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

Denis Belomestny; Alexey Naumov; Nikita Puchkin; Sergey Samsonov

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arXiv · 2206.09527

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

Denis Belomestny·Alexey Naumov·Nikita Puchkin·Sergey Samsonov

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Abstract

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.

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Denis Belomestny, Alexey Naumov, Nikita Puchkin, Sergey Samsonov. 2022-12-02. Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations. https://arxiv.org/abs/2206.09527

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