Writing study: Maximum Likelihood Reinforcement Learning

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← ICML 2026 oral writing corpus

Fahim Tajwar, Guanning Zeng, Yueer Zhou, Yuda Song, Daman Arora, Yiding Jiang, Jeff Schneider, Ruslan Salakhutdinov, Haiwen Feng, Andrea Zanette · 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

56 physical pages

Before references / appendix

≈ 9.65 page-equivalents

Main figures / tables

8 / 2

Attached appendix figures / tables

24 / 8

Attached appendix span

≈ 32.9 page-equivalents

Before references / appendix: ≈ 17.23%References / intervening material: ≈ 24.02%Attached appendix: ≈ 58.75%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Source-bound correction: Pages24–25 contain a large-title appendix contents list; those entries are not the section starts. Preserve the cover in the appendix envelope and use the actual A–M headings from pp.26–53. Pages24–25 contain a large-title appendix contents list; those entries are not the section starts. Preserve the cover in the appendix envelope and use the actual A–M headings from pp.26–53. Source-checked physical-page text restores wrapped top-level heading labels on pp. 3. 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	≈ 1.1 pp.	1.96%
2. Preliminaries	2	≈ 0.62 pp.	1.11%
3. Maximum Likelihood Reinforcement Learning (MAXRL)	3	≈ 0.8 pp.	1.43%
4. Gradient Estimators for MAXRL	3	≈ 1.7 pp.	3.04%
5. A Unifying Weight-Function View	5	≈ 0.56 pp.	1%
6. Experiments	6	≈ 3.2 pp.	5.71%
7. Related Work	9	≈ 0.52 pp.	0.93%
8. Conclusion	9	≈ 0.23 pp.	0.41%
Acknowledgements	10	≈ 0.5 pp.	0.89%
Impact Statement	10	≈ 0.11 pp.	0.2%
References	10	≈ 13.45 pp.	24.02%
Appendix: cover and contents	24	≈ 1.99 pp.	3.55%
A. Public Code Release	26	≈ 0.11 pp.	0.2%
B. Extended Related Work	26	≈ 3.89 pp.	6.95%
C. Theoretical Results	30	≈ 1 pp.	1.79%
D. Analysis of the Practical Gradient Estimator from Algorithm 1	31	≈ 2 pp.	3.57%
E. More on Unifying Weight-Function View on RL Objectives	33	≈ 2 pp.	3.57%
F. RL Implementation Details for LLM Experiments	35	≈ 1 pp.	1.79%
G. Pass@k Calculation	36	≈ 1 pp.	1.79%
H. ImageNet Experiments	37	≈ 3 pp.	5.36%
I. Maze Experiments	40	≈ 3 pp.	5.36%
J. GSM8K Experiments	43	≈ 3 pp.	5.36%
K. Qwen3 Experiments	46	≈ 3 pp.	5.36%
L. Comparison with Additional Baselines	49	≈ 4 pp.	7.14%
M. Additional Analyses	53	≈ 3.91 pp.	6.98%

All detected figure and table locations

Main text

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

Attached appendix

• Figure 8 · p. 38
• Figure 9 · p. 38
• Figure 10 · p. 40
• Figure 11 · p. 41
• Figure 12 · p. 42
• Table 3 · p. 42
• Table 4 · p. 43
• Figure 13 · p. 44
• Table 5 · p. 44
• Figure 14 · p. 45
• Table 6 · p. 46
• Figure 15 · p. 47
• Table 7 · p. 47
• Figure 16 · p. 48
• Figure 17 · p. 48
• Figure 18 · p. 48
• Figure 19 · p. 48
• Table 8 · p. 49
• Table 9 · p. 49
• Figure 20 · p. 49
• Figure 21 · p. 50
• Figure 22 · p. 50
• Figure 23 · p. 51
• Figure 24 · p. 51
• Figure 25 · p. 52
• Figure 26 · p. 52
• Figure 27 · p. 53
• Figure 28 · p. 54
• Figure 29 · p. 54
• Table 10 · p. 55
• Figure 30 · p. 55
• Figure 31 · p. 56

Argument and organization

1. Problem and contribution

   Contrasts mean pass rate with mean log pass probability, interpreting additional sampling as changing the optimized approximation rather than merely reducing estimator variance.

2. Argument sequence

   An objective expansion precedes the estimator and control variate, followed by a unifying weight-function table and increasingly realistic training regimes.

3. Evidence and verification

   The practical zero-success convention changes the unbiased target from order N to N-1, explicitly disclosed beside the algorithm; the efficiency claim assumes a perfect verifier.

Problem, claim and contribution

Contrasts mean pass rate with mean log pass probability, interpreting additional sampling as changing the optimized approximation rather than merely reducing estimator variance.

Introduction and preliminaries, p. 2 · Main paper

Argument and section order

An objective expansion precedes the estimator and control variate, followed by a unifying weight-function table and increasingly realistic training regimes.

Sections 3-6, pp. 3-8 · Main paper

Figures, captions and visual explanation

The implementation highlights its changed advantage line next to population-weight curves; model-by-benchmark pass@k panels make both single-sample quality and high-k coverage visible.

Algorithm 1/Table 2/Figure 1, p. 5; Figure 5, p. 8; both PNGs inspected · Main paper

Tables, comparisons and evidence

The practical zero-success convention changes the unbiased target from order N to N-1, explicitly disclosed beside the algorithm; the efficiency claim assumes a perfect verifier.

Section 4.3, p. 5; Section 6.4, p. 8 · Main paper

Appendix and reproducibility

Implementation specifies one update per rollout batch, and the continuous-reward experiment is labeled preliminary with broader theory deferred.

Appendix F, p. 35; M.4, pp. 55-56 · Main paper

Clarity and writing conventions

Takeaway boxes aid navigation, while precise estimator and verifier qualifications prevent the short one-line-change story from erasing consequential assumptions.

Sections 4.3 and 6.4, pp. 5, 8 · Main paper

What to reuse—and what not to infer

Transferable lesson: Trace a training objective all the way to the exact implemented estimator, explicitly recording any order, bias or protocol change introduced for practicality.

Boundary: Pass@k savings under a perfect verifier are not measured end-to-end latency; the correctness-based likelihood theory should not be automatically extended to arbitrary rewards.

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 1-8, pp. 1-9: latent correctness likelihood, compute-indexed expansion, estimators/control variates, implementation and weight-function comparisons, learning regimes and behavior diagnostics.

Appendix / supplement

Writing-focused appendix scan; code, data and external assets not executed. Appendices A-M, pp. 26-56: related work, proofs and practical estimator analysis, RL implementation, pass@k, task configurations, extra baselines and non-binary extension.

Visually inspected PDF pages

5, 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. · 56 pages