Analysis of evolution strategies with the optimal weighted recombination

Abstract

This paper studies the performance for evolution strategies with the optimal weighed recombination on spherical problems in finite dimensions. We first discuss the different forms of functions that are used to derive the optimal recombination weights and step size, and then derive an inequality that establishes the relationship between these functions. We prove that using the expectation of random variables to derive the optimal recombination weights and step size can be disappointing in terms of the expected performance of evolution strategies. We show that using the realizations of random variables is a better choice. We generalize the results to any convex functions and establish an inequality for the normalized quality gain. We prove that the normalized quality gain of the evolution strategies have a better and robust performance when they use the optimal recombination weights and the optimal step size that are derived from the realizations of random variables rather than using the expectations of random variables.