Advertisement

Fast Fitness Improvements in Estimation of Distribution Algorithms Using Belief Propagation

  • Alexander Mendiburu
  • Roberto Santana
  • Jose A. Lozano
Part of the Adaptation, Learning, and Optimization book series (ALO, volume 14)

Abstract

Factor graphs can serve to represent Markov networks and Bayesian networks models. They can also be employed to implement efficient inference procedures such as belief propagation. In this paper we introduce a flexible implementation of belief propagation on factor graphs in the context of estimation of distribution algorithms (EDAs). By using a transformation from Bayesian networks to factor graphs, we show the way in which belief propagation can be inserted within the Estimation of Bayesian Networks Algorithm (EBNA). The objective of the proposed variation is to increase the search capabilities by extracting information of the, computationally costly to learn, Bayesian network. Belief Propagation applied to graphs with cycles allows to find (with a low computational cost), in many scenarios, the point with the highest probability of a Bayesian network. We carry out some experiments to show how this modification can increase the potentialities of Estimation of Distribution Algorithms.

Keywords

Bayesian Network Evolutionary Computation Belief Propagation Variable Node Factor Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baluja, S., Davies, S.: Using optimal dependency-trees for combinatorial optimization: Learning the structure of the search space. Tech. rep., Carnegie Mellon Report, CMU-CS-97-107 (1997)Google Scholar
  2. 2.
    Bayati, M., Shah, D., Sharma, M.: Maximum weight matching via max-product belief propagation. IEEE Transactions on Information Theory 54(3), 1241–1251 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chickering, D.M., Geiger, D., Heckerman, D.: Learning Bayesian networks is NP-hard. Tech. Rep. MSR-TR-94-17, Microsoft Research, Redmond, WA (1994)Google Scholar
  4. 4.
    Coughlan, J.M., Ferreira, S.J.: Finding Deformable Shapes Using Loopy Belief Propagation. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 453–468. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Crick, C., Pfeffer, A.: Loopy belief propagation as a basis for communication in sensor networks. In: Proceedings of the 19th Annual Conference on Uncertainty in Artificial Intelligence (UAI 2003), pp. 159–166. Morgan Kaufmann Publishers (2003)Google Scholar
  6. 6.
    Deb, K., Goldberg, D.E.: Sufficient conditions for deceptive and easy binary functions. Annals of Mathematics and Artificial Intelligence 10, 385–408 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Echegoyen, C., Lozano, J.A., Santana, R., Larrañaga, P.: Exact Bayesian network learning in estimation of distribution algorithms. In: Proceedings of the 2007 Congress on Evolutionary Computation, CEC 2007, pp. 1051–1058. IEEE Press (2007)Google Scholar
  8. 8.
    Echegoyen, C., Mendiburu, A., Santana, R., Lozano, J.: Analyzing the probability of the optimum in EDAs based on Bayesian networks. In: Proceedings of the 2009 Congress on Evolutionary Computation (CEC 2009), pp. 1652–1659. IEEE Press, Trondheim (2009)CrossRefGoogle Scholar
  9. 9.
    Echegoyen, C., Mendiburu, A., Santana, R., Lozano, J.A.: Towards understanding edas based on bayesian networks through a quantitative analysis. IEEE Trans. Evolutionary Computation (accepted for publication)Google Scholar
  10. 10.
    Freeman, W.T., Pasztor, E.C., Carmichael, O.T.: Learning low-level vision. International Journal of Computer Vision 40(1), 25–47 (2000)zbMATHCrossRefGoogle Scholar
  11. 11.
    Hauschild, M., Pelikan, M., Lima, C., Sastry, K.: Analyzing probabilistic models in hierarchical BOA on traps and spin glasses. In: Thierens [48], pp. 523–530 (2007c)Google Scholar
  12. 12.
    Hauschild, M., Pelikan, M., Sastry, K., Lima, C.F.: Analyzing probabilistic models in hierarchical boa. IEEE Trans. Evolutionary Computation 13(6), 1199–1217 (2009)CrossRefGoogle Scholar
  13. 13.
    Henrion, M.: Propagating uncertainty in Bayesian networks by probabilistic logic sampling. In: Lemmer, J.F., Kanal, L.N. (eds.) Proceedings of the Second Annual Conference on Uncertainty in Artificial Intelligence, pp. 149–164. Elsevier (1988)Google Scholar
  14. 14.
    Höns, R.: Estimation of distribution algorithms and minimum relative entropy. Ph.D. thesis, University of Bonn, Bonn, Germany (2006)Google Scholar
