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

TL;DR

This paper introduces a distributed kernel ridge regression method that partitions data for parallel processing, achieving near-optimal statistical rates while significantly reducing computation time.

Contribution

It proposes a simple, scalable partitioning approach for kernel ridge regression that retains minimax optimal rates, balancing computational efficiency and statistical accuracy.

Findings

Method achieves statistical minimax rates with multiple processors.

Partitioning reduces computation time significantly.

Experimental results confirm theoretical advantages.

Abstract

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. 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, 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 processors m may grow nearly linearly for finite-rank kernels and 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.