Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866909761354596352 |
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| author | Yang, Yahong Wu, Yue Yang, Haizhao Xiang, Yang |
| author_facet | Yang, Yahong Wu, Yue Yang, Haizhao Xiang, Yang |
| contents | This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs, and obtains the generalization error in Sobolev spaces via an estimate of the pseudo-dimension of higher-order derivatives of DNNs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10766 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces Yang, Yahong Wu, Yue Yang, Haizhao Xiang, Yang Numerical Analysis This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs, and obtains the generalization error in Sobolev spaces via an estimate of the pseudo-dimension of higher-order derivatives of DNNs. |
| title | Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2310.10766 |