Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations

06/20/2022

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by   Denis Belomestny, et al.

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This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any Hölder smooth function up to a given approximation error in Hölder norms in such a way that all weights of this neural network are bounded by 1. The latter feature is essential to control generalization errors in many statistical and machine learning applications.