Runtime Analysis of the $$(\mu + 1)$$-EA on the Dynamic BinVal Function

Lengler, Johannes; Riedi, Simone

doi:10.1007/978-3-030-72904-2_6

Cited by 15 publications

(16 citation statements)

References 23 publications

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“…For all constant values ρ, ℓ ∈ (0, 1), we show that for sufficiently small χ the algorithm finds the optimum of TwoLin ρ,ℓ in time O(n log n) if started with o(n) zero-bits, but it takes superpolynomial time for large values of χ. For the symmetric case ρ = ℓ = .5 the treshold between the two regimes is at χ 0 ≈ 2.557, which is only slightly larger than the best known thresholds for the (1 + 1)-EA on monotone functions (χ 0 ≈ 2.13 [15,21]) and for general Dynamic Linear Functions and DynBV (χ 0 ≈ 1.59 [18][19][20]). Thus, we successfully identify a minimal example in which the same failure mode of the (1+1)-EA shows as for monotone functions and for the general dynamic settings, and it shows almost as early as in those settings.…”

Section: Resultsmentioning

confidence: 74%

“…Unfortunately, some other benchmarks for failure modes are rather technical, in particular in the context of monotone functions. Recently, it was discovered that the same failure modes as for monotone functions could be observed by studying certain dynamic environments, more concretely Dynamic Linear Functions and the Dynamic Binary Value function DynBV [18][19][20]. These environments are very simple, so they allow to study failure modes in greater detail.…”

Section: The Application: Twolinmentioning

confidence: 95%

“…The second step is obtained by a standard computation that can e.g. be found in the proof of [19,Lemma 2].…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

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Two-Dimensional Drift Analysis: Optimizing Two Functions Simultaneously Can Be Hard

Janett¹,

Lengler²

2022

Preprint

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In this paper we show how to use drift analysis in the case of two random variables X1, X2, when the drift is approximatively given by A • (X1, X2) T for a matrix A. The non-trivial case is that X1 and X2 impede each other's progress, and we give a full characterization of this case. As application, we develop and analyze a minimal example TwoLin of a dynamic environment that can be hard. The environment consists of two linear function f1 and f2 with positive weights 1 and n, and in each generation selection is based on one of them at random. They only differ in the set of positions that have weight 1 and n. We show that the (1 + 1)-EA with mutation rate χ/n is efficient for small χ on TwoLin, but does not find the shared optimum in polynomial time for large χ.

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Section: Resultsmentioning

confidence: 74%

Section: The Application: Twolinmentioning

confidence: 95%

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Two-Dimensional Drift Analysis: Optimizing Two Functions Simultaneously Can Be Hard

Janett¹,

Lengler²

2022

Preprint

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“…Let us explain the details of our result. The function Dynamic BinVal [29,30] is defined as follows. In each generation t, it fixes a random permutation π t : {1, .…”

Section: Our Resultsmentioning

confidence: 99%

“…That means that in each generation t, we choose a different function f t and use f t in the selection update step of Algorithm 1. We choose Dynamic BinVal or DBv [29,30], which is the binary value function BinVal, applied to a randomly selected permutation of the positions of the input string. More formally, denote by S n the symmetric group on n elements.…”

Section: The Benchmarks: Onemax and Dynamic Binvalmentioning

confidence: 99%

OneMax is not the Easiest Function for Fitness Improvements

Kaufmann¹,

Larcher²,

Lengler³

et al. 2022

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We study the (1 : s + 1) success rule for controlling the population size of the (1, λ)-EA. It was shown by Hevia Fajardo and Sudholt that this parameter control mechanism can run into problems for large s if the fitness landscape is too easy. They conjectured that this problem is worst for the OneMax benchmark, since in some wellestablished sense OneMax is known to be the easiest fitness landscape. In this paper we disprove this conjecture and show that OneMax is not the easiest fitness landscape with respect to finding improving steps. As a consequence, we show that there exists s and ε such that the selfadjusting (1, λ)-EA with (1 : s + 1)-rule optimizes OneMax efficiently when started with εn zero-bits, but does not find the optimum in polynomial time on Dynamic BinVal. Hence, we show that there are landscapes where the problem of the (1 : s + 1)-rule for controlling the population size of the (1, λ)-EA is more severe than for OneMax.

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