L¹ and L^∞ entropies of the class of Barron-regular functions.

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• Minimax learning rates for estimating binary classifiers under margin conditions
• Super-fast rates of convergence for Neural Networks Classifiers under the Hard Margin Condition

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