Recently, the runtime analysis of multi-valued estimation-of-distribution algorithms in the framework of Ben Jedidia et al. (TCS 2024) has made significant advancements. However, almost all existing analyses are limited to multi-valued objective functions that in each dimension only distinguish between two types, also called categories, of values and hence can be treated with similar methods as pseudo-Boolean problems. Only recently, Adak and Witt (GECCO 2025) have presented a first runtime analysis of a multi-valued compact genetic algorithm (cGA) on the multi-valued OneMax function G-OneMax$\colon \{0,\dots,r-1\}^n \to \mathbf{N}$ defined by G-OneMax$(x_1,\dots,x_n)=\sum_{i=1}^n {x}_i$ and truly depending on all $r$ categories. We improve their runtime result from $\textrm{O}\bigl(n r^3 \log^2( n)\log (r)\bigr)$ to $\textrm{O}\bigl(n r \log^3(n)\log^3(r)\bigr)$, both for an optimal choice of the update strength $K$. Our result matches, up to polylogarithmic factors, the existing bound for the simpler $r$-valued OneMax function depending essentially only on two values and analyzed in several previous works. To show the new bound, we use improved drift theorems for processes with high self-loop probabilities and specifically derived concentration inequalities to analyze how probability mass in the multi-valued cGA moves into successively smaller and smaller intervals of the $r$-valued frequency matrix.

翻译：摘要：最近，在Ben Jedidia等人（TCS 2024）的框架下，多值分布估计算法的运行时分析取得了显著进展。然而，几乎所有现有分析都局限于多值目标函数，这些函数在每个维度上只区分两类（也称为类别）值，因此可以用与伪布尔问题类似的方法处理。直到最近，Adak和Witt（GECCO 2025）才首次对多值紧致遗传算法（cGA）在由G-OneMax$(x_1,\dots,x_n)=\sum_{i=1}^n {x}_i$定义且真正依赖于所有$r$个类别的多值OneMax函数G-OneMax$\colon \{0,\dots,r-1\}^n \to \mathbf{N}$上进行了运行时分析。我们将其运行时结果从$\textrm{O}\bigl(n r^3 \log^2( n)\log (r)\bigr)$改进为$\textrm{O}\bigl(n r \log^3(n)\log^3(r)\bigr)$，两者均基于更新强度$K$的最优选择。我们的结果在多项式对数因子范围内与先前多项工作中分析的、本质上仅依赖于两个值的更简单的$r$值OneMax函数的现有界相匹配。为证明新界，我们使用了针对高自环概率过程的改进漂移定理，并特别推导了浓度不等式，以分析多值cGA中的概率质量如何逐步移动到$r$值频率矩阵的越来越小的区间内。