Optimal Rates of Kernel Ridge Regression under Source Condition in Large Dimensions

• URL: http://arxiv.org/abs/2401.01270v1
• Date: Tue, 2 Jan 2024 16:14:35 GMT
• Title: Optimal Rates of Kernel Ridge Regression under Source Condition in Large Dimensions
• Authors: Haobo Zhang, Yicheng Li, Weihao Lu, Qian Lin
• Abstract summary: We study the large-dimensional behavior of kernel ridge regression (KRR) where the sample size $n asymp dgamma$ for some $gamma > 0$. Our results illustrate that the curves of rate varying along $gamma$ exhibit the periodic plateau behavior and the multiple descent behavior.
• Score: 15.988264513040903
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by the studies of neural networks (e.g.,the neural tangent kernel theory), we perform a study on the large-dimensional behavior of kernel ridge regression (KRR) where the sample size $n \asymp d^{\gamma}$ for some $\gamma > 0$. Given an RKHS $\mathcal{H}$ associated with an inner product kernel defined on the sphere $\mathbb{S}^{d}$, we suppose that the true function $f_{\rho}^{*} \in [\mathcal{H}]^{s}$, the interpolation space of $\mathcal{H}$ with source condition $s>0$. We first determined the exact order (both upper and lower bound) of the generalization error of kernel ridge regression for the optimally chosen regularization parameter $\lambda$. We then further showed that when $0<s\le1$, KRR is minimax optimal; and when $s>1$, KRR is not minimax optimal (a.k.a. he saturation effect). Our results illustrate that the curves of rate varying along $\gamma$ exhibit the periodic plateau behavior and the multiple descent behavior and show how the curves evolve with $s>0$. Interestingly, our work provides a unified viewpoint of several recent works on kernel regression in the large-dimensional setting, which correspond to $s=0$ and $s=1$ respectively.

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