Toward a Theory of Evolution Strategies: Self-Adaptation

This paper analyzes the self-adaptation (SA) algorithm widely used to adapt strategy parameters of the evolution strategy (ES) in order to obtain maximal ES performance. The investigations are concentrated on the adaptation of one general mutation strength sigma (called sigma SA) in (1, lambda) ESs. The hypersphere serves as the fitness model. Starting from an introduction to the basic concept of self-adaptation, a framework for the analysis of sigma SA is developed on two levels: a microscopic level, concerning the description of the stochastic changes from one generation to the next, and a macroscopic level, describing the evolutionary dynamics of the sigma SA over time (generations). The sigma SA requires the fixing of a new strategy parameter, known as the learning parameter. The influence of this parameter on ES performance is investigated and rules for its tuning are presented and discussed. The results of the theoretical analysis are compared with ES experiments; it will be shown that applying Schwefel's tau-scaling rule guarantees the linear convergence order of the ES.