Skip to main content

Advertisement

SpringerLink
Log in

Model accuracy in the Bayesian optimization algorithm

  • Original Paper
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Evolutionary algorithms (EAs) are particularly suited to solve problems for which there is not much information available. From this standpoint, estimation of distribution algorithms (EDAs), which guide the search by using probabilistic models of the population, have brought a new view to evolutionary computation. While solving a given problem with an EDA, the user has access to a set of models that reveal probabilistic dependencies between variables, an important source of information about the problem. However, as the complexity of the used models increases, the chance of overfitting and consequently reducing model interpretability, increases as well. This paper investigates the relationship between the probabilistic models learned by the Bayesian optimization algorithm (BOA) and the underlying problem structure. The purpose of the paper is threefold. First, model building in BOA is analyzed to understand how the problem structure is learned. Second, it is shown how the selection operator can lead to model overfitting in Bayesian EDAs. Third, the scoring metric that guides the search for an adequate model structure is modified to take into account the non-uniform distribution of the mating pool generated by tournament selection. Overall, this paper makes a contribution towards understanding and improving model accuracy in BOA, providing more interpretable models to assist efficiency enhancement techniques and human researchers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Notes

  1. Given a sufficient population size (learning data set).

References

  • Ackley DH (1987) A connectionist machine for genetic hill climbing. Kluwer Academic, Boston

    Google Scholar 

  • Ahn CW, Ramakrishna RS (2008) On the scalability of the real-coded Bayesian optimization algorithm. IEEE Trans Evol Comput 12(3):307–322

    Article  Google Scholar 

  • Balakrishnan N, Nevzorov VB (2003) A primer on statistical distributions. Wiley

  • Blickle T, Thiele L (1997) A comparison of selection schemes used in genetic algorithms. Evol Comput 4(4):311–347

    Google Scholar 

  • Brindle A (1981) Genetic algorithms for function optimization. PhD thesis, University of Alberta, Edmonton, Canada (unpublished doctoral dissertation)

  • Chickering DM, Geiger D, Heckerman D (1994) Learning Bayesian networks is NP-Hard. Technical Report MSR-TR-94-17, Microsoft Research, Redmond, WA

  • Chickering DM, Heckerman D, Meek C (1997) A Bayesian approach to learning Bayesian networks with local structure. Technical Report MSR-TR-97-07, Microsoft Research, Redmond, WA

  • Cooper GF, Herskovits EH (1992) A Bayesian method for the induction of probabilistic networks from data. Mach Learn 9:309–347

    MATH  Google Scholar 

  • Cormen TH, Leiserson CE, Rivest RL (1990) Introduction to algorithms. MIT Press, Massachusetts

    MATH  Google Scholar 

  • Correa ES, Shapiro JL (2006) Model complexity vs. performance in the Bayesian optimization algorithm. In: Runarsson TP et al (eds) PPSN IX: Parallel problem solving from nature, LNCS 4193, Springer, pp 998–1007

  • Deb K, Goldberg DE (1993) Analyzing deception in trap functions. Foundations of genetic algorithms 2, pp 93–108

  • Echegoyen C, Lozano JA, Santana R, Larrañaga P (2007) Exact bayesian network learning in estimation of distribution algorithms. In: Proceedings of the IEEE congress on evolutionary computation, IEEE Press, pp 1051–1058

  • Etxeberria R, Larrañaga P (1999) Global optimization using Bayesian networks. In: Rodriguez AAO et al (eds) Second symposium on artificial intelligence (CIMAF-99), Habana, Cuba, pp 332–339

  • Friedman N, Goldszmidt M (1999) Learning Bayesian networks with local structure. Graphical Models. MIT Press, pp 421–459

  • Goldberg DE, Sastry K (2010) Genetic algorithms: the design of innovation, 2nd edn. Springer

  • Goldberg DE, Korb B, Deb K (1989) Messy genetic algorithms: motivation, analysis, and first results. Complex Syst 3(5):493–530

    MathSciNet  MATH  Google Scholar 

  • Harik GR (1995) Finding multimodal solutions using restricted tournament selection. In: Proceedings of the sixth international conference on genetic algorithms pp 24–31

  • Harik GR, Lobo FG, Goldberg DE (1999) The compact genetic algorithm. IIEEE Trans Evol Comput 3(4):287–297

    Article  Google Scholar 

  • Hauschild M, Pelikan M (2008) Enhancing efficiency of hierarchical BOA via distance-based model restrictions. In: Proceedings of the 10th international conference on parallel problem solving from nature, Springer-Verlag, pp 417–427

  • Hauschild M, Pelikan M, Sastry K, Goldberg DE (2008) Using previous models to bias structural learning in the hierarchical BOA. In: Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2008), ACM, New York, NY, USA, pp 415–422

  • Hauschild M, Pelikan M, Sastry K, Lima CF (2009) Analyzing probabilistic models in hierarchical BOA. IEEE Trans Evol Comput 13(6):1199–1217

