The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

翻译：针对特定核回归方法的泛化误差曲线，旨在确定泛化误差在不同源条件、噪声水平及正则化参数选择下的精确阶次，而非极小极大速率。本文在温和假设下，严格刻画了核梯度下降方法（以及一大类解析谱算法）在核回归中的泛化误差曲线的完整特征。由此，我们能够锐化核插值的近不一致性，并阐明具有更高资格的核回归算法的饱和效应等。借助神经正切核理论，这些结果极大地提升了对宽神经网络训练泛化行为的理解。一项新颖的技术贡献——解析函数论证方法，可能具有独立的研究价值。