A tight runtime analysis for the (1 + (λ, λ)) GA on leadingones

We conduct a rigorous runtime analysis of the (1 + (λ, λ)) evolutionary algorithm with standard parameter settings, that is, a mutation rate of <i>p</i> = λ/n and a crossover bias of <i>c</i> = 1/λ when optimizing the classic LeadingOnes benchmark function. We show that, for all λ ∈ [1..<i>n</i>/2], the runtime is Θ(<i>n</i><sup>2</sup>/λ) iterations and Θ(<i>n</i><sup>2</sup>) fitness evaluations. This is, asymptotically, the same number of iterations as for the (1 + λ) EA and the same number of fitness evaluations as for the (1 + λ) EA for any value of λ = <i>O(n)</i>. We also extend our results to parameter control techniques and prove that for any dynamic choice of λ the bound of Θ(<i>n</i><sup>2</sup>) fitness evaluations still holds.

[1]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[2]  Frank Neumann,et al.  A rigorous view on neutrality , 2007, 2007 IEEE Congress on Evolutionary Computation.

[3]  Timo Kötzing.  Concentration of First Hitting Times Under Additive Drift , 2015, Algorithmica.

[4]  Dirk Sudholt,et al.  Design and analysis of migration in parallel evolutionary algorithms , 2013, Soft Comput..

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

[6]  Benjamin Doerr,et al.  Optimal Parameter Choices Through Self-Adjustment: Applying the 1/5-th Rule in Discrete Settings , 2015, GECCO.

[7]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[8]  Benjamin Doerr,et al.  Black-Box Complexity: Breaking the O(n logn) Barrier of LeadingOnes , 2011, Artificial Evolution.

[9]  Dirk Sudholt,et al.  The choice of the offspring population size in the (1, λ) evolutionary algorithm , 2014, Theor. Comput. Sci..

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

[11]  Frank Neumann,et al.  Time Complexity Analysis of Evolutionary Algorithms on Random Satisfiable k-CNF Formulas , 2016, Algorithmica.

[12]  Frank Neumann,et al.  Runtime Analysis of Evolutionary Algorithms on Randomly Constructed High-Density Satisfiable 3-CNF Formulas , 2014, PPSN.

[13]  Benjamin Doerr,et al.  Analyzing randomized search heuristics via stochastic domination , 2019, Theor. Comput. Sci..

[14]  Benjamin Doerr,et al.  Theoretical and empirical study of the (1 + (λ, λ)) EA on the leadingones problem , 2019, GECCO.

[15]  Benjamin Doerr,et al.  Fast genetic algorithms , 2017, GECCO.

[16]  Xin Yao,et al.  Drift analysis and average time complexity of evolutionary algorithms , 2001, Artif. Intell..

[17]  Benjamin Doerr,et al.  Optimal Static and Self-Adjusting Parameter Choices for the (1+(λ,λ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( , 2017, Algorithmica.

[18]  Frank Neumann,et al.  Fast re-optimization via structural diversity , 2019, GECCO.

[19]  Benjamin Doerr,et al.  Runtime analysis of the (1 + (λ, λ)) genetic algorithm on random satisfiable 3-CNF formulas , 2017, GECCO.

[20]  Benjamin Doerr,et al.  From black-box complexity to designing new genetic algorithms , 2015, Theor. Comput. Sci..

[21]  Thomas Jansen,et al.  A method to derive fixed budget results from expected optimisation times , 2013, GECCO '13.

[22]  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 .

[23]  Dirk Sudholt,et al.  On the robustness of evolutionary algorithms to noise: refined results and an example where noise helps , 2018, GECCO.

[24]  Kurt Mehlhorn,et al.  The Query Complexity of a Permutation-Based Variant of Mastermind , 2019, Discret. Appl. Math..

[25]  Dirk Sudholt,et al.  A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms , 2011, IEEE Transactions on Evolutionary Computation.

[26]  Frank Neumann,et al.  Optimal Fixed and Adaptive Mutation Rates for the LeadingOnes Problem , 2010, PPSN.

[27]  Dirk Sudholt,et al.  When do evolutionary algorithms optimize separable functions in parallel? , 2013, FOGA XII '13.