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Covariance Matrix Adaptation Evolution Strategy without a matrix

Jarosław Arabas, Adam Stelmaszczyk, Eryk Warchulski, Dariusz Jagodziński, Rafał Biedrzycki

TL;DR

The paper introduces MF-CMA-ES, a matrix-free variant of CMA-ES that eliminates covariance matrix decomposition by maintaining an archive of past difference vectors and generating new samples as weighted combinations of archived directions. A key theoretical result (Theorem 1) shows that the sampling distribution remains Gaussian with the same covariance as standard CMA-ES, preserving distributional properties. The approach uses a history window to bound memory; its performance is analyzed against matrix-based CMA-ES without step-size adaptation and with step-size adaptation using PPMF, showing competitive results on a quadratic function and the CEC'2017 benchmark, with dimension-dependent strengths. The work demonstrates that covariance adaptation can be achieved implicitly, simplifying the algorithm and providing a stepping stone toward more efficient, scalable evolutionary strategies. Practical impact lies in enabling CMA-ES-like optimization in higher dimensions with reduced computational overhead, potentially broadening applicability in science and engineering.

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.

Covariance Matrix Adaptation Evolution Strategy without a matrix

TL;DR

The paper introduces MF-CMA-ES, a matrix-free variant of CMA-ES that eliminates covariance matrix decomposition by maintaining an archive of past difference vectors and generating new samples as weighted combinations of archived directions. A key theoretical result (Theorem 1) shows that the sampling distribution remains Gaussian with the same covariance as standard CMA-ES, preserving distributional properties. The approach uses a history window to bound memory; its performance is analyzed against matrix-based CMA-ES without step-size adaptation and with step-size adaptation using PPMF, showing competitive results on a quadratic function and the CEC'2017 benchmark, with dimension-dependent strengths. The work demonstrates that covariance adaptation can be achieved implicitly, simplifying the algorithm and providing a stepping stone toward more efficient, scalable evolutionary strategies. Practical impact lies in enabling CMA-ES-like optimization in higher dimensions with reduced computational overhead, potentially broadening applicability in science and engineering.

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.
Paper Structure (12 sections, 1 theorem, 11 equations, 8 figures, 3 tables)

This paper contains 12 sections, 1 theorem, 11 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Vanilla CMA-ES
  3. Matrix-free CMA-ES (MF-CMA-ES)
  4. Sketch of proof:
  5. Comparison of matrix-based and matrix-free CMA-ES without step-size adaptation
  6. Analysis of the convergence curves
  7. Dynamics of the covariance matrix eigenvalues
  8. Matrix-free CMA-ES with step-size adaptation
  9. Cumulative Step Adaptation (CSA)
  10. Previous Point Midpoint Fitness (PPMF)
  11. Benchmarking matrix-free and matrix-based CMA-ES with step-size adaptation
  12. Concluding remarks

Key Result

Theorem 1

Consider the random vector: where ${\bf d} _j^{(t)}$ and ${\bf p} _c^{(t)}$ are defined in Fig. alg:cmaes, lines 8 and 16, respectively. Vectors ${\bf d} _j^{(t)}$ are ordered according to the fitness of their corresponding points ${\bf x} _j^{(t)}$. Symbols ${\mathcal{N}}(0,1)$ and ${\mathcal{N}}( {\bf 0} , {\bf I} )$ stand

Figures (8)

  • Figure 1: Outline of the matrix-based CMA-ES
  • Figure 2: Outline of the matrix-free CMA-ES (MF-CMA-ES)
  • Figure 3: Averaged convergence curves obtained on 30 independent runs of CMA-ES and MF-CMA-ES with different history window sizes on function \ref{['eq:quadratic']}.
  • Figure 4: Dynamics of eigenvalues for the vanilla CMA-ES (a) and for the MF-CMA-ES with the window size $h=60$ (b) and $h=10$ (c)
  • Figure 5: Outline of the Cumulative Step Adaptation (CSA)
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1