Accelerating Quasi-Monte Carlo in Reproducing Kernel Hilbert Spaces

Quasi-Monte Carlo (QMC) methods are being adopted in machine learning due to the increasingly challenging nature of numerical integrals that are routinely encountered in contemporary applications. For integrands that are $\alpha$-times differentiable, an $\alpha$-optimal QMC algorithm converges at a best-possible rate $O(N^{-\alpha- 1/2 +\epsilon})$ where $\epsilon>0$ can be arbitrarily small. However, in applications the value of $\alpha$ can be unknown and/or a rate-optimal QMC algorithm can be unavailable. Standard practice is to employ $\alpha_L$-optimal QMC where the lower bound $\alpha_L \leq \alpha$ is known, but this does not exploit the full power of QMC when $\alpha_L < \alpha$. We present a novel solution that uses kernel methods to accelerate QMC by a factor $O(N^{-(\alpha - \alpha_L)/d})$, where $d$ is the dimension of the integral. For $d=1$ we can therefore recover optimal convergence rates. A topical application to robotic arm data demonstrates a substantial speed-up in the computation required to evaluate predictions for mechanical torques.