# A Tutorial on Bayesian Optimization

arXiv: https://arxiv.org/abs/1807.02811 · HTML: https://mlsanity.com/paper/1807.02811/

## Key facts

- Cited by 744 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2018.

## Built on by (most cited first, 30 of 744)

- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [Unexpected Improvements to Expected Improvement for Bayesian Optimization](https://mlsanity.com/paper/2310.20708/) (2023)
- [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)
- [High-Dimensional Bayesian Optimization with Sparse Axis-Aligned Subspaces](https://mlsanity.com/paper/2103.00349/) (2021)
- [Large Language Models to Enhance Bayesian Optimization](https://mlsanity.com/paper/2402.03921/) (2024)
- [Transformer Neural Processes: Uncertainty-Aware Meta Learning Via Sequence Modeling](https://mlsanity.com/paper/2207.04179/) (2022)
- [Advances and Challenges in Foundation Agents: From Brain-Inspired Intelligence to Evolutionary, Collaborative, and Safe Systems](https://mlsanity.com/paper/2504.01990/) (2025)
- [Structured Pruning for Deep Convolutional Neural Networks: A survey](https://mlsanity.com/paper/2303.00566/) (2023)
- [Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement](https://mlsanity.com/paper/2105.08195/) (2021)
- [DEUP: Direct Epistemic Uncertainty Prediction](https://mlsanity.com/paper/2102.08501/) (2021)
- [Accelerating Bayesian Optimization for Biological Sequence Design with Denoising Autoencoders](https://mlsanity.com/paper/2203.12742/) (2022)
- [Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces](https://mlsanity.com/paper/2109.10964/) (2021)
- [Recent Advances in Bayesian Optimization](https://mlsanity.com/paper/2206.03301/) (2022)
- [PFNs4BO: In-Context Learning for Bayesian Optimization](https://mlsanity.com/paper/2305.17535/) (2023)
- [InstructZero: Efficient Instruction Optimization for Black-Box Large Language Models](https://mlsanity.com/paper/2306.03082/) (2023)
- [MLCopilot: Unleashing the Power of Large Language Models in Solving Machine Learning Tasks](https://mlsanity.com/paper/2304.14979/) (2023)
- [DeepScientist: Advancing Frontier-Pushing Scientific Findings Progressively](https://mlsanity.com/paper/2509.26603/) (2025)
- [Black Box Adversarial Prompting for Foundation Models](https://mlsanity.com/paper/2302.04237/) (2023)
- [Exploration in Deep Reinforcement Learning: From Single-Agent to Multiagent Domain](https://mlsanity.com/paper/2109.06668/) (2021)
- [High-Dimensional Experimental Design and Kernel Bandits](https://mlsanity.com/paper/2105.05806/) (2021)
- [Joint Entropy Search for Maximally-Informed Bayesian Optimization](https://mlsanity.com/paper/2206.04771/) (2022)
- [High-Dimensional Bayesian Optimisation with Variational Autoencoders and Deep Metric Learning](https://mlsanity.com/paper/2106.03609/) (2021)
- [Think Global and Act Local: Bayesian Optimisation over High-Dimensional Categorical and Mixed Search Spaces](https://mlsanity.com/paper/2102.07188/) (2021)
- [Feedback Efficient Online Fine-Tuning of Diffusion Models](https://mlsanity.com/paper/2402.16359/) (2024)
- [Pareto Set Learning for Expensive Multi-Objective Optimization](https://mlsanity.com/paper/2210.08495/) (2022)
- [A Study of Bayesian Neural Network Surrogates for Bayesian Optimization](https://mlsanity.com/paper/2305.20028/) (2023)
- [AutoPEFT: Automatic Configuration Search for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2301.12132/) (2023)
- [Benchmarking the Performance of Bayesian Optimization across Multiple Experimental Materials Science Domains](https://mlsanity.com/paper/2106.01309/) (2021)
- [Combining Latent Space and Structured Kernels for Bayesian Optimization over Combinatorial Spaces](https://mlsanity.com/paper/2111.01186/) (2021)
- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)

## Abstract (arXiv)

Bayesian optimization is an approach to optimizing objective functions that take a long time (minutes or hours) to evaluate. It is best-suited for optimization over continuous domains of less than 20 dimensions, and tolerates stochastic noise in function evaluations. It builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. In this tutorial, we describe how Bayesian optimization works, including Gaussian process regression and three common acquisition functions: expected improvement, entropy search, and knowledge gradient. We then discuss more advanced techniques, including running multiple function evaluations in parallel, multi-fidelity and multi-information source optimization, expensive-to-evaluate constraints, random environmental conditions, multi-task Bayesian optimization, and the inclusion of derivative information. We conclude with a discussion of Bayesian optimization software and future research directions in the field. Within our tutorial material we provide a generalization of expected improvement to noisy evaluations, beyond the noise-free setting where it is more commonly applied. This generalization is justified by a formal decision-theoretic argument, standing in contrast to previous ad hoc modifications.