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Coevolutionary bid-based genetic programming for problem decomposition in classification

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Abstract

In this work a cooperative, bid-based, model for problem decomposition is proposed with application to discrete action domains such as classification. This represents a significant departure from models where each individual constructs a direct input-outcome map, for example, from the set of exemplars to the set of class labels as is typical under the classification domain. In contrast, the proposed model focuses on learning a bidding strategy based on the exemplar feature vectors; each individual is associated with a single discrete action and the individual with the maximum bid ‘wins’ the right to suggest its action. Thus, the number of individuals associated with each action is a function of the intra-action bidding behaviour. Credit assignment is designed to reward correct but unique bidding strategies relative to the target actions. An advantage of the model over other teaming methods is its ability to automatically determine the number of and interaction between cooperative team members. The resulting model shares several traits with learning classifier systems and as such both approaches are benchmarked on nine large classification problems. Moreover, both of the evolutionary models are compared against the deterministic Support Vector Machine classification algorithm. Performance assessment considers the computational, classification, and complexity characteristics of the resulting solutions. The bid-based model is found to provide simple yet effective solutions that are robust to wide variations in the class representation. Support Vector Machines and classifier systems tend to perform better under balanced datasets albeit resulting in black-box solutions.

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Notes

  1. Hereafter, we will use ‘class’ and ‘action’ interchangeably.

  2. As opposed to the test data employed for assessing the generalization performance of the trained classifier.

  3. Compared to a parallel implementation, training overhead increases minimally as Step 2(c) in Fig. 1 is executed |L| times more often where L is the set of class labels.

  4. An alternative approach might take the form of an auction-based model for credit assignment. Such models are based on the concept of ‘wealth’ [38]. However, it becomes increasingly difficult to establish robust mechanisms for deriving the ‘wealth’ property when the subset of exemplars from which wealth-based performance metrics are derived is continuously varying—as is the case of the competitive coevolutionary paradigm central to scaling the BGP model to large problem domains.

  5. For efficiency, micro-classifiers representing identical condition-action rules are recorded as a single macro-classifier. Each macro-classifier has an associated numerosity which is used to weigh calculations accordingly. Here, the algorithm is described in terms of micro-classifiers.

  6. Otherwise the test data would have been used to build the model compromising the independence of the test partition.

  7. All datasets with more than two classes will be ‘unbalanced’ since each class can at best (i.e., when the number of exemplars belonging to each class is the same) account for \(\frac{1}{\# of classes}\) of all exemplars.

  8. The removal of introns, which were found to account for between 60% to 90% of instructions in linear GP [51], was not performed.

  9. Given that 1,250,000 new learners are generated when training learners of each action for 50,000 generations, Step 2 of the learning algorithm, on THY roughly one in 127, 116, and 173 new learners is accepted into the population when training for action 0, 1, and 2 respectively.

  10. Although a small mutation in a program’s genotype may yield a disproportionally large change in its phenotype.

References

  1. J.R. Koza, in Genetic Programming: On the Programming of Computers by Means of Natural Selection. (MIT Press, Cambridge, MA, 1992)

    MATH  Google Scholar 

  2. W.B. Langdon, R. Poli, in Foundations of Genetic Programming (Springer-Verlag, Berlin, 2002)

    Book  MATH  Google Scholar 

  3. P.J. Angeline, J.B. Pollack, The evolutionary induction of subroutines, in Proceedings of the Fourteenth Annual Conference of the Cognitive Science Society (Lawrence Erlbaum, Hillsdale, NJ, 1992), pp. 236–241.

