Active Learning of Computer Experiment with both Quantitative and Qualitative Inputs

Anita Shahrokhian; C. Devon Lin; Xinwei Deng

REVIEW 3 major objections 5 minor 37 references

Active Learning of Computer Experiment with both Quantitative and Qualitative Inputs

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Adaptive designs beat one-shot designs for mixed-input computer experiments.

desk verdict A useful survey plus a modest, plausible first application of prediction-oriented acquisition to mixed inputs, but the empirical case needs error bars and a sensitivity check on the discrete candidate set. read the letter →

arxiv 2504.13441 v2 pith:T63VTSE7 submitted 2025-04-18 stat.ME

Anita Shahrokhian , Xinwei Deng , C. Devon Lin This is my paper · ORCID

classification stat.ME

keywords activelearningadaptivedesigncomputerexperimentsmixedquantitativeandqualitativeinputsGaussianprocessemulatorexpectedimprovementcontourestimationsequential

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 argues that adaptive, sequential designs outperform one-shot space-filling designs for computer experiments whose inputs include both quantitative and qualitative variables, across the three goals of optimization, contour estimation, and prediction. Using the easy-to-interpret Gaussian process (EzGP) emulator as the surrogate, it compares criteria originally developed for quantitative-only inputs, including expected improvement, lower confidence bound, adaptive-region, and entropy-based criteria, and shows numerically that spending the same total simulation budget adaptively produces better minima, contour locations, and predictions than running a fixed design in advance. The practical payoff is that scientists and engineers running expensive simulators can get more accuracy per simulation run by choosing where to sample next based on the current emulator. The paper also applies the EI-MC and EI-SC criteria, designed for prediction with quantitative inputs, to mixed inputs and reports that they outperform one-shot designs.

What carries the argument

The central object is the EzGP (easy-to-interpret Gaussian process) model, which writes the response as a constant mean plus an overall Gaussian process on the quantitative inputs plus a separate Gaussian process for each level of each qualitative factor, with multiplicative correlation between the qualitative and quantitative parts. This surrogate supplies the predictive mean $\hat{\mu}(w)$ and predictive variance $\hat{\sigma}^2(w)$ that every acquisition criterion consumes. The carrying mechanisms are the acquisition criteria: expected improvement for optimization and contours, lower and upper confidence bounds, the adaptive-region ARSD rule, the region-based cooperative RCC rule that splits the space into confident and uncertain zones, and the EI-MC and EI-SC criteria for prediction, each of which selects the next simulator run by maximizing a goal-specific score. The numerical comparisons then quantify how many simulation runs each criterion saves relative to a one-shot design.

What would settle it

Rerun the three examples with candidate sets of size 10, 100, and 1000 random Latin hypercube points per qualitative level, and also with a continuous optimizer for the quantitative coordinates; if one-shot designs match or beat the adaptive criteria as the candidate set grows, then the paper's superiority claim is an artifact of the coarse search grid rather than a property of the criteria.

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Extended reading notes

Core claim

The paper's central claim is that active learning is both feasible and beneficial when the input space mixes continuous quantitative variables with discrete qualitative factors, provided the surrogate is an EzGP model that decomposes the response into an overall Gaussian process plus level-specific Gaussian processes for each qualitative factor. For optimization, the EI, LCB, ARSD, and Hybrid criteria all approach the true minimum more closely than one-shot designs in three test functions, with the best method varying by example. For contour estimation, the RCC, EI-C, ECL, ARSD-C, and LCB-C criteria improve contour accuracy, measured by MC0, the mean absolute prediction error on points near the contour, over one-shot designs in almost all settings, with RCC and ARSD-C being substantially faster. For prediction, the EI-MC and EI-SC criteria, originally proposed for quantitative-only inputs, yield lower log RMSE than one-shot designs. The paper's concluding statement is that adaptive designs outperform one-shot designs for prediction in mixed-input experiments, and it leaves the development of dedicated prediction criteria for mixed inputs as an open problem.

