Self-adjusting Population Sizes for the $(1, \lambda)$-EA on Monotone Functions

TL;DR

This paper analyzes a self-adjusting population size evolutionary algorithm on monotone functions, showing efficiency depends on success rule parameters and initial proximity to the optimum, with results applicable in dynamic environments.

Contribution

It provides a comprehensive analysis of the $(1, ext{}\lambda)$-EA with adaptive population control on monotone functions, extending previous results and exploring parameter effects.

Findings

Efficient on all monotone functions for small success rule parameter s.

Superpolynomial runtime occurs for large s on all monotone functions.

Fast optimization is guaranteed if starting near the optimum.

Abstract

We study the $(1,\lambda)$-EA with mutation rate $c/n$ for $c\le 1$, where the population size is adaptively controlled with the $(1:s+1)$-success rule. Recently, Hevia Fajardo and Sudholt have shown that this setup with $c=1$ is efficient on \onemax for $s<1$, but inefficient if $s \ge 18$. Surprisingly, the hardest part is not close to the optimum, but rather at linear distance. We show that this behavior is not specific to \onemax. If $s$ is small, then the algorithm is efficient on all monotone functions, and if $s$ is large, then it needs superpolynomial time on all monotone functions. In the former case, for $c<1$ we show a $O(n)$ upper bound for the number of generations and $O(n\log n)$ for the number of function evaluations, and for $c=1$ we show $O(n\log n)$ generations and $O(n^2\log\log n)$ evaluations. We also show formally that optimization is always fast, regardless of $s$, if the algorithm starts in proximity of the optimum. All results also hold in a dynamic environment where the fitness function changes in each generation.