Significance-based estimation-of-distribution algorithms

Doerr, Benjamin; Krejca, Martin S.

doi:10.1145/3205455.3205553

Cited by 27 publications

(32 citation statements)

References 40 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.…”

mentioning

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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“…It has been observed that the stability of an algorithm can lead to difficulties when solving problems in which the fitness only gives a weak signal on what is the right value for a bit-position. In [DK18a], it was proven that the scGA, a version of the cGA artificially made stable, has a runtime of exp(Ω(min{n, K})) on the OneMax benchmark function when the hypothetical population size is K. For the convex search algorithm (CSA), an at least super-polynomial runtime was shown for the optimization of OneMax [DK18b].…”

Section: Sec:onemaxmentioning

confidence: 99%

“…Analogous results hold for the optimization of the BinaryValue function (Theorem thm:BDEforBVwAssumption 24 and Lemma lem:BVassumiBDE 25). Although stability is generally a desirable property (see [FKK16,DK18a] for examples how stable EDAs can outperform common EDAs, which are all unstable), stability can make it hard to find the optimal values of decision variables with small influence on the objective function. We take the OneMax function as an example, and prove that the expected runtime is at least exponential in the dimension when we initialize the population by setting each bit to 1 with probability 0.6 (Theorem thm:rtlargeprob 27).…”

Section: Introductionmentioning

confidence: 99%

Working principles of binary differential evolution

Doerr

Zheng

2020

Theoretical Computer Science

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We conduct a first fundamental analysis of the working principles of binary differential evolution (BDE), an optimization heuristic for binary decision variables that was derived by Gong and Tuson (2007) from the very successful classic differential evolution (DE) for continuous optimization. We show that unlike most other optimization paradigms, it is stable in the sense that neutral bit values are sampled with probability close to 1/2 for a long time. This is generally a desirable property, however, it makes it harder to find the optima for decision variables with small influence on the objective function. This can result in an optimization time exponential in the dimension when optimizing simple symmetric functions like OneMax. On the positive side, BDE quickly detects and optimizes the most important decision variables. For example, dominant bits converge to the optimal value * This is a significantly extended version of the 8-page conference paper [ZYD18]. All authors of this version have contributed equally. The authors are given in alphabetical order as common in theoretical computer science.† Work done while affiliated with Department of Computer Science and Technology in Tsinghua University and partially during a research stay atÉcole Polytechnique's computer science lab (LIX).1 in time logarithmic in the population size. This enables BDE to optimize the most important bits very fast. Overall, our results indicate that BDE is an interesting optimization paradigm having characteristics significantly different from classic evolutionary algorithms or estimation-of-distribution algorithms (EDAs).On the technical side, we observe that the strong stochastic dependencies in the random experiment describing a run of BDE prevent us from proving all desired results with the mathematical rigor that was successfully used in the analysis of other evolutionary algorithms. Inspired by mean-field approaches in statistical physics we propose a more independent variant of BDE, show experimentally its similarity to BDE, and prove some statements rigorously only for the independent variant. Such a semi-rigorous approach might be interesting for other problems in evolutionary computation where purely mathematical methods failed so far.

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Significance-based estimation-of-distribution algorithms

Cited by 27 publications

References 40 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

Working principles of binary differential evolution

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