Abstract. 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.

Noise is present in many real-world continuous optimization problems. Stochastic search algorithms such as Evolution Strategies (ESs) have been proposed as effective search methods in such contexts. In this paper, we provide a mathematical analysis of the convergence of a (1 + 1)-ES on unimodal spherical objective functions in the presence of noise. We prove for a multiplicative noise model that for a positive expected value of the noisy objective function, convergence or divergence happens depending on the infimum of the support of the noise. Moreover, we investigate convergence rates and show that log-linear convergence is preserved in presence of noise. This result is a strong theoretical foundation of the robustness of ESs with respect to noise.