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Methods for Ti­ght Analysis of Popu­lation-based Evolutionary Algorithms

Abstract : Evolutionary Algorithms (EAs for brevity) is a broad class of optimization algorithms which are inspired by the natural evolution. They are often used to solve practical problems which cannot be solved precisely in a reasonable time, since they can find satisfying solutions without spending too much computation resources. The practical efficiency of the EAs is supported by the theory of evolutionary computation which has produced a huge number of impressive results during the last two decades. These results give valuable recommendations on how to set up parameters of algorithms or even propose new EAs.Theoretical studies mostly observe how simple algorithms optimize model problems. It is hard to analyse more real-world settings, since even the most simple algorithms are often described via highly complicated stochastic processes. In particular, not so much is known about the behavior of the population-based algorithms, while the populations are believed to be essential by most practitioners. The lack of theoretical understanding of how populations work raises a risk that populations are used not in the most effective way.The existing analysis tools, however, are not suitable to give us a better understanding of populations. Hence, the main aim of this work is to develop new analysis methods which would help to deliver new runtime bounds for evolutionary algorithms and extend our knowledge of the role of populations. We propose the following analysis methods for the population-based EAs.- The method of the complete trees for delivering the lower bounds on the runtime of the population-based EAs.- The method of the analysis of the no-drift processes.- The method for delivering the precise bounds on the runtime distribution for the EAs on plateaus.- The additive drift theorem with tail bounds.With these analysis methods we perform a runtime analysis of the following algorithms.- With the method of the complete trees we derive a tight bound on the runtime of the (mu + lambda) EA on OneMax. These bounds, in particular, suggest that using parent population size mu which is O(log(n)) does not increase the asymptotical runtime and that using offspring population size lambda greater than max{mu, ln(n)} does not give a significant decrease in the expected number of iterations.- With the method of the analysis of the no-drift processes we analyse the (mu, lambda) EA on OneMax with the threshold parameter values lambda is approximately e*mu. We show that in this setting (where there is almost no drift of the number of the best offspring) the absolute population size plays a significant role and that this regime seems to be the most interesting for the practical application of the (mu, lambda) EA.- With the method for the analysis of EAs on plateaus we deliver precise estimates on the runtime of the (1 + 1) EA and (lambda + lambda) EA on Plateau_k function, demonstrating that the choice of the mutation operator does not play a significant role when an EA is traversing a plateau.We also propose a new crossover-based algorithm with non-trivial offspring population --- the heavy-tailed (1 + (lambda, lambda)) GA. Our analysis of this algorithm on OneMax, LeadingOnes and Jump_k functions reveals the efficiency of the random parameter choices from a power-law distribution. While on Jump_k this random parameter choice gives us a one-size-fits-all algorithm which relieves us from choosing the optimal static parameters (which depend on function parameter k, probably unknown in advance), on OneMax we observed a runtime which is better than the runtime for the best static parameter choice. On the LeadingOnes (with a help of the developed additive drift theorem) we showed that the asymptotical runtime is the same for any static or dynamic choice of parameters and is the same as for the most standard mutation-based algorithms.
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https://tel.archives-ouvertes.fr/tel-03080386
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Submitted on : Thursday, December 17, 2020 - 4:28:16 PM
Last modification on : Friday, December 18, 2020 - 3:42:54 AM
Long-term archiving on: : Thursday, March 18, 2021 - 8:20:18 PM

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Denis Antipov. Methods for Ti­ght Analysis of Popu­lation-based Evolutionary Algorithms. Neural and Evolutionary Computing [cs.NE]. Institut Polytechnique de Paris; ITMO University, 2020. English. ⟨NNT : 2020IPPAX069⟩. ⟨tel-03080386⟩

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