Kernel interpolation generalizes poorly

One of the most interesting problems in the recent renaissance of the studies in kernel regression might be whether the kernel interpolation can generalize well, since it may help us understand the `benign overfitting henomenon' reported in the literature on deep networks. In this paper, under mild conditions, we show that for any $\varepsilon>0$, the generalization error of kernel interpolation is lower bounded by $\Omega(n^{-\varepsilon})$. In other words, the kernel interpolation generalizes poorly for a large class of kernels. As a direct corollary, we can show that overfitted wide neural networks defined on the sphere generalize poorly.