Speeding Up Hyper-Heuristics With Markov-Chain Operator Selection and the Only-Worsening Acceptance Operator

TL;DR

This paper introduces two modifications to move-acceptance hyper-heuristics, including a Markov chain-based operator choice and an only-worsening acceptance operator, significantly improving performance on complex benchmark functions.

Contribution

It proposes a novel Markov chain approach for operator selection and the use of an only-worsening acceptance operator, both enhancing hyper-heuristic efficiency on challenging benchmarks.

Findings

Reduced runtime on Jump$_m$ functions from Ω(n^{2m-1}) to O(n^{m+1})

Achieved runtime of O(n^3 log n) on Jump functions with the only-worsening operator

Proved good runtime of O(n^{k+1} log n) on the SEQOPT$_k$ benchmark class

Abstract

The move-acceptance hyper-heuristic was recently shown to be able to leave local optima with astonishing efficiency (Lissovoi et al., Artificial Intelligence (2023)). In this work, we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic Cliff$_d$ and Jump$_m$ function classes. (i) Instead of randomly choosing between the only-improving and any-move acceptance operator, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on Jump$_m$ functions with gap parameter $m$ from $\Omega(n^{2m-1})$ to $O(n^{m+1})$. (ii) We then replace the all-moves acceptance operator with the operator that only accepts worsenings. Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to $O(n^3 \log n)$ independent of the gap size. In general, we prove a remarkably good runtime of $O(n^{k+1} \log n)$ for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT$_k$, which contains a large number of functions having $k$ successive local optima, and which contains the commonly studied Jump$_m$ and Cliff$_d$ functions for $k=2$.