When Non-Elitism Meets Time-Linkage Problems

TL;DR

This paper demonstrates that non-elitist evolutionary algorithms outperform elitist ones on time-linkage problems by escaping local optima, with theoretical proofs and empirical validation showing improved runtime and success probability.

Contribution

It provides the first theoretical analysis showing non-elitism's advantage in time-linkage problems, including runtime bounds and empirical evidence.

Findings

Non-elitist (1,λ)EA can reach the global optimum with high probability.

Elitist (1+λ)EA tends to get stuck in local optima.

The compact genetic algorithm efficiently escapes local optima with expected runtime O(n^3 log n).

Abstract

Many real-world applications have the time-linkage property, and the only theoretical analysis is recently given by Zheng, et al. (TEVC 2021) on their proposed time-linkage OneMax problem, OneMax$_{(0,1^n)}$. However, only two elitist algorithms (1+1)EA and ($\mu$+1)EA are analyzed, and it is unknown whether the non-elitism mechanism could help to escape the local optima existed in OneMax$_{(0,1^n)}$. In general, there are few theoretical results on the benefits of the non-elitism in evolutionary algorithms. In this work, we analyze on the influence of the non-elitism via comparing the performance of the elitist (1+$\lambda$)EA and its non-elitist counterpart (1,$\lambda$)EA. We prove that with probability $1-o(1)$ (1+$\lambda$)EA will get stuck in the local optima and cannot find the global optimum, but with probability $1$, (1,$\lambda$)EA can reach the global optimum and its expected runtime is $O(n^{3+c}\log n)$ with $\lambda=c \log_{\frac{e}{e-1}} n$ for the constant $c\ge 1$. Noting that a smaller offspring size is helpful for escaping from the local optima, we further resort to the compact genetic algorithm where only two individuals are sampled to update the probabilistic model, and prove its expected runtime of $O(n^3\log n)$. Our computational experiments also verify the efficiency of the two non-elitist algorithms.