# Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement

arXiv: https://arxiv.org/abs/2105.08195 · HTML: https://mlsanity.com/paper/2105.08195/

## Key facts

- Cited by 118 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 74 references that are in the map.
- First posted on arXiv 17 May 2021.

## Builds on (most cited first, 26 of 26)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Regularized Evolution for Image Classifier Architecture Search](https://mlsanity.com/paper/1802.01548/) (2018)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [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)
- [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)
- [A Flexible Framework for Multi-Objective Bayesian Optimization using Random Scalarizations](https://mlsanity.com/paper/1805.12168/) (2018)
- [Max-value Entropy Search for Multi-Objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2009.01721/) (2020)
- [An Easy-to-use Real-world Multi-objective Optimization Problem Suite](https://mlsanity.com/paper/2009.12867/) (2020)
- [Random Hypervolume Scalarizations for Provable Multi-Objective Black Box Optimization](https://mlsanity.com/paper/2006.04655/) (2020)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [Parallel Bayesian Global Optimization of Expensive Functions](https://mlsanity.com/paper/1602.05149/) (2016)
- [Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/1609.01051/) (2016)
- [Data-efficient Learning of Morphology and Controller for a Microrobot](https://mlsanity.com/paper/1905.01334/) (2019)
- [Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization](https://mlsanity.com/paper/2006.11267/) (2020)
- [Efficient Computation of Expected Hypervolume Improvement Using Box Decomposition Algorithms](https://mlsanity.com/paper/1904.12672/) (2019)
- [Multi-fidelity Bayesian Optimization with Max-value Entropy Search and its parallelization](https://mlsanity.com/paper/1901.08275/) (2019)
- [Bayesian Optimization for Policy Search via Online-Offline Experimentation](https://mlsanity.com/paper/1904.01049/) (2019)
- [Parallel Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2004.00601/) (2020)
- [Real-world Video Adaptation with Reinforcement Learning](https://mlsanity.com/paper/2008.12858/) (2020)
- [Using Well-Understood Single-Objective Functions in Multiobjective Black-Box Optimization Test Suites](https://mlsanity.com/paper/1604.00359/) (2016)
- [Thompson Sampling for Contextual Bandit Problems with Auxiliary Safety Constraints](https://mlsanity.com/paper/1911.00638/) (2019)
- [A Box Decomposition Algorithm to Compute the Hypervolume Indicator](https://mlsanity.com/paper/1510.01963/) (2015)
- [Uncrowded Hypervolume Improvement: COMO-CMA-ES and the Sofomore framework](https://mlsanity.com/paper/1904.08823/) (2019)

## Built on by (most cited first, 30 of 118)

- [Accelerating Bayesian Optimization for Biological Sequence Design with Denoising Autoencoders](https://mlsanity.com/paper/2203.12742/) (2022)
- [Multi-Objective GFlowNets](https://mlsanity.com/paper/2210.12765/) (2022)
- [Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces](https://mlsanity.com/paper/2109.10964/) (2021)
- [Pareto Set Learning for Expensive Multi-Objective Optimization](https://mlsanity.com/paper/2210.08495/) (2022)
- [Sample-efficient Multi-objective Molecular Optimization with GFlowNets](https://mlsanity.com/paper/2302.04040/) (2023)
- [LightTS: Lightweight Time Series Classification with Adaptive Ensemble Distillation -- Extended Version](https://mlsanity.com/paper/2302.12721/) (2023)
- [AutoPEFT: Automatic Configuration Search for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2301.12132/) (2023)
- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [Robust Multi-Objective Bayesian Optimization Under Input Noise](https://mlsanity.com/paper/2202.07549/) (2022)
- [Multi-Objective Hyperparameter Optimization in Machine Learning -- An Overview](https://mlsanity.com/paper/2206.07438/) (2022)
- [Bayesian Optimization over Discrete and Mixed Spaces via Probabilistic Reparameterization](https://mlsanity.com/paper/2210.10199/) (2022)
- [Personalized LLM Response Generation with Parameterized Memory Injection](https://mlsanity.com/paper/2404.03565/) (2024)
- [Bayesian Optimization with Conformal Prediction Sets](https://mlsanity.com/paper/2210.12496/) (2022)
- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [ParetoFlow: Guided Flows in Multi-Objective Optimization](https://mlsanity.com/paper/2412.03718/) (2024)
- [Latency-Aware Neural Architecture Search with Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2106.11890/) (2021)
- [BOtied: Multi-objective Bayesian optimization with tied multivariate ranks](https://mlsanity.com/paper/2306.00344/) (2023)
- [Offline Multi-Objective Optimization](https://mlsanity.com/paper/2406.03722/) (2024)
- [Faster, Cheaper, Better: Multi-Objective Hyperparameter Optimization for LLM and RAG Systems](https://mlsanity.com/paper/2502.18635/) (2025)
- [qPOTS: Efficient batch multiobjective Bayesian optimization via Pareto optimal Thompson sampling](https://mlsanity.com/paper/2310.15788/) (2023)
- [Expensive Multi-Objective Bayesian Optimization Based on Diffusion Models](https://mlsanity.com/paper/2405.08674/) (2024)
- [BOFormer: Learning to Solve Multi-Objective Bayesian Optimization via Non-Markovian RL](https://mlsanity.com/paper/2505.21974/) (2025)
- [Looper: An end-to-end ML platform for product decisions](https://mlsanity.com/paper/2110.07554/) (2021)
- [A portfolio approach to massively parallel Bayesian optimization](https://mlsanity.com/paper/2110.09334/) (2021)
- [A Robust Multi-Objective Bayesian Optimization Framework Considering Input Uncertainty](https://mlsanity.com/paper/2202.12848/) (2022)
- [PropertyDAG: Multi-objective Bayesian optimization of partially ordered, mixed-variable properties for biological sequence design](https://mlsanity.com/paper/2210.04096/) (2022)
- [Collaborative Bayesian Optimization via Wasserstein Barycenters](https://mlsanity.com/paper/2504.10770/) (2025)
- [CATransformers: Carbon Aware Transformers Through Joint Model-Hardware Optimization](https://mlsanity.com/paper/2505.01386/) (2025)
- [Sparse Bayesian Optimization](https://mlsanity.com/paper/2203.01900/) (2022)
- [Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach](https://mlsanity.com/paper/2306.05843/) (2023)

## Abstract (arXiv)

Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this uncertainty in the Pareto frontier. We argue that, even in the noiseless setting, generating multiple candidates in parallel is an incarnation of EHVI with uncertainty in the Pareto frontier and therefore can be addressed using the same underlying technique. Through this lens, we derive a natural parallel variant, qNEHVI, that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. qNEHVI is one-step Bayes-optimal for hypervolume maximization in both noisy and noiseless environments, and we show that it can be optimized effectively with gradient-based methods via sample average approximation. Empirically, we demonstrate not only that qNEHVI is substantially more robust to observation noise than existing MOBO approaches, but also that it achieves state-of-the-art optimization performance and competitive wall-times in large-batch environments.