OneMax is not the Easiest Function for Fitness Improvements

TL;DR

This paper investigates the impact of the $(1:s+1)$-rule on the self-adjusting SA-$(1,\lambda)$-EA in dynamic monotone landscapes. By contrasting OneMax with Dynamic BinVal, the authors show that OneMax is not the easiest function with respect to improving steps, contradicting prior intuition. The main result proves the existence of constants $\varepsilon>0$ and $s>0$ such that, starting at distance $\varepsilon n$ from the optimum, SA-$(1,\lambda)$-EA optimizes OneMax in $O(n)$ generations whp, while failing to achieve polynomial-time optimization on DBv, highlighting a landscape-dependent limit of parameter-control schemes. The analysis combines drift arguments, equilibrium-population-size constructions, and concentration results, and is complemented by simulations that support the separation and illustrate the dynamics of $\lambda$ and improvement probabilities. The work clarifies two distinct notions of ‘easiness’ for benchmarks and suggests broader implications for designing robust PCMs in dynamic optimization settings.

Abstract

We study the $(1:s+1)$ success rule for controlling the population size of the $(1,λ)$-EA. It was shown by Hevia Fajardo and Sudholt that this parameter control mechanism can run into problems for large $s$ if the fitness landscape is too easy. They conjectured that this problem is worst for the OneMax benchmark, since in some well-established sense OneMax is known to be the easiest fitness landscape. In this paper we disprove this conjecture and show that OneMax is not the easiest fitness landscape with respect to finding improving steps. As a consequence, we show that there exists $s$ and $\varepsilon$ such that the self-adjusting $(1,λ)$-EA with $(1:s+1)$-rule optimizes OneMax efficiently when started with $\varepsilon n$ zero-bits, but does not find the optimum in polynomial time on Dynamic BinVal. Hence, we show that there are landscapes where the problem of the $(1:s+1)$-rule for controlling the population size of the $(1, λ)$-EA is more severe than for OneMax.

Figures (5)

• Figure 1: The probability of fitness improvement with a single offspring for search points with different OneMax values. Each data point in the figure is estimated by first sampling 1000 search points of the corresponding OneMax value, then sampling 100 offspring for each of the sampled search points, and calculating the frequency of an offspring fitter than its parent. The parameters of HotTopiclengler2019general are $L=100$, $\alpha=0.25$, $\beta=0.05$, and $\varepsilon=0.05$.
• Figure 2: Smoothed average of $\lambda$, smoothed average drift, average number of generations, and average number of evaluations of the self-adjusting $(1 , \lambda)$-EA with $s=3$, $F=1.5$, and $c=1$ in 100 runs at each OneMax value when optimizing monotone functions with $n=1000$. The parameters of HotTopiclengler2019general are the same as the ones in Figure \ref{['fig:improvement']}. The average of $\lambda$ is shown in log scale. The average of $\lambda$ and the average drift are smoothed using a moving average over a window of size 15. The number of generations/evaluations is normalized such that its sum over all OneMax values is 1.
• Figure 3: The same experiments as shown in Figure \ref{['fig:average3']} using $s=2$ instead of $s=3$.
• Figure 4: Average number of generations, and average number of evaluations of the self-adjusting $(1 , \lambda)$-EA with $s=3$, $F=1.5$, and $c=1$ in 100 runs plotted against OneMax values when optimizing monotone functions with $n=1000$. The parameters of HotTopiclengler2019general are the same as the ones in Figure \ref{['fig:improvement']}. The shaded areas indicate one standard deviation above and below the respective means.
• Figure 5: The same experiments as shown in Figure \ref{['fig:deviation3']} using $s=2$ instead of $s=3$.

Theorems & Definitions (24)

• theorem thmcountertheorem: Tail Bound for Additive Drift --- kotzing2016concentration, Theorem 2
• theorem thmcountertheorem: Negative Drift Theorem --- oliveto2015improved, Theorem 2
• lemma thmcounterlemma
• definition thmcounterdefinition
• definition thmcounterdefinition: Improvement Probability, Equilibrium Population Size
• remark thmcounterremark
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof