Control Functionals for Quasi-Monte Carlo Integration

Chris Oates, Mark Girolami
Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:56-65, 2016.

Abstract

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 α, an optimal QMC rule converges at a best-possible rate O(N^-α/d). However, in applications the value of αcan 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 ≤αis 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v51-oates16, title = {Control Functionals for Quasi-Monte Carlo Integration}, author = {Oates, Chris and Girolami, Mark}, booktitle = {Proceedings of the 19th International Conference on Artificial Intelligence and Statistics}, pages = {56--65}, year = {2016}, editor = {Gretton, Arthur and Robert, Christian C.}, volume = {51}, series = {Proceedings of Machine Learning Research}, address = {Cadiz, Spain}, month = {09--11 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v51/oates16.pdf}, url = {https://proceedings.mlr.press/v51/oates16.html}, abstract = {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 α, an optimal QMC rule converges at a best-possible rate O(N^-α/d). However, in applications the value of αcan 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 ≤αis 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.} }
Endnote
%0 Conference Paper %T Control Functionals for Quasi-Monte Carlo Integration %A Chris Oates %A Mark Girolami %B Proceedings of the 19th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2016 %E Arthur Gretton %E Christian C. Robert %F pmlr-v51-oates16 %I PMLR %P 56--65 %U https://proceedings.mlr.press/v51/oates16.html %V 51 %X 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 α, an optimal QMC rule converges at a best-possible rate O(N^-α/d). However, in applications the value of αcan 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 ≤αis 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.
RIS
TY - CPAPER TI - Control Functionals for Quasi-Monte Carlo Integration AU - Chris Oates AU - Mark Girolami BT - Proceedings of the 19th International Conference on Artificial Intelligence and Statistics DA - 2016/05/02 ED - Arthur Gretton ED - Christian C. Robert ID - pmlr-v51-oates16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 51 SP - 56 EP - 65 L1 - http://proceedings.mlr.press/v51/oates16.pdf UR - https://proceedings.mlr.press/v51/oates16.html AB - 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 α, an optimal QMC rule converges at a best-possible rate O(N^-α/d). However, in applications the value of αcan 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 ≤αis 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. ER -
APA
Oates, C. & Girolami, M.. (2016). Control Functionals for Quasi-Monte Carlo Integration. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 51:56-65 Available from https://proceedings.mlr.press/v51/oates16.html.

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