{"id":"5fabcd36-5ad2-4326-852e-9f431438af25","arxiv_id":"2512.23581","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A two-stage acquisition scheme estimates profile optima of expensive black-box simulations over a control parameter, beating standard BO and profile-EI on tested benchmarks.","lead":"This paper introduces a Bayesian optimization method that maps the best achievable response of a computer simulation as a function of one chosen control parameter, while optimizing over all other inputs. It combines exploration along the control parameter with fast local exploitation, and demonstrates gains on synthetic benchmarks and a jet-engine diffuser simulation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on the modified tricands candidate set containing the nuisance minimizer; the 90% fringe rule deliberately avoids boundaries/corners, so profile optima at the edge of X_-* are systematically missed, biasing T-hat and all subsequent acquisitions.","rationale":"The reader's verdict is CONDITIONAL with moderate confidence, and I agree with the identified weakest assumption: the candidate set from modified tricands must contain the nuisance minimizer for the profile curve estimate and the acquisition rule to be reliable. The paper has genuine support: reproducible code, synthetic benchmarks with 30 re-randomized starts, and an explicit statement of the corner limitation in Section 6. My concern does not overturn the demonstrated results; rather, it sharpens the condition under which the central claim holds. The proposed benchmark isolates the candidate-set assumption from other issues (e.g., DGP coverage) and would decide whether the boundary/corner limitation materially degrades T-hat and acquisitions. This is a critique of the argument, not of the authors; no dismissal of the paper's contributions is implied.","tokens_in":18366,"tokens_out":4773,"duration_ms":46544,"concrete_test":"Run PBO on a modified 2D benchmark where the profile minimizer lies on the nuisance boundary for most x*, e.g., f(x*, x_-*) = (x_-* - 1)^2 + 10(0.5 - x*)^2 on [0,1]^2, with the existing GP setup (n=10, m=30, 30 starts). Report RMSE/MaxAD and the fraction of acquisitions with x_-* within 0.01 of the boundary. Compare with the same function shifted to an interior minimizer. If the boundary version shows materially worse RMSE/MaxAD or zero boundary acquisitions, the candidate-set assumption is the limiting factor and the paper's abstract claim needs qualification; if performance is similar, the concern is mitigated for this case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"T(x*) is estimated as the minimum over X^{tri+} = x*_grid ⊗ tricands(X_-*^n) (Eq. 6). For a fixed posterior sample, min over a finite candidate set is an upper bound on the true continuous minimum; if the minimizer is not in the candidate set, mu_T(x*) is biased upward and CI_T is miscentered. The acquisition step (Eq. 8) maximizes PEI over the same tricands, so it can never select a point on the boundary; the design cannot self-correct this miss. Section 6 acknowledges 'If the profile optima is found in a corner, our candidates could struggle to find them quickly,' but none of the Section 4 benchmarks appears to place the profile minimizer on the boundary of the nuisance space. Since optima in constrained engineering simulations often occur at boundaries, this is a load-bearing gap in the 'variety of benchmarks' claim. The method may still outperform alternatives on interior-optima problems, but the central claim as stated is conditional on candidate coverage.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18696,"tokens_out":6259,"duration_ms":61691,"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":[{"comment":"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.","section":"§3.1, Eqs. (6),(8)"},{"comment":"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'.","section":"§5, abstract"},{"comment":"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.","section":"§4.2, Figs. 10–11"},{"comment":"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.","section":"§4.1, Fig. 6; abstract"}],"minor_comments":[{"comment":"Typo: 'Dark pink lines mark the the mean' should read 'mark the mean.'","section":"§2.1, Fig. 2 caption"},{"comment":"The notation CI^wd_T(x*) is used before it is defined. Define 'width' formally in the text or in Supplement B.","section":"§3.2, Eq. (7)"},{"comment":"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.","section":"§5, Fig. 14"},{"comment":"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.","section":"§4.2"},{"comment":"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.","section":"§1"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a statistics/computer-experiments journal, and the core methodological idea is sound. The main risks are the boundary-coverage gap in the candidate set and the lack of validation in the RDC application. These are fixable with additional benchmarks/comparisons and more careful claims. No concerns about authorship or citation behavior beyond the reuse of the authors' own software packages, which is appropriately cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTwo 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.\n\nOverall, 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.","headline":"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.","tokens_in":19185,"tokens_out":4678,"would_cite":true,"duration_ms":44245,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62K05","62F15","62L05","62M20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["profile optimization","Bayesian optimization","Gaussian process surrogate","deep Gaussian process","expected improvement","profile expected improvement","triangulation candidates","computer experiments"],"falsifier":"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.","tokens_in":18270,"feed_emoji":"📉","tokens_out":6300,"duration_ms":56666,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two-stage algorithm recovers the entire optimal-response curve","feed_subtitle":"Expensive simulations with one control parameter get a full trade-off curve, not just a single optimum.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Two-stage BO recovers full trade-off curve","Profile BO maps optimal response across control range","Profile BO finds best response for every control setting","New BO variant learns entire optimal-response curve"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two-stage BO recovers full trade-off curve","Profile BO maps optimal response across control range","Profile BO finds best response for every control setting","New BO variant learns entire optimal-response curve"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000632,"raw_usage":{"total_tokens":2737,"prompt_tokens":711,"completion_tokens":2026,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":1969}},"tokens_in":455,"tokens_out":2026,"duration_ms":13533,"temperature":1.0,"reasoning_tokens":1969,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:36:05.868259+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}