Joint Entropy Search for Multi-Objective Bayesian Optimization

Ben TuAxel GandyNikolas KantasBehrang Shafei

article2022NeurIPS75 citationsBest Paper Award

Proposes Joint Entropy Search, an information-theoretic acquisition function for multi-objective Bayesian optimization that simultaneously evaluates information gain over optimal inputs and outputs to achieve superior sample efficiency across both synthetic and real-world benchmarks.

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Many critical industrial and scientific challenges—such as chemical reaction engineering, pharmaceutical manufacturing, and structural design—require balancing multiple competing goals simultaneously when testing options is noisy, slow, and expensive. While Bayesian optimization provides an efficient framework to guide these evaluations using probabilistic models, existing selection strategies present significant drawbacks. Many methods rely on arbitrary transformations of objectives or focus narrowly on improving specific geometric metrics like the hypervolume indicator, which can introduce distortion when objective scales are unfamiliar or uncalibrated.

The article aims to introduce and evaluate a new selection strategy called Joint Entropy Search for multi-objective Bayesian optimization. This approach assesses how informative a candidate trial will be by measuring the combined information gained about both the optimal input settings and their corresponding output performance trade-offs.

To establish a practical method, the authors developed analytically tractable, gradient-friendly approximations and lower bounds to calculate the joint entropy gain efficiently in sequential and parallel batch evaluations. They also introduced a generalized hypervolume metric to assess algorithm performance across targeted regions of the trade-off frontier. The credibility of the framework was tested via empirical simulations across 100 random trials on synthetic mathematical benchmarks and three noisy engineering simulations: a chemical synthesis reaction, a penicillin manufacturing process, and a multi-attribute ship design problem.

The findings show that Joint Entropy Search consistently ranks among the top-performing methods across both sequential and batch experimental settings, matching or outperforming established state-of-the-art approaches. Furthermore, the framework offers inherent theoretical robustness: unlike indicator-based methods, information-theoretic approaches remain invariant to monotonic rescaling of objectives, ensuring that performance does not depend on arbitrary problem parameterizations. In terms of computation, lower-bound approximations matched the optimization quality of more complex estimation methods while keeping acquisition time comparable to existing approaches and significantly faster than classic input-based entropy methods.

These results demonstrate that organizations can reduce the total number of physical trials required to discover optimal trade-offs, leading to lower experimentation costs and shorter development timelines in complex R&D pipelines. The framework is especially valuable in initial exploratory phases where stakeholder preferences are undefined and scale invariance prevents premature, biased decisions. However, because information-theoretic methods explore broadly rather than exploiting immediately known points, practitioners whose final decision is restricted strictly to evaluated points should pair the strategy with a greedy decision rule to query top-performing solutions directly.

Moving forward, adopting teams are recommended to utilize the analytical lower-bound formulation for practical deployments to minimize computational overhead. Future engineering work is required to extend the method's computational scalability to higher-dimensional problems, integrate explicit operational constraints, and expand support for multi-fidelity simulations.

BASF SE Imperial College London bayesian methods hypervolume multi-objective optimization problem Joint Entropy Search Multi-Objective Bayesian Optimization multi-objective acquisition functions

arXiv: 2210.02905

Paper: A Tutorial on Bayesian Optimization, Peter I. Frazier (2018). This tutorial provides a comprehensive foundation in Gaussian process surrogates and entropy-based acquisition functions that underpin Bayesian optimization selection strategies.
Paper: A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning, Eric Brochu et al. (2010). This foundational tutorial introduces the core principles of Bayesian optimization for expensive black-box functions, establishing the exploitation-exploration trade-offs central to acquisition design.
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Paper: Pymoo: Multi-Objective Optimization in Python, Julian Blank et al. (2020). This paper establishes the formal multi-objective optimization benchmarks, Pareto frontier concepts, and hypervolume evaluation frameworks adapted in multi-objective Bayesian optimization studies.
Paper: Variational Learning of Inducing Variables in Sparse Gaussian Processes, Michalis K. Titsias (2009). This paper presents the variational learning framework for Gaussian processes that enables tractable approximations and gradient-friendly bounds for surrogate modeling.
Paper: Practical Bayesian Optimization of Machine Learning Algorithms, Jasper Snoek et al. (2012). This work outlines practical methodologies for Gaussian process Bayesian optimization including parallel batch evaluation strategies.

