Analyzing the impact of mirrored sampling and sequential selection in elitist evolution strategies

This paper presents a refined single parent evolution strategy that is derandomized with mirrored sampling and/or uses sequential selection. The paper analyzes some of the elitist variants of this algorithm. We prove, on spherical functions with finite dimension, linear convergence of different strategies with scale-invariant step-size and provide expressions for the convergence rates as the expectation of some known random variables. In addition, we derive explicit asymptotic formulae for the convergence rate when the dimension of the search space goes to infinity. Convergence rates on the sphere reveal lower bounds for the convergence rate of the respective step-size adaptive strategies. We prove the surprising result that the (1+2)-ES with mirrored sampling converges at the same rate as the (1+1)-ES without and show that the tight lower bound for the (1+λ)-ES with mirrored sampling and sequential selection improves by 16% over the (1+1)-ES reaching an asymptotic value of about -0.235.