Log-Linear Convergence of the Scale-Invariant (µ/µw, lambda)-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/µw, 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.