基于度量图的贝叶斯优化，提升复杂网络上的黑箱函数优化效率。
Bayesian Optimization on Networks
- 用度量图上的随机偏微分方程定义高斯过程先验，贴合网络几何结构。
- 在光滑目标函数下建立后悔界，未知平滑性时仍可高效优化。
- 适用于通信网络反演等实际场景，数值验证效果显著。
本文研究在度量图上进行的优化问题。针对目标函数评估成本高或仅能以黑箱形式获取的应用场景，提出基于贝叶斯优化的方法，通过序列更新高斯过程代理模型来指导查询点的选择。为使代理模型适应网络几何结构，采用基于度量图上随机偏微分方程定义的Whittle-Matérn高斯过程先验。除建立对足够光滑目标函数的后悔界外，还分析了平滑性未知且使用有限元表示Whittle-Matérn先验的实际情形。数值结果表明，该算法在合成度量图上优化基准目标函数，以及在通信网络中通过最大后验估计进行贝叶斯反演方面均表现有效。
原文摘要 · Abstract (English)
This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Matérn Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Matérn prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.
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