Optimal Rate of Kernel Regression in Large Dimensions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917707886100480 |
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| author | Lu, Weihao Zhang, Haobo Li, Yicheng Xu, Manyun Lin, Qian |
| author_facet | Lu, Weihao Zhang, Haobo Li, Yicheng Xu, Manyun Lin, Qian |
| contents | We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^γ$ for some $γ>0$ ). We first build a general tool to characterize the upper bound and the minimax lower bound of kernel regression for large dimensional data through the Mendelson complexity $\varepsilon_{n}^{2}$ and the metric entropy $\bar{\varepsilon}_{n}^{2}$ respectively. When the target function falls into the RKHS associated with a (general) inner product model defined on $\mathbb{S}^{d}$, we utilize the new tool to show that the minimax rate of the excess risk of kernel regression is $n^{-1/2}$ when $n\asymp d^γ$ for $γ=2, 4, 6, 8, \cdots$. We then further determine the optimal rate of the excess risk of kernel regression for all the $γ>0$ and find that the curve of optimal rate varying along $γ$ exhibits several new phenomena including the multiple descent behavior and the periodic plateau behavior. As an application, For the neural tangent kernel (NTK), we also provide a similar explicit description of the curve of optimal rate. As a direct corollary, we know these claims hold for wide neural networks as well. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04268 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Rate of Kernel Regression in Large Dimensions Lu, Weihao Zhang, Haobo Li, Yicheng Xu, Manyun Lin, Qian Machine Learning Statistics Theory 62G08, 46E22, 68T07 We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^γ$ for some $γ>0$ ). We first build a general tool to characterize the upper bound and the minimax lower bound of kernel regression for large dimensional data through the Mendelson complexity $\varepsilon_{n}^{2}$ and the metric entropy $\bar{\varepsilon}_{n}^{2}$ respectively. When the target function falls into the RKHS associated with a (general) inner product model defined on $\mathbb{S}^{d}$, we utilize the new tool to show that the minimax rate of the excess risk of kernel regression is $n^{-1/2}$ when $n\asymp d^γ$ for $γ=2, 4, 6, 8, \cdots$. We then further determine the optimal rate of the excess risk of kernel regression for all the $γ>0$ and find that the curve of optimal rate varying along $γ$ exhibits several new phenomena including the multiple descent behavior and the periodic plateau behavior. As an application, For the neural tangent kernel (NTK), we also provide a similar explicit description of the curve of optimal rate. As a direct corollary, we know these claims hold for wide neural networks as well. |
| title | Optimal Rate of Kernel Regression in Large Dimensions |
| topic | Machine Learning Statistics Theory 62G08, 46E22, 68T07 |
| url | https://arxiv.org/abs/2309.04268 |