Large Dimensional Kernel Ridge Regression: Extending to Product Kernels

TL;DR

This paper extends the understanding of large dimensional kernel ridge regression by analyzing a broad family of kernels, revealing phenomena like saturation, multiple descent, and optimality under various conditions.

Contribution

It introduces a new class of kernels and derives convergence rates, generalizing prior findings and confirming key behaviors beyond specific restrictive settings.

Findings

Recover minimax optimality for source condition s ≤ 1

Identify saturation effect for s > 1

Observe periodic plateau and multiple-descent phenomena

Abstract

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 $n$.