# Unexpected Improvements to Expected Improvement for Bayesian Optimization

arXiv: https://arxiv.org/abs/2310.20708 · HTML: https://mlsanity.com/paper/2310.20708/

## Key facts

- Cited by 149 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 78 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 30 of 42)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Scalable Global Optimization via Local Bayesian Optimization](https://mlsanity.com/paper/1910.01739/) (2019)
- [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)
- [Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2006.05078/) (2020)
- [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)
- [Maximizing acquisition functions for Bayesian optimization](https://mlsanity.com/paper/1805.10196/) (2018)
- [Bayesian Optimization with Unknown Constraints](https://mlsanity.com/paper/1403.5607/) (2014)
- [Constrained Bayesian Optimization with Noisy Experiments](https://mlsanity.com/paper/1706.07094/) (2017)
- [Scalable Constrained Bayesian Optimization](https://mlsanity.com/paper/2002.08526/) (2020)
- [Combinatorial Bayesian Optimization using the Graph Cartesian Product](https://mlsanity.com/paper/1902.00448/) (2019)
- [The Parallel Knowledge Gradient Method for Batch Bayesian Optimization](https://mlsanity.com/paper/1606.04414/) (2016)
- [An Easy-to-use Real-world Multi-objective Optimization Problem Suite](https://mlsanity.com/paper/2009.12867/) (2020)
- [Tuning Hyperparameters without Grad Students: Scalable and Robust Bayesian Optimisation with Dragonfly](https://mlsanity.com/paper/1903.06694/) (2019)
- [Bayesian Optimization of Combinatorial Structures](https://mlsanity.com/paper/1806.08838/) (2018)
- [Bayesian Optimization with Gradients](https://mlsanity.com/paper/1703.04389/) (2017)
- [Joint Entropy Search for Maximally-Informed Bayesian Optimization](https://mlsanity.com/paper/2206.04771/) (2022)
- [Think Global and Act Local: Bayesian Optimisation over High-Dimensional Categorical and Mixed Search Spaces](https://mlsanity.com/paper/2102.07188/) (2021)
- [Benchmarking the Performance of Bayesian Optimization across Multiple Experimental Materials Science Domains](https://mlsanity.com/paper/2106.01309/) (2021)
- [GIBBON: General-purpose Information-Based Bayesian OptimisatioN](https://mlsanity.com/paper/2102.03324/) (2021)
- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [Parallel Bayesian Global Optimization of Expensive Functions](https://mlsanity.com/paper/1602.05149/) (2016)
- [Cost-aware Bayesian Optimization](https://mlsanity.com/paper/2003.10870/) (2020)
- [Efficient Nonmyopic Bayesian Optimization via One-Shot Multi-Step Trees](https://mlsanity.com/paper/2006.15779/) (2020)
- [Robust Multi-Objective Bayesian Optimization Under Input Noise](https://mlsanity.com/paper/2202.07549/) (2022)
- [Bayesian Optimization over Discrete and Mixed Spaces via Probabilistic Reparameterization](https://mlsanity.com/paper/2210.10199/) (2022)
- [Scaling Gaussian Process Regression with Derivatives](https://mlsanity.com/paper/1810.12283/) (2018)
- [Bayesian Optimization over High-Dimensional Combinatorial Spaces via Dictionary-based Embeddings](https://mlsanity.com/paper/2303.01774/) (2023)

