Abstract
Population diversity is essential for avoiding premature convergence in genetic algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the ( ${\mu +1}$ ) GA and the Jump test function. We show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to significant improvements of the expected optimization time compared to mutation-only algorithms like the (1 + 1) evolutionary algorithm. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to even larger speedups. Experiments were conducted to complement our theoretical findings and further highlight the benefits of crossover on the function class.
Highlights
Genetic Algorithms (GAs) are powerful general-purpose optimisers that perform surprisingly well in many applications, including those where the problem is not well understood to apply a tailored algorithm
In [7], we have shown that a small change to the tie-breaking rule of the (μ+1) GA to introduce many common principles of preserving diversity can lead to a sizeable advantage on the expected optimisation time of Jumpk function
We provide a novel approach loosely inspired from population genetics: we show that diversity can be created by crossover, followed by mutation
Summary
Genetic Algorithms (GAs) are powerful general-purpose optimisers that perform surprisingly well in many applications, including those where the problem is not well understood to apply a tailored algorithm. For the maximum crossover probability pc = 1, we show that on Jumpk diversity emerges naturally in a population: the interplay of crossover, followed by mutation, can serve as a catalyst for creating a diverse range of search points out of few different individuals This naturally emerging diversity allows proving a speedup of order n/log n for k ≥ 3 and standard mutation rate pm = 1/n compared to mutation-only algorithms such as the (1+1) EA. We prove that the size of the largest species is described either by an almost-fair random walk (for standard mutation rates), or by an unfair random walk that is biased toward increased diversity (for higher mutation rates) This allows us to bound the expected time until sufficient diversity is present in the population to perform a crossover that successfully generates the global optimum. The experimental results showed that the setting of high mutation rate can be as competitive as using specific diversity mechanisms from [7]
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