Combinatorial Bandit Bayesian Optimization for Tensor Outputs

2026-01-31

AI Summary

This work addresses the challenge of optimizing black-box functions with tensor-valued outputs, particularly in settings where only partial observations are available through combinatorial feedback. The authors propose a Bayesian optimization framework tailored for tensor outputs, leveraging tensor-output Gaussian processes and tensor kernel functions to capture structured dependencies among output dimensions. They introduce an upper confidence bound (UCB)-based acquisition function and, to handle partial observability, innovatively integrate combinatorial multi-armed bandits with Bayesian optimization, yielding a novel Combinatorial UCB2 criterion that jointly selects input points and subsets of outputs in a sequential manner. Theoretical analysis establishes a sublinear regret bound, and experiments on both synthetic and real-world datasets demonstrate significant improvements over existing methods, confirming the effectiveness and superiority of the proposed approach in tensor-output and combinatorial feedback scenarios.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Bayesian optimization (BO) has been widely used to optimize expensive and black-box functions across various domains. Existing BO methods have not addressed tensor-output functions. To fill this gap, we propose a novel tensor-output BO method. 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 the queried points. Furthermore, we introduce a more complex and practical problem setting, named combinatorial bandit Bayesian optimization (CBBO), where only a subset of the outputs can be selected to contribute to the objective function. To tackle this, we propose a tensor-output CBBO method, which extends TOGP to handle partially observed outputs, and accordingly design a novel combinatorial multi-arm bandit-UCB2 (CMAB-UCB2) criterion to sequentially select both the queried points and the optimal output subset. Theoretical regret bounds for the two methods are established, ensuring their sublinear performance. Extensive synthetic and real-world experiments demonstrate their superiority.

Problem

• tensor-output
• Bayesian optimization
• combinatorial bandit
• black-box optimization
• partially observed outputs

Innovation

• Tensor-output Gaussian Process
• Combinatorial Bandit Bayesian Optimization
• CMAB-UCB2
• Structured Output Optimization
• Bayesian Optimization