Generalized Heavy-tailed Mutation for Evolutionary Algorithms

TL;DR

This paper generalizes the heavy-tailed mutation operator in evolutionary algorithms using a regularly varying distribution, extending theoretical bounds and proposing a new mutation operator with promising experimental results.

Contribution

It introduces a generalized heavy-tailed mutation operator based on regularly varying distributions, extending existing theoretical bounds and demonstrating improved performance.

Findings

Generalized bounds on expected optimization time for the $(1+(mbda,mbda))$ genetic algorithm.

Proposed a new heavy-tailed mutation operator satisfying generalized conditions.

Computational experiments show promising results for the new operator.

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 smaller than what can be achieved by the $(1+(\lambda,\lambda))$ 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.