New Online Kernel Ridge Regression via Incremental Predictive Sampling

Online kernel ridge regression via existing sampling approaches, which aim at approximating the kernel matrix as accurately as possible, is independent of learning and has a cubic time complexity with respect to the sampling size for updating hypothesis. In this paper, we propose a new online kernel ridge regression via an incremental predictive sampling approach, which has the nearly optimal accumulated loss and performs efficiently at each round. We use the estimated ridge leverage score of the labeled matrix, which depends on the accumulated loss at each round, to construct the predictive sampling distribution, and use this sampling probability for the Nyströ m approximation. To avoid calculating the inverse of the approximated kernel matrix directly, we use the Woodbury formula to accelerate the computation and adopt the truncated incremental singular value decomposition to update the generalized inverse of the intersection matrix. Our online kernel ridge regression has a time complexity of $O(tmk+k^3 )$ for updating hypothesis at round t, where k is the truncated rank of the intersection matrix, and enjoys a regret bound of order $O(\sqrtT )$, where T is the time horizon. Experimental results show that the proposed online kernel ridge regression via the incremental predictive sampling performs more stably and efficiently than the online kernel ridge regression via existing online sampling approaches that directly approximate the kernel matrix.