解决多维张量输出优化难题，实现高效黑箱函数搜索。
Combinatorial Bandit Bayesian Optimization for Tensor Outputs
- 用张量核高斯过程建模输出结构依赖关系。
- 在部分输出可观测条件下，实现最优子集与点的联合选择。
- 理论保证亚线性误差，适合高维复杂系统优化。
贝叶斯优化（BO）被广泛用于优化各类昂贵且不可见的函数。然而现有方法未涵盖张量输出函数。为此，我们提出一种新型张量输出贝叶斯优化框架。首先引入具有两类张量输出核的张量输出高斯过程（TOGP），作为张量输出函数的代理模型，可有效捕捉张量内部的结构依赖。基于此，设计上置信界（UCB）获取函数以选择查询点。进一步提出更具实际意义的组合博弈贝叶斯优化（CBBO）问题设置：仅能选取张量输出的子集参与目标优化。为此，我们提出张量输出CBBO方法，将TOGP扩展至处理部分观测的张量输出，并设计新的组合多臂赌博机-UCB2（CMAB-UCB2）准则，用于序列化选择查询点与输出子集。我们为两种方法建立了理论后悔界，保证亚线性后悔。在合成及真实数据集上的大量实验验证了方法优越性。
原文摘要 · Abstract (English)
Bayesian optimization (BO) has been widely used to optimize expensive and black-box functions across various domains. However, existing BO methods have not addressed tensor-output functions. To fill this gap, we propose a novel tensor-output BO framework. Specifically, we first introduce a tensor-output Gaussian process (TOGP) with two classes of tensor-output kernels as a surrogate model of the tensor-output function, which can effectively capture the structural dependencies within the tensor. Based on it, we develop an upper confidence bound (UCB) acquisition function to select query points. Furthermore, we introduce a more practical and challenging problem setting, termed combinatorial bandit Bayesian optimization (CBBO), where only a subset of the tensor outputs can be selected to contribute to the objective. To tackle this, we propose a tensor-output CBBO method, which extends TOGP to handle partially observed tensor outputs, and accordingly design a novel combinatorial multi-arm bandit-UCB2 (CMAB-UCB2) criterion to sequentially select both the query points and the output subset. We establish theoretical regret bounds for both methods, guaranteeing sublinear regret. Extensive experiments on synthetic and real-world datasets demonstrate the superiority of our methods.
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