research-article

Control variate selection for Monte Carlo integration

Authors:

François Portier,

Johan SegersAuthors Info & Claims

Statistics and Computing, Volume 31, Issue 4

https://doi.org/10.1007/s11222-021-10011-z

Published: 01 July 2021 Publication History

Abstract

Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model with the integrand as response and the control variates as covariates. Even without special knowledge on the integrand, significant efficiency gains can be obtained if the control variate space is sufficiently large. Incorporating a large number of control variates in the ordinary least squares procedure may however result in (i) a certain instability of the ordinary least squares estimator and (ii) a possibly prohibitive computation time. Regularizing the ordinary least squares estimator by preselecting appropriate control variates via the Lasso turns out to increase the accuracy without additional computational cost. The findings in the numerical experiment are confirmed by concentration inequalities for the integration error.

References

[1]

Asuncion, A., Newman, D.: UCI machine learning repository. (2007) https://archive.ics.uci.edu/ml/index.php

[2]

Avramidis AN and Wilson JR A splitting scheme for control variates Op. Res. Lett. 1993 14 4 187-198

[3]

Belloni A and Chernozhukov V Least squares after model selection in high-dimensional sparse models Bernoulli 2013 19 2 521-547

[4]

Belomestny D, Iosipoi L, Moulines E, Naumov A, and Samsonov S Variance reduction for Markov chains with application to MCMC Stat. Comput. 2020 30 973-997

[5]

Bickel PJ, Ritov Y, and Tsybakov AB Simultaneous analysis of Lasso and Dantzig selector Ann. Stat. 2009 37 4 1705-1732

[6]

Boucheron S, Lugosi G, and Massart P Conc. Inequal. 2013 Oxford Oxford University Press

[7]

Brooks SP, Catchpole EA, and Morgan BJT Bayesian animal survival estimation Stat. Sci. 2000 15 4 357-376

[8]

Brosse, N., Durmus, A., Meyn, S., Moulines, É., Radhakrishnan, A.: Diffusion approximations and control variates for MCMC. (2018) arXiv preprint arXiv:1808.01665

[9]

Caflisch RE Monte Carlo and quasi-Monte Carlo methods Acta Numer. 1998 7 1-49

[10]

Davis, R.A., do Rêgo Sousa, T., Klüppelberg, C.: Indirect inference for time series using the empirical characteristic function and control variates. J. Time Ser. Anal. (2019)

[11]

Glasserman P Monte Carlo Methods in Financial Engineering 2013 Berlin Springer

[12]

Glynn, P.W., Szechtman, R.: Some new perspectives on the method of control variates. In: Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, pp 27–49 (2002)

[13]

Gobet E and Labart C Solving BSDE with adaptive control variate SIAM J. Numer. Anal. 2010 48 1 257-277

[14]

Gorman RP and Sejnowski TJ Analysis of hidden units in a layered network trained to classify sonar targets Neural Netw. 1988 1 1 75-89

[15]

Gower, R., Le Roux, N., Bach, F.: Tracking the gradients using the hessian: A new look at variance reducing stochastic methods. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 707–715 (2018)

[16]

Horn RA and Johnson CR Matrix Analysis 2012 Cambridge Cambridge University Press

[17]

Hsu, D., Kakade, S.M., Zhang, T.: Random design analysis of ridge regression. In: Conference on learning theory, JMLR Workshop and Conference Proceedings, pp 9–1 (2012)

[18]

Hsu D, Kakade SM, and Zhang T Random design analysis of ridge regression Found. Comput. Math. 2014 14 3 569-600

[19]

Javanmard A and Montanari A Debiasing the lasso: optimal sample size for Gaussian designs Ann. Stat. 2018 46 6A 2593-2622

[20]

Jie, T., Abbeel, P.: On a connection between importance sampling and the likelihood ratio policy gradient. In: Advances in Neural Information Processing Systems, pp 1000–1008 (2010)

[21]

Jin, C., Netrapalli, P., Ge, R., Kakade, S.M., Jordan, M.: A short note on concentration inequalities for random vectors with subGaussian norm. arXiv preprint arXiv:1902.03736 (2019)

[22]

Lebreton JD, Burnham KP, Clobert J, and Anderson DR Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies Ecol Monogr 1992 62 1 67-118

[23]

Liu, H., Feng, Y., Mao, Y., Zhou, D., Peng, J., Liu, Q.: Action-dependent control variates for policy optimization via stein identity. In: ICLR 2018 Conference (2018)

[24]

Marzolin G Polygynie du Cincle plongeur (Cinclus cinclus) dans les côtes de Lorraine Oiseau et la Revue Francaise d’Ornithologie 1988 58 4 277-286

[25]

Newey WK Convergence rates and asymptotic normality for series estimators J. Econom. 1997 79 1 147-168

[26]

Nott DJ, Drovandi CC, Mengersen K, Evans M, et al. Approximation of Bayesian predictive

p

-values with regression ABC Bayesian Anal. 2018 13 1 59-83

[27]