Load-bearing premise

The reported advantage of adaptive designs over one-shot designs assumes that maximizing each acquisition criterion over 100 random Latin hypercube candidates per qualitative level combination finds essentially the same input as maximizing over the whole design space.

Editorial extensions

If this is right

Practitioners with expensive mixed-input simulators can spend the same budget adaptively and expect better optimization, contour, and prediction accuracy than a one-shot space-filling design.
Quantitative-only criteria such as EI-MC and EI-SC transfer to mixed inputs when paired with an EzGP surrogate, giving lower RMSE than one-shot designs in all three examples.
For contour estimation, RCC and ARSD-C match or beat entropy-based and expected-improvement alternatives while requiring much less computation time.
No single optimization criterion dominates: EI converges fastest in one example, ARSD and LCB are best at the final budget in another, and Hybrid is best in the highest-dimensional example.
The main practical cost shifts to model fitting and criterion evaluation, so faster parameter estimation for mixed-input Gaussian process models would widen the applicability of these adaptive designs.

Reading between the lines

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

Editorial inference: because each acquisition step searches a fixed set of 100 random Latin hypercube candidates per qualitative level, the reported gains are demonstrated against that grid, not against a true continuous maximizer; denser or differently generated grids could change the ranking.
Editorial inference: replacing the random candidate grid with a continuous optimizer on the quantitative coordinates inside each qualitative level could reduce the computational cost the paper identifies as an open challenge, and would directly test whether the candidate-grid approximation is the bottleneck.
Editorial inference: the same EI-MC and EI-SC prediction machinery could be pointed at related mixed-input goals such as probability-of-failure estimation or simultaneous estimation of several contours, which the paper does not examine.

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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

Worth a look if you work on computer experiments with mixed inputs, but don't expect a breakthrough. The new bit is carrying EI-MC and EI-SC—two prediction-oriented acquisition functions originally built for quantitative-only inputs—over to mixed inputs with an EzGP surrogate. The chapter also pulls together recent optimization and contour-estimation adaptive designs into one place, which is genuinely useful as a survey.

The numerical work is solid in setup: three test problems, 50 replications, comparisons against one-shot space-filling designs, and code on GitHub. The tables consistently show adaptive designs beating one-shot designs, and the computation-time breakdown is a nice touch. The review of the literature is competent.

The weaknesses are clear. The comparisons are averages without standard errors or intervals, so you can't tell if a 2x improvement is meaningful. Tuning parameters—delta, epsilon, rho—are set per example, often borrowed from the authors' own earlier papers, with no sensitivity analysis. More importantly, the acquisition criteria are maximized over a finite candidate set: 100 random LHD points per qualitative level combination. The paper itself flags this as 'an open and critical challenge' in Section 5, but the abstract and concluding remarks state the superiority claim without that caveat. In Example 3 (3 quantitative inputs, 27 level combinations), 100 points per cube is pretty sparse, so the true maximizer of EI or ECL can easily fall between candidates. That could change the magnitude of the reported gains (some are enormous, like 823x in Table 4). A sensitivity analysis with different candidate-set sizes would settle this. Also, Example 3's g(z2) has identical formulas for z2=1 and z2=2—likely a typo that should be fixed.

On the novelty question: the extension is fairly routine because EI-MC and EI-SC only depend on predictive mean and variance, so mixing input types just changes the surrogate. The empirical first application is still useful, but it's not a new idea.

Overall: if you're a practitioner or a researcher building on mixed-input emulation, this chapter gives you a convenient map and a working starting point. It deserves a serious referee—the review parts are strong enough and the empirical claims, while underevidenced, are plausible. I'd send it out with the expectation that the authors add uncertainty quantification, check candidate-set sensitivity, and fix the typo. I wouldn't desk-reject it.

