A Unified Information-Theoretic Approach to Constrained Multi-Fidelity Multi-Objective Bayesian Optimization

Rikuto Matsumoto; Masanori Ishikura; Masayuki Karasuyama

cs.LGOct 7, 2026

A Unified Information-Theoretic Approach to Constrained Multi-Fidelity Multi-Objective Bayesian Optimization

Authors: Rikuto Matsumoto, Masanori Ishikura, Masayuki Karasuyama

Organizations: Nagoya Institute of Technology

Abstract

Bayesian optimization often involves multiple objectives, constraints, and fidelity levels. We address the challenge of jointly selecting where and at which fidelity to evaluate to identify the highest-fidelity feasible Pareto frontier in this combined setting. From a unified information-theoretic perspective, we measure query utility by the information gain about this frontier, provided by an observation. Since this mutual information is intractable, we derive a variational lower bound using a mixture of under- and over-truncated approximations to the Pareto-consistent region. Multi-fidelity surrogate models propagate the information to arbitrary fidelities, yielding a cost-aware acquisition function without separate heuristics for fidelity selection or constraint handling. Experiments on synthetic, benchmark, and real-world problems demonstrate effectiveness across diverse objective, constraint, and fidelity settings.

Figures & tables

Figure 1: Admissible target-fidelity regions for two objectives and one constraint. The $Z$ -axis represents the constraint $g_{1}^{(M)}$ , for which below the threshold $z$ (purple level plane) is infeasible. The $XY$ -plane corresponds to two objectives $f_{1}^{(M)}$ and $f_{2}^{(M)}$ , on which the Pareto frontier ${\cal F}^{\ast}$ is defined, depicted as the green wavy plane. (a) The distribution of $\boldsymbol{h}_{\boldsymbol{x}}^{(M)}$ (the sphere) is truncated by ${\cal F}^{\ast}$ and $g_{1}^{(M)}(\boldsymbol{x})=z$ . (b) Based on sampled $\widehat{{\cal F}}^{\ast}$ depicted as the red diamonds, the approximate truncated region is constructed by the over-truncation (the yellow zigzag plane). (c) Same as (b), the approximate truncated region is constructed by the under-truncation (the yellow zigzag plane). The truncations in (b) and (c) are two extreme approximations of the unknown truncation in (a). Equation ( 4 ) uses a mixture of their truncated distributions as the variational distribution.

Panel of Figure 2 — Figure 2: Results on GP-derived synthetic functions (10 runs average and standard error).

Panel of Figure 3 — Figure 3: Results on benchmark functions (10 runs average and standard error).

Panel of Figure 4 — Figure 4: Results on hyper-parameter selection of LightGBM (10 runs average and standard error).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

$L$	$C$	$d$	$\ell_{x}$
2	1	2	0.20
2	1	5	0.25
2	2	2	0.20
2	2	5	0.25
3	1	5	0.20
3	2	5	0.20

Appendix

Table 1: GP-generated test-function settings. Each problem is fixed across BO seeds.

Dataset	Classes	Features	Train	Validation	$d$
Waveform-5000	3	40	2500	2500	3
Dry Bean	7	16	9527	4084	4
Covertype	7	54	60000	40000	4

Appendix

Table 2: LightGBM dataset settings. All use

512

candidates,

L=2

, and

C=1

Dataset	Constraint metric	Quantile	$\theta$	Frontier size
Waveform-5000	Worst-class recall	0.70	0.7892434988	6
Dry Bean	Accuracy	0.50	0.9269098923	10
Covertype	Worst-class recall	0.70	0.5836391437	13

Appendix

Table 3: Constraint thresholds and feasible highest-fidelity pool frontier sizes. Stored full-precision thresholds are used in calculations.

Method	Median (s)	$[Q_{1},Q_{3}]$ (s)
CMF-PFEV	103.1	$[102.5,\,121.6]$
MF-OSEMO-PoF	50.2	$[50.0,\,50.2]$
MF-HVKG-PoF	26.6	$[13.7,\,30.9]$
MFMO-SEGO	13.2	$[13.0,\,13.5]$
MF-CMOBO	12.7	$[11.5,\,12.8]$
MF-JESMOC	3354.1	$[3339.6,\,3364.0]$

Appendix

Table 4: Next-query selection time on a GP-sampled synthetic problem with

L=2

objectives,

C=2

constraints,

d=5

M=2

fidelity levels, and

N=100

total observations (50 at each fidelity). Values are the median and quartiles over five seeds, measured with one CPU thread (Xeon Gold 5317).

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Sep 24, 2026, cs.LGJ/K move · Enter open · S save

MF-SCBO : Multi-fidelity Scalable Constrained Bayesian Optimization

Lucas Palazzolo, Mickaël Binois, Laëtitia Giraldi

Université Côte d’Azur, Inria, Calisto team, Sophia Antipolis, France · Université Côte d’Azur, Inria, Acumes team, CNRS, LJAD, Sophia Antipolis, France

Multi-Fidelity Optimization Bayesian Optimization Black-Box Optimization

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Many real-world optimization problems rely on expensive simulations or experiments, making the efficient use of available data essential. Multi-fidelity optimization of high-dimensional black-box functions subject to black-box constraints is increasingly relevant as the cost of objective evaluations continues to rise in applications such as machine learning, engineering, and control. To our knowledge, no existing method simultaneously addresses high-dimensionality, black-box constraints, an arbitrary number of fidelity levels, and non-nested sampling. In this work, we extend the Scalable Constrained Bayesian Optimization method to the multi-fidelity setting, resulting in the MF-SCBO method. The proposed approach is evaluated on standard benchmark functions as well as challenging problems. The experimental results demonstrate that MF-SCBO generally achieves better convergence than both the single-fidelity SCBO and the other multi-fidelity method considered in this high-dimensional and constrained settings.

Abstract

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MF-SCBO : Multi-fidelity Scalable Constrained Bayesian Optimization

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