Exploiting quotients of markov chains to derive properties of the stationary distribution of the markov chain associated to an evolutionary algorithm

Authors:
Boris Mitavskiy;Jonathan E. Rowe;Alden Wright;Lothar M. Schmitt
Affiliations:
Academic Unit of Mathematical Modelling and Genetic Epidemiology, Division of Genomic Medicine, School of Medicine, University of Sheffield, Sheffield;School of Computer Science, University of Birmingham, Birmingham, Great Britain;Computer Science, University of Montana, Missoula, MT;School of Computer Science and Engineering, The University of Aizu, Aizu-Wakamatsu City, Fukushima Prefecture, Japan
Venue:
SEAL'06 Proceedings of the 6th international conference on Simulated Evolution And Learning
Year:
2006

Citing 7
Cited 3

Theory of genetic algorithms

Theoretical Computer Science
The Simple Genetic Algorithm: Foundations and Theory

The Simple Genetic Algorithm: Foundations and Theory
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
An Extension of Geiringer's Theorem for a Wide Class of Evolutionary Search Algorithms.

Evolutionary Computation
Some results about the Markov chains associated to GPs and general EAs

Theoretical Computer Science - Foundations of genetic algorithms
A schema-based version of geiringer's theorem for nonlinear genetic programming with homologous crossover

FOGA'05 Proceedings of the 8th international conference on Foundations of Genetic Algorithms
A further result on the Markov chain model of genetic algorithms and its application to a simulated annealing-like strategy

IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics

Quotients of Markov chains and asymptotic properties of the stationary distribution of the Markov chain associated to an evolutionary algorithm

Genetic Programming and Evolvable Machines
Estimating the ratios of the stationary distribution values for markov chains modeling evolutionary algorithms

Evolutionary Computation
Geiringer theorems: from population genetics to computational intelligence, memory evolutive systems and Hebbian learning

Natural Computing: an international journal

Quantified Score

Hi-index	0.00

Visualization

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.