  Kernel Ridge Regression (KRR) is a simple yet powerful technique for
non-parametric regression whose computation amounts to solving a linear system.
This system is usually dense and highly ill-conditioned. In addition, the
dimensions of the matrix are the same as the number of data points, so direct
methods are unrealistic for large-scale datasets. In this paper, we propose a
preconditioning technique for accelerating the solution of the aforementioned
linear system. The preconditioner is based on random feature maps, such as
random Fourier features, which have recently emerged as a powerful technique
for speeding up and scaling the training of kernel-based methods, such as
kernel ridge regression, by resorting to approximations. However, random
feature maps only provide crude approximations to the kernel function, so
delivering state-of-the-art results by directly solving the approximated system
requires the number of random features to be very large. We show that random
feature maps can be much more effective in forming preconditioners, since under
certain conditions a not-too-large number of random features is sufficient to
yield an effective preconditioner. We empirically evaluate our method and show
it is highly effective for datasets of up to one million training examples.
