# PF2ES: Parallel Feasible Pareto Frontier Entropy Search for Multi-Objective Bayesian Optimization

arXiv: https://arxiv.org/abs/2204.05411 · HTML: https://mlsanity.com/paper/2204.05411/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 78 references that are in the map.
- First posted on arXiv 11 Apr 2022.

## Builds on (most cited first, 30 of 32)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (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)
- [pymoo: Multi-objective Optimization in Python](https://mlsanity.com/paper/2002.04504/) (2020)
- [Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2006.05078/) (2020)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement](https://mlsanity.com/paper/2105.08195/) (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)
- [Predictive Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/1511.05467/) (2015)
- [Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces](https://mlsanity.com/paper/2109.10964/) (2021)
- [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)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [GIBBON: General-purpose Information-Based Bayesian OptimisatioN](https://mlsanity.com/paper/2102.03324/) (2021)
- [Scalable Thompson Sampling using Sparse Gaussian Process Models](https://mlsanity.com/paper/2006.05356/) (2020)
- [Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/1609.01051/) (2016)
- [Efficient Computation of Expected Hypervolume Improvement Using Box Decomposition Algorithms](https://mlsanity.com/paper/1904.12672/) (2019)
- [A Bayesian approach to constrained single- and multi-objective optimization](https://mlsanity.com/paper/1510.00503/) (2015)
- [Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound](https://mlsanity.com/paper/2102.09788/) (2021)
- [MUMBO: MUlti-task Max-value Bayesian Optimization](https://mlsanity.com/paper/2006.12093/) (2020)
- [GPflowOpt: A Bayesian Optimization Library using TensorFlow](https://mlsanity.com/paper/1711.03845/) (2017)
- [Optimization as Estimation with Gaussian Processes in Bandit Settings](https://mlsanity.com/paper/1510.06423/) (2015)
- [Improved Max-value Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2011.01150/) (2020)
- [Output Space Entropy Search Framework for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2110.06980/) (2021)
- [Parallel Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2004.00601/) (2020)
- [A Box Decomposition Algorithm to Compute the Hypervolume Indicator](https://mlsanity.com/paper/1510.01963/) (2015)
- [Bayesian Optimisation for Constrained Problems](https://mlsanity.com/paper/2105.13245/) (2021)

## Built on by (most cited first, 5 of 5)

- [Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimisation](https://mlsanity.com/paper/2301.10123/) (2023)
- [Return of the Latent Space COWBOYS: Re-thinking the use of VAEs for Bayesian Optimisation of Structured Spaces](https://mlsanity.com/paper/2507.03910/) (2025)
- [A penalisation method for batch multi-objective Bayesian optimisation with application in heat exchanger design](https://mlsanity.com/paper/2206.13326/) (2022)
- [MOBO-OSD: Batch Multi-Objective Bayesian Optimization via Orthogonal Search Directions](https://mlsanity.com/paper/2510.20872/) (2025)
- [Learning Design-Score Manifold to Guide Diffusion Models for Offline Optimization](https://mlsanity.com/paper/2506.05680/) (2025)

## Abstract (arXiv)

We present Parallel Feasible Pareto Frontier Entropy Search (PF2ES) -- a novel information-theoretic acquisition function for multi-objective Bayesian optimization supporting unknown constraints and batch query. Due to the complexity of characterizing the mutual information between candidate evaluations and (feasible) Pareto frontiers, existing approaches must either employ crude approximations that significantly hamper their performance or rely on expensive inference schemes that substantially increase the optimization's computational overhead. By instead using a variational lower bound, PF2ES provides a low-cost and accurate estimate of the mutual information. We benchmark PF2ES against other information-theoretic acquisition functions, demonstrating its competitive performance for optimization across synthetic and real-world design problems.