Paper: Increasing the Scope as You Learn: Adaptive Bayesian Optimization in Nested Subspaces, Leonard Papenmeier et al. (2022). This paper tackles the high-dimensional scaling bottleneck in Bayesian optimization via adaptively expanding nested subspaces, extending the practical scalability of surrogate search.
Paper: Sample Efficiency Matters: A Benchmark for Practical Molecular Optimization, Wenhao Gao et al. (2022). This benchmark provides a standardized experimental testbed for evaluating the sample efficiency of Bayesian optimization and search algorithms in practical chemical and molecular discovery.
Paper: Rewards-in-Context: Multi-objective Alignment of Foundation Models with Dynamic Preference Adjustment, Rui Yang et al. (2024). This work applies multi-objective Pareto optimization principles to the dynamic alignment and trade-off calibration of foundation models across conflicting objectives.

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Abstract

Many real-world problems can be phrased as a multi-objective optimization problem, where the goal is to identify the best set of compromises between the competing objectives. Multi-objective Bayesian optimization (BO) is a sample efficient strategy that can be deployed to solve these vector-valued optimization problems where access is limited to a number of noisy objective function evaluations. In this paper, we propose a novel information-theoretic acquisition function for BO called Joint Entropy Search (JES), which considers the joint information gain for the optimal set of inputs and outputs. We present several analytical approximations to the JES acquisition function and also introduce an extension to the batch setting. We showcase the effectiveness of this new approach on a range of synthetic and real-world problems in terms of the hypervolume and its weighted variants.

Topics

bayesian methods hypervolume multi-objective optimization problem optimization constrained optimization problems

Keywords

Joint Entropy Search Multi-Objective Bayesian Optimization multi-objective acquisition functions information-theoretic acquisition functions Parallel Predictive Entropy Search Max-value Entropy Search Output Space Entropy Search efficient global optimization black-box optimization

1 Introduction
2 Preliminaries
3 Approximating JES
3.1 Estimating the conditional entropy
3.2 Batch evaluations
4 Performance criteria
5 Experiments
5.1 Benchmarks
5.2 Results and discussion
6 Conclusion
Acknowledgments and Disclosure of Funding
References
Checklist

Knowls

Knowl 1 — Joint Entropy Search Acquisition Function for Multi-Objective Bayesian Optimization
model/method
Let $f: \mathcal{X} \to \mathbb{R}^M$ be an unknown vector-valued black-box objective function over a bounded input domain $\mathcal{X} \subset \mathbb{R}^D$ , with $M$ competing objectives. Observations at an input location $x \in \mathcal{X}$ are corrupted by additive Gaussian noise $y = f(x) + \epsilon$ , where $\epsilon \sim \mathcal{N}(0, \text{diag}(\sigma(x)))$ with noise variance vector $\sigma(x) \in \mathbb{R}_{\ge 0}^M$ . Given observed data $D_n = \{(x_t, y_t)\}_{t=1}^n$ , independent Gaussian process priors are placed on each objective component $f^{(m)} \sim \mathcal{GP}(\mu_0^{(m)}, \Sigma_0^{(m)})$ .

The Pareto optimal set of inputs is defined as $X^* = \arg\max_{x \in \mathcal{X}} f(x) = \{x^* \in \mathcal{X} : \nexists x \in \mathcal{X} \text{ s.t. } f(x) \succ f(x^*)\}$ , where $\succ$ denotes strict Pareto domination, and its image in objective space is the Pareto front $Y^* = f(X^*)$ .

The Joint Entropy Search (JES) acquisition function measures the mutual information between the next observation $y$ and the joint distribution of the optimal inputs and outputs $(X^*, Y^*)$ conditioned on $D_n$ : $\alpha^{\text{JES}}(x \mid D_n) = \text{MI}(y; (X^*, Y^*) \mid x, D_n) = H[p(y \mid x, D_n)] - \mathbb{E}_{p((X^*, Y^*) \mid D_n)}\left[H[p(y \mid x, D_n, (X^*, Y^*))]\right]$ where $H[p(z)] = -\int p(z) \log p(z) dz$ denotes differential entropy. This criterion unifies input-space information gain (targeted by Predictive Entropy Search, PES) and output-space information gain (targeted by Max-value Entropy Search, MES).
Knowl 2 — Joint Entropy Search as an Upper Bound to PES and MES
theoretical result
Let $\alpha^{\text{PES}}(x \mid D_n) = \text{MI}(y; X^* \mid x, D_n)$ denote the Predictive Entropy Search acquisition function targeting the Pareto optimal set $X^*$ , and let $\alpha^{\text{MES}}(x \mid D_n) = \text{MI}(y; Y^* \mid x, D_n)$ denote the Max-value Entropy Search acquisition function targeting the Pareto front $Y^*$ , evaluated at candidate input $x \in \mathcal{X}$ given dataset $D_n$ .

