# Beyond Objective-Based Improvement: Stationarity-Aware Expected Improvement for Bayesian Optimization

arXiv: https://arxiv.org/abs/2601.21357 · HTML: https://mlsanity.com/paper/2601.21357/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 14 of 14)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Predictive Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/1511.05467/) (2015)
- [The Parallel Knowledge Gradient Method for Batch Bayesian Optimization](https://mlsanity.com/paper/1606.04414/) (2016)
- [Bayesian Optimization with Gradients](https://mlsanity.com/paper/1703.04389/) (2017)
- [Improving the Expected Improvement Algorithm](https://mlsanity.com/paper/1705.10033/) (2017)
- [Greed is Good: Exploration and Exploitation Trade-offs in Bayesian Optimisation](https://mlsanity.com/paper/1911.12809/) (2019)
- [Local policy search with Bayesian optimization](https://mlsanity.com/paper/2106.11899/) (2021)
- [Local Bayesian optimization via maximizing probability of descent](https://mlsanity.com/paper/2210.11662/) (2022)
- [Scaling Gaussian Processes with Derivative Information Using Variational Inference](https://mlsanity.com/paper/2107.04061/) (2021)
- [User Preference Meets Pareto-Optimality in Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2502.06971/) (2025)
- [Self-Adjusting Weighted Expected Improvement for Bayesian Optimization](https://mlsanity.com/paper/2306.04262/) (2023)

## Abstract (arXiv)

Bayesian Optimization (BO) is a principled framework for optimizing expensive black-box functions, with Expected Improvement (EI) among its most widely used acquisition functions. Despite its empirical success, EI is agnostic to first-order optimality conditions, relying solely on objective-value improvement. As a result, it can exhibit vanishing acquisition signals where the improvement criterion is uninformative, limiting its effectiveness in guiding search. We propose Expected Improvement via Gradient Norms (EI-GN), a novel acquisition function that extends the improvement principle to incorporate first-order stationarity, promoting sampling in regions that are both high-performing and close to stationary points. We derive a tractable closed-form expression for EI-GN and show that it remains consistent with the improvement-based acquisition framework. By embedding progress toward stationarity into the acquisition criterion, EI-GN provides a richer and more informative notion of improvement. Empirical results on standard BO benchmarks demonstrate consistent gains over baseline methods, and we further illustrate its applicability to control policy learning.