Profile Bayesian Optimization for Expensive Computer Experiments

Annie Booth; Courtney Kyger; James Braun; James Fernandez; John A. Grunenwald

REVIEW 4 major objections 5 minor 8 references

Profile Bayesian Optimization for Expensive Computer Experiments

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Profile Bayesian optimization estimates the full optimal-response curve T(x*) from limited evaluations of an expensive simulator, using a two-stage explore-then-exploit acquisition scheme.

desk verdict PBO is a sensible two-stage extension of profile-EI that works on the test problems it tries; the main open question is whether its candidate set breaks on boundary optima. read the letter →

arxiv 2512.23581 v2 pith:N5VWUAL4 submitted 2025-12-29 stat.ME

Courtney Kyger , James Fernandez , John A. Grunenwald , James Braun , Annie Booth This is my paper · ORCID

classification stat.ME MSC 62K0562F1562L0562M20

keywords profileoptimizationBayesianGaussianprocesssurrogatedeepexpectedimprovementtriangulationcandidatescomputerexperiments

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

This paper proposes profile Bayesian optimization (PBO) to solve a problem ordinary Bayesian optimization ignores: instead of finding a single global optimum over all inputs, find the optimal response as a function of one control parameter while minimizing over all other 'nuisance' parameters. The authors argue that for expensive deterministic simulators—their motivating example is a rotating detonation combustor diffuser, where each run costs minutes to hours—PBO can recover the entire trade-off curve with a few dozen to a few hundred evaluations. The method works by building GP or deep-GP surrogates, estimating the profile curve from joint posterior samples, and using a two-stage acquisition: first pick the control value where the estimated curve is most uncertain, then pick the nuisance inputs that maximize profile expected improvement on that slice. On benchmarks and the diffuser application, PBO outperforms space-filling designs, classic EI-based BO, and prior profile expected improvement alone. If correct, engineers can choose design trade-offs across the whole control range rather than only at one point.

What carries the argument

The load-bearing machinery is a two-stage acquisition loop built on joint posterior draws from a Gaussian process or deep Gaussian process surrogate. Profile expected improvement (PEI) replaces the global best observed value y_min in the standard EI formula with t_min(x*)=max(y_min, mu_T(x*)), where mu_T(x*) is the mean of the posterior minima across a candidate set for that control slice; this targets the minimum on the slice rather than the global minimum. To make PEI computable in higher dimensions, the paper introduces 'modified tricands': candidate locations formed by taking a Delaunay triangulation of the nuisance-space training inputs, placing interior candidates at simplex centers an

What would settle it

Take a two-dimensional test function whose true profile minimum lies exactly at the corner of the nuisance domain (e.g., T(x*) achieved at x_-*=1 for all x*), run PBO from many starting designs with the same budget, and compute MaxAD of the estimated profile. If the 90% fringe rule keeps candidates just inside the boundary, the method should systematically overestimate T(x*) near that corner, and adding boundary candidates should remove the bias.

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Extended reading notes

Core claim

The central claim is that the profile function T(x*) = min_{x_-*} f(x*, x_-*) — the best achievable response for each value of the control parameter — can be learned accurately from a small budget of simulator runs. PBO's acquisition rule separates exploration from exploitation: the control coordinate is chosen where the credible-interval width of the estimated profile is largest, and then the nuisance coordinates are chosen by maximizing profile expected improvement (EI computed against the estimated profile value, not the global observed minimum) using a candidate set derived from Delaunay triangulation of the current design in nuisance space. The result is a posterior distribution for the

Load-bearing premise

For any control value selected, the true nuisance-parameter minimizer must lie close to the candidate set created by Delaunay simplex centers and points placed at 90% of the distance to the boundary; if the true profile minimizer sits in a corner or between these candidates, both the estimated curve and the PEI acquisition on that slice are off.

Editorial extensions

If this is right

With a budget of 30–80 simulator evaluations, PBO gives lower profile-estimation error (RMSE and maximum absolute deviation) and tighter credible intervals than equal-sized space-filling designs, classic expected-improvement BO, and profile expected improvement used alone across the benchmarks.
The exploration stage prevents the acquisition clustering around the global optimum that the paper documents for PEI-only and EI-only methods, spreading evaluations across the whole control range.
Both GP and deep-GP surrogates work; the deep-GP version gives the best accuracy on nonstationary functions, while the choice of PBO design over a fixed design matters more than the surrogate family.
On the six-parameter rotating detonation combustor diffuser simulation, the procedure produces a profile of energy loss versus diffuser length from 150 evaluations, with uncertainty shrinking as acquisitions proceed, giving a quantitative basis for the length-versus-efficiency trade-off.

Reading between the lines

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

The same uncertainty-then-exploit template could be extended to multiple control parameters by measuring the profile-uncertainty width on a control manifold, though the paper only treats one scalar control.
A low-cost diagnostic for the boundary-candidate weakness would be to add exact boundary and corner points to the candidate set; if MaxAD falls on functions with corner optima, the 90% fringe rule is the bottleneck.
The acquisition's first stage could accept other uncertainty-aware slice-selection rules, such as Thompson-sampling-style draws of the profile curve, without changing the overall architecture.
Because the surrogates use a near-zero nugget and interpolate, the method as presented targets deterministic simulators; applying it to noisy simulations would require reworking how posterior minima and their credible intervals are computed.

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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

Quick take on Kyger et al. (arXiv:2512.23581). The paper proposes Profile Bayesian Optimization, a two-stage sequential design for recovering T(x*) = min_{x_-*} f(x*, x_-*) from an expensive simulator. The core idea is new and mostly works: first pick x* by maximizing the width of the posterior distribution of T, then pick x_-* by maximizing profile expected improvement on that slice, using a modified tricands candidate set (grid over x* crossed with tricands over the nuisance space). That is a practical fix to Ginsbourger et al.'s PEI, which had only been shown in 2D and tends to over-exploit. The synthetic results are fairly convincing: four functions, 30 random starts each, and PBO generally beats LHS, BO, and PEI in RMSE/MaxAD, with reasonable UQ for the stationary cases.

The big soft spot is the candidate set. T(x*) is computed as the minimum over X^{tri+} = x*_grid ⊗ tricands(X_-*^n). For a fixed posterior sample that minimum is an upper bound on the continuous minimum, and the acquisition step is restricted to the same tricands. Since fringe candidates are placed at 90% of the distance to the boundary, the design never actually samples a point on the boundary. If the profile minimizer sits in a corner, the estimate is biased and the acquisition cannot correct it directly (the candidate set only approaches the boundary as points accumulate nearby). The authors acknowledge this in Section 6, but none of the benchmarks appear to have boundary optima, so the 'variety of benchmarks' claim is narrower than it reads. This is a real limitation, not a reason to reject the paper — the method can still be useful on interior-optima problems — but it should be tested and, ideally, fixed by adding boundary candidates.

Two smaller issues. The DGP variants show poor CI coverage (around 0.4–0.6), meaning the UQ is overconfident; that matters because the first acquisition stage depends on uncertainty. The authors note it but do not resolve it. And the RDC-diffuser section has no ground truth and no comparator; the UQ shrinking over acquisitions is suggestive but not evidence of effectiveness. I would not call that section validation.

Overall, this is a solid methods paper with honest limitations and usable code. It would be a good candidate for a serious referee: the novelty is real, the experiments are reproducible in principle, and the boundary-candidate issue is addressable in revision. I'd send it to review, with a clear request for boundary-optima benchmarks and more careful UQ evaluation.

Referee Report

4 major / 5 minor

Summary. The paper proposes Profile Bayesian Optimization (PBO) for estimating the profile optima T(x*) = min_{x_-*} f(x*, x_-*) of an expensive deterministic black-box simulator over the full support of a single control parameter x*, with multiple nuisance parameters x_-*. The method combines GP/DGP surrogates with a modified triangulation-candidate set X^{tri+} = x*_grid ⊗ tricands(X_-*^n) and a two-stage acquisition: Eq. (7) selects the control value with the largest credible-interval width for the estimated profile, and Eq. (8) selects the nuisance point maximizing profile expected improvement along that slice. The paper compares PBO against LHS, classic BO, and PEI on four synthetic functions (Branin, Kyger3D, Kyger2D, squiggle) with 30 random starts, and presents an RDC-diffuser application with a DGP surrogate.

Significance. If the central claim holds, PBO is a practically useful contribution: it recovers the full optimal-response curve as a function of a control parameter from O(m) simulator evaluations, extends profile optimization beyond two dimensions, and accommodates nonstationary response surfaces through DGPs. The paper has clear strengths: the algorithm is explicitly stated, the code is public, the benchmark setup uses 30 re-randomized starts, and modern Vecchia-based posterior sampling is leveraged. The methodology is not circular — the benchmarks are external and no target-dependent fitted constants are smuggled in. However, two load-bearing gaps remain: the candidate set cannot represent boundary/corner minimizers, and the RDC section provides no ground truth or comparator despite the abstract's claim that the method 'proves effective' on the motivating application. These issues need to be addressed before the central claims are fully supported.

