Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates

We study a decomposition-based scalable approach to kernel ridge regression, and show that it achieves minimax optimal convergence rates under relatively mild conditions. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset using a careful choice of the regularization parameter, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimality is retained: as long as m is not too large, the partition-based estimator achieves the statistical minimax rate over all estimators using the set of N samples. As concrete examples, our theory guarantees that the number of subsets m may grow nearly linearly for finite-rank or Gaussian kernels and polynomially in N for Sobolev spaces, which in turn allows for substantial reductions in computational cost. We conclude with experiments on both simulated data and a music-prediction task that complement our theoretical results, exhibiting the computational and statistical benefits of our approach.