Writing study: Fast Escape, Slow Convergence: Learning Dynamics of Phase Retrieval under Power-Law Data

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← ICLR 2026 oral writing corpus

Guillaume Braun, Bruno Loureiro, Minh Ha Quang, Masaaki Imaizumi · Official ICLR 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Theory & formal foundations (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

43 physical pages

Before references / appendix

≈ 10.11 page-equivalents

Main figures / tables

3 / 0

Attached appendix figures / tables

2 / 1

Attached appendix span

≈ 29.82 page-equivalents

Before references / appendix: ≈ 23.51%References / intervening material: ≈ 7.14%Attached appendix: ≈ 69.35%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

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	1	≈ 2.53 pp.	5.88%
2 PROBLEM SETUP	4	≈ 0.57 pp.	1.33%
3 LOSS GEOMETRY AND EVOLUTION OF SUMMARY STATISTICS	4	≈ 2.03 pp.	4.72%
4 MAIN RESULTS: THREE-PHASE GRADIENT FLOW DYNAMICS	6	≈ 1.81 pp.	4.21%
5 PROOF OUTLINE	8	≈ 1.96 pp.	4.56%
6 DISCUSSION	10	≈ 0.32 pp.	0.74%
ACKNOWLEDGEMENTS	10	≈ 0.31 pp.	0.72%
REFERENCES	11	≈ 3.08 pp.	7.16%
A SUMMARY OF THE DIFFERENT STOPPING TIMES	14	≈ 0.28 pp.	0.65%
B PROOF OF SECTION 3	14	≈ 2.75 pp.	6.4%
C ANALYSIS OF PHASE I: ESCAPING MEDIOCRITY	17	≈ 10.15 pp.	23.6%
D ANALYSIS OF PHASE II	27	≈ 6.41 pp.	14.91%
E PHASE III: SCALING LAWS	33	≈ 7.68 pp.	17.86%
F ADDITIONAL NUMERICAL EXPERIMENTS	41	≈ 2.55 pp.	5.93%

All detected figure and table locations

Main text

• Figure 1 · p. 2
• Figure 2 · p. 6
• Figure 3 · p. 7

Attached appendix

• Table 1 · p. 14
• Figure 4 · p. 41
• Figure 5 · p. 43

Argument and organization

1. Anisotropy changes the learning trajectory

   Motivate a tractable nonlinear model rather than claim a universal deep-network law.

2. Empirical phases and theorem roadmap

   Introduce the three stages before technical stopping-time statements.

3. Proof outline and complete appendices

   Keep the main argument navigable while exposing detailed derivations.

Problem, claim and contribution

The paper separates prediction loss from sign-invariant parameter recovery MSE. That choice matters: high-variance directions can give good prediction while weak spectral directions remain unrecovered, explaining why the two convergence stories differ.

§2, p.4; §4.2, pp.7–8 · Main paper

Argument and section order

The empirical phase decomposition precedes Theorems 1–2, and the proof outline repeats those phases as headings. Appendix A's stopping-time table provides a compact index linking definitions, scaling and initialization dependence.

§§3.4–5, pp.5–10; Table 1, p.14 · Main paper

Figures, captions and visual explanation

Figure 2 places alignment, energy and MSE on one timeline with three stopping-time markers. Its right MSE axis covers a narrow interval, so the apparently large downward curve represents only a small change; this is a useful reminder to read both axes.

Figure 2, p.6 · Main paper

Tables, comparisons and evidence

Figure 4 overlays the idealized spectral relaxation with population dynamics, testing the approximation used in the derivation rather than reporting an unrelated leaderboard. Online SGD appears separately and is explicitly outside the gradient-flow guarantee.

§6, p.10; Appendix F, pp.41–43 · Main paper

Appendix and reproducibility

The long appendix exposes assumptions and derivation steps, but reading it is not proof certification. Eq. C.18 uses a convolution form although Eq. 5.2 contains Θ(t)−Θ(τ), generally not Θ(t−τ); the general-kernel reduction needs clarification.

Eq.5.2, p.19; Eqs.C.18–C.19, p.21; §C.3.2, pp.23–25 · Main paper

Clarity and writing conventions

Broad rigor claims warrant caution around visible inconsistencies: D.22 drops the positive mixed term from the preceding equality, and D.15 states a positive power-law tail even below the smallest finite eigenvalue. These require technical review, not silent reuse as settled lemmas.

Eq.D.15, p.31; Lemma 11 and Eq.D.22, p.32 · Main paper

What to reuse—and what not to infer

Transferable lesson: Introduce observables and empirical phases before theorem statements, then keep the proof roadmap and appendix headings aligned with that narrative.

Boundary: The setting is Gaussian, power-law, population quadratic regression with initialization conditions, not a proven SGD or foundation-model scaling law. Numerical illustrations do not resolve the noted derivation issues.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-09-19

Reading status

Full text + selected visuals inspected

Main text

Pages 1–10, Sections 1–6; references 11–13

Appendix / supplement

Pages 14–43, Appendices A–F: stopping times, geometry proofs, three-phase analysis and numerical experiments

Visually inspected PDF pages

6, 41

• official ICLR 2026 proceedings PDF · 43 pages