  4. M. Brameier, W. Banzhaf, Evolving teams of predictors with linear genetic programming. Genet. Program. Evolvable Mach. 2(4), 381–407 (2001)

    Article  MATH  Google Scholar 

  5. T. Soule, in Cooperative Evolution on the Intertwined Spirals Problem, ed. by E. Cantú-Paz, J.A. Foster, K. Deb, L.D. Davis, R. Roy, U.-M. O’Reilly, H.-G. Beyer, R. Standish, G. Kendall, S. Wilson, J. Wegener, D. Dasgupta, M.A. Potter, A.C. Schultz. Proceedings of the European Conference on Genetic Programming (Springer-Verlag, Berlin, 2003), pp. 434–442

    Google Scholar 

  6. R. Thomason, T. Soule, in Novel Ways of Improving Cooperation and Performance in Ensemble Classifiers, ed. by D. Thierens, H.-G. Beyer, J. Bongard, J. Branke, J.A. Clark, D. Cliff, C.B. Congdon, K. Deb, B. Doerr, T. Kovacs, S. Kumar, J.F. Miller, J. Moore, F. Neumann, M. Pelikan, R. Poli, K. Sastry, K.O. Stanley, T. Stutzle, R.A. Watson, I. Wegener. Proceedings of the Genetic and Evolutionary Computation Conference. (ACM Press, New York, NY, 2007), pp. 1708–1715

    Chapter  Google Scholar 

  7. E.D. De Jong, J.B. Pollack, Ideal evaluation from coevolution. Evol. Comput. 12, 159–192 (2004)

    Article  Google Scholar 

  8. S.G. Ficici, J.B. Pollack, Pareto Optimality in Coevolutionary Learning, ed. by J. Kelemen, P. Sosik. Proceedings of the 6th European Conference on Advances in Artificial Life (Springer-Verlag, Berlin, 2001), pp. 316–325

    Google Scholar 

  9. J. Noble, R.A. Watson, in Pareto Coevolution: Using Performance Against Coevolved Opponents in a Game as Dimensions for Pareto Selection, ed. by L. Spector, E.D. Goodman, A. Wu, W.B. Langdon, H.-M. Voigt, M. Gen, S. Sen, M. Dorigo, S. Pezeshk, M.H. Garzon, E. Burke. Proceedings of the Genetic and Evolutionary Computation Conference (Morgan Kaufmann, San Francisco, CA, 2001), pp. 493–500

    Google Scholar 

  10. M. Lemczyk, M.I. Heywood, in Training Binary GP Classifiers Efficiently: A Pareto-coevolutionary Approach, ed. by M. Ebner, M. O’Neill, A. Ekárt, L. Vanneschi, A.I. Esparcia-Alcázar. Proceedings of the European Conference on Genetic Programming (Berlin, Springer-Verlag, 2007), pp. 229–240

    Google Scholar 

  11. P. Lichodzijewski, M.I. Heywood, in Pareto-coevolutionary Genetic Programming for Problem Decomposition in Multi-class Classification, ed. by D. Thierens, H.-G. Beyer, J. Bongard, J. Branke, J.A. Clark, D. Cliff, C.B. Congdon, K. Deb, B. Doerr, T. Kovacs, S. Kumar, J.F. Miller, J. Moore, F. Neumann, M. Pelikan, R. Poli, K. Sastry, K.O. Stanley, T. Stutzle, R.A. Watson, I. Wegener. Proceedings of the Genetic and Evolutionary Computation Conference (ACM Press, New York, NY, 2007), pp. 464–471

    Chapter  Google Scholar 

  12. A. McIntyre, M.I. Heywood, in Multi-Objective Competitive Coevolution for Efficient GP Classifier Problem Decomposition. Proceedings of the International Conference on Systems, Man and Cybernetics (IEEE Press, 2007), pp. 1930–1937

  13. J.H. Holland, J.S. Reitman, Cognitive Systems Based on Adaptive Algorithms (Pattern Directed Inference Systems, 1978)

  14. S. Wilson, Classifier fitness based on accuracy. Evol. Comput. 3(2), 149–175 (1995)

    Article  Google Scholar 

  15. C.-C. Chang, C.-J. Lin. LIBSVM: A Library for Support Vector Machines, Version 2.85 (2007), Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm

  16. E. Bauer, R. Kohavi, An empirical comparison of voting classification algorithms: bagging, boosting, and variants. Mach. Learn. 36(1–2), 105–139 (1999)

