# Derivation of Closed Form of Expected Improvement for Gaussian Process Trained on Log-Transformed Objective

arXiv: https://arxiv.org/abs/2411.18095 · HTML: https://mlsanity.com/paper/2411.18095/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 12 references that are in the map.
- First posted on arXiv 27 Nov 2024.

## Builds on (most cited first, 6 of 6)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [SMAC3: A Versatile Bayesian Optimization Package for Hyperparameter Optimization](https://mlsanity.com/paper/2109.09831/) (2021)
- [Unexpected Improvements to Expected Improvement for Bayesian Optimization](https://mlsanity.com/paper/2310.20708/) (2023)
- [Tree-Structured Parzen Estimator: Understanding Its Algorithm Components and Their Roles for Better Empirical Performance](https://mlsanity.com/paper/2304.11127/) (2023)
- [Constrained Efficient Global Optimization of Expensive Black-box Functions](https://mlsanity.com/paper/2211.00162/) (2022)
- [BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization](https://mlsanity.com/paper/1910.06403/) (2019)

## Built on by (most cited first, 2 of 2)

- [mlr3mbo: Bayesian Optimization in R](https://mlsanity.com/paper/2603.29730/) (2026)
- [Scalable Bayesian Optimization of Composite Functions for Image-Based Inverse Problems in Materials Characterization](https://mlsanity.com/paper/2609.02126/) (2026)

## Abstract (arXiv)

Expected Improvement (EI) is arguably the most widely used acquisition function in Bayesian optimization. However, it is often challenging to enhance the performance with EI due to its sensitivity to numerical precision. Previously, Hutter et al. (2009) tackled this problem by using Gaussian process trained on the log-transformed objective function and it was reported that this trick improves the predictive accuracy of GP, leading to substantially better performance. Although Hutter et al. (2009) offered the closed form of their EI, its intermediate derivation has not been provided so far. In this paper, we give a friendly derivation of their proposition.