  Regularized kernel-based methods such as support vector machines (SVMs)
typically depend on the underlying probability measure $\mathrm{P}$
(respectively an empirical measure $\mathrm{D}_n$ in applications) as well as
on the regularization parameter $\lambda$ and the kernel $k$. Whereas classical
statistical robustness only considers the effect of small perturbations in
$\mathrm{P}$, the present paper investigates the influence of simultaneous
slight variations in the whole triple $(\mathrm{P},\lambda,k)$, respectively
$(\mathrm{D}_n,\lambda_n,k)$, on the resulting predictor. Existing results from
the literature are considerably generalized and improved. In order to also make
them applicable to big data, where regular SVMs suffer from their super-linear
computational requirements, we show how our results can be transferred to the
context of localized learning. Here, the effect of slight variations in the
applied regionalization, which might for example stem from changes in
$\mathrm{P}$ respectively $\mathrm{D}_n$, is considered as well.
