Projection-based restricted covariance matrix adaptation for high dimension

We propose a novel variant of the covariance matrix adaptation evolution strategy (CMA-ES) using a covariance matrix parameterized with a smaller number of parameters. The motivation of a restricted covariance matrix is twofold. First, it requires less internal time and space complexity that is desired when optimizing a function on a high dimensional search space. Second, it requires less function evaluations to adapt the covariance matrix if the restricted covariance matrix is rich enough to express the variable dependencies of the problem. In this paper we derive a computationally efficient way to update the restricted covariance matrix where the model richness of the covariance matrix is controlled by an integer and the internal complexity per function evaluation is linear in this integer times the dimension, compared to quadratic in the dimension in the CMA-ES. We prove that the proposed algorithm is equivalent to the sep-CMA-ES if the covariance matrix is restricted to the diagonal matrix, it is equivalent to the original CMA-ES if the matrix is not restricted. Experimental results reveal the class of efficiently solvable functions depending on the model richness of the covariance matrix and the speedup over the CMA-ES.