How rotational invariance of common kernels prevents generalization in high dimensions

TL;DR

This paper demonstrates that the rotational invariance of common kernels causes a bias towards low-degree polynomials in high-dimensional spaces, leading to limitations in their generalization capabilities.

Contribution

It reveals that rotational invariance induces a bias in kernel ridge regression in high dimensions, challenging existing assumptions about their universal effectiveness.

Findings

Rotational invariance causes bias towards low-degree polynomials in high dimensions.

Lower bounds on generalization error depend on kernel eigenvalue decay.

Standard kernel consistency results may not hold without considering kernel structure.

Abstract

Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for kernel regression under certain assumptions on the ground truth function and the distribution of the input data. In this paper, we show that the rotational invariance property of commonly studied kernels (such as RBF, inner product kernels and fully-connected NTK of any depth) induces a bias towards low-degree polynomials in high dimensions. Our result implies a lower bound on the generalization error for a wide range of distributions and various choices of the scaling for kernels with different eigenvalue decays. This lower bound suggests that general consistency results for kernel ridge regression in high dimensions require a more refined analysis that depends on the structure of the kernel beyond its eigenvalue decay.