Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels
This work addresses efficiency bottlenecks in kernel methods for machine learning practitioners, offering an incremental improvement over existing randomized approaches.
The paper tackles the problem of improving efficiency in randomized Fourier feature maps for kernel methods on large datasets by proposing Quasi-Monte Carlo (QMC) approximations instead of Monte Carlo, resulting in enhanced training and testing speed as demonstrated empirically.
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.