用热核统一解释组合优化中的核函数，提升搜索效率。
Omnipresent Yet Overlooked: Heat Kernels in Combinatorial Bayesian Optimization
- 基于热核构建统一框架，推导出简洁闭式表达
- 实验证明多种成功核函数本质是热核或等价形式
- 热核对最优解位置不敏感，适合复杂组合任务
贝叶斯优化（BO）在材料科学、神经网络架构搜索等组合任务中具有潜力，但需专用核函数建模。尽管已有若干组合核被提出，其内在关系尚不明确。本文系统推导并建立基于热核的统一框架，以闭式表达呈现。理论证明：多个成功组合核与热核相关或等价，并在实验中验证。分析还确认并扩展了Bounce的研究：当目标函数最优解缺乏特定结构时，某些算法性能显著下降；而热核不受最优解位置影响。此外，仅依赖热核的快速简单流程，在多个任务上达到甚至超越现有复杂或缓慢算法的性能。
原文摘要 · Abstract (English)
Bayesian Optimization (BO) has the potential to solve various combinatorial tasks, ranging from materials science to neural architecture search. However, BO requires specialized kernels to effectively model combinatorial domains. Recent efforts have introduced several combinatorial kernels, but the relationships among them are not well understood. To bridge this gap, we develop a unifying framework based on heat kernels, which we derive in a systematic way and express as simple closed-form expressions. Using this framework, we prove that many successful combinatorial kernels are either related or equivalent to heat kernels, and validate this theoretical claim in our experiments. Moreover, our analysis confirms and extends the results presented in Bounce: certain algorithms' performance decreases substantially when the unknown optima of the function do not have a certain structure. In contrast, heat kernels are not sensitive to the location of the optima. Lastly, we show that a fast and simple pipeline, relying on heat kernels, is able to achieve state-of-the-art results, matching or even outperforming certain slow or complex algorithms.
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