Oates CJ, Girolami M, and Chopin N Control functionals for Monte Carlo integration J. R. Stat. Soc. Ser. B 2017 79 3 695-718 (Statistical Methodology)(Statistical Methodology)(Statistical Methodology)(Statistical Methodology)

[28]

Owen A and Zhou Y Safe and effective importance sampling J. Am. Stat. As. 2000 95 449 135-143

[29]

Owen, A.B.: Monte Carlo Theory, Methods and Examples (2013)

[30]

Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, and Duchesnay E Scikit-learn: machine learning in Python J. Mach. Learn. Res. 2011 12 2825-2830

[31]

Portier F and Delyon B Asymptotic optimality of adaptive importance sampling Adv. Neural Inf. Process. Syst. 2018 31 3134-3144

[32]

Portier F and Segers J Monte Carlo integration with a growing number of control variates J. Appl. Probab. 2019 56 1168-1186

[33]

Ranganath, R., Gerrish, S., Blei, D.: Black box variational inference. In: Artificial Intelligence and Statistics, pp 814–822 (2014)

[34]

Rudin W Real and Complex Analysis 2006 New York Tata McGraw-Hill Education

[35]

South, L.F., Oates, C.J., Mira, A., Drovandi, C.: Regularised zero-variance control variates for high-dimensional variance reduction. arXiv preprint arXiv:1811.05073 (2018)

[36]

Tibshirani R Regression shrinkage and selection via the lasso J. R. Stat. Soc. Ser.B 1996 58 1 267-288 Methodological

[37]

Tibshirani R, Wainwright M, and Hastie T Statistical Learning with Sparsity: The Lasso and Generalizations 2015 Boca Raton Chapman and Hall/CRC

[38]

Tropp, J.A.: An introduction to matrix concentration inequalities. Foundations and Trends® in Machine Learning 8(1-2):1–230, (2015) arXiv:1501.01571

[39]

van de Geer SA and Bühlmann P On the conditions used to prove oracle results for the Lasso Electr. J. Stat. 2009 3 1360-1392

[40]

Wang, C., Chen, X., Smola, A.J., Xing, E.P.: Variance reduction for stochastic gradient optimization. In: Advances in Neural Information Processing Systems, pp 181–189 (2013)

[41]

Zhang A, Brown LD, and Cai TT Semi-supervised inference: general theory and estimation of means Ann. Stat. 2019 47 5 2538-2566

Cited By

Leluc RDieuleveut APortier FSegers JZhuman ASalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Sliced-Wasserstein estimation with spherical harmonics as control variatesProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693154(27191-27214)Online publication date: 21-Jul-2024
https://dl.acm.org/doi/10.5555/3692070.3693154
Sun ZOates CBriol FEvans RShpitser I(2023)Meta-learning control variatesProceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence10.5555/3625834.3626026(2047-2057)Online publication date: 31-Jul-2023
https://dl.acm.org/doi/10.5555/3625834.3626026
Sun ZBarp ABriol FKrause ABrunskill ECho KEngelhardt BSabato SScarlett J(2023)Vector-valued control variatesProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3619770(32819-32846)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.5555/3618408.3619770

Recommendations

Coupling control variates for Markov chain Monte Carlo

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control ...
Monte Carlo, Control Variates, and Stochastic Ordering

Using prior information about the stochastic orderings between a phenomenon of interest and other ancillary phenomena whose population parameters are known in a Monte Carlo experiment, this paper derives an unbiased control variate estimator and an ...
On the optimization of approximate control variates with parametrically defined estimators
Highlights
- Multi-model Monte Carlo methods, such as approximate control variates, can be used for estimator variance reduction.
Abstract
Multi-model Monte Carlo methods, such as multi-level Monte Carlo (MLMC) and multifidelity Monte Carlo (MFMC), allow for efficient estimation of the expectation of a quantity of interest given a set of models of varying fidelities. ...

Comments

Information & Contributors

Information

Published In

cover image Statistics and Computing

Statistics and Computing Volume 31, Issue 4

Jul 2021

272 pages

ISSN:0960-3174

Issue’s Table of Contents

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.

Publisher

Kluwer Academic Publishers

United States

Publication History

Published: 01 July 2021

Accepted: 27 March 2021

Received: 04 July 2020

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Leluc RDieuleveut APortier FSegers JZhuman ASalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Sliced-Wasserstein estimation with spherical harmonics as control variatesProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693154(27191-27214)Online publication date: 21-Jul-2024
https://dl.acm.org/doi/10.5555/3692070.3693154
Sun ZOates CBriol FEvans RShpitser I(2023)Meta-learning control variatesProceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence10.5555/3625834.3626026(2047-2057)Online publication date: 31-Jul-2023
https://dl.acm.org/doi/10.5555/3625834.3626026
Sun ZBarp ABriol FKrause ABrunskill ECho KEngelhardt BSabato SScarlett J(2023)Vector-valued control variatesProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3619770(32819-32846)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.5555/3618408.3619770
Demange-Chryst JBachoc FMorio J(2023)Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variatesStatistics and Computing10.1007/s11222-023-10270-y33:5Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1007/s11222-023-10270-y

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents