Memetic Computing

, Volume 10, Issue 3, pp 245–255 | Cite as

DICE: exploiting all bivariate dependencies in binary and multary search spaces

  • Fergal Lane
  • R. Muhammad Atif Azad
  • Conor Ryan
Regular Research Paper


Although some of the earliest Estimation of Distribution Algorithms (EDAs) utilized bivariate marginal distribution models, up to now, all discrete bivariate EDAs had one serious limitation: they were constrained to exploiting only a limited O(d) subset out of all possible \(O(d^{2})\) bivariate dependencies. As a first we present a family of discrete bivariate EDAs that can learn and exploit all \(O(d^{2})\) dependencies between variables, and yet have the same run-time complexity as their more limited counterparts. This family of algorithms, which we label DICE (DIscrete Correlated Estimation of distribution algorithms), is rigorously based on sound statistical principles, and particularly on a modelling technique from statistical physics: dichotomised multivariate Gaussian distributions. Initially (Lane et al. in European Conference on the Applications of Evolutionary Computation, Springer, 1999), DICE was trialled on a suite of combinatorial optimization problems over binary search spaces. Our proposed dichotomised Gaussian (DG) model in DICE significantly outperformed existing discrete bivariate EDAs; crucially, the performance gap increasingly widened as dimensionality of the problems increased. In this comprehensive treatment, we generalise DICE by successfully extending it to multary search spaces that also allow for categorical variables. Because correlation is not wholly meaningful for categorical variables, interactions between such variables cannot be fully modelled by correlation-based approaches such as in the original formulation of DICE. Therefore, here we extend our original DG model to deal with such situations. We test DICE on a challenging test suite of combinatorial optimization problems, which are defined mostly on multary search spaces. While the two versions of DICE outperform each other on different problem instances, they both outperform all the state-of-the-art bivariate EDAs on almost all of the problem instances. This further illustrates that these innovative DICE methods constitute a significant step change in the domain of discrete bivariate EDAs.


Dichotomised Gaussian models Bivariate estimation of distribution algorithms Combinatorial optimization 



This work was supported with the financial support of the Science Foundation Ireland grant 13/RC/2094.


