  We study Nystr\"om type subsampling approaches to large scale kernel methods,
and prove learning bounds in the statistical learning setting, where random
sampling and high probability estimates are considered. In particular, we prove
that these approaches can achieve optimal learning bounds, provided the
subsampling level is suitably chosen. These results suggest a simple
incremental variant of Nystr\"om Kernel Regularized Least Squares, where the
subsampling level implements a form of computational regularization, in the
sense that it controls at the same time regularization and computations.
Extensive experimental analysis shows that the considered approach achieves
state of the art performances on benchmark large scale datasets.
