title: The Dynamics of Cumulative Step Size Adaptation on the Ellipsoid Model.
abstract: The behavior of the [Formula: see text]-Evolution Strategy (ES) with cumulative step size adaptation (CSA) on the ellipsoid model is investigated using dynamic systems analysis. At first a nonlinear system of difference equations is derived that describes the mean value evolution of the ES. This system is successively simplified to finally allow for deriving closed-form solutions of the steady state behavior in the asymptotic limit case of large search space dimensions. It is shown that the system exhibits linear convergence order. The steady state mutation strength is calculated, and it is shown that compared to standard settings in [Formula: see text] self-adaptive ESs, the CSA control rule allows for an approximately [Formula: see text]-fold larger mutation strength. This explains the superior performance of the CSA in non-noisy environments. The results are used to derive a formula for the expected running time. Conclusions regarding the choice of the cumulation parameter c and the damping constant D are drawn.
authors.affiliation: Department of Computer Science, Research Center Process and Product Engineering, Vorarlberg University of Applied Sciences, Dornbirn, 6850, Austria hans-georg.beyer@fhv.at.
authors.forename: Hans-Georg
authors.initials: HG
authors.lastname: Beyer
authors.affiliation: Research Center Process and Product Engineering, Vorarlberg University of Applied Sciences, Dornbirn, 6850, Austria michael.hellwig@fhv.at.
authors.forename: Michael
authors.initials: M
authors.lastname: Hellwig
mesh_terms: D000465:Algorithms; D005075:Biological Evolution; D019295:Computational Biology; D003198:Computer Simulation; D016014:Linear Models; D008957:Models, Genetic; D015233:Models, Statistical; D009154:Mutation; D017711:Nonlinear Dynamics; D013269:Stochastic Processes
keywords: Evolution strategies; cumulative step size adaptation; ellipsoid model; mutation strength.