# Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Framework

arXiv: https://arxiv.org/abs/2605.09775 · HTML: https://mlsanity.com/paper/2605.09775/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 20 of 20)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces](https://mlsanity.com/paper/2109.10964/) (2021)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [No-regret Algorithms for Multi-task Bayesian Optimization](https://mlsanity.com/paper/2008.08885/) (2020)
- [On Controller Tuning with Time-Varying Bayesian Optimization](https://mlsanity.com/paper/2207.11120/) (2022)
- [Functional Data Analysis: An Introduction and Recent Developments](https://mlsanity.com/paper/2312.05523/) (2023)
- [BONSAI: Structure-exploiting robust Bayesian optimization for networked black-box systems under uncertainty](https://mlsanity.com/paper/2510.03893/) (2025)
- [Infinite-Task Learning with RKHSs](https://mlsanity.com/paper/1805.08809/) (2018)
- [Multitask Learning with No Regret: from Improved Confidence Bounds to Active Learning](https://mlsanity.com/paper/2308.01744/) (2023)
- [Data-driven adaptive building thermal controller tuning with constraints: A primal-dual contextual Bayesian optimization approach](https://mlsanity.com/paper/2310.00758/) (2023)
- [Fine-Tuning of Neural Network Approximate MPC without Retraining via Bayesian Optimization](https://mlsanity.com/paper/2512.14350/) (2025)
- [Disturbance-Adaptive Data-Driven Predictive Control: Trading Comfort Violations for Savings in Building Climate Control](https://mlsanity.com/paper/2412.09238/) (2024)
- [Preferential Bayesian Optimization with Crash Feedback](https://mlsanity.com/paper/2604.01776/) (2026)
- [Decentralized Feedback Optimization via Sensitivity Decoupling: Stability and Sub-optimality](https://mlsanity.com/paper/2311.09408/) (2023)
- [What price to pay? Auto-tuning a building MPC controller for optimal economic cost](https://mlsanity.com/paper/2501.10859/) (2025)
- [Function-on-Function Bayesian Optimization](https://mlsanity.com/paper/2511.12783/) (2025)
- [Personalized Building Climate Control with Contextual Preferential Bayesian Optimization](https://mlsanity.com/paper/2512.09481/) (2025)
- [Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual function](https://mlsanity.com/paper/2603.16481/) (2026)

## Built on by (most cited first, 1 of 1)

- [Sequential operator learning under dependent data](https://mlsanity.com/paper/2608.24426/) (2026)

## Abstract (arXiv)

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.