Control Functionals for Quasi-Monte Carlo Integration

Quasi-Monte Carlo (QMC) methods are being adopted in statistical applications due to the increasingly challenging nature of numerical integrals that are now routinely encountered. For integrands that are $\alpha$-times differentiable, an $\alpha$-optimal QMC rule converges at a best-possible rate $O(N^{-\alpha- 1/2 +\epsilon})$. However, in applications the value of $\alpha$ can be unknown and/or a rate-optimal QMC rule can be unavailable. Standard practice is to employ $\alpha_L$-optimal QMC where the lower bound $\alpha_L \leq \alpha$ is known, but in general this does not exploit the full power of QMC. We present an elegant solution that uses control functionals to accelerate $\alpha_L$-QMC by a factor $O(N^{-(\alpha - \alpha_L)/d})$, where $d$ is the dimension of the integral. For $d=1$ we therefore recover optimal convergence rates. A challenging application to robotic arm data demonstrates a substantial variance reduction in predictions for mechanical torques.