Medium step sizes are harmful for the compact genetic algorithm

Lengler, Johannes; Sudholt, Dirk; Witt, Carsten

doi:10.1145/3205455.3205576

Cited by 34 publications

(35 citation statements)

References 15 publications

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“…Recently EDAs have drawn a growing attention from the theory community of evolutionary computation [10,17,26,44,46,25,45,27,12,31]. The aim of the theoretical analyses of EDAs in general is to gain insights into the behaviour of the algorithms when optimising an objective function, especially in terms of the optimisation time, that is the number of function evaluations, required by the algorithm until an optimal solution has been found for the first time.…”

Section: Introductionmentioning

confidence: 99%

Level-Based Analysis of the Univariate Marginal Distribution Algorithm

Dang¹,

Lehre

Nguyen

2018

Algorithmica

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Estimation of Distribution Algorithms (EDAs) are stochastic heuristics that search for optimal solutions by learning and sampling from probabilistic models. Despite their popularity in real-world applications, there is little rigorous understanding of their performance. Even for the Univariate Marginal Distribution Algorithm (UMDA) -a simple population-based EDA assuming independence between decision variables -the optimisation time on the linear problem OneMax was until recently undetermined. The incomplete theoretical understanding of EDAs is mainly due to lack of appropriate analytical tools.We show that the recently developed level-based theorem for non-elitist populations combined with anticoncentration results yield upper bounds on the expected optimisation time of the UMDA. This approach results in the bound O nλ log λ + n 2 on two problems, LeadingOnes and BinVal, for population sizes λ > µ = Ω(log n), where µ and λ are parameters of the algorithm. We also prove that the UMDA with population sizes µ ∈ O ( √ n) ∩ Ω(log n) optimises OneMax in expected time O (λn), and for larger population sizes µ = Ω( √ n log n), in expected time O (λ √ n). The facility and generality of our arguments suggest that this is a promising approach to derive bounds on the expected optimisation time of EDAs.

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Section: Introductionmentioning

confidence: 99%

Level-Based Analysis of the Univariate Marginal Distribution Algorithm

Dang¹,

Lehre

Nguyen

2018

Algorithmica

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“…While rigorous runtime analyses provide deep insights into the performance of randomised search heuristics, it is highly challenging even for simple algorithms on toy functions. Most current runtime results merely concern univariate EDAs on functions like OneMax [32,51,36,53,40], LeadingOnes [15,22,37,53,38], BinVal [52,37] and Jump [26,11,12], hoping that this provides valuable insights into the development of new techniques for analysing multivariate variants of EDAs and the behaviour of such algorithms on easy parts of more complex problem spaces [13]. There are two main reasons accounted for this.…”

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confidence: 99%

On the limitations of the univariate marginal distribution algorithm to deception and where bivariate EDAs might help

Lehre

Nguyen

2019

Proceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

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We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to rigorously study the runtime of the Univariate Marginal Distribution Algorithm (UMDA) in the presence of epistasis and deception. We show that simple Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure µ/λ is extremely high, where µ and λ are the parent and offspring population sizes, respectively. More precisely, we show that the UMDA with a parent population size of µ = Ω(log n) has an expected runtime of e Ω(µ) on the DLB problem assuming any selective pressure µ λ ≥ 14 1000 , as opposed to the expected runtime of O nλ log λ + n 3 for the non-elitist (µ, λ) EA with µ/λ ≤ 1/e. These results illustrate inherent limitations of univariate EDAs against deception and epistasis, which are common characteristics of real-world problems. In contrast, empirical evidence reveals the efficiency of the bi-variate MIMIC algorithm on the DLB problem. Our results suggest that one should consider EDAs with more complex probabilistic models when optimising problems with some degree of epistasis and deception. * Preliminary version of this work will appear in the Proceedings of 15th ACM/SIGEVO Workshop on FoundationsEstimation of distribution algorithms (EDAs) [42,43,33] are a class of randomised search heuristics with many real-world applications (see [27] and references therein). Unlike traditional EAs, which define implicit models of promising solutions via genetic operations such as crossover and mutation, EDAs optimise objective functions by constructing and sampling explicit probabilistic models to generate offspring for the next iteration. The workflow of EDAs is an iterative process, where the initial model is a uniform distribution over the search space. The starting population consists of λ individuals sampled from the uniform distribution. A fitness function then scores each individual, and the algorithm selects the µ fittest individuals to update the model (where µ < λ). The procedure is repeated until some termination condition is fulfilled, which is usually a threshold on the number of iterations or on the quality of the fittest offspring [27, 19]. Many variants of EDAs have been proposed over the last decades. They differ in the way their models are represented, updated as well as sampled over iterations. In general, EDAs are categorised into two main classes: univariate and multivariate. Univariate EDAs take advantage of first-order statistics (i.e. the mean) to build a probability vector-based model and assume independence between decision variables. The probabilistic model is represented as an n-vector, where each component is called a marginal (also frequency) and n is the problem instance size. Typical univariate EDAs are compact Genetic Algorithm (cGA [25]), Univariate Marginal Distribution Algorithm (UMDA [42]) and Population-Based Incremental Learning (PBIL [3]). In contrast, multivariate EDAs apply higher-order statistics to model the correlations between decision variables of the addressed problems.Th...

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“….n µ ]}. This assumption was made, e.g., in the proof of Theorem 2 in [SW16] and in the paper [LSW18] (see the paragraph following Lemma 2.1).…”

Section: The Compact Genetic Algorithmmentioning

confidence: 99%

“…Not much is known for hypothetical population sizes below the order of √ n. It is clear that then the frequencies will reach the lower boundary of the frequency range, so working with a non-trivial lower boundary like 1 n is necessary to prevent premature convergence. The recent lower bound Ω(µ 1/3 n) valid for µ = O( √ n log n log log n ) of [LSW18] indicates that already a little below the √ n regime significantly larger runtimes occur, but with no upper bounds this regime remains largely not understood. We refer the reader to the recent survey [KW18] for more results on the runtime of the cGA on classic unimodal test functions like LeadingOnes and BinVal.…”

Section: Related Workmentioning

confidence: 99%

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An exponential lower bound for the runtime of the compact genetic algorithm on jump functions

Doerr

2019

Proceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

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In the first runtime analysis of an estimation-of-distribution algorithm (EDA) on the multi-modal jump function class, Hasenöhrl and Sutton (GECCO 2018) proved that the runtime of the compact genetic algorithm with suitable parameter choice on jump functions with high probability is at most polynomial (in the dimension) if the jump size is at most logarithmic (in the dimension), and is at most exponential in the jump size if the jump size is super-logarithmic. The exponential runtime guarantee was achieved with a hypothetical population size that is also exponential in the jump size. Consequently, this setting cannot lead to a better runtime.In this work, we show that any choice of the hypothetical population size leads to a runtime that, with high probability, is at least exponential in the jump size. This result might be the first non-trivial exponential lower bound for EDAs that holds for arbitrary parameter settings.

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Medium step sizes are harmful for the compact genetic algorithm

Abstract: We study the intricate dynamics of the Compact Genetic Algorithm (cGA) on OneMax, and how its performance depends on the step size 1/K, that determines how quickly decisions about promising bit values are fixed in the probabilistic model.

Cited by 34 publications

References 15 publications

Level-Based Analysis of the Univariate Marginal Distribution Algorithm

Level-Based Analysis of the Univariate Marginal Distribution Algorithm

On the limitations of the univariate marginal distribution algorithm to deception and where bivariate EDAs might help

An exponential lower bound for the runtime of the compact genetic algorithm on jump functions

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