Evolution strategies with cumulative step length adaptation on the noisy parabolic ridge

This paper presents an analysis of the performance of the (μ/μ,λ)-ES with isotropically distributed mutations and cumulative step length adaptation on the noisy parabolic ridge. Several forms of dependency of the noise strength on the distance from the ridge axis are considered. Closed form expressions are derived that describe the mutation strength and the progress rate of the strategy in high-dimensional search spaces. It is seen that as for the sphere model, larger levels of noise present lead to cumulative step length adaptation generating increasingly inadequate mutation strengths, and that the problem can be ameliorated to some degree by working with larger populations.