Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay
This work provides theoretical insights into generalization behavior for kernel methods and neural networks, though it appears incremental in extending existing spectral algorithm analysis.
The authors characterized generalization error curves for kernel gradient descent and analytic spectral algorithms in kernel regression, providing exact error orders rather than minimax rates. Their results sharpen understanding of kernel interpolation inconsistency and saturation effects, with applications to wide neural networks via neural tangent kernel theory.
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.