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）是解决连续黑箱优化问题最成功的方法之一。CMA-ES 的一个实用优势在于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，尤其对于多模态或含噪问题等困难任务。本研究全面探讨了学习率对 CMA-ES 性能的影响，并借助常微分方程论证了小学习率的必要性，进而讨论了理想学习率的设定。基于这些讨论，我们提出了一种维持恒定信噪比的 CMA-ES 学习率自适应机制。此外，我们通过数值实验研究了采用该学习率自适应机制的 CMA-ES 行为，并将其与固定学习率和种群规模自适应的 CMA-ES 结果进行了比较。结果表明，采用所提出学习率自适应的 CMA-ES 能在无需昂贵学习率调优的情况下，有效处理多模态和/或含噪问题。