In the first and so far only mathematical runtime analysis of an estimation-of-distribution algorithm (EDA) on a multimodal problem, Hasenohrl and Sutton (GECCO 2018) showed for any $k = o(n)$ that the compact genetic algorithm (cGA) with any hypothetical population size $\mu = \Omega(ne^{4k} + n^{3.5+\varepsilon})$ with high probability finds the optimum of the $n$-dimensional jump function with jump size $k$ in time $O(\mu n^{1.5} \log n)$. We significantly improve this result for small jump sizes $k \le \frac 1 {20} \ln n -1$. In this case, already for $\mu = \Omega(\sqrt n \log n) \cap \text{poly}(n)$ the runtime of the cGA with high probability is only $O(\mu \sqrt n)$. For the smallest admissible values of $\mu$, our result gives a runtime of $O(n \log n)$, whereas the previous one only shows $O(n^{5+\varepsilon})$. Since it is known that the cGA with high probability needs at least $\Omega(\mu \sqrt n)$ iterations to optimize the unimodal OneMx function, our result shows that the cGA in contrast to most classic evolutionary algorithms here is able to cross moderate-sized valleys of low fitness at no extra cost. For large $k$, we show that the exponential (in $k$) runtime guarantee of Hasenohrl and Sutton is tight and cannot be improved, also not by using a smaller hypothetical population size. We prove that any choice of the hypothetical population size leads to a runtime that, with high probability, is at least exponential in the jump size $k$. This result might be the first non-trivial exponential lower bound for EDAs that holds for arbitrary parameter settings.

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跳跃函数上的紧凑遗传算法的运行时

Hasenohrl 和 Sutton (GECCO 2018) 在第一次也是迄今为止唯一一次针对多模态问题的分布估计算法 (EDA) 的数学运行时分析中表明，对于任何 $k = o(n)$，紧凑遗传算法（ cGA) 以任何假设的种群大小 $\mu = \Omega(ne^{4k} + n^{3.5+\varepsilon})$ 以高概率找到具有跳跃大小 $k 的 $n$ 维跳跃函数的最优值$ 时间 $O(\mu n^{1.5} \log n)$。我们显着改善了小跳跃尺寸 $k \le \frac 1 {20} \ln n -1$ 的这个结果。在这种情况下，对于 $\mu = \Omega(\sqrt n \log n) \cap \text{poly}(n)$，cGA 的高概率运行时间仅为 $O(\mu \sqrt n) $. 对于 $\mu$ 的最小允许值，我们的结果给出了 $O(n \log n)$ 的运行时间，而前一个仅显示 $O(n^{5+\varepsilon})$。由于已知具有高概率的 cGA 至少需要 $\Omega(\mu\sqrt n)$ 迭代来优化单峰 OneMx 函数，我们的结果表明，与这里的大多数经典进化算法相比，cGA 能够跨越中等大小的低适应性山谷，无需额外费用。对于较大的 $k$，我们表明 Hasenohrl 和 Sutton 的指数（以 $k$ 为单位）运行时保证是严格的并且无法改进，也不能通过使用较小的假设种群大小来实现。我们证明，假设人口规模的任何选择都会导致运行时间，该运行时间很有可能至少是跳跃大小 $k$ 的指数。这个结果可能是 EDA 的第一个非平凡指数下限，适用于任意参数设置。