  Motivated by the studies of neural networks (e.g.,the neural tangent kernel
theory), we perform a study on the large-dimensional behavior of kernel ridge
regression (KRR) where the sample size $n \asymp d^{\gamma}$ for some $\gamma >
0$. Given an RKHS $\mathcal{H}$ associated with an inner product kernel defined
on the sphere $\mathbb{S}^{d}$, we suppose that the true function $f_{\rho}^{*}
\in [\mathcal{H}]^{s}$, the interpolation space of $\mathcal{H}$ with source
condition $s>0$. We first determined the exact order (both upper and lower
bound) of the generalization error of kernel ridge regression for the optimally
chosen regularization parameter $\lambda$. We then further showed that when
$0<s\le1$, KRR is minimax optimal; and when $s>1$, KRR is not minimax optimal
(a.k.a. he saturation effect). Our results illustrate that the curves of rate
varying along $\gamma$ exhibit the periodic plateau behavior and the multiple
descent behavior and show how the curves evolve with $s>0$. Interestingly, our
work provides a unified viewpoint of several recent works on kernel regression
in the large-dimensional setting, which correspond to $s=0$ and $s=1$
respectively.
