Upper and Lower Bounds on Unrestricted Black-Box Complexity of Jump $$_{n,\ell }$$

Buzdalov, Maxim; Kever, Mikhail; Doerr, Benjamin

doi:10.1007/978-3-319-16468-7_18

Cited by 3 publications

(1 citation statement)

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“…A preliminary version of a larger part of these results has appeared as a conference paper of Buzdalov et al (2015). Apart from making a more detailed introduction, which was impossible in the conference paper format, we have refined the lower bound for extreme jump sizes up to n + Θ( √ n) to match the upper bound (see Section 6), which seems to be impossible by a straightforward application of the matrix lower bound theorem and required substantial additional effort.…”

Section: Introductionmentioning

confidence: 99%

The Unrestricted Black-Box Complexity of Jump Functions

Buzdalov

Doerr

Kever

2016

Evolutionary Computation

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We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

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Section: Introductionmentioning

confidence: 99%

The Unrestricted Black-Box Complexity of Jump Functions

Buzdalov

Doerr

Kever

2016

Evolutionary Computation

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Theoretical and Empirical Analysis of Parameter Control Mechanisms in the (1 + (λ, λ)) Genetic Algorithm

Fajardo

Sudholt

2022

ACM Trans. Evol. Learn. Optim.

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The self-adjusting (1+( λ , λ )) GA is the best known genetic algorithm for problems with a good fitness-distance correlation as in OneMax . It uses a parameter control mechanism for the parameter λ that governs the mutation strength and the number of offspring. However, on multimodal problems, the parameter control mechanism tends to increase λ uncontrollably. We study this problem for the standard ${Jump}_k $ benchmark problem class using runtime analysis. The self-adjusting (1+( λ , λ )) GA behaves like a (1+ n ) EA whenever the maximum value for λ is reached. This is ineffective for problems where large jumps are required. Capping λ at smaller values is beneficial for such problems. Finally, resetting λ to 1 allows the parameter to cycle through the parameter space. We show that resets are effective for all ${Jump}_k $ problems: the self-adjusting (1+( λ , λ )) GA performs as well as the (1+1) EA with the optimal mutation rate and evolutionary algorithms with heavy-tailed mutation, apart from a small polynomial overhead. Along the way, we present new general methods for translating existing runtime bounds from the (1+1) EA to the self-adjusting (1+( λ , λ )) GA. We also show that the algorithm presents a bimodal parameter landscape with respect to λ on ${Jump}_k $ . For appropriate n and k , the landscape features a local optimum in a wide basin of attraction and a global optimum in a narrow basin of attraction. To our knowledge this is the first proof of a bimodal parameter landscape for the runtime of an evolutionary algorithm on a multimodal problem.

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Unbiased Black-Box Complexities of Jump Functions

Doerr

Kötzing

2015

Evolutionary Computation

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We analyze the unbiased black-box complexities of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is [Formula: see text], that is, when only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities 3 and higher are of the same order as those for the simple OneMax function. Even for the extreme jump function, in which all but the two fitness values [Formula: see text] and n are blanked out, polynomial time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a [Formula: see text] fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not only by comparing the fitnesses of the competing search points but also by taking into account the (empirical) expected fitnesses of their offspring.

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Upper and Lower Bounds on Unrestricted Black-Box Complexity of Jump $$_{n,\ell }$$

Cited by 3 publications

References 5 publications

The Unrestricted Black-Box Complexity of Jump Functions

The Unrestricted Black-Box Complexity of Jump Functions

Theoretical and Empirical Analysis of Parameter Control Mechanisms in the (1 + (λ, λ)) Genetic Algorithm

Unbiased Black-Box Complexities of Jump Functions

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