# Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound

arXiv: https://arxiv.org/abs/2102.09788 · HTML: https://mlsanity.com/paper/2102.09788/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 67 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [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)
- [Max-value Entropy Search for Multi-Objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2009.01721/) (2020)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [Predictive Entropy Search for Bayesian Optimization with Unknown Constraints](https://mlsanity.com/paper/1502.05312/) (2015)
- [A General Framework for Constrained Bayesian Optimization using Information-based Search](https://mlsanity.com/paper/1511.09422/) (2015)
- [GIBBON: General-purpose Information-Based Bayesian OptimisatioN](https://mlsanity.com/paper/2102.03324/) (2021)
- [Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian](https://mlsanity.com/paper/1605.09466/) (2016)
- [Constrained Bayesian Optimization with Max-Value Entropy Search](https://mlsanity.com/paper/1910.07003/) (2019)
- [Improved Max-value Entropy Search for Multi-objective Bayesian Optimization with Constraints](https://mlsanity.com/paper/2011.01150/) (2020)

## Built on by (most cited first, 23 of 23)

- [Joint Entropy Search for Maximally-Informed Bayesian Optimization](https://mlsanity.com/paper/2206.04771/) (2022)
- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [A General Framework for User-Guided Bayesian Optimization](https://mlsanity.com/paper/2311.14645/) (2023)
- [Bayesian Optimization for Cascade-type Multi-stage Processes](https://mlsanity.com/paper/2111.08330/) (2021)
- [Randomized Gaussian Process Upper Confidence Bound with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2302.01511/) (2023)
- [Posterior Sampling-Based Bayesian Optimization with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2311.03760/) (2023)
- [Constrained Bayesian Optimization Under Partial Observations: Balanced Improvements and Provable Convergence](https://mlsanity.com/paper/2312.03212/) (2023)
- [Constrained Multi-objective Bayesian Optimization through Optimistic Constraints Estimation](https://mlsanity.com/paper/2411.03641/) (2024)
- [PF2ES: Parallel Feasible Pareto Frontier Entropy Search for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2204.05411/) (2022)
- [Robust Entropy Search for Safe Efficient Bayesian Optimization](https://mlsanity.com/paper/2405.19059/) (2024)
- [Regret Analysis for Randomized Gaussian Process Upper Confidence Bound](https://mlsanity.com/paper/2409.00979/) (2024)
- [Regret Analysis of Posterior Sampling-Based Expected Improvement for Bayesian Optimization](https://mlsanity.com/paper/2507.09828/) (2025)
- [A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization](https://mlsanity.com/paper/2501.18756/) (2025)
- [Pareto-frontier Entropy Search with Variational Lower Bound Maximization](https://mlsanity.com/paper/2501.19073/) (2025)
- [BILBO: BILevel Bayesian Optimization](https://mlsanity.com/paper/2502.02121/) (2025)
- [Constrained Bayesian Optimization with Adaptive Active Learning of Unknown Constraints](https://mlsanity.com/paper/2310.08751/) (2023)
- [Randomized Kriging Believer for Parallel Bayesian Optimization with Regret Bounds](https://mlsanity.com/paper/2603.01470/) (2026)
- [α-PFN: Fast Entropy Search via In-Context Learning](https://mlsanity.com/paper/2606.07134/) (2026)
- [Information-Theoretic Bayesian Optimization for Bilevel Optimization Problems](https://mlsanity.com/paper/2509.21725/) (2025)
- [A Constrained Multi-Fidelity Bayesian Optimization Method](https://mlsanity.com/paper/2510.10984/) (2025)
- [Local Constrained Bayesian Optimization](https://mlsanity.com/paper/2603.07965/) (2026)
- [Towards Regret Guarantees for One-Step Lookahead Bayesian Optimization](https://mlsanity.com/paper/2606.00956/) (2026)
- [Constrained Bayesian Optimisation with Multiple Information Sources](https://mlsanity.com/paper/2607.00865/) (2026)

## Abstract (arXiv)

Max-value entropy search (MES) is one of the state-of-the-art approaches in Bayesian optimization (BO). In this paper, we propose a novel variant of MES for constrained problems, called Constrained MES via Information lower BOund (CMES-IBO), that is based on a Monte Carlo (MC) estimator of a lower bound of a mutual information (MI). Unlike existing studies, our MI is defined so that uncertainty with respect to feasibility can be incorporated. We derive a lower bound of the MI that guarantees non-negativity, while a constrained counterpart of conventional MES can be negative. We further provide theoretical analysis that assures the low-variability of our estimator which has never been investigated for any existing information-theoretic BO. Moreover, using the conditional MI, we extend CMES-IBO to the parallel setting while maintaining the desirable properties. We demonstrate the effectiveness of CMES-IBO by several benchmark functions and real-world problems.