  15. 15.
    Höns, R., Santana, R., Larrañaga, P., Lozano, J.A.: Optimization by max-propagation using Kikuchi approximations. Tech. Rep. EHU-KZAA-IK-2/07, Department of Computer Science and Artificial Intelligence, University of the Basque Country (2007), http://www.sc.ehu.es/ccwbayes/technical.htm
  16. 16.
    Kschischang, F.R., Frey, B.J., Loeliger, H.A.: Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory 47(2), 498–519 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Larrañaga, P., Etxeberria, R., Lozano, J.A., Peña, J.: Combinatorial optimization by learning and simulation of Bayesian networks. In: Proceedings of the Sixteenth Annual Conference on Uncertainty in Artificial Intelligence (UAI 2000), pp. 343–352. Morgan Kaufmann Publishers, San Francisco (2000)Google Scholar
  18. 18.
    Larrañaga, P., Lozano, J.A. (eds.): Estimation of Distribution Algorithms. A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Boston (2002)zbMATHGoogle Scholar
  19. 19.
    Lima, C.F., Pelikan, M., Lobo, F.G., Goldberg, D.E.: Loopy Substructural Local Search for the Bayesian Optimization Algorithm. In: Stützle, T., Birattari, M., Hoos, H.H. (eds.) SLS 2009. LNCS, vol. 5752, pp. 61–75. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Lima, C.F., Pelikan, M., Sastry, K., Butz, M.V., Goldberg, D.E., Lobo, F.G.: Substructural Neighborhoods for Local Search in the Bayesian Optimization Algorithm. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN 2006. LNCS, vol. 4193, pp. 232–241. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    McEliece, R.J., MacKay, D.J.C., Cheng, J.F.: Turbo Decoding as an Instance of Pearl’s ”Belief Propagation” Algorithm. IEEE Journal on Selected Areas in Communications 16(2), 140–152 (1998)CrossRefGoogle Scholar
  22. 22.
    Meltzer, T., Yanover, C., Weiss, Y.: Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation. In: ICCV, pp. 428–435. IEEE Computer Society (2005)Google Scholar
  23. 23.
    Mendiburu, A., Santana, R., Lozano, J.: Introducing belief propagation in estimation of distribution algorithms: A parallel approach. Tech. Rep. EHU-KAT-IK-11-07, Department of Computer Science and Artificial Intelligence, The University of the Basque Country (2007)Google Scholar
  24. 24.
    Mendiburu, A., Santana, R., Lozano, J.A., Bengoetxea, E.: A parallel framework for loopy belief propagation. In: Thierens [48], pp. 2843–2850 (2007c)Google Scholar
  25. 25.
    Mühlenbein, H.: The equation for response to selection and its use for prediction. Evolutionary Computation 5(3), 303–346 (1997)CrossRefGoogle Scholar
  26. 26.
    Mühlenbein, H., Höns, R.: The factorized distributions and the minimum relative entropy principle. In: Pelikan, M., Sastry, K., Cantú-Paz, E. (eds.) Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications. SCI, pp. 11–38. Springer (2006)Google Scholar
  27. 27.
    Mühlenbein, H., Mahnig, T.: FDA – a scalable evolutionary algorithm for the optimization of additively decomposed functions. Evolutionary Computation 7(4), 353–376 (1999)CrossRefGoogle Scholar
  28. 28.
    Mühlenbein, H., Paaß, G.: From Recombination of Genes to the Estimation of Distributions I. Binary Parameters. In: Ebeling, W., Rechenberg, I., Voigt, H.-M., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 178–187. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  29. 29.
    Nilsson, D.: An efficient algorithm for finding the M most probable configurations in probabilistic expert systems. Statistics and Computing 2, 159–173 (1998)CrossRefGoogle Scholar
  30. 30.
    Ochoa, A.: EBBA - Evolutionary best basis algorithm. In: Ochoa, A., Soto, M.R., Santana, R. (eds.) Proceedings of the Second International Symposium on Adaptive Systems (ISAS 1999), pp. 93–98. Editorial Academia, Havana (1999)Google Scholar
  31. 31.
    Ochoa, A., Höns, R., Soto, M., Mühlenbein, H.: A Maximum Entropy Approach to Sampling in EDA – the Single Connected Case. In: Sanfeliu, A., Ruiz-Shulcloper, J. (eds.) CIARP 2003. LNCS, vol. 2905, pp. 683–690. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  32. 32.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, Palo Alto (1988)Google Scholar
  33. 33.
    Pelikan, M., Goldberg, D.E., Cantú-Paz, E.: BOA: The Bayesian optimization algorithm. In: Banzhaf, W., Daida, J., Eiben, A.E., Garzon, M.H., Honavar, V., Jakiela, M., Smith, R.E. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 1999, Orlando FL, vol. I, pp. 525–532. Morgan Kaufmann Publishers, San Francisco (1999)Google Scholar
