Writing study: Controlled LLM Training on Spectral Sphere

ICML overviewCorpus atlasBy contribution typeVisual galleryRules & templateWriting patternsPaper studies

← ICML 2026 oral writing corpus

Tian Xie, Haoming Luo, Haoyu Tang, Hu Yiwen, Jason Klein Liu, Qingnan Ren, Yang Wang, Xin Zhao, Rui Yan, Bing Su, Chong Luo, Baining Guo · Official ICML 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: Training, objectives & optimization (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

16 physical pages

Before references / appendix

≈ 8.26 page-equivalents

Main figures / tables

9 / 3

Attached appendix figures / tables

2 / 0

Attached appendix span

≈ 4.91 page-equivalents

Before references / appendix: ≈ 51.63%References / intervening material: ≈ 17.69%Attached appendix: ≈ 30.69%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	≈ 1.05 pp.	6.56%
2. Preliminary	2	≈ 1 pp.	6.25%
3. Method	3	≈ 1.81 pp.	11.31%
4. Algorithm Details	5	≈ 0.74 pp.	4.63%
5. Infrastructure Design	5	≈ 1.42 pp.	8.88%
6. Scaling Experiments	7	≈ 1.08 pp.	6.75%
7. Discussion	8	≈ 0.74 pp.	4.63%
Acknowledgements	9	≈ 0.05 pp.	0.31%
Impact Statement	9	≈ 0.05 pp.	0.31%
References	9	≈ 2.83 pp.	17.69%
A. Appendix	12	≈ 4.91 pp.	30.69%

All detected figure and table locations

Main text

• Figure 1 · p. 1
• Figure 2 · p. 3
• Figure 3 · p. 4
• Figure 4 · p. 5
• Figure 5 · p. 6
• Figure 6 · p. 6
• Table 1 · p. 7
• Table 2 · p. 7
• Table 3 · p. 8
• Figure 7 · p. 8
• Figure 8 · p. 8
• Figure 9 · p. 8

Attached appendix

• Figure 10 · p. 14
• Figure 11 · p. 16

Argument and organization

1. Problem and contribution

   Poses steepest descent and controlled activation scaling as simultaneous requirements rather than separate optimizer heuristics.

2. Argument sequence

   Tangent-space and retraction constraints precede practical radius, scaler and module decisions; a dedicated infrastructure section bridges equations to large-scale experiments.

3. Evidence and verification

   The dense-training caption discloses whose learning rate was tuned and defines efficiency through steps to a target loss; load imbalance receives a separate metric.

Problem, claim and contribution

Poses steepest descent and controlled activation scaling as simultaneous requirements rather than separate optimizer heuristics.

Introduction and preliminary definitions, p. 2 · Main paper

Argument and section order

Tangent-space and retraction constraints precede practical radius, scaler and module decisions; a dedicated infrastructure section bridges equations to large-scale experiments.

Sections 3-6, pp. 3-8 · Main paper

Figures, captions and visual explanation

Width-specific learning-rate curves sit beside the mathematical formulation. Dense/MoE/depth panels share optimizer colors, and the task table retains several wins for MuonSphere rather than the proposed variant.

Figure 2, p. 3; Table 3 and Figures 7-9, p. 8; both PNGs inspected · Main paper

Tables, comparisons and evidence

The dense-training caption discloses whose learning rate was tuned and defines efficiency through steps to a target loss; load imbalance receives a separate metric.

Figures 7-8 and Section 6.3, p. 8 · Main paper

Appendix and reproducibility

The supplement exposes root-solver synchronization, precision/iteration choices and work still needed for GPU-native and low-precision execution.

Appendix A.2, pp. 12-14; A.6, p. 16 · Main paper

Clarity and writing conventions

Algorithmic geometry and systems constraints receive distinct sections, making it possible to distinguish formal intent from current implementation compromises.

Sections 4-5, pp. 5-6; Appendix A.6, p. 16 · Main paper

What to reuse—and what not to infer

Transferable lesson: Connect a principled optimizer to concrete module granularity, solver overhead and matched target-loss measurements.

Boundary: Step-count gains are not automatically wall-clock gains; unimplemented GPU-native/low-precision improvements and broad training-stability claims should remain separate.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-10-07

Reading status

Writing-focused manuscript scan + selected visuals inspected

Main text

Writing-focused distributed manuscript scan, not exhaustive technical/proof validation. Sections 1-7, pp. 1-9: spectral geometry, constrained update, radius/scaler/granularity choices, distributed implementation and dense/MoE/depth tests.

Appendix / supplement

Writing-focused appendix scan; code, data and external assets not executed. Appendix A.1-A.6, pp. 12-16: norm duality, root localization, decay/routing analysis, learning-rate sweep and future implementation work.

Visually inspected PDF pages

3, 8

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.

• Published ICML 2026 proceedings PDF, PMLR volume 306; retrieved 2026-10-07. Public OpenReview decision notes were not independently reverified. · 16 pages