Rates of Approximation by ReLU Shallow Neural Networks

IF 2.2 4区数学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal of complex networks 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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来源期刊

Journal of complex networks MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-

CiteScore

4.20

自引率

9.50%

发文量

期刊介绍： Journal of Complex Networks publishes original articles and reviews with a significant contribution to the analysis and understanding of complex networks and its applications in diverse fields. Complex networks are loosely defined as networks with nontrivial topology and dynamics, which appear as the skeletons of complex systems in the real-world. The journal covers everything from the basic mathematical, physical and computational principles needed for studying complex networks to their applications leading to predictive models in molecular, biological, ecological, informational, engineering, social, technological and other systems. It includes, but is not limited to, the following topics: - Mathematical and numerical analysis of networks - Network theory and computer sciences - Structural analysis of networks - Dynamics on networks - Physical models on networks - Networks and epidemiology - Social, socio-economic and political networks - Ecological networks - Technological and infrastructural networks - Brain and tissue networks - Biological and molecular networks - Spatial networks - Techno-social networks i.e. online social networks, social networking sites, social media - Other applications of networks - Evolving networks - Multilayer networks - Game theory on networks - Biomedicine related networks - Animal social networks - Climate networks - Cognitive, language and informational network

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