Convergence Analysis of the Hessian Estimation Evolution Strategy

TL;DR

This paper provides a formal convergence analysis of Hessian Estimation Evolution Strategies, proving stability and linear convergence guarantees for a minimal variant on convex quadratic problems, enhancing understanding of their theoretical properties.

Contribution

It offers the first formal proof of stability and linear convergence for the (1+4)-HE-ES on convex quadratic functions, establishing theoretical foundations for this class of algorithms.

Findings

Proves stability of covariance matrix updates in HE-ES.

Demonstrates linear convergence rate independent of problem instance.

Validates practical efficiency through theoretical guarantees.

Abstract

The class of algorithms called Hessian Estimation Evolution Strategies (HE-ESs) update the covariance matrix of their sampling distribution by directly estimating the curvature of the objective function. The approach is practically efficient, as attested by respectable performance on the BBOB testbed, even on rather irregular functions. In this paper we formally prove two strong guarantees for the (1+4)-HE-ES, a minimal elitist member of the family: stability of the covariance matrix update, and as a consequence, linear convergence on all convex quadratic problems at a rate that is independent of the problem instance.