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.

翻译：离散域中进化算法的大多数理论分析集中于二进制优化算法，尽管分类域上的黑盒优化具有大量实际应用。本文考虑一种基于概率模型的算法，该算法使用分类分布族作为其基础分布，并将样本大小设为二。我们将这一特定算法称为分类紧凑遗传算法（ccGA）。ccGA可被视为紧凑遗传算法（cGA）的扩展，后者是一种高效的二进制优化算法。我们从理论上分析了可能类别数$K$、维度数$D$和学习率$\eta$对运行时间的依赖关系。我们研究了算法在分类域上两个典型线性函数：分类OneMax（COM）和KVal的运行时间尾部界。我们推导出，以高概率计，在COM和KVal上的运行时间分别为$O(\sqrt{D} \ln (DK) / \eta)$和$\Theta(D \ln K/ \eta)$。我们的分析是对二进制域上cGA分析的一种推广。