Two-layer neural networks with values in a Banach space

• URL: http://arxiv.org/abs/2105.02095v1
• Date: Wed, 5 May 2021 14:54:24 GMT
• Title: Two-layer neural networks with values in a Banach space
• Authors: Yury Korolev
• Abstract summary: We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $mathbb Rd$-valued neural networks this corresponds to the ReLU activation function.
• Score: 1.90365714903665
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates, extending existing results for the finite-dimensional case. In the second part of the paper, we consider training such networks using a finite amount of noisy observations from the regularisation theory viewpoint. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.

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