    Article  Google Scholar 

  • Heckerman D, Geiger D, Chickering DM (1994) Learning Bayesian networks: the combination of knowledge and statistical data. Technical Report MSR-TR-94-09, Microsoft Research, Redmond, WA

  • Henrion M (1988) Propagation of uncertainty in Bayesian networks by logic sampling. In: Lemmer JF, Kanal LN (eds.) Uncertainty in artificial intelligence, Elsevier, pp 149–163

  • Japkowicz N, Stephen S (2002) The class imbalance problem: a systematic study. Intell Data Anal 6(5):429–450

    MATH  Google Scholar 

  • Johnson A, Shapiro J (2001) The importance of selection mechanisms in distribution estimation algorithms. In: Proceedings of the 5th European conference on artificial evolution, LNCS vol 2310, Springer-Verlag, London, pp 91–103

  • Kubat M, Matwin S (1997) Addressing the curse of imbalanced training sets: one-sided selection. In: Proc. 14th international Conference on Machine Learning, Morgan Kaufmann, pp 179–186

  • Larrañaga P, Lozano JA (eds) (2002) Estimation of distribution algorithms: a new tool for evolutionary computation. Kluwer Academic Publishers, Boston, MA

    MATH  Google Scholar 

  • Lima CF (2009) Substructural local search in discrete estimation of distribution algorithms. PhD thesis, University of Algarve, Portugal

  • Lima CF, Sastry K, Goldberg DE, Lobo FG (2005) Combining competent crossover and mutation operators: a probabilistic model building approach. In: Beyer H et al (eds) Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2005), ACM Press, pp 735–742

  • Lima CF, Pelikan M, Sastry K, Butz M, Goldberg DE, Lobo FG (2006) Substructural neighborhoods for local search in the Bayesian optimization algorithm. In: Runarsson TP et al (eds) PPSN IX: parallel problem solving from nature, LNCS 4193, Springer, pp 232–241

  • Lima CF, Goldberg DE, Pelikan M, Lobo FG, Sastry K, Hauschild M (2007) Influence of selection and replacement strategies on linkage learning in BOA. In: Tan KC et al (eds) IEEE Congress on evolutionary computation (CEC-2007), IEEE Press, pp 1083–1090

  • Lima CF, Pelikan M, Lobo FG, Goldberg DE (2009) Loopy substructural local search for the Bayesian optimization algorithm. In: Proceedings of the second international workshop on engineering stochastic local search algorithms (SLS-2009), LNCS Vol. 5752, Springer, pp 61–75

  • Lozano JA, Larrañaga P, Inza I, Bengoetxea E (eds) (2006) Towards a new evolutionary computation: advances on estimation of distribution algorithms. Springer, Berlin

  • Mühlenbein H (2008) Convergence of estimation of distribution algorithms for finite samples. (unpublished manuscript)

  • Mühlenbein H, Mahning T (1999) FDA—a scalable evolutionary algorithm for the optimization of additively decomposed functions. Evol Comput 7(4):353–376

    Article  Google Scholar 

  • Mühlenbein H, Schlierkamp-Voosen D (1993) Predictive models for the breeder genetic algorithm: I. Continuous parameter optimization. Evol Comput 1(1):25–49

    Article  Google Scholar 

  • Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, San Mateo, CA

    Google Scholar 

  • Pelikan M (2005) Hierarchical Bayesian optimization algorithm: toward a new generation of evolutionary algorithms. Springer

  • Pelikan M, Goldberg DE (2001) Escaping hierarchical traps with competent genetic algorithms. In: Spector L, et al (eds) Proceedings of the genetic and evolutionary computation conference (GECCO-2001), Morgan Kaufmann, San Francisco, CA, pp 511–518

  • Pelikan M, Sastry K (2004) Fitness inheritance in the bayesian optimization algorithm. In: Deb K et al (eds) Proceedings of the genetic and evolutionary computation conference (GECCO-2004), Part II, LNCS 3103, Springer, pp 48–59

  • Pelikan M, Goldberg DE, Cantú-Paz E (1999) BOA: the Bayesian optimization algorithm. In: Banzhaf W et al (eds) Proceedings of the genetic and evolutionary computation conference GECCO-99, Morgan Kaufmann, San Francisco, CA, pp 525–532

  • Pelikan M, Goldberg DE, Lobo F (2002) A survey of optimization by building and using probabilistic models. Comput Optim Appl 21(1):5–20

    Article  MathSciNet  MATH  Google Scholar 

  • Pelikan M, Sastry K, Goldberg DE (2003) Scalability of the Bayesian optimization algorithm. Int J Approx Reason 31(3):221–258

    Article  MathSciNet  Google Scholar 

  • Pelikan M, Sastry K, Cantú-Paz E (eds) (2006) Scalable optimization via probabilistic modelling: from algorithms to applications. Springer

  • Pyle D (1999) Data preparation for data mining. Morgan Kaufmann, San Francisco, CA

    Google Scholar 

  • Rissanen JJ (1978) Modelling by shortest data description. Automatica 14:465–471