Referee Report

3 major / 5 minor

Summary. This manuscript reviews active learning and adaptive design methods for computer experiments with mixed quantitative and qualitative inputs. It describes the EzGP surrogate model, surveys adaptive design criteria for optimization (EI, LCB, ARSD, Hybrid), contour estimation (RCC, EI-C, ECL, ARSD-C, LCB-C), and prediction (EI-MC, EI-SC), and reports numerical studies on three test functions with 50 simulations each, comparing adaptive designs against one-shot space-filling designs. The central empirical claim is that adaptive designs outperform one-shot designs for optimization, contour estimation, and prediction. Code and examples are made available in a GitHub repository.

Significance. If the empirical claims hold, the paper provides useful practical guidance for practitioners who need to allocate limited simulation budgets in experiments with mixed inputs: the adaptive criteria, used with the EzGP emulator, can yield substantially better optimization, contour, and prediction accuracy per run than one-shot designs. The manuscript is also valuable as a review of recent developments in this area. Strengths include the breadth of methods compared, the use of 50 replications on several test functions, and the public code repository. The reported computational times provide useful practical information. However, the main comparisons currently lack uncertainty quantification, and the acquisition-optimization step is based on a fixed finite candidate set whose adequacy is not examined; these issues affect the strength of the central claim.

major comments (3)

[Section 4, first paragraph; Section 5] The numerical studies select the next input by maximizing each acquisition criterion over a finite candidate set consisting of all qualitative level combinations crossed with 100 random Latin hypercube points per level. The paper itself concedes in Section 5 that this is only an approximation to the true maximizer and that the problem of finding optimal inputs for adaptive designs with mixed inputs is 'an open and critical challenge'. Because the reported superiority of adaptive designs over one-shot designs is entirely based on this approximation, the central empirical claim is not fully supported unless the approximation is shown to be adequate. Please provide a sensitivity analysis with respect to the number of candidate points per level (e.g., 100, 200, 500, 1000) or compare against a continuous optimizer for the quantitative dimensions, and report how the relative efficiencies in Tables 4, 6, and 8 change. Without this, the magnitude of the reported advantages, including the 823-fold improvement in Table 4, is difficult to interpret.
[Tables 1-4, 6, 8; Figures 6-8] All performance summaries are reported as averages over 50 simulations without standard errors, confidence intervals, or any measure of variability. Many of the claimed advantages rely on very small MC0 values, such as the values in Table 4 at N=17 and N=19, where relative efficiencies of 412, 823, or 364 are computed from averages that may be dominated by a few simulations or by numerical noise. Please report standard errors or confidence intervals, and ideally paired comparisons between methods, so that the reader can assess whether the differences are statistically meaningful. This is also needed for the prediction comparisons in Figures 6-8, where the boxplots are not accompanied by numeric summaries.
[Section 4.2, especially Tables 4, 6, 8; Section 3.2] The contour-estimation comparisons depend on tuning parameters epsilon and delta, which are chosen separately for each example, and the text states that epsilon should be 'carefully chosen based on the complexity of the response function'. The RCC method in particular relies on delta, and the recommended value comes from the authors' prior work. No sensitivity analysis is provided for these choices, so it is unclear whether the relative performance of RCC is robust to the 'neither too large nor too small' recommendation in Section 3.2. Please add a small sensitivity study for epsilon and delta, or provide a principled data-driven rule for setting them, so that the comparisons are not tied to example-specific tuning.

minor comments (5)

