Writing study: Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport

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← ICLR 2026 oral writing corpus

Harry Amad, Mihaela van der Schaar · 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: Learning & optimization algorithms (primary) · Theory & formal foundations.

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

30 physical pages

Before references / appendix

≈ 10.33 page-equivalents

Main figures / tables

3 / 7

Attached appendix figures / tables

2 / 8

Attached appendix span

≈ 15.89 page-equivalents

Before references / appendix: ≈ 34.43%References / intervening material: ≈ 12.6%Attached appendix: ≈ 52.97%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Caption audit 2026-09-19, P5B97gZwRb: rendered physical PDF pages 8 and 10 confirm Table 3 (Reacher surrogate rewards) and Table 6 (two-moons surrogate Wasserstein distance). Each wrapped side table has a bare label and separately extracted caption words. Positions use yMin 585.025357 / 792 plus page index 7, and 84.073357 / 792 plus page index 9.

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	≈ 1.11 pp.	3.7%
2 PRELIMINARIES	2	≈ 1.42 pp.	4.73%
3 RELATED WORKS	4	≈ 0.54 pp.	1.8%
4 NEURAL CONDITIONAL LAGRANGIAN OPTIMAL TRANSPORT	4	≈ 2.46 pp.	8.2%
5 EXPERIMENTS	7	≈ 3.52 pp.	11.73%
6 DISCUSSION	10	≈ 0.48 pp.	1.6%
REPRODUCIBILITY STATEMENT	11	≈ 0.15 pp.	0.5%
ACKNOWLEDGEMENTS	11	≈ 0.08 pp.	0.27%
REFERENCES	11	≈ 3.77 pp.	12.57%
A FURTHER APPLICATIONS OF HTI	15	≈ 3 pp.	10%
B SAMPLING ALGORITHM	18	≈ 1 pp.	3.33%
C EXPERIMENTAL DETAILS	19	≈ 9 pp.	30%
D EXTENDING TO MULTIPLE HYPERPARAMETERS	28	≈ 1 pp.	3.33%
E SPARSITY INVESTIGATION	29	≈ 1 pp.	3.33%
F LLM USAGE	30	≈ 0.89 pp.	2.97%

All detected figure and table locations

Main text

• Figure 1 · p. 7
• Table 1 · p. 7
• Table 2 · p. 8
• Figure 2 · p. 8
• Table 3 · p. 8
• Table 4 · p. 9
• Table 5 · p. 9
• Figure 3 · p. 9
• Table 6 · p. 10
• Table 7 · p. 10

Attached appendix

• Figure 4 · p. 16
• Table 8 · p. 20
• Table 9 · p. 22
• Table 10 · p. 23
• Table 11 · p. 23
• Table 12 · p. 24
• Table 13 · p. 25
• Table 14 · p. 26
• Table 15 · p. 27
• Figure 5 · p. 29

Argument and organization

1. A new inference task before a method

   Defines interpolation of conditional output distributions across one training hyperparameter, then introduces conditional transport machinery.

2. Two inductive biases and multiple demonstrations

   Density preference and learned geometry are separated before semicircle, control, forecasting and dropout experiments.

Problem, claim and contribution

Concrete reward-weight and quantile examples explain why the surrogate predicts output distributions rather than a scalar Bayesian-optimization objective; notation links the user problem to trajectory inference.

Sections 1–2.1, pages 1–2 · Main paper

Argument and section order

Primal, dual and semi-dual transport definitions are introduced in a dependency order before the learned Lagrangian, reducing the conceptual jump into nested optimization.

Sections 2.2–2.3, page 3 · Main paper

Figures, captions and visual explanation

The four-panel semicircle ablation uses identical conditions and geometry to distinguish density bias from metric learning; adjacent numerical results evaluate withheld intermediate times rather than only endpoints.

Figure 1 and Table 1, page 7 · Main paper

Forecast panels reuse the same sample columns and put methods in rows, allowing interval shape/width comparisons alongside surrogate MSE; these are comparisons to trained forecasters, not empirical coverage guarantees.

Figure 3 and Table 5, page 9 · Main paper

Tables, comparisons and evidence

Tables state metric direction and standard errors over 20 runs. The dropout experiment openly shows the identity-metric density variant beating the full model, a useful exception to the narrative of combined-component gains.

Section 5; Tables 6–7, page 10 · Main paper

Appendix and reproducibility

Sampling is separated from training, and detailed tables specify anchors, model widths, solver steps and datasets. Additional applications and multi-hyperparameter ideas are explicitly prospective rather than mixed with demonstrated results.

Appendices A–E, pages 15–29 · Main paper

What to reuse—and what not to infer

Transferable lesson: Define the new task precisely, use a controlled visual ablation to expose each inductive bias, and label hypothetical applications separately from demonstrated experiments.

Boundary: Interpolation concerns one continuous hyperparameter and relatively low-dimensional examples; the cancer experiment is an ODE simulator, not clinical or pathology evidence. Matching trained-model quantiles does not establish calibrated uncertainty, and fixed eigenvalue sum alone is not a uniform determinant lower bound.

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 1–11, Sections 1–6 and reproducibility statement

Appendix / supplement

Pages 15–30, Appendices A–F including prospective applications, sampling, full experimental settings and sparsity study

Visually inspected PDF pages

7, 9

• ICLR 2026 final proceedings · 30 pages