This paper focuses on learning rate analysis of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. Using a recently developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an integral operator approach based on second-order decomposition, we succeed in deriving almost optimal learning rates of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. A series of numerical experiments are carried out to verify our theoretical results, showing the excellent learning performance of Nystrom regularization with sequential sub-sampling in learning massive time series data. All these results extend the applicable range of Nyström regularization from i.i.d. samples to non-i.i.d. sequences.

This paper focuses on learning rate analysis of distributed kernel ridge regression (DKRR) for strong mixing sequences. Using a recently developed integral operator approach and a classical covariance inequality for Banach-valued strong mixing sequences, we succeed in deriving optimal learning rates of DKRR. As a byproduct, we deduce a sufficient condition for the mixing property to guarantee the optimal learning rates for kernel ridge regression, which fills the gap of learning rates between i.i.d. samples and strong mixing sequences. A series of numerical experiments are conducted to verify our theoretical assertions via showing excellent learning performance of DKRR in learning both toy and real world time series data. All these results extend the applicable range of distributed learning from i.i.d. samples to non-i.i.d. sequences.