In a seminal paper in 2013, Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time $(1+o(1))n \ln n/p_1$ to find the optimum of any linear function, as long as the probability $p_1$ to flip exactly one bit is $\Theta(1)$. In this paper we investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator. This situation is notably different, since the stochastic domination argument used for the lower bound by Witt no longer holds. In particular, starting closer to the optimum is not necessarily an advantage, and OneMax is no longer the easiest function for arbitrary starting positions. Nevertheless, we show that Witt's result carries over if $p_1$ is not too small, with different constraints for upper and lower bounds, and if the number of flipped bits has bounded expectation~$\chi$. Notably, this includes some of the heavy-tail mutation operators used in fast genetic algorithms, but not all of them. We also give examples showing that algorithms with unbounded $\chi$ have qualitatively different trajectories close to the optimum.

翻译：在2013年一篇开创性论文中，Witt证明了使用标准位变异（1+1）进化算法在任意线性函数上找到最优解需要时间$(1+o(1))n \ln n/p_1$，前提是恰好翻转一位的概率$p_1$满足$\Theta(1)$。本文研究当标准位变异被任意无偏变异算子替代时该结果的泛化情况。此情形与之前显著不同，因为Witt下界证明中使用的随机支配论证不再成立。具体而言，靠近最优解的初始位置未必具有优势，且OneMax不再是任意初始位置下最简单的函数。然而我们证明：当$p_1$不过小（上下界约束条件不同）且翻转位数期望$\chi$有界时，Witt的结果仍能保持。值得注意的是，这包含了快速遗传算法中部分（而非全部）重尾变异算子。我们还给出实例表明，具有无界$\chi$的算法在接近最优解时展现出本质不同的轨迹。