Estimation of Distribution Algorithms (EDAs) are one branch of Evolutionary Algorithms (EAs) in the broad sense that they evolve a probabilistic model instead of a population. Many existing algorithms fall into this category. Analogous to genetic drift in EAs, EDAs also encounter the phenomenon that updates of the probabilistic model not justified by the fitness move the sampling frequencies to the boundary values. This can result in a considerable performance loss. This paper proves the first sharp estimates of the boundary hitting time of the sampling frequency of a neutral bit for several univariate EDAs. For the UMDA that selects $\mu$ best individuals from $\lambda$ offspring each generation, we prove that the expected first iteration when the frequency of the neutral bit leaves the middle range $[\tfrac 14, \tfrac 34]$ and the expected first time it is absorbed in 0 or 1 are both $\Theta(\mu)$. The corresponding hitting times are $\Theta(K^2)$ for the cGA with hypothetical population size $K$. This paper further proves that for PBIL with parameters $\mu$, $\lambda$, and $\rho$, in an expected number of $\Theta(\mu/\rho^2)$ iterations the sampling frequency of a neutral bit leaves the interval $[\Theta(\rho/\mu),1-\Theta(\rho/\mu)]$ and then always the same value is sampled for this bit, that is, the frequency approaches the corresponding boundary value with maximum speed. For the lower bounds implicit in these statements, we also show exponential tail bounds. If a bit is not neutral, but neutral or has a preference for ones, then the lower bounds on the times to reach a low frequency value still hold. An analogous statement holds for bits that are neutral or prefer the value zero.

中文翻译：

---

分布算法估计中遗传漂移的锐界

从广义上讲，分布算法估计 (EDA) 是进化算法 (EA) 的一个分支，因为它们进化的是概率模型而不是总体。许多现有算法都属于这一类。类似于 EA 中的遗传漂变，EDA 也遇到了这样的现象，即概率模型的更新无法通过适应度证明将采样频率移动到边界值。这会导致相当大的性能损失。本文证明了对几个单变量 EDA 的中性位采样频率的边界命中时间的首次精确估计。对于每一代从 $\lambda$ 后代中选择 $\mu$ 最佳个体的 UMDA，我们证明了当中性位的频率离开中间范围 $[\tfrac 14, \tfrac 34]$ 和预期的第一次吸收在 0 或 1 中都是 $\Theta(\mu)$。对于假设种群大小为 $K$ 的 cGA，相应的命中时间为 $\Theta(K^2)$。本文进一步证明，对于参数为 $\mu$、$\lambda$ 和 $\rho$ 的 PBIL，在预期的 $\Theta(\mu/\rho^2)$ 迭代次数中，中性点的采样频率位离开区间 $[\Theta(\rho/\mu),1-\Theta(\rho/\mu)]$ 然后总是对这个位采样相同的值，即频率接近对应的边界最大速度时的值。对于这些语句中隐含的下界，我们还显示了指数尾界。如果位不是中立的，而是中立的或偏爱一个位，则达到低频值的时间的下限仍然成立。