  The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has been the
most successful Evolution Strategy at exploiting covariance information; it
uses a form of Principle Component Analysis which, under certain conditions, is
suggested to converge to the correct covariance matrix, formulated as the
inverse of the mathematically well-defined Hessian matrix. However, in
practice, there exist conditions where CMA-ES converges to the global optimum
(accomplishing its primary goal) while it does not learn the true covariance
matrix (missing an auxiliary objective), likely due to step-size deficiency.
These circumstances can involve high-dimensional landscapes with large
condition numbers. This paper introduces a novel technique entitled Forced
Optimal Covariance Adaptive Learning (FOCAL), with the explicit goal of
determining the Hessian at the global basin of attraction. It begins by
introducing theoretical foundations to the inverse relationship between the
learned covariance and the Hessian matrices. FOCAL is then introduced and
demonstrated to retrieve the Hessian matrix with high fidelity on both model
landscapes and experimental Quantum Control systems, which are observed to
possess a non-separable, non-quadratic search landscape. The recovered Hessian
forms are corroborated by physical knowledge of the systems. This study
constitutes an example for Natural Computing successfully serving other
branches of natural sciences, and introducing at the same time a powerful
generic method for any high-dimensional continuous search seeking landscape
information.
