Precise Runtime Analysis for Plateaus

To gain a better theoretical understanding of how evolutionary algorithms cope with plateaus of constant fitness, we analyze how the $(1 + 1)$~EA optimizes the $n$-dimensional $Plateau_k$ function. This function has a plateau of second-best fitness in a radius of $k$ around the optimum. As optimization algorithm, we regard the $(1 + 1)$~EA using an arbitrary unbiased mutation operator. Denoting by $\alpha$ the random number of bits flipped in an application of this operator and assuming $\Pr[\alpha = 1] = \Omega(1)$, we show the surprising result that for $k \ge 2$ the expected optimization time of this algorithm is \[\frac{n^k}{k!\Pr[1 \le \alpha \le k]}(1 + o(1)),\] that is, the size of the plateau times the expected waiting time for an iteration flipping between $1$ and $k$ bits. Our result implies that the optimal mutation rate for this function is approximately~$k/en$. Our main analysis tool is a combined analysis of the Markov chains on the search point space and on the Hamming level space, an approach that promises to be useful also for other plateau problems.