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Combinatorial Bandit Bayesian Optimization for Tensor Outputs

Jingru Huang, Haijie Xu, Jie Guo, Manrui Jiang, Chen Zhang

TL;DR

This work introduces TOBO, a Bayesian optimization framework tailored for tensor-valued outputs by deploying tensor-output Gaussian processes (TOGP) with non-separable and separable kernels that capture input- and mode-wise correlations. It uses a UCB acquisition to efficiently identify the maximizer of a scalarized tensor objective and extends to combinatorial bandit BO (TOCBBO) for partially observed outputs via a CMAB-UCB2 policy that jointly selects inputs and output subsets. The authors establish sublinear regret bounds for both TOBO and TOCBBO, and demonstrate superior performance on synthetic benchmarks and four real-world datasets, including partially observed tensor outputs. The methods offer a principled way to leverage intrinsic tensor structure in BO, with practical impact in domains where tensor-valued measurements are natural and sampling costs are high.

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.

Combinatorial Bandit Bayesian Optimization for Tensor Outputs

TL;DR

This work introduces TOBO, a Bayesian optimization framework tailored for tensor-valued outputs by deploying tensor-output Gaussian processes (TOGP) with non-separable and separable kernels that capture input- and mode-wise correlations. It uses a UCB acquisition to efficiently identify the maximizer of a scalarized tensor objective and extends to combinatorial bandit BO (TOCBBO) for partially observed outputs via a CMAB-UCB2 policy that jointly selects inputs and output subsets. The authors establish sublinear regret bounds for both TOBO and TOCBBO, and demonstrate superior performance on synthetic benchmarks and four real-world datasets, including partially observed tensor outputs. The methods offer a principled way to leverage intrinsic tensor structure in BO, with practical impact in domains where tensor-valued measurements are natural and sampling costs are high.

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.
Paper Structure (38 sections, 7 theorems, 66 equations, 4 figures, 15 tables, 4 algorithms)

This paper contains 38 sections, 7 theorems, 66 equations, 4 figures, 15 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Tensor-output Bayesian optimization
  4. Tensor-output Gaussian process
  5. Upper Confidence Bound Acquisition Strategy
  6. Tensor-output combinatorial bandit Bayesian optimization
  7. Partially observed tensor-output Gaussian process
  8. CMAB-UCB2 Acquisition Strategy
  9. Experiments
  10. Synthetic experiments
  11. Case studies
  12. Conclusion and discussion
  13. Use of LLMs
  14. The details of the tensor decomposition in Remark \ref{['remark 1']}
  15. Low-order tensors ($m \leq 3$):
  16. ...and 23 more sections

Key Result

Proposition 1

The kernel function $\mathbf{K}$ in Definitions def-non-sep ker--def-sep ker for the TOGP is symmetric and positive semi-definite on $\mathcal{X}$. Specifically: (1) $\forall\bm x,\bm x'\in\mathcal{X}$, the kernel satisfies symmetry: $\mathbf{K}(\bm x,\bm x')=\mathbf{K}(\bm x',\bm x)^\top$; (2) $\fo

Figures (4)

  • Figure 1: Each round's logarithmic instantaneous regret of different methods in the Setting (1) (L), (2) (M), and (3) (R) for BO (Top row) and CBBO (Bottom row).
  • Figure 2: Each round's optimal objective value in REEN for BO (L) and CBBO (R).
  • Figure 3: Each round’s optimal objective value in CHEM (L), MAT (M), and PRINT (R).
  • Figure 4: Each round's logarithmic instantaneous regret of different methods in the Setting (1) (Left), (2) (Middle), and (3) (Right) when $k=T/3$ (Top row) and $k=2T/3$ (Bottom row).

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 3
  • Lemma 1
  • Theorem 1
  • Proposition 2
  • ...and 13 more