How the Move Acceptance Hyper-Heuristic Copes With Local Optima: Drastic Differences Between Jumps and Cliffs

TL;DR

This work evaluates the Move Acceptance Hyper-Heuristic (MAHH) on the Jump$_m$ benchmark to test its ability to escape local optima beyond Cliff. It presents a general non-asymptotic lower bound showing that, for $m = o(n^{1/2})$, MAHH runs as slow as $\Omega(n^{2m-1}/(2m-1)!)$, and proves an exponential lower bound when $m$ scales linearly with $n$ (i.e., $m = \alpha n$, $\alpha<0.5$). An upper bound for the standard MAHH with $p = m/n$ is established as $O(n\log n + n^{2m-1}/(m!\,m^{m-2}))$, illustrating that MAHH can be slower than elitist EAs on Jump. Introducing global mutation (bitwise mutation with rate $1/n$) yields a best-of-two-worlds bound, $\mathbb{E}[T] = O(n\log n + \min\{n^{m}, n^{2m-1}/(m!\,m^{m-2})\})$, which essentially takes the better of the two competing strategies. Overall, the results show that the favorable Cliff performance of MAHH does not generalize to Jump, but combining local search with global mutation can provide robust performance across multimodal landscapes.

Abstract

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. With its $O(n^3)$ runtime, for almost all cliff widths $d,$ the MAHH massively outperforms the $Θ(n^d)$ runtime of simple elitist evolutionary algorithms (EAs). For the most prominent multimodal benchmark, the jump functions, the given runtime estimates of $O(n^{2m} m^{-Θ(m)})$ and $Ω(2^{Ω(m)})$, for gap size $m \ge 2$, are far apart and the real performance of MAHH is still an open question. In this work, we resolve this question. 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 $Ω(n^{2m-1} / (2m-1)!)$. This renders the MAHH much slower than simple elitist evolutionary algorithms with their typical $O(n^m)$ runtime. We also show that the MAHH with the global bit-wise mutation operator instead of the local one-bit operator optimizes jump functions in time $O(\min\{m n^m,\frac{n^{2m-1}}{m!Ω(m)^{m-2}}\})$, essentially the minimum of the optimization times of the $(1+1)$ EA and the MAHH. This suggests that combining several ways to cope with local optima can be a fruitful approach.

Theorems & Definitions (27)

• lemma 1: DrosteJW00
• lemma 2
• theorem 1: Multiplicative Drift Theorem DoerrJW12algo
• theorem 2: Wald's Formula Wald44
• theorem 3: General Formula for the Expected Duration of the Last Step
• proof
• theorem 4
• proof
• theorem 5
• proof