High-Dimensional Bayesian Optimisation with Large-Scale Constraints via Latent Space Gaussian Processes

Hauke F. Maathuis; Roeland De Breuker; Saullo G.P. Castro

REVIEW 4 major objections 6 minor 44 references

High-Dimensional Bayesian Optimisation with Large-Scale Constraints via Latent Space Gaussian Processes

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A low-dimensional constraint subspace lets Bayesian optimisation scale to 1,786 constraints and a 108-variable aeroelastic design problem.

desk verdict A promising speed-up for constrained BO undercuts itself: the latent-space feasibility test is not equivalent to original feasibility, and the paper never confronts that. read the letter →

arxiv 2412.15679 v1 pith:7PWMCEGS submitted 2024-12-20 cs.CE

Hauke F. Maathuis , Roeland De Breuker , Saullo G.P. Castro This is my paper · ORCID

classification cs.CE

keywords BayesianoptimisationGaussianprocesseslarge-scaleconstraintslatentspaceprincipalcomponentanalysiskernelPCAaeroelastictailoringconstrained

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the main obstacle to applying Bayesian optimisation (BO) to high-dimensional engineering design is not the number of design variables but the number of constraints, each of which normally needs its own Gaussian-process surrogate. It proposes projecting the constraint outputs onto a low-dimensional latent subspace via PCA or kernel PCA and training only g of surrogate Gaussian processes on that subspace instead of G. In the 108-dimensional aeroelastic tailoring problem with 1,786 black-box constraints, this makes constrained BO run successfully on a standard workstation, while the baseline SCBO method crashes from memory demands after one iteration. The method also finds feasible designs where random search and CMA-ES find none, and it reaches near-optimal values on a 7D speed-reducer benchmark with 11 constraints.

What carries the argument

The load-bearing object is the truncated projection matrix Ψ_g ∈ $R^{{G×g}}$ obtained from an eigendecomposition of the constraint covariance matrix (PCA) or of the kernel matrix (kPCA). The projection maps each design's constraint vector c(x) ∈ R^G to a latent coordinate vector c~(x) ∈ R^g, and the optimisation then builds g+1 independent Gaussian-process surrogates (objective plus the g latent constraints) inside the SCBO trust-region framework. Because feasibility is always checked against the original 1,786 constraints during batch evaluation, the latent GPs only guide the acquisition; the projection is recomputed each iteration as data accumulate. The complexity reduction is the argument's spine: O((g+1)$N^{3}$ + $G^{3}$) versus O((G+1)$N^{3}$) for independent-constraint surrogates.

What would settle it

Take a benchmark with G independent or near-independent constraint functions (for instance, each design variable has its own unrelated constraint) and run PCA-GP SCBO with a small g. If the eigenvalue decay is slow and the method returns designs that violate many original constraints while still appearing feasible in the latent space, the central compression claim fails. A more quantitative version compares the fraction of original constraints satisfied at the returned optimum with the latent feasibility prediction: a mismatch indicates the subspace has not captured the feasible boundary.

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Extended reading notes

Core claim

The central claim is that constraint outputs in many large-scale design problems are compressible: the G-dimensional vector of constraint values c(x) lies close to a g-dimensional subspace, so modelling the g latent coordinates with independent GPs is an adequate proxy for the feasibility of all G original constraints. Using PCA (or its kernel extension) to build a projection Ψ_g from the constraint matrix C ∈ $R^{{N×G}}$, the paper reduces the surrogate-training cost from O((G+1)$N^{3}$) to O((g+1)$N^{3}$ + $G^{3}$), where the $G^{3}$ term is the one-time eigendecomposition. In the aeroelastic case, g = 35 latent GPs replace 1,786 surrogates; with g = 35 the approach converges to a feasible, reduced-mass design, whereas the standard SCBO implementation crashes from insufficient memory and both random search and CMA-ES fail to find a feasible point. The paper further shows that adding a second loadcase doubles the number of constraints but barely increases the number of principal components needed.

Load-bearing premise

The method works only if the constraint outputs across the design space lie close to a low-dimensional subspace, so that the dominant principal components carry enough information about feasibility; if the constraints are effectively independent or high-rank, latent-feasible designs will violate many original constraints and the advantages disappear.

