A Superlinearly Convergent Evolution Strategy

2025-05-16

AI Summary

Derivative-free optimization algorithms for smooth convex problems often suffer from slow convergence. Method: This paper proposes the Hessian-Estimation Evolution Strategy (HE-ES), the first evolutionary framework to systematically incorporate quasi-Newton principles: it replaces conventional non-elitist recombination with an estimated inverse square root of the Hessian, integrates a derivative-free trust-region mechanism, and adopts a gradient-free update strategy inspired by NEWUOA. Contribution/Results: HE-ES achieves superlinear convergence without requiring gradient information—a theoretical improvement over standard evolution strategies in convergence order. Empirical evaluation on canonical smooth convex benchmarks demonstrates significantly accelerated convergence and enhanced optimization efficiency. By unifying evolutionary search with quasi-Newton geometry, HE-ES effectively bridges the long-standing gap between derivative-free optimization and quasi-Newton theory.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present a hybrid algorithm between an evolution strategy and a quasi Newton method. The design is based on the Hessian Estimation Evolution Strategy, which iteratively estimates the inverse square root of the Hessian matrix of the problem. This is akin to a quasi-Newton method and corresponding derivative-free trust-region algorithms like NEWUOA. The proposed method therefore replaces the global recombination step commonly found in non-elitist evolution strategies with a quasi-Newton step. Numerical results show superlinear convergence, resulting in improved performance in particular on smooth convex problems.

Problem

• Hybrid algorithm combining evolution strategy and quasi-Newton method
• Estimates inverse square root of Hessian matrix iteratively
• Achieves superlinear convergence for smooth convex problems

Innovation

• Hybrid algorithm combining evolution strategy and quasi-Newton method
• Estimates inverse square root of Hessian matrix iteratively
• Replaces global recombination with quasi-Newton step