Large Dimensional Kernel Ridge Regression: Extending to Product Kernels
It provides a theoretical foundation for understanding generalization in high-dimensional KRR with product kernels, addressing an open question for practitioners working with such kernels.
This paper extends the study of saturation effects and multiple descent behavior in large dimensional kernel ridge regression from inner product kernels to a broader family of product kernels, deriving convergence rates that recover key phenomena such as minimax optimality, saturation, and periodic plateaus.
Recent studies have reported $\textit{saturation effects}$ and $\textit{multiple descent behavior}$ in large dimensional kernel ridge regression (KRR). However, these findings are predominantly derived under restrictive settings, such as inner product kernels on sphere or strong eigenfunction assumptions like hypercontractivity. Whether such behaviors hold for other kernels remains an open question. In this paper, we establish a broad, new family of large dimensional kernels and derive the corresponding convergence rates of the generalization error. As a result, we recover key phenomena previously associated with inner product kernels on sphere, including: $i)$ the $\textit{minimax optimality}$ when the source condition $s\le 1$; $ii)$ the $\textit{saturation effect}$ when $s>1$; $iii)$ a $\textit{periodic plateau phenomenon}$ in the convergence rate and a $\textit {multiple-descent behavior}$ with respect to the sample size $n$.