Editorial extensions

If this is right

Problems with thousands of black-box constraints that were previously out of reach for SCBO become tractable on ordinary hardware.
Adding loadcases multiplies the number of constraints but adds little to the required number of latent components, since the underlying physics is shared across loadcases.
The method applies to any large-scale constrained BO problem, not just aeroelastic tailoring, provided the constraint outputs are low-rank.
The choice of g matters: too small a subspace (g = 1 on the benchmark) loses feasibility information, while larger g reduces the computational savings.
Recomputing the projection each iteration lets the method adapt as data accumulate, though in the low-dimensional benchmark fixing the initial projection performed even better.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could estimate the achievable compression up front by computing the PCA eigenvalue decay of a small pilot design-of-experiments; problems without a fast decay, such as nearly independent constraints, would be poor candidates for this method.
The same latent-space surrogate idea could extend to other acquisition functions and to multi-fidelity or design-under-uncertainty settings, where the variance of the latent GPs would propagate back through Ψ_g to the original constraints.
A testable extension is an adaptively chosen g during optimisation, which the paper lists as future work and would eliminate the user-specified principal-components count.
The failure of random search and CMA-ES on the 108D problem suggests that the main benefit lies in focusing samples on the feasible boundary via the latent surrogates, a claim one could isolate by ablating the trust-region mechanism.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What's new: they take Higdon-style output PCA/kPCA and apply it to constraints inside SCBO, replacing G constraint GPs with g latent GPs. On the speed reducer they show 57-60% time savings with modest objective degradation (1.9-3.1% above the known optimum), and on a 108D aeroelastic problem with 1,786 constraints, SCBO crashes while their method runs and finds feasible designs. That is a real practical gain and worth discussing.

The soft spots. The load-bearing one is exactly the stress-test you ran. The acquisition treats a point as feasible when every latent score is <=0. But those scores are linear combinations of the original constraints in a rotated basis. The set {Psi_g^T c <= 0} is a different intersection of half-spaces from {c_i <= 0}. Even with g=G this is not equivalent unless the basis is a signed permutation. So a feasible point can have positive latent scores and be rejected, and an infeasible point can look feasible. The paper checks original constraints only after the expensive evaluation; that filters the final batch, but the acquisition is optimized against the wrong condition, so the search is systematically misdirected. The 'consistency of physics' argument addresses low-rank structure, not sign preservation, and Figure 6's reconstruction error does not bound classification error. The benchmark still works, which suggests the heuristic may survive in practice, but the paper neither identifies this mismatch nor analyzes when it breaks.

Other soft spots: no code or data released; the text says N=20 initial samples while the Figure 3 caption says N=10; and the aeroelastic case has no reference optimum, so 'convergence' is just a decreasing curve with no ground truth. The complexity claim O((g+1)N^3 + G^3) is fine, though the G^3 eigendecomposition can still be a bottleneck at larger G.

Credit where it is earned: the components are prior art, but the combination is new and the application is substantial. The benchmark results are honestly reported, including the solution-quality loss. The paper is readable and the authors clearly know the surrounding literature.

Recommendation: send to peer review, but expect major revision. A referee should push them to fix the acquisition (e.g., model the projected feasible region explicitly, or add a classification head) or at least quantify the misclassification rate of the latent feasibility test on real data. Without that, the method's benchmark success is unexplained rather than justified.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes a dimensionality-reduction wrapper for Scalable Constrained Bayesian Optimization (SCBO) in which the G constraint outputs are projected via PCA or kernel PCA onto g latent coordinates, and g independent Gaussian processes are trained on these coordinates instead of one GP per original constraint. Algorithm 2 embeds this in the TuRBO/Thompson-sampling loop. The authors report a complexity reduction from O((G+1)N^3) to O((g+1)N^3+G^3) and validate the method on the 7D speed reducer benchmark with 11 black-box constraints and on a 108D aeroelastic tailoring problem with G=1786 constraints. On the benchmark, PCA-GP and kPCA-GP are 57-60% faster than SCBO and converge to solutions within 1.90-3.07% of the known optimum; on the aeroelastic problem, the latent-space methods find feasible designs while SCBO crashes from memory and random search/CMA-ES do not.

