Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Framework
For practitioners optimizing expensive black-box functions with structured outputs, this work provides a principled framework and regret guarantees, though the extension from scalar to vector-valued outputs is incremental.
Bayesian optimization with structured measurements (e.g., trajectories or spatial fields) is studied, where the objective is a linear functional of vector-valued outputs. The proposed UCB-based algorithm achieves sublinear regret and improved sample efficiency by leveraging richer information from structured outputs.
Bayesian optimization (BO) is an efficient framework for optimizing expensive black-box functions. However, it is typically formulated as learning an end-to-end mapping from inputs to scalar objectives, thereby discarding the potentially rich information whenever a structured system output is available. In this work, we study Bayesian optimization over a vector-valued operator with structured measurements, where each measurement observes multidimensional or functional outputs, e.g., trajectories or spatial fields, rather than a single scalar value. The objective is then defined as a linear functional of these measurements. This allows each observation to reveal substantially richer information about the underlying system compared to scalar observations. Assuming the unknown operator lies in a vector-valued reproducing kernel Hilbert space (RKHS), we derive high-probability concentration bounds for the kernel ridge regression (KRR) estimator directly in the measurement space, characterizing uncertainty in a general Hilbert space. Building on these results, we propose an algorithm based on the upper confidence bound (UCB) acquisition function with regret guarantees under mild assumptions, recovering sublinear rates for common kernels. Empirically, we demonstrate that leveraging structured measurements leads to improved sample efficiency by enabling efficient transfer of information across objectives and adaptation to time-varying settings.