Writing study: Asymmetric Perturbation in Solving Bilinear Saddle-Point Optimization

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← ICML 2026 oral writing corpus

Kenshi Abe, Mitsuki Sakamoto, Kaito Ariu, Atsushi Iwasaki · 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: 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

35 physical pages

Before references / appendix

≈ 8.75 page-equivalents

Main figures / tables

5 / 0

Attached appendix figures / tables

4 / 4

Attached appendix span

≈ 23.65 page-equivalents

Before references / appendix: ≈ 25%References / intervening material: ≈ 7.43%Attached appendix: ≈ 67.57%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Source-checked physical-page text restores wrapped top-level heading labels on pp. 5, 7. Heading starts and measured positions are unchanged; this is label completeness, not a new section. Newly detected unnumbered impact/singular-acknowledgment statements are retained from the size-aware source extraction; authored heading corrections are unchanged.

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	≈ 1.02 pp.	2.91%
2. Preliminaries	2	≈ 1.04 pp.	2.97%
3. Asymmetric Payoff Perturbation	3	≈ 1.96 pp.	5.6%
4. Asymmetrically Perturbed Gradient Descent Ascent	5	≈ 2.02 pp.	5.77%
5. Dilated AsymP-GDA for Extensive-Form Games	7	≈ 1.52 pp.	4.34%
6. Related Literature	8	≈ 0.46 pp.	1.31%
7. Conclusion and Limitations	9	≈ 0.35 pp.	1%
Acknowledgements	9	≈ 0.04 pp.	0.11%
Impact Statement	9	≈ 0.05 pp.	0.14%
References	9	≈ 2.6 pp.	7.43%
A. Experimental Details and Additional Experimental Results	12	≈ 2.38 pp.	6.8%
B. Impossibility Results for Symmetric Perturbation	14	≈ 1.66 pp.	4.74%
C. Independently Perturbed Game	16	≈ 0.23 pp.	0.66%
D. Parameter-Free AsymP-GDA	16	≈ 1 pp.	2.86%
E. Game Instance with Arbitrarily Small Allowable Range of µ	17	≈ 0.51 pp.	1.46%
F. Proofs of Theorem 3.1 and Corollary 3.2	18	≈ 1.05 pp.	3%
G. Proofs for Theorem 4.1	19	≈ 5.7 pp.	16.29%
H. Proofs for Theorem 4.3	24	≈ 4.86 pp.	13.89%
I. Proof of Theorem B.1	29	≈ 2.48 pp.	7.09%
J. Proof of Theorem C.1	32	≈ 2.16 pp.	6.17%
K. Proof of Theorem E.1	34	≈ 1.63 pp.	4.66%

All detected figure and table locations

Main text

• Figure 1 · p. 3
• Figure 2 · p. 4
• Figure 3 · p. 5
• Figure 4 · p. 8
• Figure 5 · p. 12

Attached appendix

• Table 1 · p. 12
• Table 2 · p. 13
• Table 3 · p. 13
• Figure 6 · p. 13
• Figure 7 · p. 13
• Figure 8 · p. 14
• Table 4 · p. 15
• Figure 9 · p. 15

Argument and organization

1. Problem and contribution

   Contrasts perturbing both players with perturbing one player, then separates original-game equilibrium recovery from merely solving the perturbed game.

2. Argument sequence

   A simple invariance phenomenon precedes algorithmic rates; the extensive-form variant is introduced only after the main theorem and proof sketch.

3. Evidence and verification

   A technical remark states that the dilated regularizer lacks the global smoothness needed for the main proof, leaving that empirical extension without an analogous convergence theorem.

Problem, claim and contribution

Contrasts perturbing both players with perturbing one player, then separates original-game equilibrium recovery from merely solving the perturbed game.

Introduction and preliminaries, p. 2 · Main paper

Argument and section order

A simple invariance phenomenon precedes algorithmic rates; the extensive-form variant is introduced only after the main theorem and proof sketch.

Sections 3-5, pp. 3-8 · Main paper

Figures, captions and visual explanation

The opening comparison marks a finite invariance region; the game panel caption explicitly charges both separate player runs to the horizontal update budget.

Figure 1, p. 3; Figure 4, p. 8; both PNGs inspected · Main paper

Tables, comparisons and evidence

A technical remark states that the dilated regularizer lacks the global smoothness needed for the main proof, leaving that empirical extension without an analogous convergence theorem.

Remark 5.1, p. 8 · Main paper

Appendix and reproducibility

An explicit family makes the admissible perturbation interval arbitrarily small; proof destinations separately support invariance, rate, parameter-free and impossibility claims.

Appendix E, pp. 17-18; F-K, pp. 18-35 · Main paper

Clarity and writing conventions

The paper distinguishes a strategy for one player from a strategy pair, and keeps parameter choice and cost accounting near the corresponding guarantee/plot.

Sections 3.1 and 5, pp. 3 and 8 · Main paper

What to reuse—and what not to infer

Transferable lesson: Pair a striking theorem with its boundary example and visibly separate proved settings from empirical extensions.

Boundary: The invariance threshold is instance-dependent and can be arbitrarily small; extensive-form curves are not a proof for the dilated variant or general Markov games.

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 1-6, pp. 1-8: asymmetric perturbation idea, bilinear setting, invariance, algorithm/rates, parameter-free extension and extensive-form experiments.

Appendix / supplement

Writing-focused appendix scan; code, data and external assets not executed. Appendices A-K, pp. 12-35: resources and tuning, additional comparisons, symmetric impossibility, parameter-free details, small-threshold counterexample and theorem-specific proofs.

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. · 35 pages