Optimal Static and Self-Adjusting Parameter Choices for the (1+(λ,λ)) Genetic Algorithm

• Published in: Algorithmica Vol. 80; no. 5; pp. 1658 - 1709
• Main Authors: Doerr, Benjamin, Doerr, Carola
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2018; Springer Nature B.V; Springer Verlag
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Bias; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Algorithms; Data Structures and Information Theory; Genetic algorithms; Mathematics of Computing; Mutation; Neural and Evolutionary Computing; Optimization; Parameters; Run time (computers); Special Issue on Genetic and Evolutionary Computation; Theory of Computation; Parameter choice; Theory of randomized search heuristics; Runtime analysis; Parameter control; Genetic algorithms
• ISSN: 0178-4617; 1432-0541; 1432-0541
• DOI: 10.1007/s00453-017-0354-9

The ( 1 + ( λ , λ ) )  genetic algorithm proposed in Doerr et al. (Theor Comput Sci 567:87–104, 2015 ) is one of the few examples for which a super-constant speed-up of the expected optimization time through the use of crossover could be rigorously demonstrated. It was proven that the expected optimization time of this algorithm on OneMax is O ( max { n log ( n ) / λ , λ n } ) for any offspring population size λ ∈ { 1 , … , n } (and the other parameters suitably dependent on λ ) and it was shown experimentally that a self-adjusting choice of λ leads to a better, most likely linear, runtime. In this work, we study more precisely how the optimization time depends on the parameter choices, leading to the following results on how to optimally choose the population size, the mutation probability, and the crossover bias both in a static and a dynamic fashion. For the mutation probability and the crossover bias depending on λ as in Doerr et al. ( 2015 ), we improve the previous runtime bound to O ( max { n log ( n ) / λ , n λ log log ( λ ) / log ( λ ) } ) . This expression is minimized by a value of λ slightly larger than what the previous result suggested and gives an expected optimization time of O n log ( n ) log log log ( n ) / log log ( n ) . We show that no static choice in the three-dimensional parameter space of offspring population, mutation probability, and crossover bias gives an asymptotically better runtime. We also prove that the self-adjusting parameter choice suggested in Doerr et al. ( 2015 ) outperforms all static choices and yields the conjectured linear expected runtime. This is asymptotically optimal among all possible parameter choices.