Generalized Heavy-tailed Mutation for Evolutionary Algorithms
This work provides incremental theoretical improvements for evolutionary algorithms, specifically targeting optimization time bounds in computational optimization problems.
The paper generalizes the heavy-tailed mutation operator for evolutionary algorithms by relaxing the power-law assumption to a regularly varying constraint, and shows that the $(1+(\\lambda,\\lambda))$ genetic algorithm can achieve $O(n)$ expected optimization time on the OneMax function class, which is asymptotically better than with any static mutation rate.
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.