  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.
