  In the setting of nonparametric regression, we propose and study a
combination of stochastic gradient methods with Nystr\"om subsampling, allowing
multiple passes over the data and mini-batches. Generalization error bounds for
the studied algorithm are provided. Particularly, optimal learning rates are
derived considering different possible choices of the step-size, the mini-batch
size, the number of iterations/passes, and the subsampling level. In comparison
with state-of-the-art algorithms such as the classic stochastic gradient
methods and kernel ridge regression with Nystr\"om, the studied algorithm has
advantages on the computational complexity, while achieving the same optimal
learning rates. Moreover, our results indicate that using mini-batches can
reduce the total computational cost while achieving the same optimal
statistical results.
