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

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 eΩ(µ) on the DLB problem assuming any selective pressure [EQUATION], as opposed to the expected runtime of O (nλ log λ + n3) 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.

[1]  Dirk Sudholt,et al.  How Crossover Speeds up Building Block Assembly in Genetic Algorithms , 2014, Evolutionary Computation.

[2]  Christopher M. Bishop,et al.  Pattern Recognition and Machine Learning (Information Science and Statistics) , 2006 .

[3]  Dirk Sudholt,et al.  Update Strength in EDAs and ACO: How to Avoid Genetic Drift , 2016, GECCO.

[4]  Per Kristian Lehre,et al.  Theoretical Analysis of Stochastic Search Algorithms , 2017, Handbook of Heuristics.

[5]  Shumeet Baluja,et al.  A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning , 1994 .

[6]  Benjamin Doerr,et al.  The Runtime of the Compact Genetic Algorithm on Jump Functions , 2019, Algorithmica.

[7]  Benjamin Doerr,et al.  Multiplicative drift analysis , 2010, GECCO.

[8]  Martin Pelikan,et al.  An introduction and survey of estimation of distribution algorithms , 2011, Swarm Evol. Comput..

[9]  Thomas Jansen,et al.  Exploring the Explorative Advantage of the Cooperative Coevolutionary (1+1) EA , 2003, GECCO.

[10]  Andrew M. Sutton,et al.  On the runtime dynamics of the compact genetic algorithm on jump functions , 2018, GECCO.

[11]  Duc-Cuong Dang,et al.  Level-Based Analysis of the Univariate Marginal Distribution Algorithm , 2018, Algorithmica.

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[13]  B. Eisenberg.  On the expectation of the maximum of IID geometric random variables , 2008 .

[14]  Carsten Witt,et al.  Runtime Analysis of the ( μ +1) EA on Simple Pseudo-Boolean Functions , 2006 .

[15]  Weijie Zheng,et al.  Working principles of binary differential evolution , 2018, GECCO.

[16]  Carsten Witt,et al.  Domino convergence: why one should hill-climb on linear functions , 2018, GECCO.

[17]  Per Kristian Lehre,et al.  Runtime analysis of the univariate marginal distribution algorithm under low selective pressure and prior noise , 2019, GECCO.

[18]  Daniel J. Simon,et al.  Evolutionary optimization algorithms : biologically-Inspired and population-based approaches to computer intelligence , 2013 .

[19]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[20]  Rolf H. Möhring,et al.  Stochastic Runtime Analysis of a Cross Entropy Algorithm for Traveling Salesman Problems , 2016, Theor. Comput. Sci..

[21]  Benjamin Doerr,et al.  Multiplicative Up-Drift , 2019, Algorithmica.

[22]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[23]  Per Kristian Lehre,et al.  University of Birmingham Level-based analysis of the population-based incremental learning algorithm , 2018 .

[24]  Benjamin Doerr,et al.  The Impact of Random Initialization on the Runtime of Randomized Search Heuristics , 2015, Algorithmica.

[25]  Dirk Sudholt,et al.  Medium step sizes are harmful for the compact genetic algorithm , 2018, GECCO.

[26]  David E. Goldberg,et al.  The compact genetic algorithm , 1999, IEEE Trans. Evol. Comput..

[27]  Andrew M. Sutton,et al.  The Compact Genetic Algorithm is Efficient Under Extreme Gaussian Noise , 2017, IEEE Transactions on Evolutionary Computation.

[28]  Carsten Witt,et al.  Lower bounds on the run time of the Univariate Marginal Distribution Algorithm on OneMax , 2020, Theor. Comput. Sci..

[29]  Per Kristian Lehre,et al.  When is an estimation of distribution algorithm better than an evolutionary algorithm? , 2009, 2009 IEEE Congress on Evolutionary Computation.

[30]  Carsten Witt,et al.  Runtime Analysis of the ( + 1) EA on Simple Pseudo-Boolean Functions , 2006, Evolutionary Computation.

[31]  Benjamin Doerr,et al.  Significance-Based Estimation-of-Distribution Algorithms , 2018, IEEE Transactions on Evolutionary Computation.

[32]  Devdatt P. Dubhashi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms: Contents , 2009 .

[33]  Benjamin Doerr,et al.  A tight runtime analysis for the cGA on jump functions: EDAs can cross fitness valleys at no extra cost , 2019, GECCO.

[34]  Per Kristian Lehre,et al.  Improved runtime bounds for the univariate marginal distribution algorithm via anti-concentration , 2017, GECCO.

[35]  Duc-Cuong Dang,et al.  Simplified Runtime Analysis of Estimation of Distribution Algorithms , 2015, GECCO.

[36]  Montgomery Slatkin,et al.  Linkage disequilibrium — understanding the evolutionary past and mapping the medical future , 2008, Nature Reviews Genetics.

[37]  Concha Bielza,et al.  A review of estimation of distribution algorithms in bioinformatics , 2008, BioData Mining.

[38]  Kenneth A. De Jong,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods on the Choice of the Offspring Population Size in Evolutionary Algorithms on the Choice of the Offspring Population Size in Evolutionary Algorithms , 2004 .

[39]  Paul A. Viola,et al.  MIMIC: Finding Optima by Estimating Probability Densities , 1996, NIPS.

[40]  Yuval Davidor,et al.  Epistasis Variance: A Viewpoint on GA-Hardness , 1990, FOGA.

[41]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[42]  A. E. Eiben,et al.  Introduction to Evolutionary Computing , 2003, Natural Computing Series.

[43]  Tobias Friedrich,et al.  EDAs cannot be Balanced and Stable , 2016, GECCO.

[44]  Thomas Jansen,et al.  On the analysis of the (1+1) evolutionary algorithm , 2002, Theor. Comput. Sci..

[45]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[46]  Ingo Wegener,et al.  Methods for the Analysis of Evolutionary Algorithms on Pseudo-Boolean Functions , 2003 .

[47]  Xin Yao,et al.  Towards an analytic framework for analysing the computation time of evolutionary algorithms , 2003, Artif. Intell..

[48]  Per Kristian Lehre,et al.  Negative Drift in Populations , 2010, PPSN.

[49]  David E. Goldberg,et al.  A Survey of Optimization by Building and Using Probabilistic Models , 2002, Comput. Optim. Appl..

[50]  Duc-Cuong Dang,et al.  Level-Based Analysis of Genetic Algorithms and Other Search Processes , 2016 .

[51]  Daniel Johannsen,et al.  Random combinatorial structures and randomized search heuristics , 2010 .

[52]  Robin Gras.  How efficient are genetic algorithms to solve high epistasis deceptive problems? , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[53]  David H. Ackley,et al.  An empirical study of bit vector function optimization , 1987 .

[54]  Per Kristian Lehre,et al.  Fitness-levels for non-elitist populations , 2011, GECCO '11.