Learning from Saturated Data: Signals Beyond Correctness for LLM Training

Hanno Hiss; Jasper Dekoninck; Martin Vechev

REVIEW 2 major objections 1 minor 41 references

Learning from Saturated Data: Signals Beyond Correctness for LLM Training

T0 review · 2 major / 1 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Quality signals from saturated data improve base LLM performance up to 18.6% on arithmetic tasks, outperforming standard fine-tuning.

desk verdict Quality signals from self-judgments and entropy give gains on saturated arithmetic data but lack direct validation there and falter on GSM8K. read the letter →

arxiv 2606.01436 v1 pith:VKVHQE3F submitted 2026-05-31 cs.CL

Hanno Hiss , Jasper Dekoninck , Martin Vechev This is my paper

classification cs.CL

keywords saturateddataLLMtrainingqualitysignalsself-judgmentstokenentropyarithmetictasksGSM8Ksupervisedfine-tuning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests whether data that models already solve with perfect accuracy can still improve training when richer quality signals replace binary correct-or-wrong labels. It replaces those labels with two proxies: the model's own pairwise judgments of which of its solutions is better, and the entropy of its token predictions as a measure of uncertainty. On a simple arithmetic task these signals produce gains of up to 18.6 percent over the untouched base model and beat ordinary supervised fine-tuning. On the harder GSM8K set the same signals give smaller or even negative results that depend on which signal is chosen. The work matters because many current benchmarks and datasets are already saturated, so any method that extracts extra value from them could stretch the life of existing training collections.

What carries the argument

Pairwise LLM self-judgments and token-level entropy used as quality signals in place of binary correctness labels inside several training algorithms.

What would settle it

An experiment that trains the same base model on the arithmetic task using these quality signals and measures no improvement or a drop relative to the base model or to standard supervised fine-tuning.

Watch

Extended reading notes

Core claim

When training exclusively on a simple arithmetic task, quality-based signals improve performance by up to 18.6% over the base model, substantially outperforming SFT. On GSM8K, however, gains are more modest and depend strongly on the quality signal. Pairwise LLM self-judgments show poor agreement with a stronger external judge and can even degrade performance below the base model. The central claim is therefore that fine-grained quality signals drawn from saturated questions can still supply useful training signal for base models, provided the signal is well calibrated to the task.

Load-bearing premise

That pairwise self-judgments and token entropy reliably mark higher-quality solutions in a way that produces measurable downstream gains.

Editorial extensions

If this is right

Quality signals enable performance gains even when every training example is already solved correctly.
On simple arithmetic the signals outperform binary-label supervised fine-tuning.
On GSM8K the outcome depends on the chosen signal, with some signals harming performance.
Saturated data retains training value once quality rather than correctness is measured.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same signals could be tested on other saturated domains such as code generation or multi-step reasoning to see whether calibration remains the main requirement.
Combining self-judgment and entropy into a single training objective might reduce the dependence on any one noisy signal.
The poor agreement between self-judgments and external judges on GSM8K suggests that external verification may still be needed for complex tasks even if self-signals work on simpler ones.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main result is that on a simple arithmetic task where the model already gets 100% correct, swapping binary correctness for pairwise self-judgments or token entropy as training signals produces up to 18.6% improvement over the base model and beats plain SFT. That is the concrete new angle: treating saturated data as still having usable variation rather than stopping at accuracy.

The work is straightforward in its setup. It takes Qwen3-1.7B-Base, applies the two signals inside several training algorithms, and reports the arithmetic gains clearly. The abstract also flags the task dependence without overclaiming, which is useful.

The soft spot is exactly where the stress-test note lands. The abstract gives no evidence that the self-judgment signal actually ranks quality on the arithmetic examples themselves. We only learn that the same signal agrees poorly with an external judge on GSM8K and can hurt performance there. Without an ablation or correlation check on the arithmetic set, it remains possible the reported gains come from incidental filtering effects rather than the intended quality signal. The circularity of the model judging its own outputs adds to the concern, and the strong dependence on which signal is chosen makes the approach look brittle.

This is for people working on data-efficient training and alternative supervision signals. The arithmetic result is worth checking in full, but the current evidence is narrow and the signal reliability is unproven on the data that matters for the headline claim.

I would send it to peer review so the experiments can be examined in detail.

