# Thinking inside the box: A tutorial on grey-box Bayesian optimization

arXiv: https://arxiv.org/abs/2201.00272 · HTML: https://mlsanity.com/paper/2201.00272/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 68 references that are in the map.
- First posted on arXiv 2 Jan 2022.

## Builds on (most cited first, 27 of 27)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [Max-value Entropy Search for Efficient Bayesian Optimization](https://mlsanity.com/paper/1703.01968/) (2017)
- [Predictive Entropy Search for Efficient Global Optimization of Black-box Functions](https://mlsanity.com/paper/1406.2541/) (2014)
- [Bayesian Optimization is Superior to Random Search for Machine Learning Hyperparameter Tuning: Analysis of the Black-Box Optimization Challenge 2020](https://mlsanity.com/paper/2104.10201/) (2021)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Maximizing acquisition functions for Bayesian optimization](https://mlsanity.com/paper/1805.10196/) (2018)
- [Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning](https://mlsanity.com/paper/1903.04703/) (2019)
- [Re-Examining Linear Embeddings for High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2001.11659/) (2020)
- [Multi-Information Source Optimization](https://mlsanity.com/paper/1603.00389/) (2016)
- [The Parallel Knowledge Gradient Method for Batch Bayesian Optimization](https://mlsanity.com/paper/1606.04414/) (2016)
- [Adversarially Robust Optimization with Gaussian Processes](https://mlsanity.com/paper/1810.10775/) (2018)
- [Distributionally Robust Bayesian Optimization](https://mlsanity.com/paper/2002.09038/) (2020)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Bayesian Optimization with Gradients](https://mlsanity.com/paper/1703.04389/) (2017)
- [Bayesian Optimization of Composite Functions](https://mlsanity.com/paper/1906.01537/) (2019)
- [Parallel Bayesian Global Optimization of Expensive Functions](https://mlsanity.com/paper/1602.05149/) (2016)
- [Efficient Nonmyopic Bayesian Optimization via One-Shot Multi-Step Trees](https://mlsanity.com/paper/2006.15779/) (2020)
- [Bayesian Optimization of Risk Measures](https://mlsanity.com/paper/2007.05554/) (2020)
- [BINOCULARS for Efficient, Nonmyopic Sequential Experimental Design](https://mlsanity.com/paper/1909.04568/) (2019)
- [Bayesian Optimization of Function Networks](https://mlsanity.com/paper/2112.15311/) (2021)
- [Bayesian Optimization with High-Dimensional Outputs](https://mlsanity.com/paper/2106.12997/) (2021)
- [Bayesian Optimization with Expensive Integrands](https://mlsanity.com/paper/1803.08661/) (2018)
- [Value-at-Risk Optimization with Gaussian Processes](https://mlsanity.com/paper/2105.06126/) (2021)
- [Efficient sequential experimental design for surrogate modeling of nested codes](https://mlsanity.com/paper/1712.01620/) (2017)
- [Bayesian Optimization with Shape Constraints](https://mlsanity.com/paper/1612.08915/) (2016)
- [Goal-oriented adaptive sampling under random field modelling of response probability distributions](https://mlsanity.com/paper/2102.07612/) (2021)

## Built on by (most cited first, 26 of 26)

- [Preference Exploration for Efficient Bayesian Optimization with Multiple Outcomes](https://mlsanity.com/paper/2203.11382/) (2022)
- [Model-based Causal Bayesian Optimization](https://mlsanity.com/paper/2211.10257/) (2022)
- [A General Recipe for Likelihood-free Bayesian Optimization](https://mlsanity.com/paper/2206.13035/) (2022)
- [Reinforced In-Context Black-Box Optimization](https://mlsanity.com/paper/2402.17423/) (2024)
- [Bayesian Optimization of Function Networks with Partial Evaluations](https://mlsanity.com/paper/2311.02146/) (2023)
- [No-Regret Constrained Bayesian Optimization of Noisy and Expensive Hybrid Models using Differentiable Quantile Function Approximations](https://mlsanity.com/paper/2305.03824/) (2023)
- [Joint Composite Latent Space Bayesian Optimization](https://mlsanity.com/paper/2311.02213/) (2023)
- [Efficient Language-instructed Skill Acquisition via Reward-Policy Co-Evolution](https://mlsanity.com/paper/2412.13492/) (2024)
- [Regret Bounds for Noise-Free Cascaded Kernelized Bandits](https://mlsanity.com/paper/2211.05430/) (2022)
- [Robust Bayesian Target Value Optimization](https://mlsanity.com/paper/2301.04344/) (2023)
- [Generating Transferable Adversarial Simulation Scenarios for Self-Driving via Neural Rendering](https://mlsanity.com/paper/2309.15770/) (2023)
- [BOIS: Bayesian Optimization of Interconnected Systems](https://mlsanity.com/paper/2311.11254/) (2023)
- [Causal Bayesian Optimization via Exogenous Distribution Learning](https://mlsanity.com/paper/2402.02277/) (2024)
- [Automated design of nonreciprocal thermal emitters via Bayesian optimization](https://mlsanity.com/paper/2409.09192/) (2024)
- [Scalable Bayesian optimization with high-dimensional outputs using randomized prior networks](https://mlsanity.com/paper/2302.07260/) (2023)
- [Trajectory-oriented optimization of stochastic epidemiological models](https://mlsanity.com/paper/2305.03926/) (2023)
- [Frozen Layers: Memory-efficient Many-fidelity Hyperparameter Optimization](https://mlsanity.com/paper/2504.10735/) (2025)
- [A Hierarchical Surrogate Model for Efficient Multi-Task Parameter Learning in Closed-Loop Control](https://mlsanity.com/paper/2508.12738/) (2025)
- [Long-run Behaviour of Multi-fidelity Bayesian Optimisation](https://mlsanity.com/paper/2312.12633/) (2023)
- [Optimal Observation-Intervention Trade-Off in Optimisation Problems with Causal Structure](https://mlsanity.com/paper/2309.02287/) (2023)
- [Grey-box Bayesian Optimization for Sensor Placement in Assisted Living Environments](https://mlsanity.com/paper/2309.05784/) (2023)
- [Constrained composite Bayesian optimization for rational synthesis of polymeric particles](https://mlsanity.com/paper/2411.10471/) (2024)
- [Fast Bayesian Optimization of Function Networks with Partial Evaluations](https://mlsanity.com/paper/2506.11456/) (2025)
- [Model-free Reinforcement Learning for Model-based Control: Towards Safe, Interpretable and Sample-efficient Agents](https://mlsanity.com/paper/2507.13491/) (2025)
- [Transferring Information Across Interventions in Causal Bayesian Optimization](https://mlsanity.com/paper/2606.01457/) (2026)
- [Amortizing Scaling Law Construction Costs](https://mlsanity.com/paper/2609.05016/) (2026)

## Abstract (arXiv)

Bayesian optimization (BO) is a framework for global optimization of expensive-to-evaluate objective functions. Classical BO methods assume that the objective function is a black box. However, internal information about objective function computation is often available. For example, when optimizing a manufacturing line's throughput with simulation, we observe the number of parts waiting at each workstation, in addition to the overall throughput. Recent BO methods leverage such internal information to dramatically improve performance. We call these "grey-box" BO methods because they treat objective computation as partially observable and even modifiable, blending the black-box approach with so-called "white-box" first-principles knowledge of objective function computation. This tutorial describes these methods, focusing on BO of composite objective functions, where one can observe and selectively evaluate individual constituents that feed into the overall objective; and multi-fidelity BO, where one can evaluate cheaper approximations of the objective function by varying parameters of the evaluation oracle.