[Section 4.3, Figures 6-8] There is an inconsistency between the budget sizes stated in the text and those given in the figure captions: Example 1 is described with N={15,21} but Figure 6 says N=30; Example 2 is described with N={30,40} but Figure 7 says N=40; Example 3 is described with N={80,100} but Figure 8 says N=100. Please make these consistent and explain which values are being plotted.
[Section 2.2.3] The sentence beginning 'The values of ymin and ymax are unknown contours in general but it can be estimated by a large input from the design domain. They suggest, 1000p design points.' is grammatically awkward and unclear; please rewrite and give the precise reference for the 1000p recommendation.
[Section 4.2] The sentence 'It is important to note that the same dataset is utilized for approximating multiple contours in the EI-MC method described in (21)' appears in the contour-estimation section, but EI-MC is described in Section 2.2.3 as a prediction method. This cross-reference is confusing and should be clarified.
[Throughout] Several equations and algorithms contain formatting artifacts that make them hard to read, for example 'Dn+1 =Dn S(wn+1,yn+1)' in Algorithm 2 and the union symbol appearing as the letter S. Please check the compiled version of the manuscript.
[Section 1, Abstract] The abstract says the paper 'propose[s] active learning approaches', but for prediction the manuscript applies existing EI-MC and EI-SC criteria to mixed inputs rather than proposing new criteria. The wording could more accurately distinguish the proposed/reviewed contributions from the transferred criteria.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; empirical claims are benchmarked against external test functions and one-shot designs.

full rationale

The paper's central claim is that adaptive designs outperform one-shot space-filling designs for optimization, contour estimation, and prediction with mixed quantitative/qualitative inputs. That claim is not circular: the optimized criteria (EI, LCB, ARSD, Hybrid, RCC, ECL, EI-MC, EI-SC) are defined from predictive means and variances of the EzGP surrogate, and the reported gains are computed numerically on analytic test functions against one-shot random LHD baselines in Tables 1–4, 6, and 8 and Figures 3–8. No fitted parameter is renamed as a prediction, and no equation reduces to its own input by construction. There is a mild self-referential element in the contour-estimation section: RCC is the authors' own method from Shahrokhian et al. (2024), the evaluation metric MC0 is taken from that same prior paper, and the delta value is chosen as 'the recommended value suggested by Shahrokhian et al. (2024)'. This is a real self-citation, but it is not load-bearing circularity because the comparison is resolved by the simulated MC0 values rather than by the citation itself; the prior work does not assert the numerical outcomes reported here. Similarly, the paper applies EI-MC and EI-SC from Yang et al. (2020) to mixed inputs; these are external prior criteria, not definitions of the paper's own conclusion. The candidate-set discretization in Section 4 (all qualitative level combinations crossed with 100 random LHD points) is a limitation openly acknowledged in Section 5 as 'an open and critical challenge for further research', but approximation error in maximizing an acquisition criterion is not circularity. Overall, the derivation chain is self-contained against external benchmarks, so no circular step meeting the required 'specific reduction' standard is present.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The paper's results depend on the choices of the EzGP surrogate, the plug-in Gaussian predictive distribution, the finite candidate set for acquisition optimization, and several tuning constants. No new entities are introduced.

free parameters (4)

rho (LCB/ARSD exploration weight) = 2
Set to 2 for optimization comparisons; controls exploration-exploitation trade-off and can change results (Section 4.1).
delta (RCC balance parameter) = 0.05 (Example 1), 0.02 (Example 2), 0.1 (Example 3)
Chosen from the authors' prior paper; the paper states delta critically balances exploration and exploitation (Section 4.2).
epsilon (MC0 band width) = 0.05 (Examples 1-2), 0.1 (Example 3)
Defines the set C0 of points counted as on the contour, directly affecting the reported MC0 values (Section 4.2).
c (number of contours in EI-MC) = 10
Set to 10 following Yang et al. (2020); affects which points are considered for prediction improvement (Section 4.3).

assumptions (3)

domain assumption The EzGP model with additive qualitative structure and Gaussian correlation adequately approximates the test simulators.
All adaptive designs in the numerical studies use the EzGP surrogate; if the simulator does not match this covariance structure, the comparison may not generalize (Section 2.1).
domain assumption The predictive distribution of the simulator output at unsampled points is Gaussian with the plug-in mean and variance from the fitted GP.
All acquisition criteria (EI, ECL, EI-MC, EI-SC) rely on this distributional form while treating estimated GP parameters as known (Section 2.2).
domain assumption A candidate set of 100 random Latin hypercube points per qualitative level combination is sufficient to approximate the true maximizer of the acquisition function over the continuous input space.
The adaptive algorithms select the next point by maximizing over this finite candidate set rather than over the continuous domain (Section 4, first paragraph).

