  In this paper, we deal with the data-driven selection of multidimensional and
possibly anisotropic bandwidths in the general framework 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 robust regression and 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. The selection rule consists of a comparison of
gradient empirical risks. It can be viewed as a nontrivial improvement of the
so-called Goldenshluger-Lepski method to nonlinear estimators. Furthermore, one
main advantage of our selection rule is the nondependency on the Hessian matrix
of the risk, usually involved in standard adaptive procedures.
