On the existence of minimizers in shallow residual ReLU neural network optimization landscapes

Steffen Dereich; Arnulf Jentzen; Sebastian Kassing

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arXiv · 2302.14690

On the existence of minimizers in shallow residual ReLU neural network optimization landscapes

Steffen Dereich·Arnulf Jentzen·Sebastian Kassing

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Abstract

In this article, we show existence of minimizers in the loss landscape for residual artificial neural networks (ANNs) with multi-dimensional input layer and one hidden layer with ReLU activation. Our work contrasts earlier results in [D. Gallon, A. Jentzen, and F. Lindner, preprint, arXiv:2211.15641, 2022] and [P. Petersen, M. Raslan, and F. Voigtlaender, Found. Comput. Math., 21 (2021), pp. 375-444] which showed that in many situations minimizers do not exist for common smooth activation functions even in the case where the target functions are polynomials. The proof of the existence property makes use of a closure of the search space containing all functions generated by ANNs and additional discontinuous generalized responses. As we will show, the additional generalized responses in this larger space are suboptimal so that the minimum is attained in the original function class.

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Steffen Dereich, Arnulf Jentzen, Sebastian Kassing. 2023-02-28. On the existence of minimizers in shallow residual ReLU neural network optimization landscapes. https://arxiv.org/abs/2302.14690

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