Tail Bounds on the Runtime of Categorical Compact Genetic Algorithm

TL;DR

This paper extends the theoretical analysis of the compact genetic algorithm to categorical domains, deriving tail bounds on runtime for two linear functions, and explores how parameters affect efficiency.

Contribution

It introduces the categorical compact genetic algorithm (ccGA), extending binary cGA analysis to categorical domains with new runtime bounds and tail probability insights.

Findings

Runtime on COM is O(√D ln(DK)/η) with high probability.

Runtime on KVal is Θ(D ln K/η).

Analysis generalizes binary cGA results to categorical domains.

Abstract

The majority of theoretical analyses of evolutionary algorithms in the discrete domain focus on binary optimization algorithms, even though black-box optimization on the categorical domain has a lot of practical applications. In this paper, we consider a probabilistic model-based algorithm using the family of categorical distributions as its underlying distribution and set the sample size as two. We term this specific algorithm the categorical compact genetic algorithm (ccGA). The ccGA can be considered as an extension of the compact genetic algorithm (cGA), which is an efficient binary optimization algorithm. We theoretically analyze the dependency of the number of possible categories $K$, the number of dimensions $D$, and the learning rate $\eta$ on the runtime. We investigate the tail bound of the runtime on two typical linear functions on the categorical domain: categorical OneMax (COM) and KVal. We derive that the runtimes on COM and KVal are $O(\sqrt{D} \ln (DK) / \eta)$ and $\Theta(D \ln K/ \eta)$ with high probability, respectively. Our analysis is a generalization for that of the cGA on the binary domain.