Elitist Black-Box Models

Doerr, Benjamin; Lengler, Johannes

doi:10.1145/2739480.2754654

Cited by 13 publications

References 33 publications

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Complexity Theory for Discrete Black-Box Optimization Heuristics

Doerr

2019

Natural Computing Series

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A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in general algorithms theory, this running time perspective is most useful when it is complemented by a meaningful complexity theory that studies the limits of algorithmic solutions.In the context of discrete black-box optimization, several black-box complexity models have been developed to analyze the best possible performance that a black-box optimization algorithm can achieve on a given problem. The models differ in the classes of algorithms to which these lower bounds apply. This way, black-box complexity contributes to a better understanding of how certain algorithmic choices (such as the amount of memory used by a heuristic, its selective pressure, or properties of the strategies that it uses to create new solution candidates) influences performance.In this chapter we review the different black-box complexity models that have been proposed in the literature, survey the bounds that have been obtained for these models, and discuss how the interplay of running time analysis and black-box complexity can inspire new algorithmic solutions to well-researched problems in evolutionary computation. We also discuss in this chapter several interesting open questions for future work.

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Complexity Theory for Discrete Black-Box Optimization Heuristics

Doerr

2019

Natural Computing Series

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OneMax in Black-Box Models with Several Restrictions

Doerr¹,

Lengler²

2016

Algorithmica

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Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the blackbox complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind.We analyze in particular the combined memory-restricted ranking-based blackbox complexity of OneMax for different memory sizes. While its isolated memoryrestricted as well as its ranking-based black-box complexity for bit strings of length n is only of order n/ log n, the combined model does not allow for algorithms being faster than linear in n, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory-and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models.Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in o(n log n) iterations.

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The $$(1+1)$$ ( 1 + 1 ) Elitist Black-Box Complexity of LeadingOnes

Doerr

Lengler

2017

Algorithmica

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One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the Ω(n log n) lower bound for unary unbiased algorithms on functions with a unique global optimum [Lehre/Witt, GECCO 2010], which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, almost no non-trivial bounds are known for other restricted models. Proving such bounds therefore remains to be one of the main challenges in black-box complexity theory.With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation-and bit-invariant version of LeadingOnes and prove that its (1+1) elitist black-box complexity is Ω(n 2 ), a bound that is matched by (1+1)-type evolutionary algorithms. The (1+1) elitist complexity of LeadingOnes is thus considerably larger than its unrestricted one, which is known to be of order n log log n [Afshani et al., 2013].

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Elitist Black-Box Models

Cited by 13 publications

References 33 publications

Complexity Theory for Discrete Black-Box Optimization Heuristics

Complexity Theory for Discrete Black-Box Optimization Heuristics

OneMax in Black-Box Models with Several Restrictions

The $$(1+1)$$ ( 1 + 1 ) Elitist Black-Box Complexity of LeadingOnes

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