Level-based Analysis of Genetic Algorithms and other Search Processes
This addresses the fundamental challenge in evolutionary computation of rigorously analyzing complex algorithms like Genetic Algorithms, which were previously hard to study, making it significant for researchers in optimization and algorithm theory.
The paper tackles the problem of analyzing time-complexity for population-based evolutionary algorithms by introducing the level-based theorem, a new analytic technique that provides upper bounds on expected time to reach target states, demonstrating it on functions like pseudo-Boolean problems and showing it is nearly optimal.
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.