Global Convergence of the (1+1) Evolution Strategy

TL;DR

This paper proves that the (1+1) evolution strategy globally converges to a critical point regardless of initial conditions, providing a comprehensive analysis of its convergence behavior across various complex optimization landscapes.

Contribution

It introduces a novel framework for analyzing global convergence of elitist, rank-based evolutionary algorithms, extending the notion of critical points to measurable functions.

Findings

Establishes conditions for guaranteed global convergence with high probability.

Identifies scenarios where premature convergence can occur.

Demonstrates applicability across diverse non-convex and rugged problems.

Abstract

We establish global convergence of the (1+1) evolution strategy, i.e., convergence to a critical point independent of the initial state. More precisely, we show the existence of a critical limit point, using a suitable extension of the notion of a critical point to measurable functions. At its core, the analysis is based on a novel progress guarantee for elitist, rank-based evolutionary algorithms. By applying it to the (1+1) evolution strategy we are able to provide an accurate characterization of whether global convergence is guaranteed with full probability, or whether premature convergence is possible. We illustrate our results on a number of example applications ranging from smooth (non-convex) cases over different types of saddle points and ridge functions to discontinuous and extremely rugged problems.