Quantitative Performance Analysis of Stopping Criteria for CMA-ES

2026-06-08

AI Summary

This study addresses the lack of systematic evaluation of stopping criteria in CMA-ES, which complicates determining when to terminate optimization reliably. For the first time, it presents a comprehensive quantitative analysis of eleven stopping criteria on the noiseless BBOB test suite. Using the optimal stopping point as a reference, the evaluation integrates function evaluation counts, alignment of stopping points, and statistical analysis of triggering sequences across multiple criteria. The results reveal that *tolflatfitness* and *tolfun* most frequently trigger first, whereas *tolfunhist* and certain criterion combinations achieve the highest stopping accuracy across most settings, often halting the search effectively before the algorithm fully stagnates.

Abstract

Covariance matrix adaptation evolution strategy (CMA-ES) is a state-of-the-art black-box optimization algorithm. In general, CMA-ES uses a portfolio of multiple stopping criteria to automatically determine when to stop the search. This mechanism aims to avoid unnecessary consumption of the function evaluation budget during stagnation. Stopping criteria play an important role in CMA-ES, particularly when restart strategies are employed. However, the effectiveness of stopping criteria in CMA-ES remains poorly understood. To address this issue, this paper investigates how the 11 stopping criteria in CMA-ES behave on the noiseless BBOB function set. The performance of the stopping criteria is quantitatively evaluated based on the optimal stopping point in terms of the number of function evaluations in a single run of CMA-ES. Our results show that, although which stopping criterion is triggered first depends significantly on the sample size $λ$ and the dimension $n$, \texttt{tolflatfitness} and \texttt{tolfun} are frequently the first criteria to be triggered among the portfolio of 11 stopping criteria. We also demonstrate that \texttt{tolfunhist} and the portfolio achieve the highest stopping accuracy in most cases. In addition, our results show that the \texttt{tolfun} and \texttt{tolfunhist} criteria are frequently triggered before CMA-ES reaches complete stagnation.

Problem

• CMA-ES
• stopping criteria
• black-box optimization
• performance analysis
• function evaluations

Innovation

• CMA-ES
• stopping criteria
• quantitative evaluation
• BBOB benchmark
• black-box optimization