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

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)