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 position. Nevertheless, we show that Witt's result carries over if $p_1$ is not too small and if the number of flipped bits has bounded expectation~$\mu$. 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 $\mu$ have qualitatively different trajectories close to the optimum.

翻译：在2013年的一篇开创性论文中，Witt证明了：在标准位突变下，(1+1)进化算法需要时间$(1+o(1))n \ln n/p_1$才能找到任意线性函数的最优解，前提是恰好翻转一位的概率$p_1 = \Theta(1)$。本文研究当标准位突变被任意无偏突变算子替代时，该结果如何推广。这种情况显著不同，因为Witt用于下界的随机占优论证不再成立。特别地，从接近最优解的位置出发未必具有优势，且OneMax函数也不再是任意起始位置下最简单的函数。尽管如此，我们证明了当$p_1$不过小且翻转位数具有有界期望$\mu$时，Witt的结果仍然成立。值得注意的是，这涵盖了快速遗传算法中使用的部分（但非全部）重尾突变算子。我们还通过实例表明，当$\mu$无界时，算法在最优解附近的轨迹存在本质差异。