Rates of Approximation by ReLU Shallow Neural Networks

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2023-07-01 DOI:10.48550/arXiv.2307.12461

Tong Mao, Ding-Xuan Zhou

引用次数: 4

Abstract

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H\"older space $W_\infty^r([-1, 1]^d)$ with rates $O((\log m)^{\frac{1}{2} +d}m^{-\frac{r}{d}\frac{d+2}{d+4}})$ when $r

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ReLU浅神经网络的近似速率

由整流线性单元(ReLU)激活的神经网络在最近的深度学习发展中起着核心作用。通过这些网络从Hölder空间逼近函数的主题对于理解诱导学习算法的效率至关重要。尽管这个问题已经在具有多层隐藏神经元的深度神经网络中得到了很好的研究，但对于只有一层隐藏神经元的浅层神经网络来说，它仍然是开放的。在本文中，我们给出了这些网络的一致逼近速率。我们证明了具有$m$隐藏神经元的ReLU浅神经网络可以在$r

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