Runtime Analysis of Competitive co-Evolutionary Algorithms for Maximin Optimisation of a Bilinear Function

TL;DR

This paper develops a mathematical framework for analyzing the runtime of competitive co-evolutionary algorithms, demonstrating polynomial-time solutions for bilinear maximin problems and identifying conditions leading to exponential runtime.

Contribution

It introduces a formal framework for analyzing co-evolutionary algorithms and proves polynomial runtime for a specific algorithm on bilinear maximin problems, advancing theoretical understanding.

Findings

pdcoea finds solutions in expected polynomial time for bilinear maximin problems.

Under certain settings, pdcoea requires exponential time with high probability.

The framework helps predict when co-evolutionary algorithms are efficient or encounter difficulties.

Abstract

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called \pdcoea, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where \pdcoea needs exponential time with overwhelmingly high probability to obtain a solution.