  1. 1.
    Baluja S, Caruana R (1995) Removing the genetics from the standard genetic algorithm. In: 12th International conference on machine learning, pp 38–46Google Scholar
  2. 2.
    Baluja S, Davies S (1997) Using optimal dependency-trees for combinational optimization. In: 14th International conference on machine learning, pp 30–38Google Scholar
  3. 3.
    Boros E, Hammer P, Tavares G (2007) Local search heuristics for quadratic unconstrained binary optimization (QUBO). J Heuristics 13(2):99–132CrossRefGoogle Scholar
  4. 4.
    Caprara A et al (2014) Generation of antipodal random vectors with prescribed non-stationary 2-nd order statistics. IEEE Trans Signal Process 62(6):1603–1612MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chow C, Liu C (1968) Approximating discrete probability distributions with dependence trees. IEEE Trans Inf Theory 14(3):462–467MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crawford J, Auton L (1996) Experimental results on the crossover point in random 3-SAT. Artif Intell 81(1):31–57MathSciNetCrossRefGoogle Scholar
  7. 7.
    De Bonet JS, Isbell CL Jr, Viola PA (1997) MIMIC: finding optima by estimating probability densities. In: Mozer MC, Jordan MI, Petsche T (eds) Advances in neural information processing systems 9. MIT Press, pp 424–430Google Scholar
  8. 8.
    Emrich L, Piedmonte M (1991) A method for generating high-dimensional multivariate binary variates. Am Stat 45(4):302–304Google Scholar
  9. 9.
    Etxeberria R, Larranaga P (1999) Global optimization using Bayesian networks. In: Second symposium on artificial intelligence (CIMAF-99), Habana, Cuba, pp 332–339Google Scholar
  10. 10.
    Gallo G, et al (1980) Quadratic knapsack problems. In: Combinatorial optimization, Springer, pp 132–149Google Scholar
  11. 11.
    Gange S (1995) Generating multivariate categorical variates using the iterative proportional fitting algorithm. Am Stat 49(2):134–138Google Scholar
  12. 12.
    Glover F, Hao JK, Kochenberger G (2011) Polynomial unconstrained binary optimisation-part 2. Int J Metaheuristics 1(4):317–354MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hansen N, Kern S (2004) Evaluating the CMA evolution strategy on multimodal test functions. In: PPSN VIII, pp 282–291Google Scholar
  14. 14.
    Harik G, Lobo F et al (1999) The compact genetic algorithm. IEEE Trans Evolut Comput 3(4):287–297CrossRefGoogle Scholar
  15. 15.
    Higham N (2002) Computing the nearest correlation matrix : a problem from finance. IMA J Numer Anal 22(3):329–343MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hyrš M, Schwarz J (2014) Multivariate Gaussian copula in estimation of distribution algorithm with model migration. In: IEEE Foundations of computational intelligence (FOCI), pp 114–119Google Scholar
  17. 17.
    Jin R, Wang S et al (2015) Generating spatial correlated binary data through a copulas method. Sci Res 3(4):206–212CrossRefGoogle Scholar
  18. 18.
    Krząkała F (2005) How many colors to color a random graph? Cavity, complexity, stability and all that. Progress Theor Phys Suppl 157:357–360CrossRefGoogle Scholar
  19. 19.
    Lane F, Azad R, Ryan C (2017) DICE: a new family of bivariate estimation of distribution algorithms based on dichotomised multivariate Gaussian distributions. In: European conference on the applications of evolutionary computation, Springer, pp 670–685Google Scholar
  20. 20.
    Larrañaga P, Etxeberria R, et al (2000) Combinatorial optimization by learning and simulation of Bayesian networks. In: 16th conference on uncertainty in artificial intelligence, pp 343–352Google Scholar
  21. 21.
    Lee A (1993) Generating random binary deviates having fixed marginal distributions and specified degrees of association. Am Stat 47(3):209–215Google Scholar
  22. 22.
    Li B, Wang X, Zhong R, Zhuang Z (2006) Continuous optimization based-on boosting Gaussian mixture model. In: IEEE 18th international conference on pattern recognition, vol 1, pp 1192–1195Google Scholar
  23. 23.
    Li R, Emmerich M, et al (2006) Mixed-integer NK landscapes. In: PPSN IX, pp 42–51Google Scholar
  24. 24.
    Macke J, Berens P et al (2009) Generating spike trains with specified correlation coefficients. Neural Comput 21(2):397–423MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mühlenbein H (1997) The equation for response to selection and its use for prediction. Evolut Comput 5(3):303–346CrossRefGoogle Scholar
  26. 26.
    Ohlsson E (1998) Sequential Poisson Sampling. J Off Stat 14(2):149Google Scholar
  27. 27.
    Pelikan M, Goldberg D, Cantú-Paz E (1999) BOA: the Bayesian optimization algorithm. GECCO 1999:525–532Google Scholar
  28. 28.
    Pelikan M, Mühlenbein H (1999) The bivariate marginal distribution algorithm. In: Advances in Soft Computing, pp 521–535Google Scholar
  29. 29.
    Qi H, Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J Matrix Anal Appl 28(2):360–385MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rosén B (1997) On sampling with probability proportional to size. J Stat Plan Inference 62(2):159–191MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang Q, Sun J, Tsang E, Ford J (2002) Estimation of distribution algorithm based on mixture: preliminary experimental results. In: The 2002 UK workshop on computational intelligence (UKCI’02). University of Birmingham, UKGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CSIS DepartmentUniversity of LimerickLimerickIreland
  2. 2.School of Computing and Digital TechnologyBirmingham City UniversityBirminghamUK

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