A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms

Sudholt, Dirk

doi:10.1109/tevc.2012.2202241

Cited by 167 publications

(153 citation statements)

References 56 publications

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“…Most of these works derive upper bounds for a specific EDA on the popular OneMax function, which counts the number of 1s in a bit string and is considered to be one of the easiest functions with a unique optimum [17,20]. The only exceptions are Friedrich et al [10], who look at general properties of EDAs, and Sudholt and Witt [18], who derive lower bounds on OneMax for the aforementioned cGA and iteration-best ACO.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Krejca

Witt

2017

Proceedings of the 14th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

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The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of Ω(µ √ n + n log n), where µ is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.

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Section: Introductionmentioning

confidence: 99%

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Krejca

Witt

2017

Proceedings of the 14th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

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“…This simple technique can sometimes provide tight upper bounds of the expected runtime. Recently in [33], an extension of the method has been shown to be able to derive tight lower bounds, the key idea is to estimate the number of fitness levels being skipped on average.…”

Section: Introductionmentioning

confidence: 99%

Runtime Analysis of Non-elitist Populations: From Classical Optimisation to Partial Information

Dang

Lehre

2016

Algorithmica

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Although widely applied in optimisation, relatively little has been proven rigorously about the role and behaviour of populations in randomised search processes. This paper presents a new method to prove upper bounds on the expected optimisation time of population-based randomised search heuristics that use non-elitist selection mechanisms and unary variation operators. Our results follow from a detailed drift analysis of the population dynamics in these heuristics. This analysis shows that the optimisation time depends on the relationship between the strength of the selective pressure and the degree of variation introduced by the variation operator. Given limited variation, a surprisingly weak selective pressure suffices to optimise many functions in expected polynomial time. We derive upper bounds on the expected optimisation time of nonelitist Evolutionary Algorithms (EA) using various selection mechanisms, including fitness proportionate selection. We show that EAs using fitness proportionate selection can optimise standard benchmark functions in expected polynomial time given a sufficiently low mutation rate.As a second contribution, we consider an optimisation scenario with partial information, where fitness values of solutions are only partially available. We prove that non-elitist EAs under a set of specific conditions can optimise benchmark functions in expected polynomial time, even when vanishingly little information about the fitness values of individual solutions or populations is available. To our knowledge, this is the first runtime analysis of randomised search heuristics under partial information.

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“…This statement generalises to the class of all evolutionary algorithms that only use standard bit mutation for variation [33] as well as higher mutation rates and stochastic dominance [37]. On the other hand, He et al [10] showed that the highly deceptive function Trap is the hardest function for the (1+1) EA.…”

Section: Introductionmentioning

confidence: 94%

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

Corus

Jansen

et al. 2016

Algorithmica

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Understanding which function classes are easy and which are hard for a given algorithm is a fundamental question for the analysis and design of bio-inspired search heuristics. A natural starting point is to consider the easiest and hardest functions for an algorithm. For the (1+1) EA using standard bit mutation (SBM) it is well known that OneMax is an easiest function with unique optimum while Trap is a hardest. In this paper we extend the analysis of easiest function classes to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. We define a function MinBlocks and prove that it is an easiest function for the (1+1) EA using CHM, presenting both a runtime and a fixed budget analysis. Since MinBlocks is, up to a factor of 2, a hardest function for standard bit mutations, we consider the effects of combining both operators into a hybrid algorithm. We rigorously prove that by combining the advantages of k operators, several hybrid algorithmic schemes have optimal asymptotic performance on the easiest functions for each individual operator. In particular, the hybrid algorithms using CHM and SBM have optimal asymptotic performance on both OneMax and MinBlocks. We then investigate easiest functions for hybrid schemes and show that an easiest function for a hybrid algorithm is not just a trivial weighted combination of the respective easiest functions for each operator.

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A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms

Cited by 167 publications

References 56 publications

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Runtime Analysis of Non-elitist Populations: From Classical Optimisation to Partial Information

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

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