# Function-on-Function Bayesian Optimization

arXiv: https://arxiv.org/abs/2511.12783 · HTML: https://mlsanity.com/paper/2511.12783/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 16 Nov 2025.

## Builds on (most cited first, 10 of 10)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [A Flexible Framework for Multi-Objective Bayesian Optimization using Random Scalarizations](https://mlsanity.com/paper/1805.12168/) (2018)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Mixed-Variable Bayesian Optimization](https://mlsanity.com/paper/1907.01329/) (2019)
- [No-regret Algorithms for Multi-task Bayesian Optimization](https://mlsanity.com/paper/2008.08885/) (2020)
- [MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysis](https://mlsanity.com/paper/2306.02831/) (2023)
- [Function-on-function kriging, with applications to 3D printing of aortic tissues](https://mlsanity.com/paper/1910.01754/) (2019)
- [A Multi-Level Simulation Optimization Approach for Quantile Functions](https://mlsanity.com/paper/1901.05768/) (2019)
- [A Reproducing Kernel Hilbert Space Approach to Functional Calibration of Computer Models](https://mlsanity.com/paper/2107.08288/) (2021)
- [Nested Bayesian Optimization for Computer Experiments](https://mlsanity.com/paper/2112.09797/) (2021)

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

- [Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Framework](https://mlsanity.com/paper/2605.09775/) (2026)

## Abstract (arXiv)

Bayesian optimization (BO) has been widely used to optimize expensive and gradient-free objective functions across various domains. However, existing BO methods have not addressed the objective where both inputs and outputs are functions, which increasingly arise in complex systems as advanced sensing technologies. To fill this gap, we propose a novel function-on-function Bayesian optimization (FFBO) framework. Specifically, we first introduce a function-on-function Gaussian process (FFGP) model with a separable operator-valued kernel to capture the correlations between function-valued inputs and outputs. Compared to existing Gaussian process models, FFGP is modeled directly in the function space. Based on FFGP, we define a scalar upper confidence bound (UCB) acquisition function using a weighted operator-based scalarization strategy. Then, a scalable functional gradient ascent algorithm (FGA) is developed to efficiently identify the optimal function-valued input. We further analyze the theoretical properties of the proposed method. Extensive experiments on synthetic and real-world data demonstrate the superior performance of FFBO over existing approaches.