New study reveals which CMA-ES stopping criteria trigger first

A new paper by Ryoji Tanabe, accepted at PPSN 2026, provides the first quantitative analysis of the 11 stopping criteria in CMA-ES (Covariance Matrix Adaptation Evolution Strategy), a state-of-the-art black-box optimization algorithm. The study systematically evaluates each criterion's behavior on the noiseless BBOB function set, a standard benchmark for continuous optimization. Key findings: which criterion fires first varies significantly with the sample size λ and dimension n, but two criteria – tolflatfitness and tolfun – are consistently the first to trigger among the portfolio. This is critical because premature stopping can waste function evaluations, while late stopping may consume budget during stagnation.

The research also reveals that tolfunchist and the overall portfolio achieve the highest stopping accuracy in most test cases, meaning they halt the search closest to the optimal stopping point in terms of function evaluations. Importantly, the tolfun and tolfunchist criteria are frequently triggered before CMA-ES reaches complete stagnation, making them valuable for restart strategies that restart the algorithm with different parameters. The study provides practitioners with concrete guidance on which criteria to prioritize for efficient global optimization, potentially saving significant compute time in engineering and machine learning hyperparameter tuning.