AI referee report
on “Minimax Excess Variance with Smooth Pairwise Interactions in Randomized Experiments”
The verified characterization provides a substantial theoretical contribution with largely faithful scope, while extensive repetition and insufficiently developed econometric interpretation require substantial revision for publication.
Summary
The paper characterizes minimax scaled Horvitz–Thompson excess variance under independent uniform covariates and a centered, pooled order-two Sobolev prognostic class. Its logarithm-free comparison, uniform constants, smoothness-independent assignment construction, and independent-prior converse form a meaningful contribution. The central claims closely track the verification contract, including the fixed-smoothness dimension frontier and the necessary-condition transfer to bounded-density classes. Publication requires a substantially more focused presentation and clearer explanation of the substantive modeling choices.
Strengths
- The matched finite-sample comparison identifies interaction-count dependence throughout the subcritical and saturated regimes, including the endpoint s=1.
- A single fair assignment kernel attains the comparison simultaneously across smoothness values on all original experimental units.
- The independent coefficient prior preserves the fixed-function population comparison and establishes a converse for the full admissible design class.
- The causal interpretation carefully distinguishes arbitrary square-integrable outcome laws, their law-specific benchmarks, and the Gaussian calibration.
- The dedicated early related-work section engages competing rates, sampling budgets, function classes, and estimator choices.
Findings
The manuscript repeatedly presents essentially the same characterization and causal interpretation. The long main capacity theorem is followed by another complete characterization in the appendix, while the Gaussian completion, dimension frontier, and precision implications recur across setup, main results, discussion, and proof assembly. This repetition obscures the hierarchy of contributions and makes the paper disproportionately difficult to navigate.
Suggested fix
Make one capacity statement the organizing result. Group its conclusions into clearly identified parts, present causal transfer and precision implications once, and replace repeated explanations with targeted cleveref references. Give the appendix proof assembly a short dependency roadmap. Shorten the causal-completion definition's exhaustive reference list to the few definitions needed to identify its objects. Preserve all verified conclusions and theorem-local verification disclosures during consolidation.
The sentence “We work with its centered version, whose integral over the cube is zero” can suggest that centering the actual conditional midpoint mean is an innocuous normalization. The verified causal characterization instead specifies an outcome-law class whose conditional midpoint mean itself has zero population integral. For the fixed Horvitz–Thompson estimator and the stated assignment class, this distinction is substantive.
Suggested fix
Replace the sentence with an explicit affirmative specification: “The outcome-law class specifies a conditional potential-outcome midpoint mean with population integral zero.” Explain briefly why this zero-intercept condition belongs to the model, and retain the discussion of unknown intercepts in Limitations and future work.
The detailed positioning emphasizes discrepancy, spectral methods, and the closest capacity comparison. The econometric motivation would benefit from a concise connection to matching, stratification, and regression adjustment, particularly because the paper fixes an unadjusted Horvitz–Thompson estimator and evaluates population excess above a law-specific benchmark.
Suggested fix
Add one focused paragraph connecting this criterion to matching and stratification efficiency results and to regression adjustment. Identify the estimator, model, and comparison criterion in each relevant reference, with accurate locators. Explain which experimental-design question the interaction-count characterization answers within its stated uniform-covariate model.
The setup interleaves function-class and assignment material: conditional fairness appears inside Canonical components and regularity, conditional assignment independence precedes the formal Sobolev-class definition, and explanatory paragraphs refer to first and second assignment requirements across intervening material. This organization makes otherwise precise definitions harder to follow.
Suggested fix
Complete the canonical-component, regularity, and Sobolev-class presentation together. Then present fairness, conditional assignment independence, and the admissible design class together, followed by the loss and minimax criterion. Keep the existing labels and use cleveref for subsequent references.
The repeated numerical calibration gives considerable space to the arbitrary Gaussian benchmark 4.01 and extremely conservative sample-size certificates. A worked interpretation of the pooled component budget would better explain the statistical meaning of the result: the class permits different allocations across main effects and pairs, and its dimension frontier describes worst-case performance over that class.
Suggested fix
Retain one concise Gaussian calibration and add a worked class example showing how a fixed pooled budget can be allocated across a few components or many pair effects. Explain the worst-case meaning of the interaction-count dependence and distinguish it from a sample-size prediction for an individual prognosis. Use the existing definitions and bounds without asserting additional guarantees.
The phrases “rather than a claim of sharp numerical calibration” and “they should not be read as a sharp numerical calibration” frame the interpretation through an absent claim outside the labelled limitations section.
Suggested fix
Use an affirmative description such as: “These constants provide conservative finite-sample certificates of the interaction-count and tolerance scaling.” Retain the displayed tolerance restriction and numerical constant gap.
The phrase “Its sample-dependent cutoff and smoothness-dependent normalization are specified next” is ambiguous because the prior must remain independent of the realized covariate sample. The cutoff depends on sample size and dimension.
Suggested fix
Write “Its sample-size- and dimension-dependent cutoff and smoothness-dependent normalization are specified next,” and retain the explicit statement that coefficient signs are drawn before observing covariates.
Questions for the authors
- What experimental interpretation motivates a conditional midpoint mean with zero population integral when the estimator is fixed to unit-weight Horvitz–Thompson?
- Which matching or stratification comparison best communicates the practical significance of the interaction-count frontier under the stipulated uniform-covariate model?
Review history
| Round | Date | Score | Findings |
|---|---|---|---|
| 1 | 3 Oct 2026 | 6.7 | 6 |
| 2 | 3 Oct 2026 | 6.7 | 5 |
| 3 · current | 4 Oct 2026 | 7.0 | 7 |
Every referee draw is kept: scores are noisy, so the history is shown rather than only the last or the best pass.
Read against paper.tex sha256 04ddca4d1924.