Step length adaptation is central to evolutionary algorithms in real-valued search spaces. This paper contrasts several step length adaptation algorithms for evolution strategies on a family of ridge functions. The algorithms considered are cumulative step length adaptation, a variant of mutative self-adaptation, two-point adaptation, and hierarchically organized strategies. In all cases, analytical results are derived that yield insights into scaling properties of the algorithms. The influence of noise on adaptation behavior is investigated. Similarities and differences between the adaptation strategies are discussed.

Organising evolution strategies hierarchically has been proposed as a means for adapting strategy parameters such as step lengths. Experimental research has shown that on ridge functions, hierarchically organised strategies can significantly outperform strategies that rely on mutative selfadaptation. This paper presents a first theoretical analysis of the behaviour of a hierarchically organised evolution strategy. Quantitative results are derived for the parabolic ridge that describe the dependence on the length of the isolation periods of the mutation strength and the progress rate. The issue of choosing an appropriate length of the isolation periods is discussed and comparisons with recent results for cumulative step length adaptation are drawn.