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Evolutionary Computation

Spring 2016, Vol. 24, No. 1, Pages 25-57
(doi: 10.1162/EVCO_a_00142)
© 2016 Massachusetts Institute of Technology
The Dynamics of Cumulative Step Size Adaptation on the Ellipsoid Model
Article PDF (982.47 KB)
Abstract

The behavior of the (μ/μI,λ)-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 σ self-adaptive ESs, the CSA control rule allows for an approximately μ-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.