For any convex combination weight $\beta \in [0, 1]$ , the Joint Entropy Search acquisition function $\alpha^{\text{JES}}(x \mid D_n) = \text{MI}(y; (X^*, Y^*) \mid x, D_n)$ provides an upper bound to both utilities: $\alpha^{\text{JES}}(x \mid D_n) \ge \beta \alpha^{\text{PES}}(x \mid D_n) + (1 - \beta) \alpha^{\text{MES}}(x \mid D_n)$
Knowl 3 — Invariance of Joint Entropy Search to Monotonic Reparameterizations
theoretical result
Let $f: \mathcal{X} \to \mathbb{R}^M$ be a vector-valued black-box objective function, and let $g(y) = (g_1(y^{(1)}), \dots, g_M(y^{(M)}))^{\top}$ be an objective transformation where each $g_m: \mathbb{R} \to \mathbb{R}$ is a strictly monotonically increasing function acting exclusively on the $m$ -th objective component. Such transformations preserve Pareto ordering relations in the objective space.

The information-theoretic acquisition functions Predictive Entropy Search (PES), Max-value Entropy Search (MES), and Joint Entropy Search (JES) are strictly invariant to any such monotonic reparameterization: $\alpha^{\text{JES}}(x \mid D_n) = \text{MI}(y; (X^*, Y^*) \mid x, D_n) = \text{MI}(g(y); (X^*, g(Y^*)) \mid x, D_n)$ $\alpha^{\text{PES}}(x \mid D_n) = \text{MI}(y; X^* \mid x, D_n) = \text{MI}(g(y); X^* \mid x, D_n)$ $\alpha^{\text{MES}}(x \mid D_n) = \text{MI}(y; Y^* \mid x, D_n) = \text{MI}(g(y); g(Y^*) \mid x, D_n)$ Consequently, unlike hypervolume indicator-based acquisition functions, information-theoretic criteria are inherently scale-agnostic across objectives.
Knowl 4 — Analytic First and Second Central Moments of the Conditional Skew-Normal Distribution
theoretical result
Let $D_{n*} = D_n \cup (X^*, Y^*)$ denote the dataset augmented with a discrete sample of the optimal Pareto points $(X^*, Y^*)$ . Under independent Gaussian process priors for each objective $m \in \{1, \dots, M\}$ , the posterior predictive distribution at candidate input $x$ is Gaussian with mean $\mu_{n*}^{(m)}(x)$ and marginal variance $\Sigma_{n*}^{(m)}(x, x)$ , with observation noise variance $\sigma^{(m)}(x)$ .

Let the Pareto-dominated region in objective space be decomposed into $J$ disjoint hyperrectangles $\mathcal{D}(Y^*) = \bigcup_{j=1}^J B_j = \bigcup_{j=1}^J \prod_{m=1}^M (l_j^{(m)}, u_j^{(m)}]$ . Define the standardized coordinates for any $z \in \mathbb{R}$ : $\gamma_m(z) = \frac{z - \mu_{n*}^{(m)}(x)}{\sqrt{\Sigma_{n*}^{(m)}(x, x)}}$ For each hyperrectangle $j \in \{1, \dots, J\}$ and objective $m \in \{1, \dots, M\}$ , define: $W_{j,m} = \Phi\left(\gamma_m(u_j^{(m)})\right) - \Phi\left(\gamma_m(l_j^{(m)})\right)$ $G_{j,m} = \phi\left(\gamma_m(u_j^{(m)})\right) - \phi\left(\gamma_m(l_j^{(m)})\right)$ $V_{j,m} = \gamma_m(u_j^{(m)})\phi\left(\gamma_m(u_j^{(m)})\right) - \gamma_m(l_j^{(m)})\phi\left(\gamma_m(l_j^{(m)})\right)$ where $\Phi$ and $\phi$ are the standard normal CDF and PDF, respectively, with $W_j = \prod_{m=1}^M W_{j,m}$ and total cumulative probability $W = \sum_{j=1}^J W_j = p(f(x) \preceq Y^* \mid x, D_{n*})$ .

