Asymptotic normality of support vector machine variants and other regularized kernel methods

作者：Hable Robert^*

来源：Journal of Multivariate Analysis, 2012, 106: 92-117.

DOI：10.1016/j.jmva.2011.11.004

• Nonparametric regression
• Support vector machine
• Regularized kernel method
• Asymptotic normality
• Hadamard-differentiability
• Functional delta-method

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 L, it is shown that the difference between the estimator, i.e. the empirical SVM f(L,Dn,lambda Dn), and the theoretical SVM f(L,P,lambda 0) is asymptotically normal with rate root n. That is, root n(f(L,Dn,lambda Dn) - f(L,P,lambda 0)) converges weakly to a Gaussian process in the reproducing kernel Hilbert space. As common in real applications, the choice of the regularization parameter D-n in f(L,Dn,lambda Dn) may depend on the data. The proof is done by an application of the functional delta-method and by showing that the SVM-functional P rightharpoonarrow fL,P,lambda is suitably Hadamard-differentiable.

• 出版日期2012-4