Referee Report

2 major / 1 minor

Summary. The paper claims that saturated training data (where all solutions have perfect empirical correctness) can still yield useful training signal via fine-grained quality indicators: pairwise LLM self-judgments and token-level entropy. On a simple arithmetic task with Qwen3-1.7B-Base these signals produce up to 18.6% gains over the base model and outperform SFT; on GSM8K the gains are modest, highly signal-dependent, and self-judgments exhibit poor agreement with an external judge, sometimes degrading performance below the base model.

Significance. If the quality signals are shown to be reliable on the arithmetic data, the work would be significant for continued progress on saturated benchmarks by moving beyond binary correctness. The arithmetic gains are the strongest element, but the explicit dependence on signal choice and the GSM8K degradation indicate limited generalizability without further calibration. No machine-checked proofs or parameter-free derivations are present.

major comments (2)

[Abstract] Abstract: the central 18.6% gain on arithmetic is reported to arise from pairwise self-judgments and token entropy, yet the abstract itself documents poor agreement between self-judgments and a stronger external judge on GSM8K. No corresponding agreement metric, correlation with external judgment, or ablation against random ranking is provided for the arithmetic task, so it remains possible that the observed gains result from incidental data selection rather than the quality signal.
[Abstract] Abstract: the claim that quality-based training 'substantially outperform[s] SFT' on arithmetic rests on the signals being informative when all answers are correct, but the manuscript provides no validation that self-judgments or entropy actually rank solution quality on that specific dataset.

minor comments (1)

The abstract would be clearer if it stated the number of training runs, presence or absence of error bars, and the precise training objectives used to incorporate the quality signals.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for stronger direct validation of the quality signals on the arithmetic task. We agree that the current evidence relies primarily on downstream performance gains and that explicit agreement metrics, correlations, and random-ranking ablations would make the claims more robust. We will revise the manuscript to incorporate these elements.

read point-by-point responses

Referee: [Abstract] Abstract: the central 18.6% gain on arithmetic is reported to arise from pairwise self-judgments and token entropy, yet the abstract itself documents poor agreement between self-judgments and a stronger external judge on GSM8K. No corresponding agreement metric, correlation with external judgment, or ablation against random ranking is provided for the arithmetic task, so it remains possible that the observed gains result from incidental data selection rather than the quality signal.

Authors: We acknowledge the validity of this observation. The abstract already highlights the signal-dependent and sometimes negative results on GSM8K, but the arithmetic results lack parallel validation. In the revised version we will add (i) pairwise agreement statistics between self-judgments and an external judge on a held-out sample of the arithmetic data, (ii) Spearman or Pearson correlations between each signal and external quality ratings, and (iii) a random-ranking ablation that trains on the same number of examples selected uniformly at random. These additions will directly test whether the observed 18.6 % gains exceed what would be expected from incidental selection. revision: yes
Referee: [Abstract] Abstract: the claim that quality-based training 'substantially outperform[s] SFT' on arithmetic rests on the signals being informative when all answers are correct, but the manuscript provides no validation that self-judgments or entropy actually rank solution quality on that specific dataset.

Authors: We agree that downstream gains alone do not constitute direct proof that the signals rank solution quality. While the performance lift over both the base model and SFT is consistent with the signals carrying useful information, we will strengthen the manuscript by including an explicit validation section for the arithmetic dataset. This will report agreement rates with external judgments, rank correlations between self-judgment scores / entropy and external quality labels, and, where feasible, qualitative examples illustrating the distinctions the signals capture among mathematically correct solutions. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper reports an empirical study: quality signals (self-judgments and entropy) are defined, incorporated into training algorithms, and evaluated via measured performance gains on arithmetic and GSM8K tasks. No derivation, equation, or result is shown to reduce by construction to its inputs; the 18.6% gain is an observed outcome, not a fitted or renamed quantity. The abstract itself flags poor self-judgment agreement on GSM8K, treating signal reliability as an empirical question rather than assuming it. No self-citations, uniqueness theorems, or ansatzes are load-bearing. The work is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only review yields minimal detail on parameters or background assumptions; standard supervised fine-tuning assumptions are implicit but not enumerated.

assumptions (1)

domain assumption LLM self-judgments and token entropy serve as valid proxies for solution quality
Invoked when incorporating the signals into training algorithms

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0 comments

Cite this review

Pith. "Pith review of Learning from Saturated Data: Signals Beyond Correctness for LLM Training." pith.science (2026). https://pith.science/paper/VKVHQE3F

