# Exponential Slowdown for Larger Populations: The $(\mu+1)$-EA on Monotone Functions

## TL;DR

This paper demonstrates that increasing the population size in the $(+1)$-EA can cause exponential slowdown on monotone functions, due to decreased selective pressure and accumulation of unfavorable mutations, contrasting with other benchmarks.

## Contribution

It proves that for any mutation rate, a sufficiently large population size can cause superpolynomial optimization time on monotone functions, highlighting a counter-intuitive negative effect of larger populations.

## Key findings

- Larger populations can slow down optimization on monotone functions.
- Increased population size can lead to accumulation of unfavorable mutations.
- Counter-intuitive effect where larger populations hinder rather than help optimization.

## Abstract

Pseudo-Boolean monotone functions are unimodal functions which are trivial to optimize for some hillclimbers, but are challenging for a surprising number of evolutionary algorithms (EAs). A general trend is that EAs are efficient if parameters like the mutation rate are set conservatively, but may need exponential time otherwise. In particular, it was known that the $(1+1)$-EA and the $(1+\lambda)$-EA can optimize every monotone function in pseudolinear time if the mutation rate is $c/n$ for some $c<1$, but they need exponential time for some monotone functions for $c>2.2$. The second part of the statement was also known for the $(\mu+1)$-EA. In this paper we show that the first statement does not apply to the $(\mu+1)$-EA. More precisely, we prove that for every constant $c>0$ there is a constant integer $\mu_0$ such that the $(\mu+1)$-EA with mutation rate $c/n$ and population size $\mu_0\le\mu\le n$ needs superpolynomial time to optimize some monotone functions. Thus, increasing the population size by just a constant has devastating effects on the performance. This is in stark contrast to many other benchmark functions on which increasing the population size either increases the performance significantly, or affects performance mildly. The reason why larger populations are harmful lies in the fact that larger populations may temporarily decrease selective pressure on parts of the population. This allows unfavorable mutations to accumulate in single individuals and their descendants. If the population moves sufficiently fast through the search space, such unfavorable descendants can become ancestors of future generations, and the bad mutations are preserved. Remarkably, this effect only occurs if the population renews itself sufficiently fast, which can only happen far away from the optimum. This is counter-intuitive since usually optimization gets harder as we approach the optimum.