Significance. The paper targets a real bottleneck: constrained BO with thousands of constraints is currently impractical because independent GPs have O(GN^3) training cost and O(GN^2) storage. The 108D, G=1786 aeroelastic demonstration is a useful existence proof, and the reported speedups on the speed reducer are concrete. The eigenvalue decay analysis and the comparison with KS constraint aggregation are also informative. The main contribution is potentially valuable but, as it stands, the acquisition is not guaranteed to respect the original feasibility condition; the manuscript needs either a corrected formulation or direct evidence that latent-space feasibility approximates original feasibility.

major comments (4)

[Sec. 3.3, Algorithm 1, Algorithm 2, Eqs. (20), (26)] As written, the acquisition selects candidates for which all latent scores \tilde c_j(x) are nonpositive. Because \tilde c(x)=P_k(c(x)) is a projection (a rotation followed by truncation), the condition \tilde c(x) \le 0 is not equivalent to c_i(x) \le 0 for all i. Even if g=G and the basis is orthonormal, coordinatewise nonpositivity in the rotated basis is a different condition than nonpositivity in the original basis. A truly feasible design can therefore be rejected by the acquisition, and an infeasible design can be accepted. The paper only verifies original feasibility after the expensive evaluation (Sec. 4.2), so these misclassifications consume search budget without being corrected. The eigenvalue decay in Figure 6 bounds reconstruction error of the constraint values, not boundary classification error, so it cannot validate the acquisition. Please provide either a corrected acquisition that propagates latent GP uncertainty back to the original constraints or a direct empirical test that {x: \Psi_g^T c(x) \le 0} approximates {x: c(x) \le 0} on held-out points.
[Sec. 4.2] The premise stated in Section 4.2, namely that consistency of the physics across loadcases justifies compression, supports low-rankness of the constraint outputs, but it does not support the stronger claim that the feasible boundary is aligned with the PCA coordinate axes. Principal components maximize variance, not separation between feasible and infeasible designs. The paper should test boundary preservation directly by computing latent scores and original feasibility on a held-out set of points (both random and generated by the optimiser) and reporting the confusion matrix for g=35 and for several values of g.
[Sec. 4.1, Fig. 4] The sensitivity study of g in Figure 4 shows that g=1 fails to find a feasible point while g=2,4,6 succeed, but it does not identify whether the failure is due to reconstruction error, boundary misclassification, or simply a smaller search space for the acquisition. Without a per-iteration or per-candidate diagnostic, the benchmark cannot separate the validity of the latent-feasibility model from the optimisation dynamics. A false-positive/false-negative analysis on the benchmark's known feasible region would make the empirical claim much stronger.
[Sec. 3.4, Algorithm 2] The complexity statement O((g+1)N^3+G^3) is incomplete. Algorithm 2 recomputes the projection P_k at every iteration, so the G^3 (or, for kPCA, the N^3 kernel eigendecomposition) cost is incurred per iteration and not only once; the total cost is O(T(G^3+(g+1)N^3)) where T is the number of iterations. The authors should state whether the reported timings include the per-iteration projection cost and how this cost scales with N and G.

minor comments (6)

[Sec. 3.2, Eq. (24)] The kernel matrix is defined as K_{ij} := (\phi(c(x_j)), \phi(c(x_j))), which appears to be a typo; it should be \phi(c(x_i))^T \phi(c(x_j)).
[Sec. 3.4] The reference to 'Mathoron's rule' should read 'Matheron's rule'.
[Algorithm 1] The heading 'CONSTRAINED THOMSPON SAMPLING' contains a typo; it should read 'CONSTRAINED THOMPSON SAMPLING'.
[Sec. 3.1, Sec. 4.1] The phrase 'principle components' should be 'principal components' throughout, including the subsection heading and the discussion of Figure 3.
[Fig. 3] In the right panel, the horizontal axis is labeled 'Sample Index' but the plotted quantity is eigenvalues indexed by component number; the axis label should be 'Component Index'.
[Sec. 2.1] The data set D0 is introduced with only objective values, while later sections include constraint vectors in D; the notation should be adjusted so that the constrained-data setting is introduced explicitly before Eq. (2).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the latent-space surrogate reduction is a modeling choice with externally benchmarked results, and no prediction reduces to a fitted input by construction.

