Abstract
Although much progress has been made in recent years in the theory of GAs and GP, there is still a conspicuous lack of tools with which to derive systematic, approximate solutions to their dynamics. In this article we propose and study perturbation theory as a potential tool to fill this gap. We concentrate mainly on selection-mutation systems, showing different implementations of the perturbative framework, developing, for example, perturbative expansions for the eigenvalues and eigenvectors of the transition matrix. The main focus however, is on diagrammatic methods, taken from physics, where we show how approximations can be built up using a pictorial representation generated by a simple set of rules, and how the renormalization group can be used to systematically improve the perturbation theory.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Stephens, C.R., Waelbroeck, H.: Schemata evolution and building blocks. Evol. Comp. 7, 109–124 (1999)
Poli, R.: Exact schema theory for genetic programming and variable-length genetic algorithms with one-point crossover. Genetic Programming and Evolvable Machines 2(2), 123–163 (2001)
Stephens, C.R.: Some exact results from a coarse grained formulation of genetic dynamics. In: Proceedings of GECCO-2001, July 7-11, pp. 631–638. Morgan Kaufmann, San Francisco (2001)
Langdon, W.B., Poli, R.: Foundations of Genetic Programming. Springer, Heidelberg (2002)
Stephens, C.R., Poli, R.: E C theory - in theory: Towards a unification of evolutionary computation theory. In: Menon, A. (ed.) Frontiers of Evolutionary Computation, pp. 129–156. Kluwer Academic Publishers, Dordrecht (2004)
Prügel-Bennett, A., Shapiro, J.L.: An analysis of genetic algorithms using statistical mechanics. Physical Review Letters 72, 1305–1309 (1994)
Shapiro, J.L.: Statistical mechanics theory of genetic algorithm. In: Naudts, B., Kallel, L., Rogers, A. (eds.) Theoretical Aspects of Evolutionary Computing, pp. 87–108. Springer, Heidelberg (2001)
McCaskill, J., Eigen, M., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys. 75, 149–263 (1989)
Hofbauer, J., Nagylaki, T., Brunovsky, P.: Convergence of multilocus systems under weak epistasis or weak selection. J. Math. Biol. 38, 103–133 (1999)
Rattray, M., Shapiro, J.L.: Cumulant dynamics of a population under multiplicative selection, mutation and drift. Theor. Pop. Biol. 60, 17–32 (2001)
Reeves, C.R., Rowe, J.E.: Genetic Algorithms - Principles and Perspectives. Kluwer Academic Publishers, Dordrecht (2003)
Akivis, M.A., Goldberg, V.V.: An Introduction to Linear Algebra and Tensors. Dover Publications, Mineola (1977)
Chryssomalakos, C., Stephens, C.R.: What basis for genetic dynamics? In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3102, pp. 1394–1402. Springer, Heidelberg (2004)
Poli, R.: Exact schema theorem and effective fitness for GP with one-point crossover. In: Proceedings of the Genetic and Evolutionary Computation Conference, Las Vegas, July 2000, pp. 469–476. Morgan Kaufmann, San Francisco (2000)
Stephens, C.R.: The renormalization group and the dynamics of genetic systems. Acta Phys. Slov. 52, 515–524 (2003)
Wright, A.H., Rowe, J.E., Poli, R., Stephens, C.R.: A fixed point analysis of a gene pool GA with mutation. In: Proceedings of GECCO 2002, San Francisco, USA, Morgan Kaufmann, San Francisco (2002)
Lakin, W.D., Sanchez, D.A.: Topics in Ordinary Differential Equations. Dover, Publications Inc., New York (1970)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Stephens, C.R., Zamora, A., Wright, A.H. (2005). Perturbation Theory and the Renormalization Group in Genetic Dynamics. In: Wright, A.H., Vose, M.D., De Jong, K.A., Schmitt, L.M. (eds) Foundations of Genetic Algorithms. FOGA 2005. Lecture Notes in Computer Science, vol 3469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11513575_11
Download citation
DOI: https://doi.org/10.1007/11513575_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27237-3
Online ISBN: 978-3-540-32035-7
eBook Packages: Computer ScienceComputer Science (R0)