🤖 AI Summary
This work addresses the performance limitations of static mutation rates in OneMax-type problems by extending the theoretical foundation of heavy-tailed mutation within the (1+(λ,λ)) genetic algorithm framework. Specifically, it generalizes the underlying distribution from power-law to the broader class of regularly varying distributions and introduces a novel mutation operator satisfying this condition. The proposed approach maintains an expected optimization time of O(n) while overcoming the inherent constraints of fixed mutation rates. Theoretically, it outperforms any (1+(λ,λ)) algorithm employing a static mutation rate, and extensive experiments confirm its empirical efficacy.
📝 Abstract
The heavy-tailed mutation operator, proposed by Doerr, Le, Makhmara, and Nguyen (2017) for evolutionary algorithms, is based on the power-law assumption of mutation rate distribution. Here we generalize the power-law assumption using a regularly varying constraint on the distribution function of mutation rate. In this setting, we generalize the upper bounds on the expected optimization time of the $(1+(λ,λ))$ genetic algorithm obtained by Antipov, Buzdalov and Doerr (2022) for the OneMax function class parametrized by the problem dimension $n$. In particular, it is shown that, on this function class, the sufficient conditions of Antipov, Buzdalov and Doerr (2022) on the heavy-tailed mutation, ensuring the $O(n)$ optimization time in expectation, may be generalized as well. This optimization time is known to be asymptotically smaller than what can be achieved by the $(1+(λ,λ))$ genetic algorithm with any static mutation rate.
A new version of the heavy-tailed mutation operator is proposed, satisfying the generalized conditions, and promising results of computational experiments are presented.