Optimal Static and Self-Adjusting Parameter Choices for the $$(1+(\lambda ,\lambda ))$$ ( 1 + ( λ , λ ) ) Genetic Algorithm

Abstract

The (1 + (λ, λ)) genetic algorithm (GA) proposed in [Doerr, Doerr, and Ebel. From black-box complexity to designing new genetic algorithms. Theoretical Computer Science (2015)] is one of the few examples for which a super-constant speed-up of the expected optimization time through the use of crossover could be rigorously demonstrated. It was proven that the expected optimization time of this algorithm on OneMax is O(max{n log(n)/λ, λn}) for any offspring population size λ ∈ {1, . . . , n} (and the other parameters suitably dependent on λ) and it was shown experimentally that a self-adjusting choice of λ leads to a better, most likely linear, runtime.In this work, we study more precisely how the optimization time depends on the parameter choices, leading to the following results on how to optimally choose the population size, the mutation probability, and the crossover bias both in a static and a dynamic fashion.For the mutation probability and the crossover bias depending on λ as in [DDE15], we improve the previous runtime bound to O(max{n log(n)/λ, nλ log log(λ)/ log(λ)}). This expression is minimized by a value of λ slightly larger than what the previous result suggested and gives an expected optimization time of O n log(n) log log log(n)/ log log(n) .We show that no static choice in the three-dimensional parameter space of offspring population, mutation probability, and crossover bias gives an asymptotically better runtime.Results presented in this work are based on [12][13][14]. B. DoerŕEcole Polytechnique, LIX -UMR 7161,

Self-Adaptation in Nonelitist Evolutionary Algorithms on Discrete Problems With Unknown Structure

“…Proof (of Theorem 2). We partition the search space Y into the sets A (j, ) from Definition 4, where j ∈ [0..k − 1] and ∈ [d j ], along with A (k,1) , and define A ≥(j, ) as in (12).…”

Hyper-parameter tuning for the (1 + ( λ, λ )) GA

“…The performance curves for success rates ≥ 4 seem to be roughly U-shaped with different values of A in which the minimum is obtained. It could be worthwhile to extend the mathematical analysis of the dyn (1, 1, 1, A, b) presented in [DD18a] in order to identify the precise relationship.…”

“…Next we turn our attention to the five-dimensional (1 + (λ, λ)) GA variant dyn(α, β, γ, A, b), in which not only the update strengths A and b are configurable, but also the dependence of p = αλ/n, λ 2 = nint(βλ), c = γ/λ. The dependencies of the parameters on λ are based on a theoretical result proven in [DD18a], where it is shown that any static configuration with λ 2 = λ 1 (i.e., A = b = β = 1) that achieves optimal asymptotic expected performance must necessarily satisfy p = Θ(λ/n) and γ = Θ(1/λ).…”

“…We also allow λ 1 = λ 2 , which was not the case in [DDE15]. In line with suggestions from [DDE15,DD18a] we set p = k/n, and optimize for integer k ∈ {1, . .…”

“…In [DD18a] some general advice on choosing the hyper-parameters is given, but their influence on the explicit running time is not discussed, mostly due to missing precision in the available results, which state the asymptotic linear order only, but not the leading constants or lower order terms. Also the update strength F for which the linear running time is obtained is only shown to exist, but not made explicit in [DD18a].…”