Asymptotic Normality of Support Vector Machine Variants and Other Regularized Kernel Methods

TL;DR

This paper proves that regularized kernel methods, including support vector machines, are asymptotically normal under certain conditions, enabling better understanding of their statistical properties in nonparametric classification and regression.

Contribution

It establishes the asymptotic normality of empirical SVM estimators in reproducing kernel Hilbert spaces for smooth loss functions, using the functional delta-method.

Findings

Empirical SVMs are asymptotically normal with rate √n.

The standardized difference converges to a Gaussian process.

Results hold even when the regularization parameter depends on data.

Abstract

In nonparametric classification and regression problems, regularized kernel methods, in particular support vector machines, attract much attention in theoretical and in applied statistics. In an abstract sense, regularized kernel methods (simply called SVMs here) can be seen as regularized M-estimators for a parameter in a (typically infinite dimensional) reproducing kernel Hilbert space. For smooth loss functions, it is shown that the difference between the estimator, i.e.\ the empirical SVM, and the theoretical SVM is asymptotically normal with rate $\sqrt{n}$. That is, the standardized difference converges weakly to a Gaussian process in the reproducing kernel Hilbert space. As common in real applications, the choice of the regularization parameter may depend on the data. The proof is done by an application of the functional delta-method and by showing that the SVM-functional is suitably Hadamard-differentiable.