Bandwidth selection in kernel empirical risk minimization via the gradient

Michaël Chichignoud (1) , Sébastien Loustau (2)

1 SfS - Seminar for Statistics [ETH Zürich]
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

In this paper, we deal with the data-driven selection of multidimensional and (possibly) anisotropic bandwidths in the general problem of kernel empirical risk minimization. We propose a universal selection rule, which leads to optimal adaptive results in a large variety of statistical models such as nonparametric regression or statistical learning with errors-in-variables. These results are stated in the context of smooth loss functions, where the gradient of the risk appears as a good criterion to measure the performance of our estimators. This turns out to be helpful to derive excess risk bounds - with fast rates of convergence - in noisy clustering as well as adaptive minimax results for pointwise and global estimation in robust nonparametric regression. The selection rule consists of a comparison of the gradient empirical risks. It can be viewed as a non-trivial improvement of the so-called GL method (Goldenshluger and Lepski, 2011) to non-linear estimators. Another main advantage of our selection rule is the non-dependency on the smallest eigenvalue of the Hessian matrix of the risk, which is a changing and unknown parameter determined by the underlying model.

Mots clés: Adaptivity; Bandwidth Selection; ERM; Robust Regression; Statistical Learning; Errors-in-variables

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Michaël Chichignoud, Sébastien Loustau. Bandwidth selection in kernel empirical risk minimization via the gradient. 2014. ⟨hal-00937026⟩