A Geometrical Analysis of Kernel Ridge Regression and its Applications
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Gavrilopoulos, Georgios Lecué, Guillaume Shang, Zong |
| author_facet | Gavrilopoulos, Georgios Lecué, Guillaume Shang, Zong |
| contents | We obtain upper bounds for the estimation error of Kernel Ridge Regression (KRR) for all non-negative regularization parameters, offering a geometric perspective on various phenomena in KRR. As applications: 1. We address the multiple descent problem, unifying the proofs of arxiv:1908.10292 and arxiv:1904.12191 for polynomial kernels and we establish multiple descent for the upper bound of estimation error of KRR under sub-Gaussian design and non-asymptotic regimes. 2. For a sub-Gaussian design vector and for non-asymptotic scenario, we prove a one-sided isomorphic version of the Gaussian Equivalent Conjecture. 3. We offer a novel perspective on the linearization of kernel matrices of non-linear kernel, extending it to the power regime for polynomial kernels. 4. Our theory is applicable to data-dependent kernels, providing a convenient and accurate tool for the feature learning regime in deep learning theory. 5. Our theory extends the results in arxiv:2009.14286 under weak moment assumption.
Our proof is based on three mathematical tools developed in this paper that can be of independent interest: 1. Dvoretzky-Milman theorem for ellipsoids under (very) weak moment assumptions. 2. Restricted Isomorphic Property in Reproducing Kernel Hilbert Spaces with embedding index conditions. 3. A concentration inequality for finite-degree polynomial kernel functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Geometrical Analysis of Kernel Ridge Regression and its Applications Gavrilopoulos, Georgios Lecué, Guillaume Shang, Zong Statistics Theory We obtain upper bounds for the estimation error of Kernel Ridge Regression (KRR) for all non-negative regularization parameters, offering a geometric perspective on various phenomena in KRR. As applications: 1. We address the multiple descent problem, unifying the proofs of arxiv:1908.10292 and arxiv:1904.12191 for polynomial kernels and we establish multiple descent for the upper bound of estimation error of KRR under sub-Gaussian design and non-asymptotic regimes. 2. For a sub-Gaussian design vector and for non-asymptotic scenario, we prove a one-sided isomorphic version of the Gaussian Equivalent Conjecture. 3. We offer a novel perspective on the linearization of kernel matrices of non-linear kernel, extending it to the power regime for polynomial kernels. 4. Our theory is applicable to data-dependent kernels, providing a convenient and accurate tool for the feature learning regime in deep learning theory. 5. Our theory extends the results in arxiv:2009.14286 under weak moment assumption. Our proof is based on three mathematical tools developed in this paper that can be of independent interest: 1. Dvoretzky-Milman theorem for ellipsoids under (very) weak moment assumptions. 2. Restricted Isomorphic Property in Reproducing Kernel Hilbert Spaces with embedding index conditions. 3. A concentration inequality for finite-degree polynomial kernel functions. |
| title | A Geometrical Analysis of Kernel Ridge Regression and its Applications |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2404.07709 |