While most theoretical run time analyses of discrete randomized search heuristics provide bounds on the expected number of evaluations to find the global optimum, we consider the anytime performance of evolutionary and estimation-of-distribution algorithms. For this purpose, we analyze the fixed-target run time of various algorithms using BinVal as fitness function and bound the run time to optimize the most significant $k \in o(n)$ bits of a bit string with length $n$. We analyze the run times such that they hold not only for a fixed $k$, but simultaneously for all $k \in o(n)$. For the standard (1+1) EA with fixed mutation rate $1/n$, we show that the fixed-target run time for all $k \in o(n)$ is in $Θ(n \log k)$. Using an EDA instead, we get an expected number of evaluations of $Θ(k \log n)$ for the sig-cGA. Replacing in the standard (1+1) EA the fixed mutation rate with a self-adjusting rate, we show that the fixed-target run time for $k \in o(n)$ and a constant $\varepsilon >0$ arbitrarily close to zero is in $\mathcal{O}\left(k^{1+\varepsilon}\right)$ for this algorithm. In particular, this run time is independent of $n$, holds simultaneously for all $k \in o(n)$, and is close to the run time of $Θ(k \log k)$ for the (1+1) EA with the best fixed mutation rate if $k$ is known.

翻译：尽管大多数离散随机搜索启发式算法的理论运行时间分析都提供了找到全局最优解的期望评估次数界，但我们考虑了进化算法和分布估计算法的随时性能。为此，我们以BinVal为适应度函数分析了多种算法的固定目标运行时间，并界定了优化长度为n的位串中最显著的k ∈ o(n)位的运行时间。我们分析的运行时间不仅适用于固定的k，而且同时适用于所有k ∈ o(n)。对于固定突变率为1/n的标准(1+1)EA，我们证明所有k ∈ o(n)的固定目标运行时间为Θ(n log k)。若使用EDA，sig-cGA的期望评估次数为Θ(k log n)。在标准(1+1)EA中，将固定突变率替换为自适应突变率后，我们证明对于k ∈ o(n)且任意接近于零的常数ε > 0，该算法的固定目标运行时间为O(k^{1+ε})。特别地，该运行时间与n无关，同时适用于所有k ∈ o(n)，并且接近已知k时采用最佳固定突变率的(1+1)EA的运行时间Θ(k log k)。