Covariance Matrix Adaptation Evolution Strategy without a matrix

TL;DR

This paper introduces a matrix-free version of CMA-ES that eliminates the need for covariance matrix decomposition, simplifying the algorithm and maintaining or improving optimization performance in high-dimensional spaces.

Contribution

The authors propose a novel matrix-free CMA-ES that uses an archive of difference vectors to generate new solutions, avoiding covariance matrix decomposition.

Findings

Matrix-free CMA-ES achieves comparable results to standard CMA-ES on quadratic functions.

With step-size adaptation, matrix-free CMA-ES converges faster and often yields better results.

Reducing archive size to recent populations does not impair optimization efficiency.

Abstract

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.