Under the local non-domination condition $f(x) \preceq Y^*$ , the first and second central moments of $p(y \mid x, D_{n*}, f(x) \preceq Y^*)$ are given analytically by: $\mathbb{E}[y^{(m)} \mid x, D_{n*}, f(x) \preceq Y^*] = \mu_{n*}^{(m)}(x) - \frac{\sqrt{\Sigma_{n*}^{(m)}(x, x)}}{W} \sum_{j=1}^J W_j \frac{G_{j,m}}{W_{j,m}}$ For $m \neq m'$ : $\text{Cov}\left(y^{(m)}, y^{(m')} \mid x, D_{n*}, f(x) \preceq Y^*\right) = \frac{\sqrt{\Sigma_{n*}^{(m)}(x, x)}\sqrt{\Sigma_{n*}^{(m')}(x, x)}}{W} \sum_{j=1}^J W_j \frac{G_{j,m}}{W_{j,m}} \left(\frac{G_{j,m'}}{W_{j,m'}} - \frac{1}{W} \sum_{j'=1}^J W_{j'} \frac{G_{j',m'}}{W_{j',m'}}\right)$ For $m = m'$ : $\text{Var}\left(y^{(m)} \mid x, D_{n*}, f(x) \preceq Y^*\right) = \Sigma_{n*}^{(m)}(x, x) + \sigma^{(m)}(x) - \frac{\Sigma_{n*}^{(m)}(x, x)}{W} \left(\sum_{j=1}^J W_j \frac{V_{j,m}}{W_{j,m}} + \frac{1}{W}\left(\sum_{j=1}^J W_j \frac{G_{j,m}}{W_{j,m}}\right)^2\right)$
Knowl 5 — Moment-Matched Analytical Lower Bounds for Joint Entropy Search
model/method
The expectation in the Joint Entropy Search acquisition function is approximated using $S$ Monte Carlo samples $(X_s^*, Y_s^*) \sim p((X^*, Y^*) \mid D_n)$ : $\hat{\alpha}^{\text{JES}}(x \mid D_n) = H[p(y \mid x, D_n)] - \frac{1}{S} \sum_{s=1}^S h((X_s^*, Y_s^*); x, D_n)$ where the initial entropy under independent Gaussian processes is: $H[p(y \mid x, D_n)] = \frac{M}{2}\log(2\pi e) + \frac{1}{2}\sum_{m=1}^M \log\left(\Sigma_n^{(m)}(x, x) + \sigma^{(m)}(x)\right)$ Because the conditional distribution $p(y \mid x, D_{n*}, f(x) \preceq Y^*)$ is skew normal without an analytical entropy formula, upper bounding its entropy via Gaussian moment matching yields two analytical, differentiable lower bounds for JES:
1. JES-LB (Full covariance moment matching): $h^{\text{JES-LB}}((X^*, Y^*); x, D_n) = \frac{M}{2}\log(2\pi e) + \frac{1}{2} \log \det \text{Var}\left(y \mid x, D_{n*}, f(x) \preceq Y^*\right)$
2. JES-LB2 (Diagonal covariance moment matching): $h^{\text{JES-LB2}}((X^*, Y^*); x, D_n) = \frac{M}{2}\log(2\pi e) + \frac{1}{2} \sum_{m=1}^M \log \text{Var}\left(y^{(m)} \mid x, D_{n*}, f(x) \preceq Y^*\right)$ where the moments are computed via exact closed-form expressions.
Knowl 6 — Joint Entropy Search Acquisition Function Estimation
algorithm
The Joint Entropy Search (JES) acquisition function estimation evaluates candidate points $x \in \mathcal{X}$ using cached Pareto samples and analytical conditional entropy estimates. Input-independent quantities—sampling function paths, identifying Pareto optimal sets and fronts, box-decomposing dominated space, and conditioning Gaussian processes on the augmented samples—are precomputed once per Bayesian optimization step.
Input: Candidate input $x \in \mathcal{X}$ , dataset $D_n = \{(x_t, y_t)\}_{t=1}^n$ , number of Monte Carlo samples $S$ .
Output: Acquisition value $\hat{\alpha}^{\text{JES}}(x \mid D_n)$ .
Compute initial predictive entropy $h_0 = H[p(y \mid x, D_n)]$ .
for $s = 1, \dots, S$ do
    Sample a continuous function path $f_s \sim p(f \mid D_n)$ .
    Compute the Pareto optimal points $X_s^* = \arg\max_{x' \in \mathcal{X}} f_s(x')$ and Pareto front $Y_s^* = f_s(X_s^*)$ .
    Compute the hyperrectangle box decomposition of the dominated region $\mathcal{D}(Y_s^*) = \bigcup_{j=1}^J B_j$ .
    Condition Gaussian processes on the augmented dataset $D_n \cup (X_s^*, Y_s^*)$ .
    Compute the conditional entropy estimate $h_s = h((X_s^*, Y_s^*); x, D_n)$ .
end for
return $h_0 - \frac{1}{S} \sum_{s=1}^S h_s$
