# Multi-fidelity Bayesian Optimization with Max-value Entropy Search and its parallelization

arXiv: https://arxiv.org/abs/1901.08275 · HTML: https://mlsanity.com/paper/1901.08275/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2019.

## Built on by (most cited first, 20 of 20)

- [Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement](https://mlsanity.com/paper/2105.08195/) (2021)
- [GIBBON: General-purpose Information-Based Bayesian OptimisatioN](https://mlsanity.com/paper/2102.03324/) (2021)
- [Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimisation](https://mlsanity.com/paper/2301.10123/) (2023)
- [VolcanoML: Speeding up End-to-End AutoML via Scalable Search Space Decomposition](https://mlsanity.com/paper/2107.08861/) (2021)
- [Bayesian Optimization for Cascade-type Multi-stage Processes](https://mlsanity.com/paper/2111.08330/) (2021)
- [Output Space Entropy Search Framework for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2110.06980/) (2021)
- [Constrained Bayesian Optimization Under Partial Observations: Balanced Improvements and Provable Convergence](https://mlsanity.com/paper/2312.03212/) (2023)
- [Batch Multi-Fidelity Bayesian Optimization with Deep Auto-Regressive Networks](https://mlsanity.com/paper/2106.09884/) (2021)
- [Rectified Max-Value Entropy Search for Bayesian Optimization](https://mlsanity.com/paper/2202.13597/) (2022)
- [Investigating Bayesian optimization for expensive-to-evaluate black box functions: Application in fluid dynamics](https://mlsanity.com/paper/2207.09154/) (2022)
- [Bayesian Optimization Augmented with Actively Elicited Expert Knowledge](https://mlsanity.com/paper/2208.08742/) (2022)
- [Efficient Hyperparameter Optimization for Physics-based Character Animation](https://mlsanity.com/paper/2104.12365/) (2021)
- [Information-theoretic Inducing Point Placement for High-throughput Bayesian Optimisation](https://mlsanity.com/paper/2206.02437/) (2022)
- [HyperJump: Accelerating HyperBand via Risk Modelling](https://mlsanity.com/paper/2108.02479/) (2021)
- [Multi-Fidelity Reinforcement Learning for Time-Optimal Quadrotor Re-planning](https://mlsanity.com/paper/2403.08152/) (2024)
- [Information-theoretic Bayesian Optimization: Survey and Tutorial](https://mlsanity.com/paper/2502.06789/) (2025)
- [Constrained multi-fidelity Bayesian optimization with automatic stop condition](https://mlsanity.com/paper/2503.01126/) (2025)
- [Efficient End-to-End AutoML via Scalable Search Space Decomposition](https://mlsanity.com/paper/2206.09423/) (2022)
- [Multifidelity Cross-validation](https://mlsanity.com/paper/2407.01495/) (2024)
- [Constrained Bayesian Optimisation with Multiple Information Sources](https://mlsanity.com/paper/2607.00865/) (2026)

## Abstract (arXiv)

In a standard setting of Bayesian optimization (BO), the objective function evaluation is assumed to be highly expensive. Multi-fidelity Bayesian optimization (MFBO) accelerates BO by incorporating lower fidelity observations available with a lower sampling cost. In this paper, we focus on the information-based approach, which is a popular and empirically successful approach in BO. For MFBO, however, existing information-based methods are plagued by difficulty in estimating the information gain. We propose an approach based on max-value entropy search (MES), which greatly facilitates computations by considering the entropy of the optimal function value instead of the optimal input point. We show that, in our multi-fidelity MES (MF-MES), most of additional computations, compared with usual MES, is reduced to analytical computations. Although an additional numerical integration is necessary for the information across different fidelities, this is only in one dimensional space, which can be performed efficiently and accurately. Further, we also propose parallelization of MF-MES. Since there exist a variety of different sampling costs, queries typically occur asynchronously in MFBO. We show that similar simple computations can be derived for asynchronous parallel MFBO. We demonstrate effectiveness of our approach by using benchmark datasets and a real-world application to materials science data.