On the Covariance-Hessian Relation in Evolution Strategies
This provides theoretical insight into why Evolution Strategies work, addressing a foundational problem in optimization for researchers, but it is incremental as it builds on prior work.
The paper proves that in Evolution Strategies with isotropic Gaussian mutations on quadratic objectives, the covariance matrix of selected individuals becomes proportional to the inverse Hessian as population size increases, generalizing prior results and confirming statistical learning as inherent to these methods. Numerical validation supports the findings, including extensions to (μ,λ)-selection.
We consider Evolution Strategies operating only with isotropic Gaussian mutations on positive quadratic objective functions, and investigate the covariance matrix when constructed out of selected individuals by truncation. We prove that the covariance matrix over $(1,λ)$-selected decision vectors becomes proportional to the inverse of the landscape Hessian as the population-size $λ$ increases. This generalizes a previous result that proved an equivalent phenomenon when sampling was assumed to take place in the vicinity of the optimum. It further confirms the classical hypothesis that statistical learning of the landscape is an inherent characteristic of standard Evolution Strategies, and that this distinguishing capability stems only from the usage of isotropic Gaussian mutations and rank-based selection. We provide broad numerical validation for the proven results, and present empirical evidence for its generalization to $(μ,λ)$-selection.