Level-based Analysis of Genetic Algorithms and other Search Processes

TL;DR

This paper introduces the level-based theorem, a new analytical technique for population-based evolutionary algorithms, providing upper bounds on their expected optimization time across various problems.

Contribution

The paper presents the level-based theorem, a novel method for analyzing population-based EAs, applicable to complex algorithms like GAs and EDA, with nearly optimal bounds.

Findings

The theorem applies to non-elitist processes with independent offspring sampling.

It simplifies analysis for GAs and EAs on multiple benchmark problems.

The bounds provided are nearly tight and often straightforward to verify.

Abstract

Understanding how the time-complexity of evolutionary algorithms (EAs) depend on their parameter settings and characteristics of fitness landscapes is a fundamental problem in evolutionary computation. Most rigorous results were derived using a handful of key analytic techniques, including drift analysis. However, since few of these techniques apply effortlessly to population-based EAs, most time-complexity results concern simplified EAs, such as the (1+1) EA. This paper describes the level-based theorem, a new technique tailored to population-based processes. It applies to any non-elitist process where offspring are sampled independently from a distribution depending only on the current population. Given conditions on this distribution, our technique provides upper bounds on the expected time until the process reaches a target state. We demonstrate the technique on several pseudo-Boolean functions, the sorting problem, and approximation of optimal solutions in combinatorial optimisation. The conditions of the theorem are often straightforward to verify, even for Genetic Algorithms and Estimation of Distribution Algorithms which were considered highly non-trivial to analyse. Finally, we prove that the theorem is nearly optimal for the processes considered. Given the information the theorem requires about the process, a much tighter bound cannot be proved.