The (1 + 1)-ES: Overvaluation

Abstract

It seems reasonable to start a discussion of the performance of evolution strategies in the presence of noise with the most simple strategy variants and then to proceed to increasingly more complex strategies. The (1 + 1)-ES is arguably the most simple evolution strategy. In its most basic form it is a simple stochastic hill climber. A parent x generates offspring candidate solution x+o z that replaces the parent in the next time step if and only if it appears superior in terms of its measured fitness. In this chapter, we analyze the local performance of the (1 + 1)-ES in the spherical environment in the limit of infinite search space dimensionality. We will see that the results provide a good understanding of the behavior of the (1 + 1)-ES on an N-dimensional sphere provided that N is not too small. The analysis of the local performance of the (1 + 1)-ES in the spherical environment is of particular interest as it is known from Beyer [27] that in the absence of noise its efficiency cannot be surpassed by any (µ/µ, À)-ES. The local performance of the (1 + 1)-ES in the spherical environment has been analyzed by Beyer [18] under the assumption that the parental fitness is reevaluated along with that of the offspring in every time step. This assumption is likely not to hold for most implementations of the (1 + 1)-ES as it requires two rather than one fitness function evaluations per time step. In the following sections, the consequences of failure to reevaluate the parental fitness are ex­plored. Rather than giving a full derivation of the results, we focus on the main steps and refer to Appendix C for details of the calculations.