Writing study: Uniform-in-Time Weak Propagation of Chaos in Shallow Neural Networks
← NeurIPS 2026 oral writing corpus
Margalit Glasgow, Joan Bruna · Official NeurIPS 2026 oral record
Figure, table and section ledger
Editorial contribution type: Theory & formal analysis (primary).
Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.
- Canonical PDF
- 46 physical pages
- Before references / appendix
- ≈ 27.81 page-equivalents
- Main figures / tables
- 2 / 0
- Attached appendix figures / tables
- 3 / 0
- Attached appendix span
- ≈ 14.47 page-equivalents
Source-bound correction: Top-level heading and reference/appendix boundaries checked against source PDF text. Bibliography titles/author initials, numbered contribution/property bullets, diagram steps, equations and TOC entries are not substantive headings. Section spans are page-coordinate estimates, not ink area or prose-only length. This is an author manuscript mirror, not a verified accepted-version template. Source layout uses physical single-column y/page-height, including equation-heavy pages. Restored real Figure 2 caption on physical p.28: math extraction splits its figure label from the numeral and text.
Section-by-section lengths and source pages
The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.
| Top-level heading | Starts on PDF page | Estimated span | Share of entire PDF |
|---|---|---|---|
| 1 Introduction | 2 | ≈ 3.41 pp. | 7.41% |
| 2 Setting and Preliminaries | 5 | ≈ 2.93 pp. | 6.37% |
| 3 Technical Approach | 8 | ≈ 1.79 pp. | 3.89% |
| 4 Formal Statement and Proof of Main Result on Uniform in Time PoC | 10 | ≈ 16.89 pp. | 36.72% |
| 5 Experiments | 27 | ≈ 0.49 pp. | 1.07% |
| 6 Conclusions and Future Work | 27 | ≈ 1.01 pp. | 2.2% |
| References | 28 | ≈ 3.72 pp. | 8.09% |
| A Proofs of Assumptions 2, 3 and Lemma 2 | 32 | ≈ 11.83 pp. | 25.72% |
| B Supplemental Experimental Details for Misspecified Sobolev single-index | 44 | ≈ 2.64 pp. | 5.74% |
All detected figure and table locations
Main text
Attached appendix
Argument and organization
- 1 Introduction
Explains why short-time bounds do not transfer mean-field learning indefinitely.
- 2 Setting and Preliminaries
Defines networks, coupling and fluctuation dynamics.
- 3 Technical Approach
Provides proof intuition before full assumptions.
- 4 Formal Statement and Proof of Main Result on Uniform in Time PoC
States and proves uniform-in-time bounds with comparison remarks.
- 5 Experiments
Illustrates decay assumptions on synthetic targets.
- 6 Conclusions and Future Work
Discusses weaker assumptions, slower rates and wider model classes.
Problem, claim and contribution
The introduction explains why the usual exponential-in-time Gronwall control is insufficient for finite networks tracking a mean-field limit. An informal theorem states the new long-time guarantee before the full notation and assumptions.
PDF pp.2–4, Introduction and informal Theorem 1 · Main paper
Argument and section order
A contents map leads through coupled dynamics, a technical approach, the formal theorem and proofs, then a compact empirical check. This mirror devotes substantial main-text space to proof development rather than using the standard compact conference outline.
PDF pp.1–2 contents; Sections 2–6, pp.5–28 · Main paper
Figures, captions and visual explanation
Figure 1 places the loss curve beside the integral of its square root, which is the quantity appearing in the theorem assumption. Theorem 2, assumptions, corollary, and comparison remarks are co-located on p.13; the plot thus tests an identifiable condition rather than decorating the theory.
PDF p.13, Theorem 2 and Corollary 6; p.27, Figure 1 · Main paper
Tables, comparisons and evidence
Synthetic targets vary smoothness and include a setting where the required decay condition is not satisfied. The text carefully treats this as an illustration of assumptions; observed convergence on finite trajectories is not a proof that the condition holds for all neural networks.
PDF pp.27–28, Section 5 and Figures 1–2 · Main paper
Appendix and reproducibility
Appendix A supplies concentration and dynamical-lemma proofs, with a decomposition illustration near Lemma 2. Appendix B gives the misspecified Sobolev example, target functions, and additional loss/integral curves.
PDF pp.32–46, Appendices A–B; Figures 3–5 · Main paper
Clarity and writing conventions
The conclusion explicitly asks whether the decay assumption can be weakened and notes that many tasks exhibit slower rates or burn-in behavior. This prevents a uniform-in-time statement from reading as an unconditional practical guarantee.
PDF p.28, Conclusions and Future Work · Main paper
What to reuse—and what not to infer
Transferable lesson: When theory depends on a nontrivial condition, visualize that condition itself and retain examples that delimit when it fails.
Boundary: This is a 46-page arXiv presentation with a long main proof, not evidence that NeurIPS permits a 28-page submission body. The writing scan did not verify the lengthy proofs, and the author manuscript is not confirmed identical to the accepted revision.
Apply this lesson using the argument and evidence worksheet.
Inspection record and sources
- Checked
- 2026-10-02
- Reading status
- Writing-focused manuscript scan + selected visuals inspected
- Main text
- Writing-oriented source scan of framing, argument roadmap, selected substantive passages, method/theorem statements, main figure/table captions, evidence and limits across PDF pp.1–28. Two selected main-body pages were actually inspected as rendered images. This is not exhaustive line-by-line technical verification or experiment reproduction. Title and all author names compared against official program metadata and manuscript first page; matching authors, with capitalization, middle initials or expanded given names where applicable.
- Appendix / supplement
- Appendix map, proof opening roles, decomposition-caption and experimental-detail openings were inspected; full concentration/proof verification was outside the writing audit.
- Visually inspected PDF pages
- 13, 27
This is a writing-focused scan of the manuscript and the stated supplementary scope: framing, organization, visual explanation, evidence presentation and wording. It is not a line-by-line proof audit, reproduction of results or exhaustive scientific review.