针对函数输入输出的优化难题，提出新型贝叶斯优化框架。
Function-on-Function Bayesian Optimization
- 用可分离算子核的函数型高斯过程建模输入输出函数关系。
- 设计加权算子标量化上置信度策略，实现高效最优函数搜索。
- 适用于复杂传感系统中的函数级优化，适合科研与工程应用。
贝叶斯优化（BO）广泛应用于各类昂贵且无梯度的优化问题。然而，现有方法尚未解决输入和输出均为函数的优化任务，这类问题在先进传感技术推动的复杂系统中日益普遍。为此，本文提出一种新型函数-函数贝叶斯优化（FFBO）框架。首先引入具有可分离算子值核的函数-函数高斯过程（FFGP）模型，直接在函数空间中捕捉函数输入与输出间的相关性。相比传统高斯过程，该模型更适配函数型数据。基于FFGP，采用加权算子标量化策略定义标量上置信度（UCB）采集函数。进一步提出可扩展的函数梯度上升算法（FGA），高效寻找最优函数输入。理论分析验证了方法性质。在合成数据与真实世界数据上的大量实验表明，FFBO显著优于现有方法。
原文摘要 · Abstract (English)
Bayesian optimization (BO) has been widely used to optimize expensive and gradient-free objective functions across various domains. However, existing BO methods have not addressed the objective where both inputs and outputs are functions, which increasingly arise in complex systems as advanced sensing technologies. To fill this gap, we propose a novel function-on-function Bayesian optimization (FFBO) framework. Specifically, we first introduce a function-on-function Gaussian process (FFGP) model with a separable operator-valued kernel to capture the correlations between function-valued inputs and outputs. Compared to existing Gaussian process models, FFGP is modeled directly in the function space. Based on FFGP, we define a scalar upper confidence bound (UCB) acquisition function using a weighted operator-based scalarization strategy. Then, a scalable functional gradient ascent algorithm (FGA) is developed to efficiently identify the optimal function-valued input. We further analyze the theoretical properties of the proposed method. Extensive experiments on synthetic and real-world data demonstrate the superior performance of FFBO over existing approaches.
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