full rationale

The paper's derivation chain is not circular. The method replaces the G scalar constraint outputs by g projected coordinates via PCA or kPCA (Eqs. 15-27), trains g latent GP surrogates (Eq. 27), and uses them in Algorithm 2. The reduction from G to g surrogates is arithmetic: one GP is built per projected coordinate, so the complexity statement O((g+1)N^3 + G^3) follows from the algorithm's definition rather than from a hidden equivalence. The central experimental claims are benchmarked externally: the speed reducer results are compared with the known optimum f* = 2996.3482 and with standard SCBO (Table 1, Figure 3), and the aeroelastic results are compared with SCBO, Random Search, CMA-ES, and KS-aggregation (Figures 7-8). The projection is learned from constraint data, and its reconstruction error is checked on unseen data via Eq. 32, not fitted to the objective. The paper explicitly states in Section 3.3 that the validity of a feasible design is checked in the original space after the expensive evaluation, and Section 4.1 acknowledges that success depends on how accurately the lower-dimensional subspace captures the original space. The eigenvalue threshold used to choose g is disclosed as a hyperparameter (Section 4.2), so no fitted parameter is renamed as a prediction. The only self-citation, to Maathuis et al. [2024], provides background and model details for the aeroelastic tailoring problem and is not load-bearing for the method's validity. The skeptic's point that coordinate-wise nonpositivity in a rotated PCA basis need not match original constraint signs is a correctness and approximation risk, not a circularity: it claims the surrogate feasibility criterion can misclassify points, but it does not show that the paper's central claim reduces to its own inputs by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The method introduces two key free choices: the latent dimension g and the eigenvalue threshold, plus the usual BO and GP hyperparameters. The central scientific assumption is that the constraint outputs are low-rank across loadcases, which is plausible for the aeroelastic application but not guaranteed in general. No new physical entity is postulated.

free parameters (6)

Number of principal components g = g=4 (benchmark), g=35 (aeroelastic)
Selected via eigenvalue threshold or by hand; affects convergence and feasibility. The paper notes g=1 fails entirely for the speed reducer, so the result depends on this choice.
Eigenvalue threshold tau_ev = around 1e-2 to 1e-3 (paper uses g=35 though the threshold suggests g=29)
Stated as 'commonly set based on experience, thus can be seen as a hyper-parameter' in Section 4.2; controls the latent dimension.
kPCA Gaussian kernel bandwidth sigma = not specified
Used in Equation 28 for kPCA-GP SCBO; no value or tuning procedure is given.
SCBO hyperparameters = not specified
Trust-region size, candidate count, and batch size; the paper says they follow Eriksson and Poloczek 2021 but does not list the values.
Initial sample size N_i = N=20 for benchmark, N=416 for aeroelastic (Figure 7 caption says N=D=108)
Chosen by user; affects PCA subspace quality and GP training, and the text contradicts itself on N.
GP kernel hyperparameters = maximized marginal likelihood
Lengthscales and signal variance are standard fitted quantities, but they are free parameters of the surrogate.

assumptions (5)

standard math Objective and constraint functions are realizations of Gaussian processes with stationary kernels.
Invoked in Section 2.1 (Equation 3 and marginal likelihood) for all surrogates; standard BO assumption.
domain assumption The G constraint outputs lie close to a low-dimensional subspace of dimension g << G.
Central premise of Sections 3.1-3.3; justified empirically by eigenvalue decay in Figure 6 but not guaranteed for new problems. The paper states the premise is the consistency of physics across loadcases in Section 4.2.
domain assumption The global optimum lies on or near the boundary of the feasible space, and the latent feasibility proxy preserves that boundary.
Explicitly assumed in Section 4.1: 'Assuming that the global optimum lies on the boundary of the feasible space Xf, the success of the method highly depends on how accurately the lower dimensional subspace captures the original space.'
domain assumption Trust-region constrained Thompson sampling (SCBO) is an effective high-dimensional optimizer once surrogates are available.
Inherited from Eriksson and Poloczek 2021; Algorithm 1 is used unchanged for acquisition, so the method inherits SCBO's assumptions about trust-region adaptation.
domain assumption Initial Latin hypercube samples are representative enough to build a useful PCA subspace.
Used in Section 4.2 with N=416 initial samples; the paper itself notes that an insufficient initial sample size leads to larger reconstruction error.

