基于负乘性漂移的非精英进化算法的下界
Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift
摘要：
非精英群体为基础的进化算法的一个体面的数量的下界已经显示了现在。它们中的大多数在技术上要求很高，因为（很难避免）使用负漂移定理--将预期的进展从目标转化为高命中时间的一般结果。我们提出了一个简单的负漂移定理乘法漂移的情况下，它可以简化现有的分析。我们更详细地讨论了Lehre（PPSN 2010）的\emph{negative drift in populations}方法，这是证明离散搜索空间的基于非精英变异的进化算法的运行时间下界的最通用工具之一。结合其他论证，我们得到了另一种更简单的证明，这也加强和简化了这种方法。特别是，现在只需验证先前结果的五个技术条件中的三个。我们得到的下界是明确的，而不仅仅是渐近的。这允许计算具体算法的具体下限，但也使我们能够表明，超多项式运行时已经出现时，再现率仅低于阈值的$(1 - ω(n^{-1/2}))$因素。对于使用具有随机突变率的标准位突变的算法的特殊情况（在超随机语言中称为均匀混合），我们证明了Dang和Lehre（PPSN 2016）所述的结果，并将其扩展到$Θ(1/n)$以外的突变率，其中包括Doerr，Le，Makhmara和Nguyen提出的重尾突变算子（GECCO 2017）。最后，我们应用我们的方法和一个新的支配参数，以显示一个指数下界的运行时间的变异只有简单的遗传算法\onemax为任意人口规模。
英文摘要：
A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems -- general results which translate an expected progress away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) \emph{negative drift in populations} method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a $(1 - ω(n^{-1/2}))$ factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than $Θ(1/n)$, which includes the heavy-tailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally apply our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on \onemax for arbitrary population size.
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