The covariance matrix adaptation evolution strategy (CMA-ES) is a stochastic search algorithm using a multivariate normal distribution for continuous black-box optimization. In addition to strong empirical results, part of the CMA-ES can be described by a stochastic natural gradient method and can be derived from information geometric optimization (IGO) framework. However, there are some components of the CMA-ES, such as the rank-one update, for which the theoretical understanding is limited. While the rank-one update makes the covariance matrix to increase the likelihood of generating a solution in the direction of the evolution path, this idea has been difficult to formulate and interpret as a natural gradient method unlike the rank-$\mu$ update. In this work, we provide a new interpretation of the rank-one update in the CMA-ES from the perspective of the natural gradient with prior distribution. First, we propose maximum a posteriori IGO (MAP-IGO), which is the IGO framework extended to incorporate a prior distribution. Then, we derive the rank-one update from the MAP-IGO by setting the prior distribution based on the idea that the promising mean vector should exist in the direction of the evolution path. Moreover, the newly derived rank-one update is extensible, where an additional term appears in the update for the mean vector. We empirically investigate the properties of the additional term using various benchmark functions.

翻译：协方差矩阵自适应进化策略（CMA-ES）是一种采用多元正态分布的随机搜索算法，用于连续黑盒优化问题。除了优异的实证效果外，CMA-ES的部分机制可通过随机自然梯度方法进行描述，并能从信息几何优化（IGO）框架推导得出。然而，CMA-ES的某些组件（如秩一更新）在理论理解上仍存在局限。虽然秩一更新能使协方差矩阵在进化路径方向上增加生成解的可能性，但不同于秩-$\mu$更新，这一思想始终难以用自然梯度方法进行形式化描述与解释。本研究从带先验分布的自然梯度视角，为CMA-ES中的秩一更新提供了新的理论阐释。首先，我们提出最大后验IGO（MAP-IGO）——这是将先验分布纳入考量的扩展版IGO框架。随后，基于"有潜力的均值向量应存在于进化路径方向"的思想设定先验分布，从MAP-IGO推导出秩一更新机制。值得注意的是，新推导的秩一更新具有可扩展性，其均值向量更新式中将出现附加项。我们通过多种基准函数对附加项的特性进行了实证研究。