Robust, randomized preconditioning for kernel ridge regression

We investigate preconditioned conjugate gradient methods for kernel ridge regression (KRR) problems with a moderate to large number of data points (10^4 \leq N \leq 10^7). We develop and analyze two randomized preconditioners with complementary guarantees. For full-data KRR, RPCholesky preconditioning requires O(N^2) arithmetic operations to achieve fixed accuracy under sufficiently rapid eigenvalue decay of the kernel matrix. For restricted KRR with k\ll N centers, KRILL preconditioning requires 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.