CausalSmith — CSWP-2026-009 · ← back to the paper · arXiv · GitHub

AI referee report

on “Minimax Mean Squared Error for Low-order Network Interference under Bernoulli Assignment”

Overall score 7.8/10 9 findings — 6 minor · 3 nit

The verified mathematical core delivers a sharp and useful minimax calibration, while the manuscript needs targeted prose repairs for comparison scope, cross-reference hygiene, and a few stale or confusing exposition points.

The paper may have been revised since this report. One automated referee’s estimate — not peer review. Produced by a single automated referee in the pipeline's review stage. It can be wrong, and it is not peer review.

Summary

The paper establishes a finite-population minimax MSE frontier for known bounded-degree low-order polynomial interference under common Bernoulli assignment, with matched clipped-SNIPE upper bounds, complete-block lower constructions, exact complete-block SNIPE risks, and a block-local linear benchmark. The contribution is technically meaningful for design-based network experiments and is well aligned with the verified statements. Publication is plausible after focused revisions that sharpen the comparison to prior SNIPE work, clean up presentation, and repair a few contract violations.

Strengths

Sharp two-envelope minimax frontier with explicit dependence on the complete-block score energy and the out-degree overlap factor.
Useful exact complete-block calculations and local-linear benchmark that clarify the geometry behind SNIPE weights.
The paper states fixed-parameter constants and boundary cases carefully in most places, including fair-coin cancellations and degree-zero conventions.
Dedicated related-work section engages the main interference and SNIPE literatures early in the manuscript.

Findings

minor Related work

The comparison to Cortez-Rodriguez, Eichhorn, and Yu is mostly careful, but the sentence "their SNIPE analysis supplies design-unbiasedness and a stated worst-case worst-case variance upper bound whose degree dependence enters through $d^{β}$ " contains a typo and leaves the exact comparator too imprecise for a central novelty claim.

Suggested fix

Replace the sentence with a precise description of the cited theorem's assumptions, normalization, and displayed degree factor; remove the duplicated "worst-case" and state that the present result calibrates the bounded known-graph class against that stated upper bound.

minor Main results

The theorem map says "Exact complete-block SNIPE risk" and "canonical unprojected SNIPE on complete directed blocks" but the surrounding prose also says the result is "globally" exact for that estimator. A reader could miss that exactness is for the worst-case risk of the fixed estimator when $d ∣ n$ , not the exact all-estimator minimax constant.

Suggested fix

Add one clarifying sentence after the theorem map: the exact equality concerns canonical unprojected SNIPE's worst-case risk on the fixed block graph and the global supremum over the two classes when $d ∣ n$ ; the all-measurable minimax leading constant is addressed only by the rate bounds.

minor Verification note

The sentence "where a result from the literature is a formal dependency, it enters as a published input rather than a Lean-checked step" in the author footnote is stale relative to the verification contract for this submission, which records no external formal dependencies for the displayed objects.

Suggested fix

Update the author footnote and verification-note wording to match the current contract: the displayed formal statements and proofs are Lean-verified, and cited work supplies econometric framing, terminology, and positioning for these objects.

minor Appendices

Several appendix proof paragraphs present long informal derivations for theorem statements already declared verified. This is acceptable, but the main text does not give readers a short proof roadmap before entering very lengthy proof blocks.

Suggested fix

At the start of the appendix, add a compact roadmap listing which verified declarations establish the representer identity, overlap count, Hellinger bound, minimax frontier, bounded-outcome frontier, local-linear benchmark, and fair-coin specialization.

minor Setup and assumptions

The score definition uses $\overset{ˉ}{β}_{d}$ in the outer sum for every unit and then explains the empty-sum convention for units with $∣ N_{i} ∣ < \overset{ˉ}{β}_{d}$ . This is correct, but it is less transparent than the Lean declaration, which sums to $min {β, ∣ N_{i} ∣}$ .

Suggested fix

Rewrite Definition 7 with outer limit $min {β, ∣ N_{i} ∣}$ , then mention that this equals the displayed block limit when $∣ N_{i} ∣ = d$ .

minor Related work

The related-work section cites many strands, but the closest comparison would benefit from a small displayed rate table or paragraph aligning assumptions across the prior SNIPE theorem, the present coefficient-mass class, and the bounded-outcome class.

Suggested fix

Add a short comparison table with columns for graph knowledge, assignment design, outcome class, estimator, target, and risk or variance rate; keep the text explicit that the present result sharpens the bounded-class calibration rather than the prior paper's unrestricted theorem.

nit global

House typography requires sentence case headings. The theorem and definition titles mostly comply, but title-like item labels such as "Coefficient-mass minimax frontier" and table headers are acceptable; ensure no remaining generated headings outside the excerpt are Title Case.

Suggested fix

Run a heading pass over the full source and keep section/subsection titles in sentence case, preserving proper nouns and math symbols.

nit Related work

The phrase "a stated worst-case worst-case variance upper bound" is a copy-editing error.

Suggested fix

Delete one occurrence of "worst-case".

nit Main results

The local-linear theorem includes the striking distance-equals-two construction, but the surrounding discussion gives only one sentence of interpretation. Readers may not immediately understand why this result belongs beside the minimax frontier.

Suggested fix

Add one sentence before or after the theorem explaining that the construction separates asymptotic risk optimality from normalized pointwise closeness to the canonical representer within the block-local unbiased linear class.

Questions for the authors

Can the closest-prior-work comparison state the exact theorem number, normalization, and degree factor from Cortez-Rodriguez, Eichhorn, and Yu so readers can verify the sharpening directly?
Do the authors want the exact complete-block unprojected-SNIPE equality to be highlighted as an estimator-risk identity rather than as evidence about the all-measurable minimax constant?

Review history

Round	Date	Score	Findings
1	—	8.0	6
2	—	8.0	5
3	—	7.0	10
4	—	8.0	8
5	—	8.1	6
6	—	8.0	8
7	—	7.0	10
8	—	6.8	10
9	—	7.0	8
10 · current	—	7.8	9

Every referee draw is kept: scores are noisy, so the history is shown rather than only the last or the best pass.