In recent work, Lissovoi, Oliveto, and Warwicker (Artificial Intelligence (2023)) proved that the Move Acceptance Hyper-Heuristic (MAHH) leaves the local optimum of the multimodal CLIFF benchmark with remarkable efficiency. The $O(n^3)$ runtime of the MAHH, for almost all cliff widths $d\ge 2,$ is significantly better than the $\Theta(n^d)$ runtime of simple elitist evolutionary algorithms (EAs) on CLIFF. In this work, we first show that this advantage is specific to the CLIFF problem and does not extend to the JUMP benchmark, the most prominent multi-modal benchmark in the theory of randomized search heuristics. We prove that for any choice of the MAHH selection parameter $p$, the expected runtime of the MAHH on a JUMP function with gap size $m = O(n^{1/2})$ is at least $\Omega(n^{2m-1} / (2m-1)!)$. This is significantly slower than the $O(n^m)$ runtime of simple elitist EAs. Encouragingly, we also show that replacing the local one-bit mutation operator in the MAHH with the global bit-wise mutation operator, commonly used in EAs, yields a runtime of $\min\{1, O(\frac{e\ln(n)}{m})^m\} \, O(n^m)$ on JUMP functions. This is at least as good as the runtime of simple elitist EAs. For larger values of $m$, this result proves an asymptotic performance gain over simple EAs. As our proofs reveal, the MAHH profits from its ability to walk through the valley of lower objective values in moderate-size steps, always accepting inferior solutions. This is the first time that such an optimization behavior is proven via mathematical means. Generally, our result shows that combining two ways of coping with local optima, global mutation and accepting inferior solutions, can lead to considerable performance gains.

翻译：在近期工作中，Lissovoi、Oliveto和Warwicker（《人工智能》（2023））证明了移动接受超启发式算法（MAHH）能以显著效率逃离多模态CLIFF基准测试的局部最优解。对于几乎所有悬崖宽度$d\ge 2$，MAHH的$O(n^3)$时间复杂度显著优于简单精英进化算法（EAs）在CLIFF问题上的$\Theta(n^d)$时间复杂度。本研究首先表明，这一优势仅限于CLIFF问题，并未延伸至随机搜索启发式理论中最主要的多模态基准测试——JUMP基准。我们证明，对于MAHH选择参数$p$的任何取值，在间隙尺寸$m = O(n^{1/2})$的JUMP函数上，MAHH的期望时间复杂度至少为$\Omega(n^{2m-1} / (2m-1)!)$，这显著慢于简单精英EAs的$O(n^m)$时间复杂度。值得鼓舞的是，我们还证明将MAHH中的局部单比特变异算子替换为EAs常用的全局按位变异算子后，在JUMP函数上可获得$\min\{1, O(\frac{e\ln(n)}{m})^m\} \, O(n^m)$的时间复杂度，该性能至少与简单精英EAs相当。对于更大的$m$值，该结果证明了相较于简单EAs的渐近性能提升。如我们的证明所揭示，MAHH受益于其以中等步长穿越低目标值区域并始终接受劣质解的能力。这是首次通过数学方法证明此类优化行为。总体而言，我们的研究表明，将应对局部最优解的两种策略——全局变异与接受劣质解相结合，可带来显著的性能提升。