Log-linear convergence of the scale-invariant (μ/μ_w, λ)-ES and optimal μ for intermediate recombination for large population sizes

Evolution Strategies (ESs) are population-based methods well suited for parallelization. In this paper, we study the convergence of the (μ/μ _w ,λ)-ES, an ES with weighted recombination, and derive its optimal convergence rate and optimal μ especially for large population sizes. First, we theoretically prove the log-linear convergence of the algorithm using a scale-invariant adaptation rule for the step-size and minimizing spherical objective functions and identify its convergence rate as the expectation of an underlying random variable. Then, using Monte-Carlo computations of the convergence rate in the case of equal weights, we derive optimal values for μ that we compare with previously proposed rules. Our numerical computations show also a dependency of the optimal convergence rate in ln (λ) in agreement with previous theoretical results.