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Erschienen in:
Density Estimation by Monte Carlo and Quasi-Monte Carlo Kapitel lesen Erstes Kapitel lesen

2022 | OriginalPaper | Buchkapitel

Scalable Control Variates for Monte Carlo Methods Via Stochastic Optimization

verfasst von : Shijing Si, Chris. J. Oates, Andrew B. Duncan, Lawrence Carin, François-Xavier Briol

Erschienen in: Monte Carlo and Quasi-Monte Carlo Methods

Verlag: Springer International Publishing

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Abstract

Control variates are a well-established tool to reduce the variance of Monte Carlo estimators. However, for large-scale problems including high-dimensional and large-sample settings, their advantages can be outweighed by a substantial computational cost. This paper considers control variates based on Stein operators, presenting a framework that encompasses and generalizes existing approaches that use polynomials, kernels and neural networks. A learning strategy based on minimizing a variational objective through stochastic optimization is proposed, leading to scalable and effective control variates. Novel theoretical results are presented to provide insight into the variance reduction that can be achieved, and an empirical assessment, including applications to Bayesian inference, is provided in support.

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Vorheriges Kapitel On the Selection of Random Field Evaluation Points in the p-MLQMC Method
Nächstes Kapitel Simulation of Conditional Expectations Under Fast Mean-Reverting Stochastic Volatility Models
Fußnoten
1
To simplify presentation in the paper, we always assume \(\mathcal {U}\) is a maximal set of functions for which \(\mathcal {L}u\) is well-defined and \(\Pi [\mathcal {L}u] = 0\).
 
2
We emphasize that MC can be evaluated at negligible cost and we are not advocating that our methods should be preferred for this task.
 
