Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels

TL;DR

This paper introduces Quasi-Monte Carlo feature maps for shift-invariant kernels, improving the efficiency of kernel methods on large datasets by reducing approximation error through low-discrepancy sequences.

Contribution

It proposes a novel QMC-based approach for kernel feature maps, including a new discrepancy measure and adaptive sequence learning, enhancing approximation accuracy over traditional Monte Carlo methods.

Findings

QMC methods outperform Monte Carlo in kernel approximation accuracy.

Adaptive QMC sequences further improve efficiency and precision.

Empirical results validate the theoretical advantages of QMC techniques.

Abstract

We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e.g., Gaussian kernel). In this paper, we propose to use Quasi-Monte Carlo (QMC) approximations instead, where the relevant integrands are evaluated on a low-discrepancy sequence of points as opposed to random point sets as in the Monte Carlo approach. We derive a new discrepancy measure called box discrepancy based on theoretical characterizations of the integration error with respect to a given sequence. We then propose to learn QMC sequences adapted to our setting based on explicit box discrepancy minimization. Our theoretical analyses are complemented with empirical results that demonstrate the effectiveness of classical and adaptive QMC techniques for this problem.