Bayesian Monte Carlo for the Global Optimization of Expensive Functions
Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
Pages 249 - 254
Abstract
In the last decades enormous advances have been made possible for modelling complex (physical) systems by mathematical equations and computer algorithms. To deal with very long running times of such models a promising approach has been to replace them by stochastic approximations based on a few model evaluations. In this paper we focus on the often occuring case that the system modelled has two types of inputs x = (xc, xe) with xc representing control variables and xe representing environmental variables. Typically, xc needs to be optimised, whereas xe are uncontrollable but are assumed to adhere to some distribution. In this paper we use a Bayesian approach to address this problem: we specify a prior distribution on the underlying function using a Gaussian process and use Bayesian Monte Carlo to obtain the objective function by integrating out environmental variables. Furthermore, we empirically evaluate several active learning criteria that were developed for the deterministic case (i.e., no environmental variables) and show that the ALC criterion appears significantly better than expected improvement and random selection.
References
[1]
G. Box, S. Bisgaard, and C. Fung, 'An explanation and critique of Taguchi's contributions to quality engineering', Quality and Reliability Engineering International, 4, 123-131, (1988).
[2]
P.B. Chang, B.J. Williams, K.S.B. Bhalla, T.W. Belknap, T.J. Santner, W.I. Notz, and D.L. Bartel, 'Design and analysis of robust total joint replacements: finite element model experiments with enviromental variables', J. Biomechanical Engineering, 123, 239-246, (2001).
[3]
P.B. Chang, B.J. Williams, W.I. Notz, T.J. Santner, and D.L. Bartel, 'Robust optimization of total joint replacements incorporating environmental variables', J. Biomech. Eng., 121, 304-310, (1999).
[4]
D.A. Cohn, 'Neural networks exploration using optimal experimental design', Neural Networks, 6(9), 1071-1083, (1996).
[5]
K. Crombecq and D. Gorissen, 'A novel sequential design strategy for global surrogate modeling', in Proceedings of the 41th Conference on Winter Simulation, pp. 731-742, (2009).
[6]
D.E. Finkel, 'DIRECT optimization algorithm user guide', Technical report, Center for Research in Scientific Computation, North Carolina State University, (2003).
[7]
M. Frean and P. Boyle, 'Using Gaussian processes to optimize expensive functions', in Proceedings of the 21st Australasian Joint Conference on Artificial Intelligence: Advances in Artificial Intelligence, volume 5360 of LNAI, pp. 258-267, (2008).
[8]
R.L. Iman, J.M. Davenport, and D.K. Zeigler, 'Latin hypercube sampling (program users guide)', Technical Report SAND79-1473, Sandia National Laboratories, Albuquerque, NM, (1980).
[9]
D.R. Jones, 'A taxonomy of global optimization methods based on response surfaces', Journal of Global Optimization, 21, 345-383, (2001).
[10]
D.R. Jones, 'The DIRECT global optimization algorithm', in Encyclopedia of Optimization, 725-735, Springer, (2009).
[11]
D.R. Jones, M. Schonlau, and J.W. Welch, 'Efficient global optimization of expensive black-box functions', Journal of Global Optimization, 13, 455-492, (1998).
[12]
A. Kumar, P.B. Nair, A.J. Keane, and S. Shahpar, 'Robust design using Bayesian Monte Carlo', International Journal for Numerical Methods in Engineering, 73, 1497-1517, (2008).
[13]
D. Lizotte, T. Wang, M. Bowling, and D. Schuurmans, 'Gaussian process regression for optimization', in NIPS Workshop on Value of Information, (2005).
[14]
D. Lizotte, T. Wang, M. Bowling, and D. Schuurmans, 'Automatic gait optimization with gaussian process regression', in International Joint Conference on Artificial Intelligence (IJCAI-07), (2007).
[15]
D.J. Lizotte, Practical Bayesian optimization, Ph.D. dissertation, University of Alberta, 2008.
[16]
D.J.C. MacKay, 'Information-based objective functions for active data selection', Neural Computation, 4(4), 589-603, (1992).
[17]
M.D. MacKay, R.J. Beckman, and W.J Conover, 'A comparison of three methods for selecting values of input variables in the analysis of output from a computer code', Technometrics, 21(2), 239-245, (1979).
[18]
R.H. Myers, A.I. Khuri, and G. Vining, 'Response surface alternatives to the Taguchi robust parameter design approach', The American Statistician, 46(2), 131-139, (1992).
[19]
Marin O., Designing computer experiments to estimate integrated respose functions, Ph.D. dissertation, The Ohio State University, 2005.
[20]
A. O'Hagan, 'Bayes-Hermite quadrature', Journal of Statistical Planning and Inference, 29, 245-260, (1991).
[21]
M.A. Osborne, R. Garnett, and S.J. Roberts, 'Gaussian processes for global optimization', in 3rd International Conference on Learning and Intelligent Optimization (LION3), Trento, Italy, (January 2009).
[22]
W. Ponweiser, T. Wagner, and M. Vincze, 'Clustered multiple generalized expected improvement: A novel infill sampling criterion for surrogate models', in Congress on Evolutionary Computation, pp. 3515- 3522, (2008).
[23]
C.E. Rasmussen and Z. Ghahramani, 'Bayesian Monte Carlo', in Advances in Neural Information Processing Systems, volume 15, pp. 505- 512. MIT Press, (2003).
[24]
C.E. Rasmussen and C.K.I. Williams, Gaussian Processes for Machine Learning, MIT Press, Cambridge, MA, 2006.
[25]
M.J. Sasena, P. Papalambros, and P. Goovaerts, 'Exploration of metamodeling sampling criteria for constrained global optimization', Eng. Opt., 34, 263-278, (2002).
[26]
M. Schonlau, W.J. Welch, and D.R. Jones, Global versus local search in constrained optimization of computer models, volume 34 of Lecture Notes - Monograph Series, 11-25, IMS, 1998.
[27]
S. Seo, M. Wallat, T. Graepel, and K. Obermayer, 'Gaussian process regression: active data selection and test point rejection', in Proc. International Joint Conf. Neural Networks, volume 3, pp. 241-246, (2000).
[28]
B. Settles, 'Active learning literature survey', Technical Report Computer Sciences Technical Report 1648, University of Wisconsin-Madison, (January 2009).
[29]
W.J. Welch, T.-K. Yu, S.M. Kang, and J. Sacks, 'Computer experiments for quality control by parameter design', J. Quality Technology, 22, 15- 22, (1990).
[30]
B.J. Williams, T.J. Santner, and W.I. Notz, 'Sequential design of computer experiments to minimize integrated response functions', Statistica Sinica, 10, 1133-1152, (2000).
- Bayesian Monte Carlo for the Global Optimization of Expensive Functions
Recommendations
Variational Bayesian Monte Carlo
NIPS'18: Proceedings of the 32nd International Conference on Neural Information Processing SystemsMany probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient ...
Bayesian Monte Carlo
NIPS'02: Proceedings of the 15th International Conference on Neural Information Processing SystemsWe investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation of prior knowledge, such as smoothness of the integrand, into the estimation. In a simple problem we show ...
Pseudo-marginal Hamiltonian Monte Carlo
Bayesian inference in the presence of an intractable likelihood function is computationally challenging. When following a Markov chain Monte Carlo (MCMC) approach to approximate the posterior distribution in this context, one typically either uses MCMC ...
Comments
Information & Contributors
Information
Published In
Publisher
IOS Press
Netherlands
Publication History
Published: 04 August 2010
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 25 Dec 2024