Knowl 7 — Batch Joint Entropy Search via Submodular Entropy Upper Bound
model/method
Evaluating the joint acquisition utility for a batch of $q$ candidate points $x^{[1:q]} = (x^{[1]}, \dots, x^{[q]}) \in \mathcal{X}^q$ directly requires high-dimensional cumulative normal distributions. To maintain tractability, the joint conditional entropy is upper-bounded by the sum of individual marginal conditional entropies: $H\left[p\left(y^{[1:q]} \mid x^{[1:q]}, D_{n*}, f(\mathcal{X}) \preceq Y^*\right)\right] \le \sum_{i=1}^q H\left[p\left(y^{[i]} \mid x^{[i]}, D_{n*}, f(x^{[i]}) \preceq Y^*\right)\right]$ This yields the $q$ -batch lower bound JES acquisition function: $\hat{\alpha}^{\text{qLB-JES}}(x^{[1:q]} \mid D_n) = H\left[p\left(y^{[1:q]} \mid x^{[1:q]}, D_n\right)\right] - \frac{1}{S} \sum_{s=1}^S \sum_{i=1}^q h\left((X_s^*, Y_s^*); x^{[i]}, D_n\right)$ where the initial batch joint entropy is: $H\left[p\left(y^{[1:q]} \mid x^{[1:q]}, D_n\right)\right] = \frac{Mq}{2}\log(2\pi e) + \frac{1}{2} \sum_{m=1}^M \log \det\left(\Sigma_n^{(m)}(x^{[1:q]}, x^{[1:q]}) + \text{diag}\left(\sigma^{(m)}(x^{[1:q]})\right)\right)$ Because $\hat{\alpha}^{\text{qLB-JES}}$ is submodular, it can be optimized efficiently by selecting batch candidates greedily in sequence.
Knowl 8 — Empirical Performance of Joint Entropy Search Across Multi-Objective Benchmarks
empirical result
The JES acquisition function was evaluated across 100 random initial seeds on four vector-valued benchmark problems corrupted with additive Gaussian noise:
1. ZDT2 synthetic benchmark ( $D=6, M=2$ , $10\%$ noise).
2. SnAr chemical reaction optimization ( $D=4, M=2$ , $3\%$ noise, space-time yield vs environmental impact).
3. Penicillin pharmaceutical manufacturing ( $D=7, M=3$ , $1\%$ noise, yield, carbon dioxide release, fermentation time).
4. Marine bulk carrier design ( $D=6, M=4$ , $0.5\%$ noise, annual cargo, transportation cost, ship weight, constraint penalty).
Across sequential optimization iterations ( $q=1$ ), JES-LB consistently achieved lower mean logarithm hypervolume discrepancy $\log(d_{\text{HV}})$ than competitive multi-objective baselines, including TSEMO, ParEGO, NParEGO, NEHVI, PES, and MES-LB. In batch evaluations with batch sizes $q \in \{1, 2, 4, 8\}$ , the submodular batch estimator LB-JES-LB consistently matched or outperformed batch NEHVI and LB-MES-LB across all test problems. Furthermore, the computationally cheaper lower-bound entropy estimators (JES-LB and JES-LB2) achieved optimization performance equivalent to expensive Monte Carlo integral estimates while having substantially lower acquisition wall times.
Knowl 9 — Computational Scalability Bottlenecks of Joint Entropy Search
limitation
Joint Entropy Search encounters three main computational bottlenecks:
1. Dominated space box decomposition: Computing the box decomposition $\mathcal{D}(Y^*)$ requires $O(|Y^*|^{\lfloor M/2 \rfloor + 1})$ operations, which scales poorly as the number of objectives $M$ and Pareto front points $|Y^*|$ grow.
2. Augmented GP posterior updates: Evaluating conditional variances after augmenting the dataset with the Pareto set $(X^*, Y^*)$ scales as $O((n + |Y^*|)^2)$ per candidate point, compared to $O(n^2)$ for Max-value Entropy Search (MES).
3. Sample path optimization: Estimating the Pareto set $X_s^*$ for each sampled function path $f_s \sim p(f \mid D_n)$ requires executing a multi-objective optimizer across $S$ posterior draws at every Bayesian optimization iteration.

