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What is Learnable in Valiant's Theory of the Learnable? 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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 · 2022-06-20 · via math.ST updates on arXiv.org

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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