# Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces

arXiv: https://arxiv.org/abs/2205.13627 · HTML: https://mlsanity.com/paper/2205.13627/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 56 references that are in the map.
- First posted on arXiv 26 May 2022.

## Builds on (most cited first, 12 of 12)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces](https://mlsanity.com/paper/1902.03229/) (2019)
- [High-Dimensional Experimental Design and Kernel Bandits](https://mlsanity.com/paper/2105.05806/) (2021)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Bayesian Optimization of Risk Measures](https://mlsanity.com/paper/2007.05554/) (2020)
- [Experimental Design for Overparameterized Learning with Application to Single Shot Deep Active Learning](https://mlsanity.com/paper/2009.12820/) (2020)
- [The Impact of Data on the Stability of Learning-Based Control- Extended Version](https://mlsanity.com/paper/2011.10596/) (2020)
- [Reproducing kernel Hilbert space based estimation of systems of ordinary differential equations](https://mlsanity.com/paper/1311.3576/) (2013)
- [Towards Machine Wald](https://mlsanity.com/paper/1508.02449/) (2015)
- [Sequential Bayesian optimal experimental design for structural reliability analysis](https://mlsanity.com/paper/2007.00402/) (2020)
- [Numerical Analysis of Sparse Initial Data Identification for Parabolic Problems](https://mlsanity.com/paper/1905.01226/) (2019)

## Built on by (most cited first, 9 of 9)

- [Active Exploration via Experiment Design in Markov Chains](https://mlsanity.com/paper/2206.14332/) (2022)
- [Near-Optimal Multi-Agent Learning for Safe Coverage Control](https://mlsanity.com/paper/2210.06380/) (2022)
- [Likelihood Ratio Confidence Sets for Sequential Decision Making](https://mlsanity.com/paper/2311.04402/) (2023)
- [Transition Constrained Bayesian Optimization via Markov Decision Processes](https://mlsanity.com/paper/2402.08406/) (2024)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Active Few-Shot Fine-Tuning](https://mlsanity.com/paper/2402.15441/) (2024)
- [Managing Temporal Resolution in Continuous Value Estimation: A Fundamental Trade-off](https://mlsanity.com/paper/2212.08949/) (2022)
- [Indirect Query Bayesian Optimization with Integrated Feedback](https://mlsanity.com/paper/2412.13559/) (2024)
- [Experimental Design for Multi-Channel Imaging via Task-Driven Feature Selection](https://mlsanity.com/paper/2210.06891/) (2022)

## Abstract (arXiv)

Optimal experimental design seeks to determine the most informative allocation of experiments to infer an unknown statistical quantity. In this work, we investigate the optimal design of experiments for em estimation of linear functionals in reproducing kernel Hilbert spaces (RKHSs). This problem has been extensively studied in the linear regression setting under an estimability condition, which allows estimating parameters without bias. We generalize this framework to RKHSs, and allow for the linear functional to be only approximately inferred, i.e., with a fixed bias. This scenario captures many important modern applications, such as estimation of gradient maps, integrals, and solutions to differential equations. We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise, enabling us to certify estimation with bounded error with high probability.