Control Functionals for Quasi-Monte Carlo Integration

By Chris J. Oates, Mark A. Girolami. 2015

29 citations recorded by the source corpus.

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 with $d$-dimensions and derivatives of order $\alpha$, an optimal QMC rule converges at a best-possible rate $O(N^{-\alpha/d})$. 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. One solution is to trade-off numerical integration with functional approximation. This strategy is explored herein and shown to be well-suited to modern statistical computation. A challenging application to robotic arm data demonstrates a substantial variance reduction in predictions for mechanical torques.

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “Because of this nice property, kernel quadratures have been studied extensively in recent years [7, 3, 5, 2, 17] and have started to find applications in machine learning and statistics [23, 14, 12, 6].”
Control functionals for Monte Carlo integration (2014, methodology): “Indeed we can prove sharper results under stronger conditions that include boundedness of Ω. Details are reserved for a future publication (Oates, Cockayne, Briol and Girolami, 2016).”
Action-depedent Control Variates for Policy Optimization via Stein's Identity (2017, background): “However, the bias introduced this way is often negligible in practice (see e.g., Section 2.3.4 of Oates & Girolami (2016)).”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “ that a well-known trade-o between random sampling and deterministic approximation has been studied on several separate occasions by authors including Bakhvalov (1959); Heinrich (1995); Giles (2013); Oates and Girolami (2016). Our starting point is, instead, to establish a trade-o between random sam”
Monte Carlo with determinantal point processes (2016, methodology): “Finally, we note that nonparametric control variates have also been studied for QMC and randomized QMC [Oates and Girolami, 2016].”
Measuring Sample Quality with Kernels (2017, methodology): “In particular, our kernel recommendations could be incorporated into the Monte Carlo control functionals framework of Oates et al. (2016b); Oates & Girolami (2015), the variational inference approaches of Liu & Wang (2016); Liu & Feng (2016); Ranganath et al. (2016), and the Stein generative…”
Measuring Sample Quality with Diffusions (2016, background): “While our chief motivation is sample quality measurement, our work is also directly applicable to a variety of inferential tasks that currently rely on the Langevin operator, including control variate design [66, 67], one sample hypothesis testing [? 57], variational inference [56], and importance s”