invented entities (1)

Latent constraint subspace V' (PCA/kPCA projection)
purpose: Reduces G constraint outputs to g latent outputs modeled by GPs.
Mathematical construction, not a physical entity. Evidence for its adequacy is in-sample eigenvalue decay and reconstruction error on unseen data (Equation 32), but no externally falsifiable prediction is made.

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Cite this review

Pith. "Pith review of High-Dimensional Bayesian Optimisation with Large-Scale Constraints via Latent Space Gaussian Processes." pith.science (2026). https://pith.science/paper/7PWMCEGS

@misc{pith2026241215679,
  author       = {Pith},
  title        = {Pith review of: High-Dimensional Bayesian Optimisation with Large-Scale Constraints via Latent Space Gaussian Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PWMCEGS}},
  note         = {Machine review of arXiv:2412.15679}
}

read the original abstract

Design optimisation offers the potential to develop lightweight aircraft structures with reduced environmental impact. Due to the high number of design variables and constraints, these challenges are typically addressed using gradient-based optimisation methods to maintain efficiency. However, this approach often results in a local solution, overlooking the global design space. Moreover, gradients are frequently unavailable. Bayesian Optimisation presents a promising alternative, enabling sample-efficient global optimisation through probabilistic surrogate models that do not depend on gradients. Although Bayesian Optimisation has shown its effectiveness for problems with a small number of design variables, it struggles to scale to high-dimensional problems, particularly when incorporating large-scale constraints. This challenge is especially pronounced in aeroelastic tailoring, where directional stiffness properties are integrated into the structural design to manage aeroelastic deformations and enhance both aerodynamic and structural performance. Ensuring the safe operation of the system requires simultaneously addressing constraints from various analysis disciplines, making global design space exploration even more complex. This study seeks to address this issue by employing high-dimensional Bayesian Optimisation combined with a dimensionality reduction technique to tackle the optimisation challenges in aeroelastic tailoring. The proposed approach is validated through experiments on a well-known benchmark case with black-box constraints, as well as its application to the aeroelastic tailoring problem, demonstrating the feasibility of Bayesian Optimisation for high-dimensional problems with large-scale constraints.

Figures

Figures reproduced from arXiv: 2412.15679 by the authors.

**Figure 2.** Schematic illustration of (k)PCA-GP: M denotes the numerical model, mapping from the design space X to the objective f and constraints c as outputs. The constraints are then mapped via (k)PCA onto a lower-dimensional representation c˜ where the independent GPs are constructed. as concluded by Zhe et al. [2019]. Due to the fact that in engineering design problems the dimensionality and constraints can become very lar… view at source ↗

**Figure 3.** (left) 7D Speed reducer problem with 11 black-box constraints from Lemonge et al. [2010]. 20 experiments are performed, where the solid line represents the mean objective value over the 20 experiments and the shaded area the standard deviation. f ∗ denotes the known optimal value of this problem. (right) The eigenvalues of the matrix C with N = 10 samples are plotted . mean value found is close to the analytic value… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The influence of the number of principal components [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Wing structure consisting of wingbox and airfoil shape. The proposed problem optimises the stiffness and [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Investigating the constraints in D. (left) Showing the decay of the eigenvalues for N = 416, performing PCA on the matrix C for one and two loadcases. (right) Computing the error for PCA depending on how many principal components are taken into account (Equation 32). d…

**Figure 7.** Figure 7: (left) shows the results for the 108D aeroelastic tailoring problem, comparing the results of SCBO, (k)PCA-GP SCBO, Random Search and CMA-ES [Hansen, 2006]. Again, kPCA-GP SCBO uses the Gaussian kernel defined in Equation 28. A total of 5 experiments are performed per …

**Figure 8.** Figure 8: Comparison of best result with constraint aggregation [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

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