    Article  Google Scholar 

  17. G. Folino, C. Pizzuti, G. Spezzano, GP ensembles for large-scale data classification. IEEE Trans. Evol. Comput. 10(5), 604–616 (2006)

    Article  Google Scholar 

  18. H. Iba, in Bagging, Boosting and Bloating in Genetic Programming, ed. by W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, R.E. Smith. Proceedings of the Genetic and Evolutionary Computation Conference (Morgan Kaufmann, San Francisco, CA, 1999), pp. 1053–1060

    Google Scholar 

  19. K. Imamura, T. Soule, R.B. Heckendorn, J.A. Foster, Behavioral diversity and a probabilistically optimal GP ensemble. Genet. Program. Evolvable Mach. 4(3), 235–253 (2003)

    Article  Google Scholar 

  20. A.R. McIntyre, M.I. Heywood, in MOGE:GP Classification Problem Decomposition Using Multi-objective Optimization, ed. by M. Keijzer, M. Cattolico, D. Arnold, V. Babovic, C. Blum, P. Bosman, M.V. Butz, C. Coello Coello, D. Dasgupta, S.G. Ficici, J. Foster, A. Hernandez-Aguirre, G. Hornby, H. Lipson, P. McMinn, J. Moore, G. Raidl, R. Franz, C. Ryan, D. Thierens. Proceedings of the Genetic and Evolutionary Computation Conference (ACM Press, New York, NY, 2006), pp. 863–870

    Chapter  Google Scholar 

  21. P.L. Lanzi, R.L. Riolo, Recent Trends in Learning Classifier Systems research. Advances in Evolutionary Computing: Theory and Applications (2003), pp. 955–988

  22. M. Butz, S.W. Wilson, in An Algorithmic Description of XCS, ed. by P.L. Lanzi, W. Stolzmann, S.W. Wilson. IWLCS ’00: Revised Papers from the Third International Workshop on Advances in Learning Classifier Systems (Springer-Verlag, 2001, Berlin), pp. 253–272

    Google Scholar 

  23. C. Stone, L. Bull, For real! XCS with continuous-valued inputs. Evol. Comput. 11(3), 299–336 (2003)

    Article  Google Scholar 

  24. S.W. Wilson, in Get Real! XCS With Continuous-Valued Inputs, ed. by L.P. Lanzi, W. Stolzmann, S.W. Wilson. Learning Classifier Systems, From Foundations to Applications. (Springer-Verlag, Berlin, 2000), pp. 209–222

    Chapter  Google Scholar 

  25. A.J. Bagnall, G.C. Cawley, in Learning Classifier Systems for Data Mining: A Comparison of XCS with Other Classifiers for the Forest Cover Data Set, ed. by D.C. Wunsch, M. Hasselmo, K. Venayagamoorthy, D. Wang. Proceedings of the International Joint Conference on Neural Networks, vol 3 (IEEE Press, 2003), pp. 1802–1807

  26. E. Bernado-Mansilla, J.M. Garrell-Guiu, Accuracy-based learning classifier systems: models, analysis and applications to classification tasks. Evol. Comput. 11, 209–238 (2003)

    Article  Google Scholar 

  27. C. Gathercole, P. Ross, in Dynamic Training Subset Selection for Supervised Learning in Genetic Programming, ed. by Y. Davidor, H.-P. Schwefel, R. Männer. Proceedings the Third Conference on Parallel Problem Solving from Nature (Springer-Verlag, Berlin, 1994), pp. 312–321

    Google Scholar 

  28. F.H. Bennett III, J.R. Koza, J. Shipman, O. Stiffelman, in Building a Parallel Computer System for $18,000 that Performs a Half Peta-flop Per Day, ed. by W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, R.E. Smith. Proceedings of the Genetic and Evolutionary Computation Conference (Morgan Kaufmann, San Francisco, CA, 1999), pp. 1484–1490

    Google Scholar 

  29. H. Juillé, J.B. Pollack, Massively parallel genetic programming. Adv. Genet. Program. 2, 339–357 (1996)

    Google Scholar 

  30. R. Curry, P. Lichodzijewski, M.I. Heywood, Scaling Genetic Programming to large datasets using hierarchical dynamic subset selection. IEEE Trans. Syst. Man. Cybern. B Cybern. 37(4), 1065–1073 (2007)

