  The covariance matrix adaptation evolution strategy (CMA-ES) is one of the
most successful methods for solving continuous black-box optimization problems.
A practically useful aspect of the CMA-ES is that it can be used without
hyperparameter tuning. However, the hyperparameter settings still have a
considerable impact on performance, especially for difficult tasks, such as
solving multimodal or noisy problems. This study comprehensively explores the
impact of learning rate on the CMA-ES performance and demonstrates the
necessity of a small learning rate by considering ordinary differential
equations. Thereafter, it discusses the setting of an ideal learning rate.
Based on these discussions, we develop a novel learning rate adaptation
mechanism for the CMA-ES that maintains a constant signal-to-noise ratio.
Additionally, we investigate the behavior of the CMA-ES with the proposed
learning rate adaptation mechanism through numerical experiments, and compare
the results with those obtained for the CMA-ES with a fixed learning rate and
with population size adaptation. The results show that the CMA-ES with the
proposed learning rate adaptation works well for multimodal and/or noisy
problems without extremely expensive learning rate tuning.
