Log-Linear Convergence of the Scale-Invariant (μ/μ w ,λ)-ES and Optimal μ for Intermediate Recombination for Large Population Sizes

Abstract

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.

Mirrored sampling in evolution strategies with weighted recombination

“…To obtain the convergence in expectation as defined in (1), we take the expectation in (5). For a more detailed argumentation why the expectation exists and for the independence of the random variables ln X k / X k , we refer to [8].…”

“…To do so we need to verify that the random variables are uniformly integrable. For this quite technical step we refer to [8].…”

Quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions

“…A formal proof of this result is presented in Theorem 2 of [29] relying on the uniform integrability of some random variable proved in [30]. Consider the optimal recombination weights that maximizeφ ∞ in (6).…”

“…Note that the function G(α) ∈ O(α ln(1/α)) as α → 0. Then, (18) reads sup m∈R N \{0} φ (m,σ) − ϕ(σ, (w k ), e m , A) ∈σλO(α ln(1/α)) L 1 + c m αL 2 + c m α(λ − 1)L 3 .Tr(A 2 ) It implies that the RHS of (31) divided byσ is in O(α ln(1/α)) ⊆ o(α 1− ) for any > 0 under the conditionσ C. Since α → 0 as σ/ m → 0, (31) implies(30).…”

“…The quality gain analysis and the progress rate analysis in the above listed references rely on a geometric intuition of the algorithm in the infinite dimensional search space and on various approximations. On the other hand, the rigorous derivation of the progress rate (or convergence rate of the algorithm with step-size proportional to the distance to the optimum) on the sphere function provided for instance in [17,20,29,30] only holds on spherical functions and provides solely a limit without a bound between the finite dimensional convergence rate and its asymptotic limit. The result of this paper is different in that we consider the general weighted recombination on the general convex quadratic objective and cover finite dimensional cases as well as the limit N → ∞.…”