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：我们提出了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原期所耗费的时间。这立即给出了Garnier、Kallel和Schoenauer（1999）在Needle问题上关于(1+1)进化算法预期运行时间经典估计的简洁证明。我们还采用该方法分析(1+1)进化算法在一个新基准上的运行时间，该基准由n/ℓ个有效大小为2^ℓ-1的平原组成，这些平原需要以LeadingOnes方式顺序优化。利用我们的新方法，我们确定了静态和适应度依赖突变率下的精确预期运行时间。我们还确定了渐近最优的静态和适应度依赖突变率。对于ℓ = 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)。这些此前仅针对单实例问题LeadingOnes得到证明的结果，因此在更广泛的意义上成立。我们预期LeadingOnes的其他重要结果也存在类似推广。同时我们乐观地认为，我们的傅里叶分析方法也可应用于其他平原问题。