Merge Kernel for Bayesian Optimization on Permutation Space

• URL: http://arxiv.org/abs/2507.13263v2
• Date: Fri, 18 Jul 2025 02:45:58 GMT
• Title: Merge Kernel for Bayesian Optimization on Permutation Space
• Authors: Zikai Xie, Linjiang Chen,
• Abstract summary: We propose a novel framework for generating kernel functions on permutation space based on sorting algorithms.<n>The proposed kernel consistently outperforms the state-of-the-art Mallows kernel across various permutation optimization benchmarks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Bayesian Optimization (BO) algorithm is a standard tool for black-box optimization problems. The current state-of-the-art BO approach for permutation spaces relies on the Mallows kernel-an $\Omega(n^2)$ representation that explicitly enumerates every pairwise comparison. Inspired by the close relationship between the Mallows kernel and pairwise comparison, we propose a novel framework for generating kernel functions on permutation space based on sorting algorithms. Within this framework, the Mallows kernel can be viewed as a special instance derived from bubble sort. Further, we introduce the \textbf{Merge Kernel} constructed from merge sort, which replaces the quadratic complexity with $\Theta(n\log n)$ to achieve the lowest possible complexity. The resulting feature vector is significantly shorter, can be computed in linearithmic time, yet still efficiently captures meaningful permutation distances. To boost robustness and right-invariance without sacrificing compactness, we further incorporate three lightweight, task-agnostic descriptors: (1) a shift histogram, which aggregates absolute element displacements and supplies a global misplacement signal; (2) a split-pair line, which encodes selected long-range comparisons by aligning elements across the two halves of the whole permutation; and (3) sliding-window motifs, which summarize local order patterns that influence near-neighbor objectives. Our empirical evaluation demonstrates that the proposed kernel consistently outperforms the state-of-the-art Mallows kernel across various permutation optimization benchmarks. Results confirm that the Merge Kernel provides a more compact yet more effective solution for Bayesian optimization in permutation space.

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