We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the $(1+1)$ evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the $(1+1)$ evolutionary algorithm on a new benchmark consisting of $n/\ell$ plateaus of effective size $2^\ell-1$ which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For $\ell = o(n)$, the optimal static mutation rate is approximately $1.59/n$. The optimal fitness dependent mutation rate, when the first $k$ fitness-relevant bits have been found, is asymptotically $1/(k+1)$. These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.

翻译：我们提出一种基于离散傅里叶分析的新方法，用于分析进化算法在平板上花费的时间。这直接给出了对Garnier、Kallel和Schoenauer（1999）关于$(1+1)$进化算法在Needle问题上期望运行时间的经典估计的简洁证明。我们还利用该方法分析了$(1+1)$进化算法在一个由$n/\ell$个有效尺寸为$2^\ell-1$的平板组成的新基准上的运行时间，这些平板需要以LeadingOnes的方式顺序优化。通过我们的新方法，我们确定了静态和适应度依赖突变率下的精确期望运行时间。我们还确定了渐近最优的静态和适应度依赖突变率。对于$\ell = o(n)$，最优静态突变率约为$1.59/n$。当已找到前$k$个适应度相关比特时，最优适应度依赖突变率渐近为$1/(k+1)$。这些结果目前仅针对单实例问题LeadingOnes得到证明，因此适用于更广泛的问题类别。我们预期类似的扩展对于LeadingOnes的其他重要结果也成立。此外，我们相信我们的傅里叶分析方法同样可以应用于其他平板问题。