## Built on by (most cited first, 30 of 149)

- [Vanilla Bayesian Optimization Performs Great in High Dimensions](https://mlsanity.com/paper/2402.02229/) (2024)
- [Standard Gaussian Process is All You Need for High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2402.02746/) (2024)
- [A General Framework for User-Guided Bayesian Optimization](https://mlsanity.com/paper/2311.14645/) (2023)
- [Understanding High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2502.09198/) (2025)
- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [Adaptive Kernel Design for Bayesian Optimization Is a Piece of CAKE with LLMs](https://mlsanity.com/paper/2509.17998/) (2025)
- [Cost-aware Bayesian Optimization via the Pandora's Box Gittins Index](https://mlsanity.com/paper/2406.20062/) (2024)
- [Bayesian optimization as a flexible and efficient design framework for sustainable process systems](https://mlsanity.com/paper/2401.16373/) (2024)
- [The Vizier Gaussian Process Bandit Algorithm](https://mlsanity.com/paper/2408.11527/) (2024)
- [TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusion](https://mlsanity.com/paper/2509.25171/) (2025)
- [Posterior Inference with Diffusion Models for High-dimensional Black-box Optimization](https://mlsanity.com/paper/2502.16824/) (2025)
- [Faster, Cheaper, Better: Multi-Objective Hyperparameter Optimization for LLM and RAG Systems](https://mlsanity.com/paper/2502.18635/) (2025)
- [We Still Don't Understand High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2512.00170/) (2025)
- [Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networks](https://mlsanity.com/paper/2404.03099/) (2024)
- [Expensive Multi-Objective Bayesian Optimization Based on Diffusion Models](https://mlsanity.com/paper/2405.08674/) (2024)
- [Adaptive Acquisition Selection for Bayesian Optimization with Large Language Models](https://mlsanity.com/paper/2602.07904/) (2026)
- [Global Optimization of Gaussian Process Acquisition Functions Using a Piecewise-Linear Kernel Approximation](https://mlsanity.com/paper/2410.16893/) (2024)
- [Robust Gaussian Processes via Relevance Pursuit](https://mlsanity.com/paper/2410.24222/) (2024)
- [Reasoning BO: Enhancing Bayesian Optimization with Long-Context Reasoning Power of LLMs](https://mlsanity.com/paper/2505.12833/) (2025)
- [Simplifying Bayesian Optimization Via In-Context Direct Optimum Sampling](https://mlsanity.com/paper/2505.23913/) (2025)
- [BoFire: Bayesian Optimization Framework Intended for Real Experiments](https://mlsanity.com/paper/2408.05040/) (2024)
- [EARL-BO: Reinforcement Learning for Multi-Step Lookahead, High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2411.00171/) (2024)
- [Constrained Multi-objective Bayesian Optimization through Optimistic Constraints Estimation](https://mlsanity.com/paper/2411.03641/) (2024)
- [NeST-BO: Fast Local Bayesian Optimization via Newton-Step Targeting of Gradient and Hessian Information](https://mlsanity.com/paper/2510.05516/) (2025)
- [Agentic Bayesian Optimization through Surrogate-Augmented Autoresearch](https://mlsanity.com/paper/2608.00316/) (2026)
- [Voronoi Candidates for Bayesian Optimization](https://mlsanity.com/paper/2402.04922/) (2024)
- [QuaCK-TSF: Quantum-Classical Kernelized Time Series Forecasting](https://mlsanity.com/paper/2408.12007/) (2024)
- [Optimizing Posterior Samples for Bayesian Optimization via Rootfinding](https://mlsanity.com/paper/2410.22322/) (2024)
- [Model Fusion through Bayesian Optimization in Language Model Fine-Tuning](https://mlsanity.com/paper/2411.06710/) (2024)
- [Posterior Inference in Latent Space for Scalable Constrained Black-box Optimization](https://mlsanity.com/paper/2507.00480/) (2025)

## Abstract (arXiv)

Expected Improvement (EI) is arguably the most popular acquisition function in Bayesian optimization and has found countless successful applications, but its performance is often exceeded by that of more recent methods. Notably, EI and its variants, including for the parallel and multi-objective settings, are challenging to optimize because their acquisition values vanish numerically in many regions. This difficulty generally increases as the number of observations, dimensionality of the search space, or the number of constraints grow, resulting in performance that is inconsistent across the literature and most often sub-optimal. Herein, we propose LogEI, a new family of acquisition functions whose members either have identical or approximately equal optima as their canonical counterparts, but are substantially easier to optimize numerically. We demonstrate that numerical pathologies manifest themselves in "classic" analytic EI, Expected Hypervolume Improvement (EHVI), as well as their constrained, noisy, and parallel variants, and propose corresponding reformulations that remedy these pathologies. Our empirical results show that members of the LogEI family of acquisition functions substantially improve on the optimization performance of their canonical counterparts and surprisingly, are on par with or exceed the performance of recent state-of-the-art acquisition functions, highlighting the understated role of numerical optimization in the literature.