A Revisit of Infinite Population Models for Evolutionary Algorithms on Continuous Optimization Problems

TL;DR

This paper critically examines the foundations of infinite population models in evolutionary algorithms for continuous optimization, identifies gaps in previous proofs, and introduces a rigorous analytical framework to establish convergence results for key operators.

Contribution

It corrects and extends prior theoretical work by developing a mathematically rigorous convergence framework and proving the validity of infinite population models for mutation and recombination operators.

Findings

Previous convergence proofs were incomplete or flawed.

Exchangeability assumptions do not yield valid transition equations.

The new framework proves convergence of models for mutation and recombination operators.

Abstract

Infinite population models are important tools for studying population dynamics of evolutionary algorithms. They describe how the distributions of populations change between consecutive generations. In general, infinite population models are derived from Markov chains by exploiting symmetries between individuals in the population and analyzing the limit as the population size goes to infinity. In this paper, we study the theoretical foundations of infinite population models of evolutionary algorithms on continuous optimization problems. First, we show that the convergence proofs in a widely cited study were in fact problematic and incomplete. We further show that the modeling assumption of exchangeability of individuals cannot yield the transition equation. Then, in order to analyze infinite population models, we build an analytical framework based on convergence in distribution of random elements which take values in the metric space of infinite sequences. The framework is concise and mathematically rigorous. It also provides an infrastructure for studying the convergence of the stacking of operators and of iterating the algorithm which previous studies failed to address. Finally, we use the framework to prove the convergence of infinite population models for the mutation operator and the $k$-ary recombination operator. We show that these operators can provide accurate predictions for real population dynamics as the population size goes to infinity, provided that the initial population is identically and independently distributed.