Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation

TLDR

Rotation invariance, achieved by coordinate‑system‑independent mutation adaptation, is essential for ES applicability to complex fitness functions. The paper introduces a coordinate‑system‑independent formulation for adapting arbitrary zero‑mean normal mutation distributions. The formulation allows ES to correctly scale mutations, guarantees rotation invariance, enables interpretable and user‑modifiable mutation distributions, and provides better control over overall mutation variance.

Abstract

A new formulation for coordinate system independent adaptation of arbitrary normal mutation distributions with zero mean is presented. This enables the evolution strategy (ES) to adapt the correct scaling of a given problem and also ensures invariance with respect to any rotation of the fitness function (or the coordinate system). Especially rotation invariance, here resulting directly from the coordinate system independent adaptation of the mutation distribution, is an essential feature of the ES with regard to its general applicability to complex fitness functions. Compared to previous work on this subject, the introduced formulation facilitates an interpretation of the resulting mutation distribution, making sensible manipulation by the user possible (if desired). Furthermore it enables a more effective control of the overall mutation variance (expected step length).