major comments (4)

[§3.1, Eqs. (6),(8)] T(x*) is estimated by minimizing each posterior draw over the finite candidate set X^{tri+}; this is an upper bound on the true continuous minimum, and the bound is biased upward whenever the nuisance minimizer is absent from that set. Because tricands place fringe candidates at 90% of the distance to the boundary, profile optima on or near the boundary/corners are systematically excluded. Eq. (8) maximizes PEI over the same tricands, so acquisitions cannot self-correct this miss. Section 6 acknowledges the corner problem, but the paper presents no benchmark in which the profile minimizer lies on the boundary. The 'variety of benchmarks' claim is therefore conditional on interior optima. Please add a benchmark with a boundary profile optimum and/or include boundary points in the candidate set, and temper the claim accordingly.
[§5, abstract] The abstract and conclusion state that the procedure 'proves effective in our motivating application against state-of-the-art multi-objective optimization,' but Section 5 contains no comparator and no ground truth. Figure 14 shows only that the credible interval narrows as acquisitions are added; this is evidence of decreasing posterior uncertainty, not of accuracy of the profile estimate. The RDC application is central to the paper's motivation, so this validation gap is load-bearing. Either add a comparator (e.g., PEI or BO on the same evaluation budget) or restrict the claim to 'provides a plausible estimate with quantified uncertainty'.
[§4.2, Figs. 10–11] In the nonstationary benchmarks, the DGP-based variants have CI coverage around 0.4–0.6, far below the nominal 95%, even though point accuracy is good. Since the paper lists 'thorough uncertainty quantification' as a contribution and constructs CI_T(x*) from posterior minima, this low coverage weakens the UQ claim. Section 6 suggests that more posterior samples might help, but no sensitivity analysis is provided. Please either improve the uncertainty estimates and report coverage at the improved setting, or explicitly present the nonstationary results as point-estimate accuracy improvements and flag the coverage gap as an open problem.
[§4.1, Fig. 6; abstract] On the Branin benchmark, PBO and PEI perform equally well, so the abstract's unqualified statement that PBO 'outperforms traditional BO and profile optimization methods' is stronger than the evidence. The improvement over PEI is clear in the higher-dimensional/nonstationary examples (Kyger3D, squiggle) but not in 2D Branin. Please qualify the claim to reflect where the advantage is demonstrated.

minor comments (5)

[§2.1, Fig. 2 caption] Typo: 'Dark pink lines mark the the mean' should read 'mark the mean.'
[§3.2, Eq. (7)] The notation CI^wd_T(x*) is used before it is defined. Define 'width' formally in the text or in Supplement B.
[§5, Fig. 14] The y-axis scales differ across the three panels, which makes the narrowing of the credible interval harder to judge. A common y-axis or normalized scale would improve interpretability.
[§4.2] The decision to drop BO and PEI as comparators for the nonstationary functions is reasonable, but should be stated as a limitation: the nonstationary sections demonstrate PBO versus LHS designs, not PBO versus the profile-optimization competitors.
[§1] The assumption of a unique profile optimum for each x* should be stated early, since Section 6 discusses this as a limitation and it affects the interpretation of the posterior-minimum estimator.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained and benchmarked externally.

full rationale

The paper's central claim is that the proposed two-stage PBO procedure estimates the profile curve T(x*) and outperforms alternatives on external benchmarks. The estimation of T(x*) is a Monte Carlo minimum over joint posterior samples evaluated on a candidate set; no parameter is fitted to the profile curve and then renamed as a prediction. The acquisition rule (Eq. 7) selects the control value with largest posterior CI width, and (Eq. 8) selects the nuisance value maximizing PEI on that slice; both are computed from the surrogate posterior, while the benchmark truth T(x*) and the RDC simulator are external to the fitted model. Citations to tricands (Gramacy et al. 2022), PEI (Ginsbourger et al. 2014), and software packages (deepgp, GpGp, GPvecchia) are ordinary method/implementation references, not a self-citation chain that forces the central result. The acknowledged limitation that 90% fringe candidates avoid boundaries is a coverage/correctness concern, not a circular reduction: the target is not defined in terms of the candidate set, and the finite-candidate minimum is an approximation rather than an equation-level identity. No load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The method does not introduce new physical or statistical entities; it relies on standard GP/DGP assumptions, the prior PEI criterion, and a heuristic Delaunay-based candidate set. The main free parameters are hand-chosen design constants (fringe fraction, sample counts, grid sizes) and MCMC settings, plus the unexplored size of the x*-LHS candidate set.

free parameters (6)

