This paper addresses the analysis of covariance matrix self-adaptive Evolution Strategies (CMSA-ES) on a subclass of quadratic functions subject to additive Gaussian noise: the noisy ellipsoid model. To this end, it is demonstrated that the dynamical systems approach from the context of isotropic mutations can be transferred to ES that also control the covariance matrix. Theoretical findings such as the component-wise quadratic progress rate or the self-adaptation response function can thus be reused for the CMSA-ES analysis. By deriving the steady state quantities approached on the noisy ellipsoid model for constant population size, a detailed description of the asymptotic CMSA-ES behavior is obtained. By providing self-adaptive ES with a population control mechanism, despite noise disturbances, the algorithm is able to realize continuing progress towards the optimum. Regarding the population control CMSA-ES (pcCMSA-ES), the analytical findings allow to specify its asymptotic long-term behavior and to identify influencing parameters. The finally obtained convergence rate matches the theoretical lower bound of all comparison-based direct search algorithms.

本文讨论了在具有加性高斯噪声的二次函数子类上的协方差矩阵自适应演化策略（CMSA-ES）的分析：带噪椭圆模型。为此，已经证明，从各向同性突变的角度出发，动力学系统方法可以转移到还控制协方差矩阵的ES上。因此，理论发现（例如逐分量二次进度率或自适应响应函数）可以重新用于CMSA-ES分析。通过推导在嘈杂的椭球模型上获得的恒定数量的稳态量，可以得到渐近CMSA-ES行为的详细描述。通过为自适应ES提供人口控制机制，尽管受到噪声干扰，该算法能够实现朝着最优方向不断发展。关于人口控制CMSA-ES（pcCMSA-ES），分析结果可指定其渐近长期行为并确定影响参数。最终获得的收敛速度与所有基于比较的直接搜索算法的理论下限相匹配。