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

Alexey Naumov; Denis Belomestny; Nikita Puchkin; Sergey Samsonov

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

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arxiv 2206.09527 v2 pith:ETV7IERE submitted 2022-06-20 math.NA cs.NAmath.STstat.MLstat.TH

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 This is my paper

classification math.NA cs.NAmath.STstat.MLstat.TH

keywords neuralapproximationdeepfunctionldernetworknetworkspiecewise-polynomial

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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 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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