Literatur
1.
Andradóttir, S., Heyman, D.P., Ott, T.J.: Variance reduction through smoothing and control variates for Markov chain simulations. ACM Trans. Model. Comput. Simul. 3(3), 167–189 (1993)
2.
Assaraf, R., Caffarel, M.: Zero-variance principle for Monte Carlo algorithms. Phys. Rev. Lett. 83(23), 4682 (1999)
3.
Baker, J., Fearnhead, P., Fox, E.B., Nemeth, C.: Control variates for stochastic gradient MCMC. Stat. Comput. 29, 599–615 (2019)
4.
Barbour, A.D.: Stein’s method and Poisson process convergence. J. Appl. Probab. 25, 175–184 (1988)
5.
Barp, A., Briol, F.X., Duncan, A.B., Girolami, M., Mackey, L.: Minimum Stein discrepancy estimators. In: Neural Information Processing Systems, pp. 12964–12976 (2019)
6.
7.
Belomestny, D., Iosipoi, L., Moulines, E., Naumov, A., Samsonov, S.: Variance reduction for Markov chains with application to MCMC. Stat. Comput. 30, 973–997 (2020)
8.
Belomestny, D., Iosipoi, L., Zhivotovskiy, N.: Variance reduction via empirical variance minimization: convergence and complexity. Doklady Math. 98, 494–497 (2018)
9.
Belomestny, D., Moulines, E., Shagadatov, N., Urusov, M.: Variance Reduction for MCMC Methods Via Martingale Representations (2019). arXiv:​1903.​0737
10.
Briol, F.X., Oates, C.J., Girolami, M., Osborne, M.A., Sejdinovic, D.: Probabilistic integration: a role in statistical computation? (with discussion). Stat. Sci. 34(1), 1–22 (2019)
11.
12.
Chen, L.H.Y., Goldstein, L., Shao, Q.M.: Normal Approximation by Stein’s Method. Springer, Berlin (2010)
13.
Chen, W.Y., Barp, A., Briol, F.X., Gorham, J., Girolami, M., Mackey, L., Oates, C.J.: Stein point Markov chain Monte Carlo. In: International Conference on Machine Learning, PMLR 97, pp. 1011–1021 (2019)
14.
Chen, W.Y., Mackey, L., Gorham, J., Briol, F.X., Oates, C.J.: Stein points. In: Proceedings of the International Conference on Machine Learning, PMLR 80:843–852 (2018)
15.
Chwialkowski, K., Strathmann, H., Gretton, A.: A kernel test of goodness of fit. Int. Conf. Mach. Learn. 48, 2606–2615 (2016)
16.
Dellaportas, P., Kontoyiannis, I.: Control variates for estimation based on reversible Markov chain Monte Carlo samplers. J. R. Stat. Soc. Ser. B: Stat. Methodol. 74(1), 133–161 (2012)
17.
Friel, N., Mira, A., Oates, C.J.: Exploiting multi-core architectures for reduced-variance estimation with intractable likelihoods. Bayesian Anal. 11(1), 215–245 (2014)
18.
Genz, A.: Testing multidimensional integration routines. In: Proceedings of the International Conference on Tools, Methods and Languages for Scientific and Engineering Computation, pp. 81–94 (1984)
19.
Gorham, J., Duncan, A., Mackey, L., Vollmer, S.: Measuring sample quality with diffusions. Ann. Appl. Probab. 29(5), 2884–2928 (2019)
20.
Gorham, J., Mackey, L.: Measuring sample quality with Stein’s method. In: Advances in Neural Information Processing Systems, pp. 226–234 (2015)
21.
Gorham, J., Mackey, L.: Measuring sample quality with kernels. In: Proceedings of the International Conference on Machine Learning, pp. 1292–1301 (2017)
22.
Grathwohl, W., Choi, D., Wu, Y., Roeder, G., Duvenaud, D.: Backpropagation through the void: Optimizing control variates for black-box gradient estimation. In: International Conference on Learning Representations (2018)
23.
Greensmith, E., Bartlett, P.L., Baxter, J.: Variance reduction techniques for gradient estimates in reinforcement learning. J. Mach. Learn. Res. 5, 1471–1530 (2004)
24.
Hammer, H., Tjelmeland, H.: Control variates for the Metropolis-Hastings algorithm. Scand. J. Stat. 35(3), 400–414 (2008)
25.
Henderson, S.G., Glynn, P.W.: Approximating martingales for variance reduction in Markov process simulation. Math. Oper. Res. 27(2), 253–271 (2002)
26.
Hickernell, F.J., Lemieux, C., Owen, A.B.: Control variates for quasi-Monte Carlo. Stat. Sci. 20(1), 1–31 (2005)
27.
Kennedy, M.C., Hagan, A.O.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B: Stat. Methodol. 63(3), 425–464 (2001)
28.
29.
Ley, C., Swan, Y.: Parametric Stein operators and variance bounds. Braz. J. Probab. Stat. 30(2) (2016)
30.
Liu, H., Feng, Y., Mao, Y., Zhou, D., Peng, J., Liu, Q.: Action-dependent control variates for policy optimization via Stein’s identity. In: International Conference on Learning Representation (2018)