Fringe candidate distance fraction = 0.90
Fringe candidates are placed at 90% of the distance to the boundary (Section 3.1); no sensitivity analysis is given, and the paper acknowledges corner optima may be missed.
Number of posterior samples per T estimate = 1000
Empirical quantiles/minima distributions use 1,000 joint posterior draws (Sections 2.1, 4.2); coverage is sensitive to this choice.
Size of x*-LHS candidate set = not specified
Eq. (7) selects among x*_LHS but the number of 1D LHS candidates per iteration is never stated; this controls exploration resolution.
Final control-parameter grid size = 50
The final profile estimate uses a 'semi-dense grid of 50 points' for x* (Section 3.1); profile resolution depends on this choice.
Vecchia conditioning set size = 40
Joint posterior samples use Vecchia approximation with conditioning set size 40 (Section 4.1); affects the fidelity of minima distributions.
DGP MCMC iterations and thinning = 10,000 initial / 2,000 updates, 50% burn-in, thin to 100
Used for DGP posteriors (Section 4.2); no convergence diagnostics are shown, and the paper notes resulting overconfidence.

assumptions (6)

domain assumption Each x* has a unique profile minimum over nuisance parameters.
Stated in Section 6 as a design condition; if multiple discontiguous nuisance regions tie for the minimum, per-slice minima from posterior samples may be unstable.
domain assumption Nuisance minima are captured by Delaunay triangulation component centers and 90% fringe candidates.
Section 3.1/Eq. (6); heuristic from Gramacy et al. 2022; authors note corner optima could be missed.
domain assumption GP/DGP posterior with small nugget and standard kernels faithfully models the deterministic simulator.
Section 2.1; stationarity relaxed via DGP but still a modeling assumption.
domain assumption Vecchia-approximated joint posterior samples provide reliable empirical minima distributions.
Sections 2.1 and 4.1; approximation quality depends on conditioning set size 40.
standard math PEI criterion with t_min = max(y_min, mu_T) is a suitable improvement measure for profile optimization.
Adopted from Ginsbourger et al. 2014; not re-derived, but treated as a valid baseline criterion.
domain assumption Empirical distribution of per-sample minima across 1,000 joint draws accurately represents T(x*) uncertainty.
Used throughout; DGP coverage results show it can be overconfident.

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Cite this review

Pith. "Pith review of Profile Bayesian Optimization for Expensive Computer Experiments." pith.science (2026). https://pith.science/paper/N5VWUAL4

@misc{pith2026251223581,
  author       = {Pith},
  title        = {Pith review of: Profile Bayesian Optimization for Expensive Computer Experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N5VWUAL4}},
  note         = {Machine review of arXiv:2512.23581}
}

read the original abstract

We propose a novel Bayesian optimization (BO) procedure aimed at identifying the "profile optima" of a deterministic black-box computer simulation that has a single control parameter and multiple nuisance parameters. The profile optima capture the optimal response values as a function of the control parameter. Our objective is to identify these optima across the entire plausible range of the control parameter. Classic BO, which targets a single optimum over all parameters, does not explore the entire control parameter range. Instead, we develop a novel two-stage acquisition scheme to balance exploration across the control parameter and exploitation of the profile optima, leveraging deep and shallow Gaussian process surrogates to facilitate uncertainty quantification. We are motivated by a computer simulation of a diffuser in a rotating detonation combustion engine, which returns the energy lost through diffusion as a function of various design parameters. We aim to identify the lowest possible energy loss as a function of the diffuser's length; understanding this relationship will enable well-informed design choices. Our "profile Bayesian optimization" procedure outperforms traditional BO and profile optimization methods on a variety of benchmarks and proves effective in our motivating application against state-of-the-art multi-objective optimization.

Figures

Figures reproduced from arXiv: 2512.23581 by the authors.

**Figure 2.** shows µ10(Xp) (solid blue) and its 95% credible interval (dashed blue) along the slice x ⋆ = 0.48, where Xp contains a fine and evenly-spaced grid of x −⋆ values in one dimension. The GP offers nonlinear predictions with effective UQ. Posterior uncertainty is lower near the training data—a GP hallmark. 0.0 0.2 0.4 0.6 0.8 1.0 10 30 50 70 Nuisance Parameter, x− * Y Truth Mean + UQ Samples 0.0 0.2 0.4 0.6 0.8 1.0 10 3… view at source ↗