    Article  Google Scholar 

  31. D. Song, M.I. Heywood, A.N. Zincir-Heywood, Training Genetic Programming on half a million patterns: An example from anomaly detection. IEEE Trans. Evol. Comput. 9(3), 225–239 (2005)

    Article  Google Scholar 

  32. C.W.G. Lasarczyk, P.W.G. Dittrich, W.W.G. Banzhaf, Dynamic subset selection based on a fitness case topology. Evol. Comput. 12(2), 223–242 (2004)

    Article  Google Scholar 

  33. D.-Y. Cho, B.-T. Zhang, in Genetic Programming with Active Data Selection, ed. by B. McKay, X. Yao, C.S. Newton, J.-H. Kim, T. Furuhashi. Simulated Evolution and Learning: Second Asia-Pacific Conference on Simulated Evolution and Learning (Springer-Verlag, London, UK, 1998), pp. 146–153

    Google Scholar 

  34. K. Tumer, D. Wolpert, A survey of collectives. Collectives and the Design of Complex Systems (2004), pp. 1–42

  35. L. Panait, S. Luke, Cooperative Multi-agent Learning: The State of the art. Auton. Agent. Multi. Agent. Syst. 11(3), 387–434 (2005)

    Article  Google Scholar 

  36. L. Panait, S. Luke, R.P. Wiegand, Biasing coevolutionary search for optimal multiagent behaviors. IEEE Trans. Evol. Comput. 10(6), 629–645 (2006)

    Article  Google Scholar 

  37. M. Potter, K. de Jong, Cooperative coevolution: An architecture for evolving coadapted subcomponents. Evol. Comput. 8(1), 1–29 (2000)

    Article  Google Scholar 

  38. P. Lichodzijewski, M.I. Heywood, in GP Classifier Problem Decomposition Using First-price and Second-price Auctions, ed. by M. Ebner, M. O’Neill, A. Ekárt, L. Vanneschi, A.I. Esparcia-Alcázar. Proceedings of the European Conference on Genetic Programming (Springer-Verlag, Berlin, 2007), pp. 137–147

    Google Scholar 

  39. M. Laumanns, L. Thiele, K. Deb, E. Zitzler, Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)

    Article  Google Scholar 

  40. M. Brameier, W. Banzhaf, Linear Genetic Programming (Springer, New York, NY, 2007)

    MATH  Google Scholar 

  41. S.W. Wilson, in Generalization in the XCS Classifier System, ed. by J.R. Koza, W. Banzhaf, K. Chellapilla, K. Deb, M. Dorigo, D. Fogel, M. Garzon, D.E. Goldberg, H. Iba, R. Riolo. Genetic Programming 1998: Proceedings of the Third Annual Conference (Morgan Kaufmann, San Francisco, CA, 1998), pp. 665–674

    Google Scholar 

  42. A. Orriols-Puig, E. Bernado-Mansilla, in Bounding XCS’s Parameters for Unbalanced Datasets, ed. by M. Keijzer, M. Cattolico, D. Arnold, V. Babovic, C. Blum, P. Bosman, M.V. Butz, C. Coello Coello, D. Dasgupta, S.G. Ficici, J. Foster, A. Hernandez-Aguirre, G. Hornby, H. Lipson, P. McMinn, J. Moore, G. Raidl, R. Franz, C. Ryan, D. Thierens. Proceedings of the Genetic and Evolutionary Computation Conference (ACM Press, New York, NY, 2006), pp. 1561–1568

    Chapter  Google Scholar 

  43. M.V. Butz. Documentation of XCS+TS C-Code 1.2. IlliGAL Technical Report 200323 (2003)

  44. T. Kovacs, in Deletion Schemes for Classifier Systems, ed. by W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, R.E. Smith. Proceedings of the Genetic and Evolutionary Computation Conference (Morgan Kaufmann, San Francisco, CA, 1999), pp. 329–336

