Chaos propagation in genetic algorithms: An optimal transport approach

Abstract

Genetic algorithms are high-level heuristic optimization methods which enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions following three mechanisms: selection, mutation, and crossover. In this paper, we look at the algorithm as an interacting particle system and show that it is described by a Boltzmann-type equation in the many particles limit. Specifically, we prove a propagation of chaos result with a novel technique that leverages the optimal transport formulation of the bounded Lipschitz norm and naturally incorporates the crossover mechanism into the analysis. The convergence admits a rate with respect to the number of particles, corresponding to the optimal rate in the Wasserstein-1 distance.