**Figure 3.** Left/Center: EI and PEI (purple/low, yellow/high) for the Branin function. White circles mark training data locations. Black line/circles mark the true profile/global optima. Right: The same slice shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Left: Projection of 3D LHS onto 2D nuisance parameter space. Center: Tricands (red stars) of 10-point LHS (black points). Dashed black lines represent the Delaunay triangulation. Right: Tricands repeated on 6 slices for x ⋆ ∈ {0, 0.2, 0.4, 0.6, 0.8, 1} (colors used for…

**Figure 5.** Figure 5: offers an inside look at this process for the Branin function. Starting with Tˆ(x ⋆ ) estimated from our 10-point LHS (left panel here, also shown earlier in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Results for the Branin function across 30 repetitions with [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Results for the Kyger3D function across 30 repetitions with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: PBO and PEI estimation of T(x ⋆ ) for the Kyger3D function. Black stars indicate initial LHS points, and colored circles indicate acquired points. The solid black line denotes the true T(x ⋆ ), the solid colored lines denote µT (x ⋆ ), and the dashed colored lines deno…

**Figure 9.** Figure 9: Estimation of T(x ⋆ ) for the Kyger2D function using GP/DGP surrogates with LHS and PBO. LHS−GP LHS−2DGP PBO−GP PBO−2DGP 0.00 0.05 0.10 0.15 0.20 Method RMSE LHS−GP LHS−2DGP PBO−GP PBO−2DGP 0.0 0.1 0.2 0.3 0.4 0.5 Method MaxAD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.4 0.5 0.…

**Figure 10.** Figure 10: Results for the Kyger2D function across 30 repetitions with [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: Results for the squiggle function across 30 repetitions with [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

**Figure 12.** Figure 12: Left: Schematic of a detonation wave traveling through an RDC. Right: Parameterized geometry for a 2D slice of the diffuser. Flow speed and temperature are reduced upon exiting the diffuser. The diffuser, which will mix the hot exhaust and cold bypass-air downstream o…

**Figure 13.** Figure 13: Pressure loss (P) with Mach number (M) and temperature (T) for two runs of the 6D diffuser [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 14.** Figure 14: Diffuser results from an original 50-point LHS (left), after 50 acquisitions (center), and after [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

**Figure 15.** Figure 15: shows estimated Tˆ(x ⋆ ) for all four comparators on the 2D Branin function exercise of Section 4.1. All methods start with the same initial LHS of size n = 10 and proceed with 20 acquisitions for a total of m = 30. 0.0 0.2 0.4 0.6 0.8 1.0 −20 −10 0 10 20 30 LHS Contr…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 3 linked inside Pith

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Berk, J., Nguyen, V., Gupta, S., Rana, S., and Venkatesh, S. (2018). “Exploration enhanced expected im- provement for Bayesian optimization.” Injoint european conference on machine learning and knowledge discovery in databases, 621–637. Springer. Booth, A. S. (2025a). “Deep Gaussian Processes with Gradients.”arXiv preprint arXiv:2512.18066. — (2025b).deep...

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Springer. Sauer, A., Cooper, A., and Gramacy, R. B. (2023a). “Non-stationary Gaussian process surrogates.”arXiv preprint arXiv:2305.19242. — (2023b). “Vecchia-approximated deep Gaussian processes for computer experiments.”Journal of Com- putational and Graphical Statistics, 32, 3, 824–837. Sauer, A., Gramacy, R. B., and Higdon, D. (2023c). “Active learnin...

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The (stationary) Kyger3D function is defined as: f(x) = exp (−x1 −cos(2πx 1)) + sin (2πx3) + exp (−x3 ∗sin(2πx 1)) + cos (2πx2)−exp (−x 2 ∗cos(2πx 1)) forx 1, x2 ∈[0,1]

+sforx 1 ∈[−5,10]x 2 ∈[0,15]. The (stationary) Kyger3D function is defined as: f(x) = exp (−x1 −cos(2πx 1)) + sin (2πx3) + exp (−x3 ∗sin(2πx 1)) + cos (2πx2)−exp (−x 2 ∗cos(2πx 1)) forx 1, x2 ∈[0,1]. The (nonstationary) Kyger2D function is defined as: f(x) = sin x2 1 + 1 + 2X i=1 (xi −0.5) 2 ! (cos (x2) + 1.5) e{4− x1 3 } − (x1 −0.1) 2X i=1 x2 i forx 1 ∈[...

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Katzfuss, M. and Guinness, J. (2021). “A general framework for Vecchia approximations of Gaussian processes.”Statistical Science, 36, 1, 124–141. Katzfuss, M., Guinness, J., Gong, W., and Zilber, D. (2020). “Vecchia approximations of Gaussian-process predictions.”Journal of Agricultural, Biological and Environmental Statistics, 25, 3, 383–414. Katzfuss, M...

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