Combinatorial Bandit Bayesian Optimization for Tensor Outputs

TL;DR

This paper introduces a novel Bayesian optimization framework for tensor-output functions, utilizing tensor-output Gaussian processes and a combinatorial bandit approach to efficiently optimize expensive black-box functions with tensor outputs.

Contribution

It proposes a tensor-output Gaussian process model and a combinatorial bandit Bayesian optimization method, addressing the challenge of optimizing tensor-output functions with partial observations.

Findings

The methods achieve sublinear regret bounds.

Experiments show superior performance on synthetic and real datasets.

The framework effectively captures tensor structural dependencies.

Abstract

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.