# On the Covariance-Hessian Relation in Evolution Strategies

arXiv: https://arxiv.org/abs/1806.03674 · HTML: https://mlsanity.com/paper/1806.03674/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2018.

## Built on by (most cited first, 3 of 3)

- [Synergistic eigenanalysis of covariance and Hessian matrices for enhanced binary classification](https://mlsanity.com/paper/2402.09281/) (2024)
- [Addressing Unboundedness in Quadratically-Constrained Mixed-Integer Problems](https://mlsanity.com/paper/2405.05978/) (2024)
- [Correlated Mutations for Integer Programming](https://mlsanity.com/paper/2506.22526/) (2025)

## Abstract (arXiv)

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.