Standard steady state genetic algorithms can hillclimb faster than evolutionary algorithms using standard bit mutation

Abstract

Genetic Algorithms (GAs) use principles of natural selection to evolve a population of candidate solutions obtained by the recombination of individuals of the current generation. Albeit their huge popularity, providing natural examples where standard GAs provably outperform more traditional mutation-based heuristics has turned out to be a tedious task. In this paper we rigorously prove that a standard steady state (μ+1) GA outperforms any evolutionary algorithm, that relies only on standard bit mutation (SBM) as variation operator, for hillclimbing the classical OneMax benchmark function. In particular, we show that the GA is 25% faster by providing an upper bound of (3/4)en ln n on its expected runtime versus the en ln n expected function evaluations required by any algorithm using only SBM. To achieve the result, we devise a mathematical framework which extends the classical artificial fitness levels method by coupling each level with a Markov chain. This Markov chain allows to bound the improvement probabilities of the current population based on its diversity. In turn it can be appreciated how larger populations sustain diversity for a longer time, effectively giving crossover more chances of finding improved solutions. Since diversity is created via mutation, higher rates than the standard 1/n mutation rate, lead to better upper bounds on the expected runtime. This paper summarises the work that appeared in [1]1.