The dynamics of a Genetic Algorithm for simple random Ising systems

Abstract

A formalism is presented for analysing Genetic Algorithms. It is used to study a simple Genetic Algorithm consisting of selection, mutation and crossover which is searching for the ground states of simple random Ising-spin systems: a random-field ideal paramagnet and a spin-glass chain. The formalism can also be applied to other population based search techniques and to biological models of micro-evolution. To make the problem tractable, it is assumed that the population dynamics can be described by a few macroscopic order parameters and that the remaining microscopic degrees of freedom can be averaged out. The macroscopic quantities that are used are the cumulants of the distribution of fitnesses (or energies) in the population. A statistical mechanics model is presented which describes the population configuration in terms of the cumulants, this is used to derive equations of motion for the cumulants. Predictions of the theory are compared with experiments and are shown to predict the average time to convergence and the average fitness of the final population accurately. A simplified version of the equations is produced by keeping only leading nonlinear terms, and truncating the cumulant expansion. This is shown to give a novel description of the role of genetic operators in search, e.g. it is argued that an important role of crossover is to reduce the skewness of the population.