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Exploiting Quotients of Markov Chains to Derive Properties of the Stationary Distribution of the Markov Chain Associated to an Evolutionary Algorithm

  • Boris Mitavskiy
  • Jonathan E. Rowe
  • Alden Wright
  • Lothar M. Schmitt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4247)

Abstract

In this work, a method is presented for analysis of Markov chains modeling evolutionary algorithms through use of a suitable quotient construction. Such a notion of quotient of a Markov chain is frequently referred to as “coarse graining” in the evolutionary computation literature. We shall discuss the construction of a quotient of an irreducible Markov chain with respect to an arbitrary equivalence relation on the state space. The stationary distribution of the quotient chain is “coherent” with the stationary distribution of the original chain. Although the transition probabilities of the quotient chain depend on the stationary distribution of the original chain, we can still exploit the quotient construction to deduce some relevant properties of the stationary distribution of the original chain. As one application, we shall establish inequalities that describe how fast the stationary distribution of Markov chains modelling evolutionary algorithms concentrates on the uniform populations as the mutation rate converges to 0. Further applications are discussed.

Keywords

Markov Chain Evolutionary Algorithm Stationary Distribution Coarse Graining Markov Chain Model 
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.

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References

  1. 1.
    Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs (unpublished book) (2001), http://www.stat.berkeley.edu/~aldous/RWG/book.html
  2. 2.
    Coffey, S.: An Applied Probabilist’s Guide to Genetic Algorithms. A Thesis Submitted to The University of Dublin for the degree of Master in Science (1999)Google Scholar
  3. 3.
    Davis, T., Principe, J.: A Simulated Annealing Like Convergence Theory for the Simple Genetic Algorithm. In: Belew, R., Bookers, L. (eds.) Proceedings of the Fourth International Confernence on Genetic Algorithms, pp. 174–181. Morgan Kaufmann, San Francisco (1991)Google Scholar
  4. 4.
    Mitavskiy, B., Rowe, J.: An Extension of Geiringer Theorem for a wide class of evolutionary search algorithms. Evolutionary Computation (to appear)Google Scholar
  5. 5.
    Mitavskiy, B.: A schema-based version of Geiringer’s Theorem for nonlinear Genetic Programming with homologous crossover. Foundations of Genetic Algorithms 8, pp. 156–175. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Mitavskiy, B., Rowe, J.: Some Results about the Markov Chains Associated to GPs and General EAs. Theoretical Computer Science (accepted)Google Scholar
  7. 7.
    Schmitt, L.: Theory of Genetic Algorithms. Theor. Computer Science 259, 1–61 (2001)MATHCrossRefGoogle Scholar
  8. 8.
    Schmitt, L.: Theory of Genetic Algorithms II: Models for Genetic Operators over the String-Tensor representation of Populations and Convergence to Global Optima for Arbitrary Fitness Function under Scaling. Theoretical Computer Science 310, 181–231 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Radcliffe, N.: The algebra of genetic algorithms. Annals of Mathematics and Artificial Intelligence 10, 339–384 (1994), http://users.breathemail.net/njr/papers/amai94.pdf MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Suzuki, J.: A Further Result on the Markov Chain Model of Genetic Algorithm and Its application to Simulated Annealing-Like strategy. IEEE Transactions on Systems, Man and Cybernatics 28(1) (1998)Google Scholar
  11. 11.
    Vose, M.: The simple genetic algorithm: foundations and theory. MIT Press, Cambridge (1999)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Boris Mitavskiy
    • 1
  • Jonathan E. Rowe
    • 2
  • Alden Wright
    • 3
  • Lothar M. Schmitt
    • 4
  1. 1.Academic Unit of Mathematical Modelling and Genetic Epidemiology, Division of Genomic Medicine, School of MedicineUniversity of SheffieldSheffield
  2. 2.School of Computer ScienceUniversity of BirminghamBirminghamGreat Britain
  3. 3.Computer ScienceUniversity of MontanaMissoulaUSA
  4. 4.School of Computer Science and EngineeringThe University of AizuAizu-Wakamatsu City, Fukushima PrefectureJapan

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