    Article  MATH  Google Scholar 

  • Santana R, Larrañaga P, Lozano JA (2005) Interactions and dependencies in estimation of distribution algorithms. In: Proceedings of the IEEE congress on evolutionary computation, IEEE Press, pp 1418–1425

  • Santana R, Larrañaga P, Lozano JA (2008) Protein folding in simplified models with estimation of distribution algorithms. IEEE Trans Evol Comput 12(4):418–438

    Article  Google Scholar 

  • Sastry K (2001) Evaluation-relaxation schemes for genetic and evolutionary algorithms. Master’s thesis, University of Illinois at Urbana-Champaign, Urbana, IL

  • Sastry K, Goldberg DE (2004) Designing competent mutation operators via probabilistic model building of neighborhoods. In: Deb K et al (eds) Proceedings of the genetic and evolutionary computation conference (GECCO-2004), Part II, LNCS 3103, Springer, pp 114–125

  • Sastry K, Pelikan M, Goldberg DE (2004) Efficiency enhancement of genetic algorithms via building-block-wise fitness estimation. In: Proceedings of the IEEE international conference on evolutionary computation, pp 720–727

  • Sastry K, Abbass HA, Goldberg DE, Johnson DD (2005) Sub-structural niching in estimation distribution algorithms. In: Beyer H, et al (eds) Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2005), ACM Press

  • Sastry K, Lima CF, Goldberg DE (2006) Evaluation relaxation using substructural information and linear estimation. In: Keijzer M et al (eds) Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2006), ACM Press, pp 419–426

  • Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464

    Article  MATH  Google Scholar 

  • Thierens D (1999) Scalability problems of simple genetic algorithms. Evol Comput 7(1):45–68

    Article  Google Scholar 

  • Thierens D, Goldberg DE (1993) Mixing in genetic algorithms. In: Forrest S (ed) Proceedings of the Fifth international conference on genetic algorithms, Morgan Kaufmann, San Mateo, CA, pp 38–45

  • Weiss GM, Provost F (2003) Learning when training data are costly: the effect of class distribution on tree induction. J Artif Intell Res 19:315–354

    MATH  Google Scholar 

  • Wu H, Shapiro JL (2006) Does overfitting affect performance in estimation of distribution algorithms. In: Keijzer M et al (eds) Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2006), ACM Press, pp 433–434

  • Yu TL, Goldberg DE (2004) Dependency structure matrix analysis: Offline utility of the dependency structure matrix genetic algorithm. In: Deb K et al (eds) Proceedings of the genetic and evolutionary computation conference (GECCO-2004), Part II, LNCS 3103, Springer, pp 355–366

  • Yu TL, Sastry K, Goldberg DE (2007a) Population size to go: Online adaptation using noise and substructural measurements. In: Lobo FG, et al (eds) Parameter setting in evolutionary algorithms, Springer, pp 205–224

  • Yu TL, Sastry K, Goldberg DE, Pelikan M (2007b) Population sizing for entropy-based model building in genetic algorithms. In: Thierens D, et al (eds) Proceedings of the ACM SIGEVO genetic and evolutionary computation conference (GECCO-2007), ACM Press, pp 601–608

  • Yu TL, Goldberg DE, Sastry K, Lima CF, Pelikan M (2009) Dependency structure matrix, genetic algorithms, and effective recombination. Evol Comput 17(4):595–626

    Article  Google Scholar 

Download references

Acknowledgments

This work was sponsored by the Portuguese Foundation for Science and Technology under grants SFRH-BD-16980-2004 and PTDC-EIA-67776-2006, the BBSRC under grant BB/D0196131, the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant FA9550-06-1-0096, and the National Science Foundation under NSF CAREER grant ECS-0547013. Fernando Lobo is also a member of the Center for Environmental and Sustainability Research (CENSE) and acknowledges its support. The work was also supported by the High Performance Computing Collaboratory sponsored by Information Technology Services, the Research Award and the Research Board at the University of Missouri in St. Louis. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation thereon.

Author information

Authors and Affiliations

  1. Centre for Plant Integrative Biology, University of Nottingham, Sutton Bonington Campus, Loughborough, LE12 5RD, UK

    Claudio F. Lima

  2. Department of Electronics and Informatics Engineering, University of Algarve, Campus de Gambelas, 8000-117, Faro, Portugal

    Fernando G. Lobo

  3. Department of Mathematics and Computer Science, 320 CCB, University of Missouri at St. Louis, St. Louis, MO, 63121, USA

    Martin Pelikan

  4. Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    David E. Goldberg

Authors
  1. Claudio F. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando G. Lobo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Pelikan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David E. Goldberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudio F. Lima.

Additional information

The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research, the National Science Foundation, or the US Government.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lima, C.F., Lobo, F.G., Pelikan, M. et al. Model accuracy in the Bayesian optimization algorithm. Soft Comput 15, 1351–1371 (2011). https://doi.org/10.1007/s00500-010-0675-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-010-0675-y

Keywords

Search

Navigation