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, are thus true in a much broader respect. 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台上花费的时间。该方法立即为Garnier、Kallel和Schoenauer（1999）关于$(1+1)$进化算法在Needle问题上的预期运行时间的经典估计提供了简洁证明。我们还利用此方法分析了$(1+1)$进化算法在新基准问题上的运行时间，该基准由$n/\ell$个有效大小为$2^\ell-1$的平台组成，需以LeadingOnes方式顺序优化。通过新方法，我们确定了静态突变率和适应度依赖突变率下的精确预期运行时间，并给出了渐近最优的静态突变率和适应度依赖突变率。当$\ell = o(n)$时，最优静态突变率约为$1.59/n$；当找到前$k$个适应度相关位时，最优适应度依赖突变率渐近为$1/(k+1)$。这些结果此前仅对单实例问题LeadingOnes成立，现已在更广泛意义上得到验证。我们预期LeadingOnes的其他重要结果也可类似推广，并乐观地认为该傅里叶分析方法能应用于其他平台问题。