Combinatorial Bandit Bayesian Optimization for Tensor Outputs

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.