A General Dichotomy of Evolutionary Algorithms on Monotone Functions

• Published in: IEEE transactions on evolutionary computation Vol. 24; no. 6; pp. 995 - 1009
• Main Author: Lengler, Johannes
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Computational and artificial intelligence; Evolutionary algorithms; Evolutionary computation; Genetic algorithms; Heuristic algorithms; Monotone functions; Mutation; Parameters; Runtime; Sociology; Statistics
• ISSN: 1089-778X; 1941-0026
• DOI: 10.1109/TEVC.2019.2917014

It is known that the (1 + 1)-EA with mutation rate <inline-formula> <tex-math notation="LaTeX">c/n </tex-math></inline-formula> optimizes every monotone function efficiently if <inline-formula> <tex-math notation="LaTeX">c < 1 </tex-math></inline-formula>, and needs exponential time on some monotone functions (HotTopic functions) if <inline-formula> <tex-math notation="LaTeX">c\geq 2.2 </tex-math></inline-formula>. We study the same question for a large variety of algorithms, particularly for the <inline-formula> <tex-math notation="LaTeX">(1 + \lambda) </tex-math></inline-formula>-EA, <inline-formula> <tex-math notation="LaTeX">(\mu + 1) </tex-math></inline-formula>-EA, <inline-formula> <tex-math notation="LaTeX">(\mu + 1) </tex-math></inline-formula>-GA, their "fast" counterparts, and for the <inline-formula> <tex-math notation="LaTeX">(1 + (\lambda,\lambda)) </tex-math></inline-formula>-GA. We find that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the <inline-formula> <tex-math notation="LaTeX">(1 + (\lambda,\lambda)) </tex-math></inline-formula>-GA, this dichotomy is in the parameter <inline-formula> <tex-math notation="LaTeX">c\gamma </tex-math></inline-formula>, which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in <inline-formula> <tex-math notation="LaTeX">m_{2}/m_{1} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">m_{1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m_{2} </tex-math></inline-formula> are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> nor by the offspring population size <inline-formula> <tex-math notation="LaTeX">\lambda </tex-math></inline-formula>. The picture changes completely if crossover is allowed. The genetic algorithms <inline-formula> <tex-math notation="LaTeX">(\mu + 1) </tex-math></inline-formula>-GA and <inline-formula> <tex-math notation="LaTeX">(\mu + 1) </tex-math></inline-formula>-fGA are efficient for arbitrary mutations strengths if <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> is large enough.