Linear analysis of genetic algorithms

Abstract

We represent simple and fitness-scaled genetic algorithms by Markov chains on probability distributions over the set of all possible populations of a fixed finite size. Analysis of this formulation yields new insight into the geometric properties of the three phase mutation, crossover, and fitness selection of a genetic algorithm by representing them as stochastic matrices acting on the state space. This indicates new methods using mutation and crossover as the proposal scheme for simulated annealing. We show by explicit estimates that for small mutation rates a genetic algorithm asymptotically spends most of its time in uniform populations regardless of crossover rate. The simple genetic algorithm converges in the following sense: there exists a fully positive limit probability distribution over populations. This distribution is independent of the choice of initial population. We establish strong ergodicity of the underlying inhomogeneous Markov chain for genetic algorithms that use any of a large class of fitness scaling methods including linear fitness scaling, sigma-truncation, and power law scaling. Our analysis even allows for variation in mutation and crossover rates according to a pre-determined schedule, where the mutation rate stays bounded away from zero. We show that the limit probability distribution of such a process is fully positive at all populations of uniform fitness. Consequently, genetic algorithms that use the above fitness scalings do not converge to a population containing only optimal members. This answers a question of G. Rudolph (IEEE Trans. on Neural Networks 5 (1994) 96–101). For a large set of fitness scaling methods, the limit distribution depends on the pre-order induced by the fitness function f, i. e. c ⩾ c′ ↦ f( c) ⩾ f( c′) on possible creatures c and c′, and not on the particular values assumed by the fitness function.