Jump functions are the {most-studied} non-unimodal benchmark in the theory of randomized search heuristics, in particular, evolutionary algorithms (EAs). They have significantly improved our understanding of how EAs escape from local optima. However, their particular structure -- to leave the local optimum one can only jump directly to the global optimum -- raises the question of how representative such results are. For this reason, we propose an extended class $\textsc{Jump}_{k,\delta}$ of jump functions that contain a valley of low fitness of width $\delta$ starting at distance $k$ from the global optimum. We prove that several previous results extend to this more general class: for all {$k \le \frac{n^{1/3}}{\ln{n}}$} and $\delta < k$, the optimal mutation rate for the $(1+1)$~EA is $\frac{\delta}{n}$, and the fast $(1+1)$~EA runs faster than the classical $(1+1)$~EA by a factor super-exponential in $\delta$. However, we also observe that some known results do not generalize: the randomized local search algorithm with stagnation detection, which is faster than the fast $(1+1)$~EA by a factor polynomial in $k$ on $\textsc{Jump}_k$, is slower by a factor polynomial in $n$ on some $\textsc{Jump}_{k,\delta}$ instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.
翻译:跳跃函数是随机搜索超常理论{ 随机搜索超常理论{ 特别是进化算法( EAs) 中的非单一模式基准。 它们极大地提高了我们对EAs如何逃离本地opima的理解。 然而, 它们的特殊结构 -- -- 离开本地最佳功能只能直接跳到全球最佳 -- 提出了这种结果如何具有代表性的问题。 为此, 我们提议在跳跃函数中增加一个包含宽度为美元=delta$的低谷值, 特别是进化算法( EAs) 。 我们证明, 之前的一些结果延伸到了这个更普通的类别 : $=k\le = le forest leg, $( 1+1) 美元=+ 美元=+ deltaxn} 。 当( 1+1) 美元为更低的低位值 。 快速的 美元= 美元= 美元= 美元= 美元= 美元= 美元= 美元= 美元= 美元=1) ( 美元= 美元= 美元=YEA) 更快速的运行速度比古典 美元= 美元= 美元= 美元= 美元= 美元= = = = = = = = 快速的 = = = = = = = = = = = = = = = = =