    Google Scholar 

  45. M. Butz, K. Sastry, D.E. Goldberg, in Tournament Selection in XCS, ed. by E. Cantú-Paz, J.A. Foster, K. Deb, L.D. Davis, R. Roy, U.-M. O’Reilly, H.-G. Beyer, R. Standish, G. Kendall, S. Wilson, J. Wegener, D. Dasgupta, M.A. Potter, A.C. Schultz. Proceedings of the Genetic and Evolutionary Computation Conference (Springer-Verlag, Berlin, 2003), pp. 1857–1869

    Google Scholar 

  46. S. Hettich, S.D. Bay, The UCI KDD Archive (University of California, Dept. of Information and Comp. Science, Irvine, CA, 1999),http://kdd/ics/uci/edu

  47. D.J. Newman, S. Hettich, C.L. Blake, C.J. Merz, UCI Repository of Machine Learning Databases (University of California, Dept. of Information and Comp. Science, Irvine, CA, 1998) http://www.ics.uci.edu/∼mlearn/mlrepository.html

  48. N. Japkowicz, in Why Question Machine Learning Evaluation Methods? (An Illustrative Review of the Shortcomings of Current Methods), ed. by C. Drummond, W. Elazmeh, N. Japkowicz. AAAI-2006 Workshop on Evaluation Methods for Machine Learning, Technical Report WS-06-06 (AAAI Press, Menlo Park, CA, 2006), pp. 6–11

    Google Scholar 

  49. G. Weiss, F. Provost, The Effect of Class Distribution on Classifier Learning. Technical Report ML-TR 43 (Department of Computer Science, Rutgers University, 2001)

  50. R.-E. Fan, P.-H. Chen, C.-J. Lin, Working set selection using second order information for training support vector machines. J. Mach. Learn. Res. 6, 1889–1918 (2005)

    MathSciNet  Google Scholar 

  51. M. Brameier, W. Banzhaf, A comparison of linear genetic programming and neural networks in medical data mining. IEEE Trans. Evol. Comput. 5(1), 17–26 (2001)

    Article  Google Scholar 

  52. R.A. Watson, J.B. Pollack, in Coevolutionary Dynamics in a Minimal Substrate, ed. by L. Spector, E.D. Goodman, A. Wu, W.B. Langdon, H.-M. Voigt, M. Gen, S. Sen, M. Dorigo, S. Pezeshk, M.H. Garzon, E. Burke. Proceedings of the Genetic and Evolutionary Computation Conference (Morgan Kaufmann, San Francisco, CA, 2001), pp. 702–709

    Google Scholar 

  53. M. Lemczyk, M.I. Heywood, in Pareto-coevolutionary Genetic Programming Classifier, ed. by M. Keijzer, M. Cattolico, D. Arnold, V. Babovic, C. Blum, P. Bosman, M.V. Butz, C. Coello Coello, D. Dasgupta, S.G. Ficici, J. Foster, A. Hernandez-Aguirre, G. Hornby, H. Lipson, P. McMinn, J. Moore, G. Raidl, R. Franz, C. Ryan, D. Thierens. Proceedings of the Genetic and Evolutionary Computation Conference (ACM Press, New York, NY, 2006), pp. 945–946

    Chapter  Google Scholar 

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Acknowledgements

This work was conducted while Peter Lichodzijewski held a Precarn Graduate Scholarship and a Killam Postgraduate Scholarship. Malcolm. I. Heywood would like to thank the Natural Sciences and Engineering Research Council of Canada, The Mathematics of Information Technology and Complex Systems network, and the Canadian Foundation for Innovation for their financial support.

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  1. Faculty of Computer Science, Dalhousie University, 6050 University Ave., Halifax, NS, Canada, B3H 1W5

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Lichodzijewski, P., Heywood, M.I. Coevolutionary bid-based genetic programming for problem decomposition in classification. Genet Program Evolvable Mach 9, 331–365 (2008). https://doi.org/10.1007/s10710-008-9067-9

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