Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability $\eta(x)=\mathbb{P}(Y=1\mid X=x)$, we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of $2\eta(x)-1$ and apply the method to real datasets.

本篇论文研究了核分类器的统计性能，考虑了核回归理论的最新进展，在一些条件概率的基础上得出了核分类器分类超额风险的上界，并获得了对Sobolev空间的极小下界，表明所提出的分类器的最优性，同时将理论结果推广到超参数化神经网络分类器的泛化误差，并提出了一种估计插值平滑性的简单方法并应用于真实数据集。

Sobolev空间中核分类器的最优性