首次证明高斯核岭回归在固定超参数下具有多项式收敛率。
Uniform convergence for Gaussian kernel ridge regression
- 提出统一与L²范数下的收敛性分析框架
- 在固定宽度参数下实现多项式收敛速率
- 为非参数回归中固定超参的高斯核方法提供理论支持
本文首次建立了高斯核岭回归（Gaussian kernel ridge regression, KRR）在固定超参数下，统一收敛与L²范数下的多项式收敛速率。此前，高斯核KRR的统一收敛率尚无已知结果，该研究填补了这一理论空白。同时，当高斯核的宽度参数固定时，也证明了多项式级别的L²收敛率。这拓展了对平滑核的理解，因为以往类似设定下仅知亚多项式收敛率。这些结果共同为在非参数回归中使用固定超参数的高斯核岭回归提供了新的理论依据。
原文摘要 · Abstract (English)
This paper establishes the first polynomial convergence rates for Gaussian kernel ridge regression (KRR) with a fixed hyperparameter in both the uniform and the $L^{2}$-norm. The uniform convergence result closes a gap in the theoretical understanding of KRR with the Gaussian kernel, where no such rates were previously known. In addition, we prove a polynomial $L^{2}$-convergence rate in the case, where the Gaussian kernel's width parameter is fixed. This also contributes to the broader understanding of smooth kernels, for which previously only sub-polynomial $L^{2}$-rates were known in similar settings. Together, these results provide new theoretical justification for the use of Gaussian KRR with fixed hyperparameters in nonparametric regression.
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