Faster kernel ridge regression using sketching and preconditioning

Haim Avron, Kenneth L. Clarkson, David P. Woodruff

Research output: Contribution to journalArticlepeer-review

63 Scopus citations

Abstract

Kernel ridge regression is a simple yet powerful technique for nonparametric 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.

Original languageEnglish
Pages (from-to)1116-1138
Number of pages23
JournalSIAM Journal on Matrix Analysis and Applications
Volume38
Issue number4
DOIs
StatePublished - 2017

Funding

FundersFunder number
Defense Advanced Research Projects Agency
Air Force Research LaboratoryFA8750-12-C-0323

    Keywords

    • Kernel ridge regression
    • Preconditioning
    • Random features

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