Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a highly effective optimization technique. A primary challenge when applying CMA-ES in high dimensionality is sampling from a multivariate normal distribution with an arbitrary covariance matrix, which involves its decomposition. The cubic complexity of this process is the main obstacle to applying CMA-ES in highdimensional spaces. We introduce a version of CMA-ES that uses no covariance matrix at all. In the proposed matrix-free CMA-ES, an archive stores the vectors of differences between individuals and the midpoint, normalized by the step size. New individuals are generated as the weighted combinations of the vectors from the archive. We prove that the probability distribution of individuals generated by the proposed method is identical to that of the standard CMA-ES. Experimental results show that reducing the archive size to store only a fixed number of the most recent populations is sufficient, without compromising optimization efficiency. The matrix-free and matrix-based CMA-ES achieve comparable results on the quadratic function when the step-size adaptation is turned off. When coupled with the step-size adaptation method, the matrix-free CMA-ES converges faster than the matrix-based, and usually yields the results of a comparable or superior quality, according to the results obtained for the CEC'2017 benchmark suite. Presented approach simplifies the algorithm, offers a novel perspective on covariance matrix adaptation, and serves as a stepping stone toward even more efficient methods.

翻译：协方差矩阵自适应进化策略（CMA-ES）是一种高效的优化技术。在高维空间中应用CMA-ES时面临的一个主要挑战是从具有任意协方差矩阵的多元正态分布中采样，这一过程涉及矩阵分解。该过程的立方复杂度是CMA-ES应用于高维空间的主要障碍。我们提出了一种完全不使用协方差矩阵的CMA-ES版本。在所提出的无矩阵CMA-ES中，一个存档库存储了个体与中点之间的差值向量（经步长归一化）。新个体通过存档库中向量的加权组合生成。我们证明了所提方法生成的个体概率分布与标准CMA-ES相同。实验结果表明，将存档库大小缩减至仅存储固定数量的最近种群即可满足要求，且不会影响优化效率。当关闭步长自适应机制时，无矩阵与基于矩阵的CMA-ES在二次函数上获得可比的结果。结合步长自适应方法后，根据CEC'2017基准测试集的结果，无矩阵CMA-ES比基于矩阵的版本收敛更快，并且通常能产生相当或更优的结果。所提出的方法简化了算法，为协方差矩阵自适应提供了新的视角，并为开发更高效的方法奠定了基础。