Robust, randomized preconditioning for kernel ridge regression

Abstract

We investigate preconditioned conjugate gradient methods for kernel ridge regression (KRR) problems with a moderate to large number of data points (\(10^4 \le N \le 10^7\)). We develop and analyze two randomized preconditioners with complementary guarantees. For full-data KRR, RPCholesky preconditioning requires \(\mathcal {O}(N^2)\) arithmetic operations for fixed accuracy under sufficiently rapid eigenvalue decay of the kernel matrix. For restricted KRR with \(k\ll N\) centers, KRILL preconditioning requires \(\mathcal {O}((N+k^2)k\log k)\) operations with no eigenvalue-decay assumption. Experiments on benchmark and scientific data sets demonstrate the robustness of both methods relative to existing preconditioners.