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.

翻译：协方差矩阵自适应进化策略（CMA-ES）是解决连续黑盒优化问题最成功的方法之一。CMA-ES的一个实用优势在于其无需超参数调优即可使用。然而，超参数设置仍对性能有显著影响，尤其是在处理困难任务时，例如解决多模态或含噪声问题。本研究全面探讨了学习率对CMA-ES性能的影响，并通过考虑常微分方程论证了采用较小学习率的必要性。随后，本文讨论了理想学习率的设置方式。基于这些讨论，我们为CMA-ES开发了一种新颖的学习率自适应机制，该机制能保持恒定的信噪比。此外，我们通过数值实验研究了采用所提学习率自适应机制的CMA-ES的行为，并将其结果与固定学习率的CMA-ES及采用种群规模自适应的CMA-ES的结果进行了比较。结果表明，采用所提学习率自适应的CMA-ES在多模态和/或含噪声问题上表现良好，且无需极其耗时的学习率调优过程。