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

• URL: http://arxiv.org/abs/2206.09527v2
• Date: Fri, 2 Dec 2022 10:01:35 GMT
• Title: Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations
• Authors: Denis Belomestny, Alexey Naumov, Nikita Puchkin, Sergey Samsonov
• Abstract summary: We derive the required depth, width, and sparsity of a deep neural network to approximate any H"older smooth function up to a given approximation error in H"older norms. The latter feature is essential to control generalization errors in many statistical and machine learning applications.
• Score: 2.15145758970292
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 H\"{o}lder smooth function up to a given approximation error in H\"{o}lder 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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