On the convergence of a class of estimation of distribution algorithms

We investigate the global convergence of estimation of distribution algorithms (EDAs). In EDAs, the distribution is estimated from a set of selected elements, i.e., the parent set, and then the estimated distribution model is used to generate new elements. In this paper, we prove that: 1) if the distribution of the new elements matches that of the parent set exactly, the algorithms will converge to the global optimum under three widely used selection schemes and 2) a factorized distribution algorithm converges globally under proportional selection.

[1]  Paul A. Viola,et al.  MIMIC: Finding Optima by Estimating Probability Densities , 1996, NIPS.

[2]  Kalyanmoy Deb,et al.  A Comparative Analysis of Selection Schemes Used in Genetic Algorithms , 1990, FOGA.

[3]  H. Muhlenbein,et al.  The Factorized Distribution Algorithm for additively decomposed functions , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[4]  Xin Yao,et al.  Towards an analytic framework for analysing the computation time of evolutionary algorithms , 2003, Artif. Intell..

[5]  Prügel-Bennett,et al.  Analysis of genetic algorithms using statistical mechanics. , 1994, Physical review letters.

[6]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[7]  Byoung-Tak Zhang.  A Bayesian framework for evolutionary computation , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[8]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[9]  Heinz Mühlenbein,et al.  The Equation for Response to Selection and Its Use for Prediction , 1997, Evolutionary Computation.

[10]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[11]  Günter Rudolph,et al.  Convergence analysis of canonical genetic algorithms , 1994, IEEE Trans. Neural Networks.

[12]  Lothar Thiele,et al.  A Mathematical Analysis of Tournament Selection , 1995, ICGA.

[13]  László Györfi,et al.  A Probabilistic Theory of Pattern Recognition , 1996, Stochastic Modelling and Applied Probability.

[14]  Dirk Thierens,et al.  Advancing continuous IDEAs with mixture distributions and factorization selection metrics , 2001 .

[15]  J. A. Lozano,et al.  Analyzing the PBIL Algorithm by Means of Discrete Dynamical Systems , 2000 .

[16]  L. Darrell Whitley,et al.  Test driving three 1995 genetic algorithms: New test functions and geometric matching , 1995, J. Heuristics.

[17]  Xin Yao,et al.  Drift analysis and average time complexity of evolutionary algorithms , 2001, Artif. Intell..

[18]  Markus H ohfeld,et al.  Random keys genetic algorithm with adaptive penalty function for optimization of constrained facility layout problems , 1997, Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97).

[19]  Iwona Karcz-Duleba,et al.  Dynamics of infinite populations evolving in a landscape of uni and bimodal fitness functions , 2001, IEEE Trans. Evol. Comput..

[20]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[21]  R. Elston.  The mathematical theory of quantitative genetics , 1982 .

[22]  Shumeet Baluja,et al.  A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning , 1994 .

[23]  Sewall Wright,et al.  Genetic and Biometric Foundations , 1968 .

[24]  John H. Holland,et al.  Building Blocks, Cohort Genetic Algorithms, and Hyperplane-Defined Functions , 2000, Evolutionary Computation.

[25]  Yong Gao,et al.  Comments on "Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. I. Basic properties of selection and mutation" [and reply] , 1998, IEEE Trans. Neural Networks.

[26]  William F. Punch,et al.  Global search in combinatorial optimization using reinforcement learning algorithms , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[27]  Michael D. Vose,et al.  Modeling genetic algorithms with Markov chains , 1992, Annals of Mathematics and Artificial Intelligence.

[28]  J. N. R. Jeffers,et al.  Graphical Models in Applied Multivariate Statistics. , 1990 .

[29]  D. Goldberg,et al.  BOA: the Bayesian optimization algorithm , 1999 .

[30]  Thomas Bäck,et al.  Selective Pressure in Evolutionary Algorithms: A Characterization of Selection Mechanisms , 1994, International Conference on Evolutionary Computation.

[31]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[32]  Heinz Mühlenbein,et al.  Schemata, Distributions and Graphical Models in Evolutionary Optimization , 1999, J. Heuristics.

[33]  Xin Yao,et al.  From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms , 2002, IEEE Trans. Evol. Comput..

[34]  Francesco Palmieri,et al.  Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation , 1994, IEEE Trans. Neural Networks.

[35]  R. Cerf.  Asymptotic convergence of genetic algorithms , 1998, Advances in Applied Probability.

[36]  Heinz Mühlenbein,et al.  Predictive Models for the Breeder Genetic Algorithm I. Continuous Parameter Optimization , 1993, Evolutionary Computation.

[37]  Heinz Mühlenbein,et al.  FDA -A Scalable Evolutionary Algorithm for the Optimization of Additively Decomposed Functions , 1999, Evolutionary Computation.

[38]  Christopher R. Stephens,et al.  Schemata Evolution and Building Blocks , 1999, Evolutionary Computation.

[39]  Michael D. Vose,et al.  The simple genetic algorithm - foundations and theory , 1999, Complex adaptive systems.

[40]  Stephen Marshall,et al.  Convergence Criteria for Genetic Algorithms , 2000, SIAM J. Comput..

[41]  Lothar Thiele,et al.  A Comparison of Selection Schemes Used in Evolutionary Algorithms , 1996, Evolutionary Computation.

[42]  Shumeet Baluja,et al.  Fast Probabilistic Modeling for Combinatorial Optimization , 1998, AAAI/IAAI.

[43]  Chak-Kuen Wong,et al.  A new model of simulated evolutionary computation-convergence analysis and specifications , 2001, IEEE Trans. Evol. Comput..

[44]  Riccardo Poli,et al.  Exact Schema Theory for Genetic Programming and Variable-Length Genetic Algorithms with One-Point Crossover , 2001, Genetic Programming and Evolvable Machines.