How Crossover Speeds up Building Block Assembly in Genetic Algorithms

摘要

We reinvestigate a fundamental question: How effective is crossover in genetic algorithms in combining building blocks of good solutions? Although this has been discussed controversially for decades, we are still lacking a rigorous and intuitive answer. We provide such answers for royal road functions and OneMax, where every bit is a building block. For the latter, we show that using crossover makes every (mu+lambda) genetic algorithm at least twice as fast as the fastest evolutionary algorithm using only standard bit mutation, up to small-order terms and for moderate mu and lambda. Crossover is beneficial because it can capitalize on mutations that have both beneficial and disruptive effects on building blocks: crossover is able to repair the disruptive effects of mutation in later generations. Compared to mutation-based evolutionary algorithms, this makes multibit mutations more useful. Introducing crossover changes the optimal mutation rate on OneMax from 1 to (1 + root 5)/2 . 1 approximate to 1.618. This holds both for uniform crossover and k-point crossover. Experiments and statistical tests confirm that our findings apply to a broad class of building block functions.