On the Theoretical Capacity of Evolution Strategies to Statistically Learn the Landscape Hessian

TL;DR

This paper analyzes how Evolution Strategies can statistically learn the local landscape Hessian by examining the covariance matrix of selected solutions, revealing eigenvector alignment with the Hessian.

Contribution

It provides a theoretical analysis showing the covariance matrix from ESs shares eigenvectors with the Hessian and offers an analytic approximation for large populations.

Findings

Covariance matrix shares eigenvectors with the Hessian

Analytic approximation for large population size

Numerical validation of theoretical results

Abstract

We study the theoretical capacity to statistically learn local landscape information by Evolution Strategies (ESs). Specifically, we investigate the covariance matrix when constructed by ESs operating with the selection operator alone. We model continuous generation of candidate solutions about quadratic basins of attraction, with deterministic selection of the decision vectors that minimize the objective function values. Our goal is to rigorously show that accumulation of winning individuals carries the potential to reveal valuable information about the search landscape, e.g., as already practically utilized by derandomized ES variants. We first show that the statistically-constructed covariance matrix over such winning decision vectors shares the same eigenvectors with the Hessian matrix about the optimum. We then provide an analytic approximation of this covariance matrix for a non-elitist multi-child $(1,\lambda)$-strategy, which holds for a large population size $\lambda$. Finally, we also numerically corroborate our results.