31.
Liu, Q., Lee, J.D.: Black-box importance sampling. In: Proceedings of the International Conference on Artificial Intelligence and Statistics, pp. 952–961 (2017)
32.
Liu, Q., Lee, J.D., Jordan, M.I.: A kernelized Stein discrepancy for goodness-of-fit tests and model evaluation. In: International Conference on Machine Learning, pp. 276–284 (2016)
33.
Liu, Q., Wang, D.: Stein variational gradient descent: a general purpose Bayesian inference algorithm. In: Advances in Neural Information Processing Systems (2016)
34.
Liu, S., Kanamori, T., Jitkrittum, W., Chen, Y.: Fisher efficient inference of intractable models. In: Neural Information Processing Systems, pp. 8793–8803 (2019)
35.
Mira, A., Solgi, R., Imparato, D.: Zero variance Markov chain Monte Carlo for Bayesian estimators. Stat. Comput. 23(5), 653–662 (2013)
36.
37.
Newton, N.J.: Variance reduction for simulated diffusions. SIAM J. Appl. Math. 54(6), 1780–1805 (1994)
38.
Oates, C.J., Cockayne, J., Briol, F.X., Girolami, M.: Convergence rates for a class of estimators based on Stein’s identity. Bernoulli 25(2), 1141–1159 (2019)
39.
Oates, C.J., Girolami, M., Chopin, N.: Control functionals for Monte Carlo integration. J. R. Stat. Soc. B: Stat. Methodol. 79(3), 695–718 (2017)
40.
Oates, C.J., Papamarkou, T., Girolami, M.: The controlled thermodynamic integral for Bayesian model comparison. J. Am. Stat. Assoc. (2016)
41.
O’Hagan, A.: Bayes-Hermite quadrature. J. Stat. Plan. Inference 29, 245–260 (1991)
42.
Paisley, J., Blei, D., Jordan, M.: Variational Bayesian inference with stochastic search. In: International Conference on Machine Learning (2012)
43.
Papamarkou, T., Mira, A., Girolami, M.: Zero variance differential geometric Markov chain Monte Carlo algorithms. Bayesian Anal. 9(1), 97–128 (2014)
44.
Pardoux, E., Vertennikov, A.Y.: On the Poisson equation and diffusion approximation. I. Ann. Probab. 29(3), 1061–1085 (2001)
45.
Portier, F., Segers, J.: Monte Carlo integration with a growing number of control variates. J. Appl. Probab. 56(4), 1168–1186 (2019)
46.
Ranganath, R., Altosaar, J., Tran, D., Blei, D.M.: Operator variational inference. In: Advances in Neural Information Processing Systems, pp. 496–504 (2016)
47.
Ranganath, R., Gerrish, S., Blei, D.M.: Black box variational inference. In: Artificial Intelligence and Statistics, pp. 814–822 (2014)
48.
Riabiz, M., Chen, W., Cockayne, J., Swietach, P., Niederer, S.A., Mackey, L., Oates, C.J.: Optimal thinning of MCMC output (2020). arXiv:​2005.​03952
49.
Ross, N.: Fundamentals of Stein’s method. Probab. Surv. 8, 210–293 (2011)
50.
Si, S., Oates, C.J., Duncan, A.B., Carin, L., Briol, F.X.: Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization (2020). arXiv:​2006.​07487
51.
South, L.F., Karvonen, T., Nemeth, C., Girolami, M., Oates, C.J.: Semi-exact control functionals from Sard’s method (2020). arXiv:​2002.​00033
52.
South, L.F., Oates, C.J., Mira, A., Drovandi, C.: Regularised zero-variance control variates for high-dimensional variance reduction (2019). arXiv:​1811.​05073
53.
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of 6th Berkeley Symposium on Mathematical Statistics and Probability, pp. 583–602. University of California Press (1972)
54.
Wan, R., Zhong, M., Xiong, H., Zhu, Z.: Neural control variates for Monte Carlo variance reduction. In: Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 533–547 (2019)
55.
Wang, C., Chen, X., Smola, A., Xing, E.P.: Variance reduction for stochastic gradient optimization. In: Advances in Neural Information Processing Systems, pp. 181–189 (2013)
56.
Yang, J., Liu, Q., Rao, V., Neville, J.: Goodness-of-fit testing for discrete distributions via Stein discrepancy. In: International Conference on Machine Learning, pp. 5561–5570 (2018)
Metadaten
Titel
Scalable Control Variates for Monte Carlo Methods Via Stochastic Optimization
verfasst von
Shijing Si
Chris. J. Oates
Andrew B. Duncan
Lawrence Carin
François-Xavier Briol
Copyright-Jahr
2022
Buch
Monte Carlo and Quasi-Monte Carlo Methods

Print ISBN: 978-3-030-98318-5

Electronic ISBN: 978-3-030-98319-2

Copyright-Jahr: 2022

DOI
https://doi.org/10.1007/978-3-030-98319-2_10

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