# Multi-objective Bayesian Optimization using Pareto-frontier Entropy

arXiv: https://arxiv.org/abs/1906.00127 · HTML: https://mlsanity.com/paper/1906.00127/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2019.

## Built on by (most cited first, 30 of 57)

- [Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement](https://mlsanity.com/paper/2105.08195/) (2021)
- [Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces](https://mlsanity.com/paper/2109.10964/) (2021)
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- [Pareto Set Learning for Expensive Multi-Objective Optimization](https://mlsanity.com/paper/2210.08495/) (2022)
- [Combining Latent Space and Structured Kernels for Bayesian Optimization over Combinatorial Spaces](https://mlsanity.com/paper/2111.01186/) (2021)
- [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)
- [Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound](https://mlsanity.com/paper/2102.09788/) (2021)
- [Multi-objective Asynchronous Successive Halving](https://mlsanity.com/paper/2106.12639/) (2021)
- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [Bayesian Optimization for Cascade-type Multi-stage Processes](https://mlsanity.com/paper/2111.08330/) (2021)
- [Multi-Objective Learning to Predict Pareto Fronts Using Hypervolume Maximization](https://mlsanity.com/paper/2102.04523/) (2021)
- [Randomized Gaussian Process Upper Confidence Bound with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2302.01511/) (2023)
- [Latency-Aware Neural Architecture Search with Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2106.11890/) (2021)
- [Fair and Green Hyperparameter Optimization via Multi-objective and Multiple Information Source Bayesian Optimization](https://mlsanity.com/paper/2205.08835/) (2022)
- [Offline Multi-Objective Optimization](https://mlsanity.com/paper/2406.03722/) (2024)
- [TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusion](https://mlsanity.com/paper/2509.25171/) (2025)
- [Multi-Objective Bayesian Optimization with Active Preference Learning](https://mlsanity.com/paper/2311.13460/) (2023)
- [Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2406.08799/) (2024)
- [qPOTS: Efficient batch multiobjective Bayesian optimization via Pareto optimal Thompson sampling](https://mlsanity.com/paper/2310.15788/) (2023)
- [BOFormer: Learning to Solve Multi-Objective Bayesian Optimization via Non-Markovian RL](https://mlsanity.com/paper/2505.21974/) (2025)
- [Leveraging Trust for Joint Multi-Objective and Multi-Fidelity Optimization](https://mlsanity.com/paper/2112.13901/) (2021)
- [Constrained Multi-objective Bayesian Optimization through Optimistic Constraints Estimation](https://mlsanity.com/paper/2411.03641/) (2024)
- [In-Context Multi-Objective Optimization](https://mlsanity.com/paper/2512.11114/) (2025)
- [Bayesian Optimization For Multi-Objective Mixed-Variable Problems](https://mlsanity.com/paper/2201.12767/) (2022)
- [MBORE: Multi-objective Bayesian Optimisation by Density-Ratio Estimation](https://mlsanity.com/paper/2203.16912/) (2022)
- [PF2ES: Parallel Feasible Pareto Frontier Entropy Search for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2204.05411/) (2022)
- [Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty](https://mlsanity.com/paper/2301.11588/) (2023)
- [Knowledge Gradient for Multi-Objective Bayesian Optimization with Decoupled Evaluations](https://mlsanity.com/paper/2302.01310/) (2023)
- [Preference-Aware Constrained Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2303.13034/) (2023)

## Abstract (arXiv)

This paper studies an entropy-based multi-objective Bayesian optimization (MBO). The entropy search is successful approach to Bayesian optimization. However, for MBO, existing entropy-based methods ignore trade-off among objectives or introduce unreliable approximations. We propose a novel entropy-based MBO called Pareto-frontier entropy search (PFES) by considering the entropy of Pareto-frontier, which is an essential notion of the optimality of the multi-objective problem. Our entropy can incorporate the trade-off relation of the optimal values, and further, we derive an analytical formula without introducing additional approximations or simplifications to the standard entropy search setting. We also show that our entropy computation is practically feasible by using a recursive decomposition technique which has been known in studies of the Pareto hyper-volume computation. Besides the usual MBO setting, in which all the objectives are simultaneously observed, we also consider the "decoupled" setting, in which the objective functions can be observed separately. PFES can easily adapt to the decoupled setting by considering the entropy of the marginal density for each output dimension. This approach incorporates dependency among objectives conditioned on Pareto-frontier, which is ignored by the existing method. Our numerical experiments show effectiveness of PFES through several benchmark datasets.