Generalized heavy-tailed mutation for evolutionary algorithms

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 $(1+(\lambda,\lambda))$ 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 faster than what can be achieved by the $(1+(\lambda,\lambda))$ genetic algorithm with any static mutation rate.