On the Use of the "Mutate Large, But Inherit Small Principle" in Global and Noisy Optimization

Rescaled mutations have been proven to improve the convergence behavior of Evolution Strategies (ES) in noisy or highly multimodal fitness landscapes. The basic principle is "mutate large, but inherit small", meaning that the step toward the new parental centroid is reduced by a factor 1/κ. This paper investigates the impact of κ on the progress rate of a (μ/μI, λ)-ES in a highly multimodal and noisy scenario. For the second-order progress rate, the gain part scales inverse linearly and the loss part scales inverse quadratically with κ. This relationship is applied to the Rastrigin function in the limit of large dimensionality. Furthermore, a κ-dependent first-order progress rate for the noisy sphere is derived, yielding the aforementioned scaling behavior. The findings suggest that as κ increases, the interval of the normalized noise strength where positive progress is possible gets larger. Consequently, the search space can be explored with a larger mutation strength, leading to larger success rates in the Rastrigin landscape and to a smaller steady-state distance in the noisy sphere case.