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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.07312 |
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| _version_ | 1866912182617243648 |
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| author | García, Jonathan Petersen, Philipp |
| author_facet | García, Jonathan Petersen, Philipp |
| contents | We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07312 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High-dimensional classification problems with Barron regular boundaries under margin conditions García, Jonathan Petersen, Philipp Machine Learning Probability 68T05, 62C20, 41A25, 41A46 We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set. |
| title | High-dimensional classification problems with Barron regular boundaries under margin conditions |
| topic | Machine Learning Probability 68T05, 62C20, 41A25, 41A46 |
| url | https://arxiv.org/abs/2412.07312 |