The mutation strength adaptation properties of a multi-recombinative $(\mu/\mu_I, \lambda)$-ES are studied for isotropic mutations. To this end, standard implementations of cumulative step-size adaptation (CSA) and mutative self-adaptation ($\sigma$SA) are investigated experimentally and theoretically by assuming large population sizes ($\mu$) in relation to the search space dimensionality ($N$). The adaptation is characterized in terms of the scale-invariant mutation strength on the sphere in relation to its maximum achievable value for positive progress. %The results show how the different $\sigma$-adaptation variants behave as $\mu$ and $N$ are varied. Standard CSA-variants show notably different adaptation properties and progress rates on the sphere, becoming slower or faster as $\mu$ or $N$ are varied. This is shown by investigating common choices for the cumulation and damping parameters. Standard $\sigma$SA-variants (with default learning parameter settings) can achieve faster adaptation and larger progress rates compared to the CSA. However, it is shown how self-adaptation affects the progress rate levels negatively. Furthermore, differences regarding the adaptation and stability of $\sigma$SA with log-normal and normal mutation sampling are elaborated.

翻译：本文研究了各向同性变异条件下多重组$(\mu/\mu_I, \lambda)$进化策略的变异强度自适应特性。通过假设较大种群规模$(\mu)$相对于搜索空间维度$(N)$的情况，对累积步长自适应（CSA）与变异自适应（$\sigma$SA）的标准实现进行了实验与理论分析。自适应特性通过球面上尺度不变的变异强度与其可实现最大正向进展值的关系来表征。标准CSA变体在球面上表现出显著不同的自适应特性与进展速率，其速度随$\mu$或$N$的变化而增减，这通过研究累积参数与阻尼参数的常见选择得以验证。标准$\sigma$SA变体（采用默认学习参数设置）相比CSA可实现更快的自适应与更高的进展速率，但研究表明自适应性会对进展速率水平产生负面影响。此外，本文详细阐述了采用对数正态与正态变异采样的$\sigma$SA在自适应性与稳定性方面的差异。