Genetic Algorithms, Noise, and the Sizing of Populations

This paper considers the e ect of stochasticity on the quality of convergence of genetic algorithms (GAs). In many problems, the variance of building-block tness or so-called collateral noise is the major source of variance, and a population-sizing equation is derived to ensure that average signalto-collateral-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. The sizing relation is modi ed to permit the inclusion of other sources of stochasticity, such as the noise of selection, the noise of genetic operators, and the explicit noise or nondeterminism of the objective function. In a test suite of ve functions, the sizing relation proves to be a conservative predictor of average correct convergence, as long as all major sources of noise are considered in the sizing calculation. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior. At low population sizes the GA makes many errors of decision, and the quality of convergence is largely left to the vagaries of chance or the serial xup of awed results through mutation or other serial injection of diversity. At large population sizes, GAs can reliably discriminate between good and bad building blocks, and parallel processing and recombination of building blocks lead to quick solution of even di cult deceptive problems. Additionally, the paper outlines a number of extensions to this work, including the development of more re ned models of the relation between generational average error and ultimate convergence quality, the development of online methods for sizing populations via the estimation of population-sizing parameters, and the investigation of population sizing in the context of niching and other schemes designed for use in problems with high-cardinality solution sets. The paper also discusses how these results may one day lead to rigorous proofs of convergence for recombinative GAs operating on problems of bounded deception.