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arXiv · 2404.07709

A Geometrical Analysis of Kernel Ridge Regression and its Applications

Georgios Gavrilopoulos·Guillaume Lecué·Zong Shang
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Abstract

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.

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Georgios Gavrilopoulos, Guillaume Lecué, Zong Shang. 2024-10-08. A Geometrical Analysis of Kernel Ridge Regression and its Applications. https://arxiv.org/abs/2404.07709

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