Fourier分析遇上运行时分析：平台期上的精确运行时间
Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus
AI总结：
提出基于离散Fourier分析的新方法分析进化算法在平台期上的运行时间，证明了Needle问题的经典估计，并在新基准上确定了精确运行时间及最优变异率。
摘要：
我们提出了一种基于离散Fourier分析的新方法，来分析进化算法在平台期上花费的时间。这立即为Garnier、Kallel和Schoenauer（1999）提出的关于$(1+1)$进化算法在Needle问题上的期望运行时间的经典估计提供了一个简明的证明。我们还使用该方法分析了$(1+1)$进化算法在一个新基准上的运行时间，该基准由$n/\ell$个有效大小为$2^\ell-1$的平台期组成，这些平台期必须以LeadingOnes方式依次优化。使用我们的新方法，我们确定了静态和适应度相关变异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相关变异率。对于$\ell = o(n)$，最优静态变异率约为$1.59/n$。当找到前$k$个适应度相关位时，最优的适应度相关变异率渐近为$1/(k+1)$。因此，这些迄今为止仅针对单实例问题LeadingOnes证明的结果，对于更广泛的问题类别也成立。我们期望LeadingOnes的其他重要结果也有类似的扩展成立。我们也乐观地认为，我们的Fourier分析方法可以应用于其他平台期问题。
英文摘要：
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.
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