Modeling genetic algorithms with interacting particle systems

Abstract

We present in this work a natural Interacting Particle System (IPS) approach for modeling and studying the asymptotic behavior of Genetic Algorithms (GAs). In this model, a population is seen as a distribution (or measure) on the search space, and the Genetic Algorithm as a measure valued dynamical system. This model allows one to apply recent convergence results from the IPS literature for studying the convergence of genetic algorithms when the size of the population tends to infinity. We first review a number of approaches to Genetic Algorithms modeling and related convergence results. We then describe a general and abstract discrete time Interacting Particle System model for GAs, and we propose a brief review of some recent asymptotic results about the convergence of the N-IPS approximating model (offinite N-sized-population GAs) towards the IPS model (of infinite population GAs),including law of large number theorems, IL p-mean and exponential bounds as well aslarge deviations principles. Finally, the impact of modeling Genetic Algorithms with our interacting particle system approach is detailed on different classes of generic genetic algorithms including mutation, cross-over and proportionate selection. We explore the connections between Feynman-Kac distribution flows and the simple genetic algorithm. This Feynman-Kac representation of the infinite population model is then used to develop asymptotic stability and uniform convergence results with respect to the time parameter. Keywords: Genetic algorithms, Interacting particle systems, asymptotical convergence, Feynman-Kac formula, measure valued processes.