Faster Kernel Ridge Regression Using Sketching and Preconditioning

TL;DR

This paper introduces a preconditioning method using random feature maps to accelerate Kernel Ridge Regression, making it scalable and efficient for large datasets with up to one million examples.

Contribution

The paper presents a novel preconditioning technique based on random feature maps that significantly speeds up solving KRR linear systems for large-scale data.

Findings

Effective preconditioning reduces computation time.

Method scales to datasets with up to one million examples.

Random features improve convergence with fewer features.

Abstract

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.