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Cite this review

Pith. "Pith review of Active Learning of Computer Experiment with both Quantitative and Qualitative Inputs." pith.science (2026). https://pith.science/paper/T63VTSE7

@misc{pith2026250413441,
  author       = {Pith},
  title        = {Pith review of: Active Learning of Computer Experiment with both Quantitative and Qualitative Inputs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T63VTSE7}},
  note         = {Machine review of arXiv:2504.13441}
}

read the original abstract

Computer experiments refer to the study of real systems using complex simulation models. They have been widely used as alternatives to physical experiments. Design and analysis of computer experiments have attracted great attention in past three decades. The bulk of the work, however, often focus on experiments with only quantitative inputs. In recent years, research on design and analysis for computer experiments have gain momentum. Statistical methodology for design, modeling and inference of such experiments have been developed. In this chapter, we review some of those key developments, and propose active learning approaches for modeling, optimization, contour estimation of computer experiments with both types of inputs. Numerical studies are conducted to evaluate the performance of the proposed methods in comparison with other existing methods.

Figures

Figures reproduced from arXiv: 2504.13441 by the authors.

**Figure 1.** Adaptive designs versus one-shot designs for Example 1 for (a) EI (b) LCB (c) ARSD (d) Hybrid (e) one-shot for n0 = 9; N = 15 for one simulation. 3.2 Contour Estimation This subsection discusses adaptive design approaches for contour estimation in computer experiments involving both quantitative and qualitative input variables. To the best of our knowledge, Shahrokhian et al. (2024) is the only work that considers a… view at source ↗

**Figure 2.** Adaptive designs versus one-shot designs for Example 1 for (a) RCC, (b) ECL, (c) EI-C, (d) ARSD-C (e) LCB-C, (f) one-shot for n0 = 9; N = 19 for one simulation when a = 1.2. variables with a (nearly) balanced sample across all level combinations of the qualitative factors. The random LHDs are generated using the randomLHS function from the R package lhs (Carnell, 2022). To model the computer experiment with mixed i… view at source ↗

**Figure 3.** The average of the best obtained minimum values of the true response surface in Example 1 for the adaptive designs ARSD, EI, LCB, Hybrid and one-shot designs for n0 = 9 and N = 15 over 50 simulations [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The boxplot of obtained minimum values of the response in Example 2 for the adaptive designs ARSD, EI, LCB, Hybrid and one-shot designs for n0 = 9 and N = 18 over 50 simulations [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: The boxplot of obtained minimum values of the response in Example 3 for the adaptive designs ARSD, EI, LCB, Hybrid and one-shot designs for n0 = 9 and N = 18 over 50 simulations [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: (a) The boxplot of the log of RMSEs; (b) The average of total time in minutes; (c) The average of update time in minutes; (d) The average of criterion time in minutes; for n0 = 10 and N = 30 over 50 simulations in Example 1. n=30 n=40 −5 −4 −3 −2 log(RMSE) log(RMSE) n=…

**Figure 7.** Figure 7: (a) The boxplot of the log of RMSEs; (b) The average of total time in minutes; (c) The average of update time in minutes; (d) The average of criterion time in minutes; for n0 = 20 and N = 40 over 50 simulations in Example 2. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]

**Figure 8.** Figure 8: (a) The boxplot of the log of RMSEs; (b) The average of total time in minutes; (c) The average of update time in minutes; (d) The average of criterion time in minutes; for n0 = 30 and N = 100 over 50 simulations in Example 3. 5 Concluding Remarks Adaptive designs are a…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Using confidence bounds for exploitation-exploration trade-offs

Peter Auer. Using confidence bounds for exploitation-exploration trade-offs. Journal of Machine Learning Research, 3: 0 397--422, 2002

work page 2002

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