Writing study: The Coverage Principle: How Pre-Training Enables Post-Training

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← ICLR 2026 oral writing corpus

Fan Chen, Audrey Huang, Noah Golowich, Sadhika Malladi, Adam Block, Jordan Ash, Akshay Krishnamurthy, Dylan Foster · 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) · Learning & optimization algorithms.

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

69 physical pages

Before references / appendix

≈ 10.2 page-equivalents

Main figures / tables

2 / 0

Attached appendix figures / tables

0 / 1

Attached appendix span

≈ 50.81 page-equivalents

Before references / appendix: ≈ 14.78%References / intervening material: ≈ 11.58%Attached appendix: ≈ 73.64%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	≈ 1.9 pp.	2.75%
2 PROBLEM SETUP	3	≈ 0.93 pp.	1.35%
3 CROSS-ENTROPY AND COVERAGE: SCALING LAWS AND LIMITATIONS	4	≈ 1.69 pp.	2.45%
4 NEXT-TOKEN PREDICTION IMPLICITLY OPTIMIZES COVERAGE	6	≈ 2 pp.	2.9%
5 STOCHASTIC GRADIENT DESCENT THROUGH THE LENS OF COVERAGE	8	≈ 1.32 pp.	1.91%
6 INTERVENTIONS FOR BETTER COVERAGE	9	≈ 1.36 pp.	1.97%
ACKNOWLEDGEMENTS	10	≈ 0.24 pp.	0.35%
REPRODUCIBILITY STATEMENT	11	≈ 0.09 pp.	0.13%
REFERENCES	11	≈ 7.99 pp.	11.58%
A DISCUSSION AND FUTURE WORK	19	≈ 0.68 pp.	0.99%
B RELATED WORK	19	≈ 1.29 pp.	1.87%
C COMPARISON TO CLASSICAL GENERALIZATION BOUNDS FOR MLE	21	≈ 0.96 pp.	1.39%
D EXPERIMENTS	22	≈ 5 pp.	7.25%
E PROPERTIES OF THE COVERAGE PROFILE	27	≈ 1 pp.	1.45%
F SUPPORTING RESULTS	28	≈ 5 pp.	7.25%
G ADDITIONAL RESULTS	33	≈ 4.08 pp.	5.91%
H TECHNICAL TOOLS	37	≈ 1.92 pp.	2.78%
I PROOFS FROM SECTION 3	39	≈ 0.56 pp.	0.81%
J PROOFS FROM SECTION 4	39	≈ 6.19 pp.	8.97%
K PROOFS FOR AUTOREGRESSIVE LINEAR MODELS	45	≈ 20.65 pp.	29.93%
L PROOFS FROM SECTION 6	66	≈ 3.5 pp.	5.07%

All detected figure and table locations

Main text

• Figure 1 · p. 2
• Figure 2 · p. 5

Attached appendix

• Table 1 · p. 24

Argument and organization

1. Counterexample to cross-entropy as downstream predictor

   Motivates a different statistical object before stating guarantees.

2. Coverage generalization and optimizer interventions

   Moves from general MLE classes to restricted linear autoregressive models.

3. Proof roadmap and experimental construction

   Separates interpretations, illustrations and formal derivations.

Problem, claim and contribution

A graph-reasoning example shows why improving likelihood need not improve best-of-N performance. Coverage is then defined as a density-ratio tail probability: smaller is better. This definition prevents conflation with the everyday meaning of coverage or with Pass@N itself.

Figure 1, Definition 1.1 and Appendix B footnote 5, pp.2, 20 · Main paper

Argument and section order

The paper first explains a weak KL-based bound, then a Bernoulli missing-mass example, before its main theorem. Fine and coarse complexity terms receive separate interpretations, making a long bound serve a conceptual argument rather than interrupt it.

Sections 3–4, pp.4–7 · Main paper

Figures, captions and visual explanation

Figure 1 compares checkpoint selection and two predictive quantities; Figure 2 varies horizon while holding the mechanism visible. The dual horizontal axes in Figure 1 require deliberate reading, and shaded bands in Figure 2 are seed quantiles, not confidence intervals.

Figures 1–2, pp.2, 5; Appendix D.3.2, p.26 · Main paper

Tables, comparisons and evidence

Formal guarantees assume realizability or explicitly stated alternatives. SGD results use fixed features and convex parameter constraints. The experiments are constructed graph tasks using small GPT2-like transformers, not a demonstrated universal diagnostic for production language models.

Assumptions 2.1–2.2; Sections 5–6; Appendix D, pp.3–4, 8–10, 22–26 · Main paper

Appendix and reproducibility

The appendix separates interpretation, properties, auxiliary results and proofs, then provides an organization paragraph for the long autoregressive analysis. Its distillation result requires access to all teacher next-token probabilities and is a theoretical intervention, not an empirical compression benchmark.

Appendix contents; G.4, K.1 and K.7, pp.17–18, 35, 45–46, 63–65 · Main paper

Clarity and writing conventions

Appendix A explicitly limits conclusions about general RL, prompt-distribution shift and semantic reasoning. Cross-reference checking still matters: Theorem G.2 points to update (10), although its specialized estimator is defined by (35)–(37).

Appendices A.1–A.2 and G.4, pp.19, 35 · Main paper

What to reuse—and what not to infer

Transferable lesson: Introduce a counterexample and interpretable special case before the main theorem; label illustrative experiments, assumptions and proof dependencies separately.

Boundary: The necessity statement for best-of-N is worst-case over rewards, not necessity for every fixed task. Reading the full proof appendix is not independent mathematical certification or evidence that the bounds hold for unrestricted transformers.

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; references 11–16

Appendix / supplement

Pages 17–69, contents and Appendices A–L, including experiment construction, all theorem proofs, distillation normalization and tournaments

Visually inspected PDF pages

2, 5

• official ICLR 2026 proceedings PDF · 69 pages