On the Covariance-Hessian Relation in Evolution Strategies

TL;DR

This paper proves that in isotropic Gaussian mutation-based Evolution Strategies, the covariance matrix of selected individuals approximates the inverse Hessian of the objective function as population size grows, confirming the landscape learning capability.

Contribution

It generalizes previous results by showing the covariance-Hessian relation holds for larger populations and broader selection schemes, highlighting inherent landscape learning in Evolution Strategies.

Findings

Covariance matrix becomes proportional to inverse Hessian with increasing population size

The relation holds for $(1,\lambda)$-selection and empirically extends to $(,)$-selection

Numerical validation confirms the theoretical results

Abstract

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 $(1,\lambda)$-selected decision vectors becomes proportional to the inverse of the landscape Hessian as the population-size $\lambda$ 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 $(\mu,\lambda)$-selection.