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

Doerr, Benjamin

doi:10.1145/3299904.3340304

Cited by 20 publications

(15 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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...

show abstract

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

View full text Add to dashboard Cite

show abstract

“…We refer to Lengler (2020, Section 2.4.3) for more details. Another approach to negative drift was used in Antipov et al (2019) and Doerr (2019bDoerr ( , 2020a. There the original process was transformed suitably (via an exponential function), but in a way that the drift of the new process still is negative or at most a small constant.…”

Section: Related Workmentioning

confidence: 99%

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

Doerr

2021

Evolutionary Computation

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…While the basic approach is simple and natural, the non-trivial part is finding a rescaling function д which both gives at most a slow progress towards the target and gives a large difference д(b) − д(a). The rescalings used in [5] and [17] were both of exponential type, that is, д was roughly speaking an exponential function. By construction, they only led to lower bounds exponential in b − a, and in both cases the lower bound was not tight (apart from being exponential).…”

Section: Negative Driftmentioning

confidence: 99%

“…Asymptotically better runtimes can be achieved when using crossover, though this is not as easy as one might expect. Since these results and runtime analyses for even more distant algorithms are less relevant for this work, for reasons of space we direct the reader to the original works [4, 9-12, 19, 30, 34, 39, 46, 52, 56] or the more detailed overview in [18,Section 2.3].…”

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

Self Cite

View full text Add to dashboard Cite

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;

show abstract

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

Cited by 20 publications

References 42 publications

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

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

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

Does comma selection help to cope with local optima?

Contact Info

Product

Resources

About