A tight runtime analysis for the (μ + λ) EA

Antipov, Denis; Doerr, Benjamin; Fang, Jiefeng; Hetet, Tangi

doi:10.1145/3205455.3205627

Cited by 29 publications

(22 citation statements)

References 41 publications

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“…To prove upper bounds, in Witt (2006), Chen et al (2009), Lehre (2011), Dang and Lehre (2016a), Corus et al (2018), Antipov et al (2018), and Doerr and Kötzing (2019), implicitly or explicitly potential functions were used that build on the fitness of the best individual in the population and the number of individuals having this fitness. Regarding only the current-best individuals, these potential functions might not be suitable for lower bound proofs.…”

Section: Our "Negative Drift In Populations" Resultsmentioning

confidence: 99%

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

Doerr

2021

Evolutionary Computation

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A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems — general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a [Formula: see text] factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than [Formula: see text], which includes the heavytailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on ONEMAX for arbitrary population size.

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Section: Our "Negative Drift In Populations" Resultsmentioning

confidence: 99%

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

Doerr

2021

Evolutionary Computation

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“…Runtime results are available for several other metaheuristics for optimization problems over discrete domains, for example evolutionary algorithms (EAs) (Droste et al 2002;Giel and Wegener 2003;Wegener 2002;Antipov et al 2018Antipov et al , 2019 and ant colony optimization (ACO) (Doerr et al 2007;Neumann and Witt 2007;Sudholt and Thyssen 2012). Most of the results relevant to this work concern the binary PSO algorithm and the ð1 þ 1Þ-EA algorithm.…”

Section: Related Workmentioning

confidence: 99%

Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

et al. 2021

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Meta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a meta-heuristic for a large class of optimization problems over discrete domains based on the particle swarm optimization (PSO) paradigm. We provide a comprehensive formal analysis of the performance of this algorithm on certain “easy” reference problems in a black-box setting, namely the sorting problem and the problem OneMax. In our analysis we use a Markov model of the proposed algorithm to obtain upper and lower bounds on its expected optimization time. Our bounds are essentially tight with respect to the Markov model. We show that for a suitable choice of algorithm parameters the expected optimization time is comparable to that of known algorithms and, furthermore, for other parameter regimes, the algorithm behaves less greedy and more explorative, which can be desirable in practice in order to escape local optima. Our analysis provides a precise insight on the tradeoff between optimization time and exploration. To obtain our results we introduce the notion of indistinguishability of states of a Markov chain and provide bounds on the solution of a recurrence equation with non-constant coefficients by integration.

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“…In [57], the runtime of the (µ + 1) EA on OneMax and LeadingOnes, among others, was studied. The runtime of the (µ + λ) EA with both non-trivial parent and offspring population sizes was determined in [3].…”

Section: The (µ λ) Eamentioning

confidence: 99%

“…Consequently, none of the parents ever lies in the gap, and thus the optimum can only be generated from an individual below the gap, which happens with probability at most p k . For the lower bound, we invoke the recent analysis of how the (µ + λ) EA optimizes OneMax [3] to estimate the time to find the first individual on the local optimum. With an estimate of the takeover time from [54], we estimate the time to have the whole population on the local optimum.…”

Section: The Jump Function Classmentioning

confidence: 99%

Does comma selection help to cope with local optima?

Doerr

2020

Proceedings of the 2020 Genetic and Evolutionary Computation Conference

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One hope of using non-elitism in evolutionary computation is that it aids leaving local optima. We perform a rigorous runtime analysis of a basic non-elitist evolutionary algorithm (EA), the (µ, λ) EA, on the most basic benchmark function with a local optimum, the jump function. We prove that for all reasonable values of the parameters and the problem, the expected runtime of the (µ, λ) EA is, apart from lower order terms, at least as large as the expected runtime of its elitist counterpart, the (µ + λ) EA (for which we conduct the first runtime analysis to allow this comparison). Consequently, the ability of the (µ, λ) EA to leave local optima to inferior solutions does not lead to a runtime advantage. We complement this lower bound with an upper bound that, for broad ranges of the parameters, is identical to our lower bound apart from lower order terms. This is the first runtime result for a non-elitist algorithm on a multi-modal problem that is tight apart from lower order terms. CCS CONCEPTS • Theory of computation → Theory and algorithms for application domains; Theory of randomized search heuristics;

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A tight runtime analysis for the (μ + λ) EA

Cited by 29 publications

References 41 publications

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax

Does comma selection help to cope with local optima?

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