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.

翻译：在近期核回归研究复兴中最有趣的问题之一，或许在于核插值是否能够具有良好的泛化性能，因为这可能有助于理解深度网络文献中报道的"良性过拟合现象"。在本文中，我们证明在温和条件下，对于任意$\varepsilon>0$，核插值的泛化误差存在下界$\Omega(n^{-\varepsilon})$。换言之，对于一大类核函数，核插值的泛化性能较差。作为直接推论，我们可以证明定义在球面上的过参数化宽神经网络泛化性能不佳。