Writing study: Second-Order Smooth Planning with Optimal-Transport Bellman Smoothing

← ICML 2026 oral writing corpus

Tuan Quang Dam · Official ICML 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: 40 physical pages
Before references / appendix: ≈ 9.04 page-equivalents
Main figures / tables: 0 / 2
Attached appendix figures / tables: 0 / 0
Attached appendix span: ≈ 29.92 page-equivalents

Before references / appendix: ≈ 22.6%References / intervening material: ≈ 2.6%Attached appendix: ≈ 74.8%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Include the source appendix title and contents on p.11, before notation and Appendix A on p.12. This is a physical supporting-material span, not prose-only proof length. Wrapped main heading labels restored from the immediately following same-size source line in the physical PDF. Heading start positions are unchanged. 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	≈ 1.28 pp.	3.2%
2. Related Work	2	≈ 0.82 pp.	2.05%
3. Background: Regularized Bellman Operators and SmoothCruiser	3	≈ 0.67 pp.	1.68%
4. Curvature-Driven Planning: General Theory	4	≈ 0.68 pp.	1.7%
5. OT-Smoothed Bellman Aggregators over Actions	4	≈ 1.34 pp.	3.35%
6. Second-Order SmoothCruiser with OT-Smoothed Aggregators	6	≈ 0.66 pp.	1.65%
7. Complexity Analysis for SecondOrderSmoothCruiser	6	≈ 0.78 pp.	1.95%
8. OT-GapE: Matching ε−2 via Confidence-Bound Trajectory Planning	7	≈ 1.54 pp.	3.85%
9. Gap-Dependent Extensions	9	≈ 0.49 pp.	1.23%
10. Discussion and Limitations	9	≈ 0.3 pp.	0.75%
Impact Statement	9	≈ 0.05 pp.	0.13%
Acknowledgments	9	≈ 0.11 pp.	0.27%
References	10	≈ 1.04 pp.	2.6%
Appendix: title and contents	11	≈ 1.29 pp.	3.23%
A. Curvature–complexity theory	12	≈ 4.76 pp.	11.9%
B. OT-smoothed aggregator: closed form and derivatives	17	≈ 9.2 pp.	23%
C. Gap-dependent analysis	26	≈ 7.1 pp.	17.75%
D. Bias control and worst-case complexity	33	≈ 7.56 pp.	18.9%

All detected figure and table locations

Main text

Attached appendix

No recognized caption labels.

Argument and organization

Problem and claim
Begins with the number of simulator calls needed to estimate one root value, then ties the proposed improvement to Taylor-remainder order. The resource and output being guaranteed are defined before the complexity headline.
Argument sequence
A general curvature model precedes a concrete OT-smoothed operator and estimator. Gap-dependent extensions are explicitly separated from the main worst-case result rather than silently substituting a different setting.
Evidence and verification
The guarantee fixes OT parameters independently of target accuracy and states which constants remain hidden. A dedicated paragraph distinguishes oracle calls from dense finite-action arithmetic overhead, so the bound is not reported as measured wall-clock acceleration.

Problem, claim and contribution

Begins with the number of simulator calls needed to estimate one root value, then ties the proposed improvement to Taylor-remainder order. The resource and output being guaranteed are defined before the complexity headline.

Abstract and Introduction, pp. 1–2 · Main paper

Argument and section order

A general curvature model precedes a concrete OT-smoothed operator and estimator. Gap-dependent extensions are explicitly separated from the main worst-case result rather than silently substituting a different setting.

Sections 4–9, pp. 4–9 · Main paper

Figures, captions and visual explanation

There is no decorative architecture figure: Table 1 aligns the setting, backup, remainder and oracle complexity, while Table 2 maps each contribution to its consequence and section. Algorithm 3 visibly separates linear and quadratic correction blocks.

Tables 1–2, physical p. 3; Algorithm 3, physical p. 7; rasters inspected · Main paper

Tables, comparisons and evidence

The guarantee fixes OT parameters independently of target accuracy and states which constants remain hidden. A dedicated paragraph distinguishes oracle calls from dense finite-action arithmetic overhead, so the bound is not reported as measured wall-clock acceleration.

Assumption 5.2, p. 4; Theorem 7.2, p. 6; oracle-complexity discussion, p. 7 · Main paper

Appendix and reproducibility

A contents page and notation table precede theorem-indexed proofs. The bias-control derivation distinguishes an ideal unclipped estimator from recursive and clipped implementation terms, making its proof dependencies inspectable.

Appendix contents pp. 11–12; Appendix D, pp. 33–40 · Main paper

Clarity and writing conventions

The discussion openly calls the contribution primarily theoretical and keeps regularization bias and vanishing-temperature constants separate from the fixed-parameter complexity improvement.

Section 10, physical p. 9 · Main paper

What to reuse—and what not to infer

Transferable lesson: For a theory paper, a assumptions-and-guarantees table and executable-looking pseudocode can do the explanatory work of a figure; keep the exact cost model visible.

Boundary: The writing scan does not validate the proofs, and an improved oracle exponent is neither a runtime benchmark nor an unregularized-objective guarantee without its bias analysis.

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 scan of Sections 1–10, pp. 1–9: oracle setting, assumptions, comparison tables, algorithms, fixed-parameter guarantees and limitations.
Appendix / supplement: Structural scan of the contents, notation and Appendices A–D, pp. 11–40, with selected checks of theorem-to-proof pointers, estimator assumptions and bias accounting. No independent proof verification.
Visually inspected PDF pages: 3, 7

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 proceedings PDF, PMLR volume 306, retrieved 2026-10-07 · 40 pages