  34. 34.
    Pelikan, M., Goldberg, D.E., Lobo, F.: A survey of optimization by building and using probabilistic models. IlliGAL Report No. 99018, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, IL (1999)Google Scholar
  35. 35.
    Pelikan, M., Sastry, K., Goldberg, D.E.: Sporadic model building for efficiency enhancement of the hierarchical BOA. Genetic Programming and Evolvable Machines 9(1), 53–84 (2008)CrossRefGoogle Scholar
  36. 36.
    Pereira, F.B., Machado, P., Costa, E., Cardoso, A., Ochoa, A., Santana, R., Soto, M.R.: Too busy to learn. In: Proceedings of the 2000 Congress on Evolutionary Computation, CEC 2000, pp. 720–727. IEEE Press, La Jolla Marriott Hotel La Jolla (2000)CrossRefGoogle Scholar
  37. 37.
    Potetz, B.: Efficient belief propagation for vision using linear constraint nodes. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE (2007)Google Scholar
  38. 38.
    Richardson, T.J., Urbanke, R.L.: The capacity of low-density parity-check codes under message-passing decoding. IEEE Transactions on Information Theory 47(2), 599–618 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Santana, R.: Advances in probabilistic graphical models for optimization and learning: Applications in protein modelling. Ph.D. thesis (2006)Google Scholar
  40. 40.
    Santana, R., Larrañaga, P., Lozano, J.A.: The Role of a Priori Information in the Minimization of Contact Potentials by Means of Estimation of Distribution Algorithms. In: Marchiori, E., Moore, J.H., Rajapakse, J.C. (eds.) EvoBIO 2007. LNCS, vol. 4447, pp. 247–257. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  41. 41.
    Santana, R., Larrañaga, P., Lozano, J.A.: Protein folding in simplified models with estimation of distribution algorithms. IEEE Transactions on Evolutionary Computation (2008) (in Press)Google Scholar
  42. 42.
    Santana, R., Larrañaga, P., Lozano, J.A.: Learning factorizations in estimation of distribution algorithms using affinity propagation. Evolutionary Computation 18(4), 515–546 (2010)CrossRefGoogle Scholar
  43. 43.
    Sastry, K., Goldberg, D.E.: Designing Competent Mutation Operators Via Probabilistic Model Building of Neighborhoods. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3103, pp. 114–125. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  44. 44.
    Sastry, K., Lima, C., Goldberg, D.E.: Evaluation relaxation using substructural information and linear estimation. In: Proceedings of the 8th annual Conference on Genetic and Evolutionary Computation, GECCO 2006, pp. 419–426. ACM Press, New York (2006)CrossRefGoogle Scholar
  45. 45.
    Sastry, K., Pelikan, M., Goldberg, D.: Efficiency enhancement of genetic algorithms via building-block-wise fitness estimation. In: Proceedings of the 2004 Congress on Evolutionary Computation, CEC 2004, pp. 720–727. IEEE Press, Portland (2004)Google Scholar
  46. 46.
    Shakya, S.: DEUM: A framework for an estimation of distribution algorithm based on markov random fields. Ph.D. thesis, The Robert Gordon University, Aberdeen, UK (2006)Google Scholar
  47. 47.
    Soto, M.R.: A single connected factorized distribution algorithm and its cost of evaluation. Ph.D. thesis, University of Havana, Havana, Cuba (2003) (in Spanish)Google Scholar
  48. 48.
    Thierens, D. (ed.): Proceedings of Genetic and Evolutionary Computation Conference, GECCO 2007, London, England, UK, July 7-11. Companion Material. ACM (2007)Google Scholar
  49. 49.
    Wainwright, M., Jaakkola, T., Willsky, A.: Tree consistency and bounds on the performance of the max-product algorithm and its generalizations. Statistics and Computing 14, 143–166 (2004)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Yanover, C., Weiss, Y.: Finding the M most probable configurations using loopy belief propagation. In: Thrun, S., Saul, L., Schölkopf, B. (eds.) Advances in Neural Information Processing Systems, vol. 16, p. 289. MIT Press, Cambridge (2004)Google Scholar
  51. 51.
    Yedidia, J.S., Freeman, W.T., Weiss, Y.: Constructing free energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory 51(7), 2282–2312 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  • Alexander Mendiburu
    • 1
  • Roberto Santana
    • 1
  • Jose A. Lozano
    • 1
  1. 1.Intelligent Systems GroupThe University of the Basque Country (UPV/EHU)San-SebastianSpain

Personalised recommendations