Mutation Strength Adaptation of the $(\mu/\mu_I, \lambda)$-ES for Large Population Sizes on the Sphere Function

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.