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We establish optimal convergence rates for a decomposition-based scalableapproach to kernel ridge regression. The method is simple to describe: itrandomly partitions a dataset of size N into m subsets of equal size, computesan independent kernel ridge regression estimator for each subset, then averagesthe local solutions into a global predictor. This partitioning leads to asubstantial reduction in computation time versus the standard approach ofperforming kernel ridge regression on all N samples. Our two main theoremsestablish that despite the computational speed-up, statistical optimality isretained: as long as m is not too large, the partition-based estimator achievesthe statistical minimax rate over all estimators using the set of N samples. Asconcrete examples, our theory guarantees that the number of processors m maygrow nearly linearly for finite-rank kernels and Gaussian kernels andpolynomially in N for Sobolev spaces, which in turn allows for substantialreductions in computational cost. We conclude with experiments on bothsimulated data and a music-prediction task that complement our theoreticalresults, exhibiting the computational and statistical benefits of our approach.

Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates