Weighted multirecombination evolution strategies

Abstract

Weighted recombination is a means for improving the local search performance of evolution strategies. It aims to make effective use of the information available, without significantly increasing computational costs per time step. In this paper, the potential speed-up resulting from using rank-based weighted multirecombination is investigated. Optimal weights are computed for the infinite-dimensional sphere model, and comparisons with the performance of strategies that do not make use of weighted recombination are presented. It is seen that unlike strategies that rely on unweighted recombination and truncation selection, weighted multirecombination evolution strategies are able to improve on the serial efficiency of the (1 + 1)-ES on the sphere. The implications of the use of weighted recombination for noisy optimization are studied, and parallels to the use of rescaled mutations are drawn. The significance of the findings is investigated in finite-dimensional search spaces.

σ-Self-Adaptive Weighted Multirecombination Evolution Strategy with Scaled Weights on the Noisy Sphere

“…In order to obtain maximal (normalized) quality gain ∆ * per generation (note, ∆ is the expected fitness gain per generation), optimal weights ω l,λ must be used. The following choice of scaled weights has been shown to be optimal in noisy fitness environments [2] …”

“…An asymptotically exact formula for normalized quality gain of the (λ) opt -ES with the choice of weights (3) for 1 The noisy quadratic sphere reads f (y) = ŷ −y 2 + , whereŷ ∈ R N is the optimizer and ∼ N (0, σ ) is an additive normally distributed noise term. 2 Note, σ * = σN/R, R is parental distance to the optimizer. the noisy sphere model has been obtained in [2] using several simplifications (consideration of the asymptotic behavior for N → ∞, assumption that the normalized mutation strength σ * is of O (1), Taylor series expansion)…”

“…The σ-self-adaptive weighted multirecombination ES, short (λ) opt -σSA-ES, has been proposed in [1] as a new ES which takes advantage of Arnold's weighted multirecombination [2] and σ-self-adaptation [3] at the same time. It has been shown, considering the sphere model, that the (λ) opt -σSA-ES can outperform Arnold's (λ) opt -ES with cumulative step size adaptation (CSA) which was regarded as the most efficient ES with isotropic mutations [1].…”

“…Usage of scaled weights allows for larger optimal mutation strengths [2]. It can be beneficial in noisy fitness environments as the ES will work with larger mutations [6], although strongly scaled weights can deteriorate the maximal attainable progress rate in the non-noisy case (in finitedimensional search spaces).…”

“…The superscript (l; λ) refers to the lth-best of the λ offspring (the lth-smallest for minimization). Weights ω l,λ are dependent on the rank of the individual in the set of all offspring individuals [2]. The new parent state is obtained in lines 12 and 13.…”