Coverage note — Generalized hypervolume weighting equations from Appendix K and detailed numerical hyperparameter tables of benchmark optimizers from Appendix L were omitted as supplementary implementation details.

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Citation

MLA

Tu, B., et al. “Joint Entropy Search for Multi-Objective Bayesian Optimization”. Advances in Neural Information Processing Systems, vol. 35, 2022, pp. 9922–38, https://proceedings.neurips.cc/paper_files/paper/2022/file/4086fe59dc3584708468fba0e459f6a7-Paper-Conference.pdf.

APA

Tu, B., Gandy, A., Kantas, N., & Shafei, B. (2022). Joint Entropy Search for Multi-Objective Bayesian Optimization. Advances in Neural Information Processing Systems, 35, 9922–9938. https://proceedings.neurips.cc/paper_files/paper/2022/file/4086fe59dc3584708468fba0e459f6a7-Paper-Conference.pdf

Chicago

Tu, B., A. Gandy, N. Kantas, and B. Shafei. 2022. “Joint Entropy Search for Multi-Objective Bayesian Optimization”. Advances in Neural Information Processing Systems 35: 9922–38. https://proceedings.neurips.cc/paper_files/paper/2022/file/4086fe59dc3584708468fba0e459f6a7-Paper-Conference.pdf.

Harvard

Tu, B. et al. (2022) “Joint Entropy Search for Multi-Objective Bayesian Optimization”, Advances in Neural Information Processing Systems. Curran Associates, Inc., pp. 9922–9938. Available at: https://proceedings.neurips.cc/paper_files/paper/2022/file/4086fe59dc3584708468fba0e459f6a7-Paper-Conference.pdf.

Vancouver

1. Tu B, Gandy A, Kantas N, Shafei B (2022) Joint Entropy Search for Multi-Objective Bayesian Optimization. In: Advances in Neural Information Processing Systems. Curran Associates, Inc., pp 9922–9938

BibTeX

@inproceedings{tu2022joint,
  title = {Joint Entropy Search for Multi-Objective Bayesian Optimization},
  author = {Tu, Ben and Gandy, Axel and Kantas, Nikolas and Shafei, Behrang},
  year = {2022},
  booktitle = {Advances in Neural Information Processing Systems},
  publisher = {Curran Associates, Inc.},
  volume = {35},
  pages = {9922-9938},
  url = {https://proceedings.neurips.cc/paper_files/paper/2022/file/4086fe59dc3584708468fba0e459f6a7-Paper-Conference.pdf}
}

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Joint Entropy Search for Multi-Objective Bayesian Optimization

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Knowls

Knowl 1 — Joint Entropy Search Acquisition Function for Multi-Objective Bayesian Optimization

Knowl 2 — Joint Entropy Search as an Upper Bound to PES and MES

Knowl 3 — Invariance of Joint Entropy Search to Monotonic Reparameterizations

Knowl 4 — Analytic First and Second Central Moments of the Conditional Skew-Normal Distribution

Knowl 5 — Moment-Matched Analytical Lower Bounds for Joint Entropy Search

Knowl 6 — Joint Entropy Search Acquisition Function Estimation

Knowl 7 — Batch Joint Entropy Search via Submodular Entropy Upper Bound

Knowl 8 — Empirical Performance of Joint Entropy Search Across Multi-Objective Benchmarks

Knowl 9 — Computational Scalability Bottlenecks of Joint Entropy Search

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