@misc{pith2026260601436,
  author       = {Pith},
  title        = {Pith review of: Learning from Saturated Data: Signals Beyond Correctness for LLM Training},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKVHQE3F}},
  note         = {Machine review of arXiv:2606.01436}
}

abstract

The growing capabilities of large language models (LLMs) have led to the saturation of many benchmarks and training datasets used to improve them. Motivated by this, we investigate whether questions solved with perfect empirical accuracy can nevertheless be used to improve downstream performance. To do so, we replace binary correctness with two sources of more fine-grained quality signals: (1) pairwise LLM self-judgments, in which the model evaluates the relative quality of its own solutions, and (2) token-level entropy, where token-level uncertainty is used as a proxy for solution quality. We incorporate these signals into several training algorithms and evaluate them on Qwen3-1.7B-Base. When training exclusively on a simple arithmetic task, quality-based signals improve performance by up to $18.6\%$ over the base model, substantially outperforming SFT. On GSM8K, however, gains are more modest and depend strongly on the quality signal. For instance, self-judgments show poor agreement with a stronger external judge and can even degrade performance below the base model. Overall, our results suggest that quality-based training can extract useful signal from saturated questions for base models, but that applying such signals to more complex tasks requires careful calibration and further study.

Figures

Figures reproduced from arXiv: 2606.01436 by the authors.

**Figure 1.** Quality signals on saturated questions. (1) Correctness: when all sampled completions are correct, binary reward provides no contrast. (2) LLM-as-a-Judge: a judge runs all pairwise comparisons, the win count wi ranks completions by judged quality. (3) Inverse Entropy: mean token-level entropy along each completion gives a judge-free quality proxy. 1. Easy-to-hard transfer on chain sum: σ-RRHF with inverse-entropy-ra… view at source ↗

**Figure 2.** DPO training dynamics on Dhard for random vs. judge-selected pair construction. 0 10−2 10−1 0.5 1 34 36 38 40 42 Rank loss weight λ Accuracy (%) (a) pass@1 0 10−2 10−1 0.5 1 71 73 75 77 79 Rank loss weight λ Accuracy (%) (b) pass@8 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Rank loss weight ablation on Dhard chain sum problems. The x-axis is log-scaled; λ= 0 (no rank loss) is placed at the left as a reference point. Pass@1 peaks at λ= 10−2 and degrades monotonically for larger weights. Pass@8 peaks at λ= 10−1 with no clear monotonic trend, but some rank loss has an effect on generalization to OOD questions. Responses were ranked with the Self-judge. C.3. σ-RRHF Component Ablation To un… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Implicit reward margin rˆ(x, yw) − rˆ(x, yl) over training on GSM8K. Inverse entropy reaches the highest margin; Qwen3-30B grows more steadily; Qwen3-1.7B-Base (self-judge) stays near zero. Faint lines are raw per-step values; bold lines are rolling averages [PITH_FUL…

**Figure 5.** Figure 5: Distribution of within-question Spearman ρ between each pair of scorers, for D ′ strict chain sum (left) and Dsaturated GSM8K (right). Row labels indicate the scorer pair. Dashed red line: mean (ρ¯); solid orange line: median (ρ˜). The strong judge and inverse entropy …

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Reference graph

Works this paper leans on

41 extracted references · 4 canonical work pages

[1]

Lost in the Middle: How Language Models Use Long Contexts

URLhttp://arxiv.org/abs/2602.03876. Cobbe, K., Kosaraju, V ., Bavarian, M., Chen, M., Jun, H., Kaiser, L., Plappert, M., Tworek, J., Hilton, J., Nakano, R., Hesse, C., and Schulman, J. Training Verifiers to Solve Math Word Problems, 2021. URL http://arxiv. org/abs/2110.14168. DeepSeek-AI, Guo, D., Yang, D., Zhang, H., Song, J., Zhang, R., Xu, R., Zhu, Q.,...

work page Pith review doi:10.1162/tacl 2021
[2]

Tulu 3: Pushing Frontiers in Open Language Model Post-Training

URL https://github.com/huggingface/ math-verify. Lambert, N., Morrison, J., Pyatkin, V ., Huang, S., Ivison, H., Brahman, F., Miranda, L. J. V ., Liu, A., Dziri, N., Lyu, S., Gu, Y ., Malik, S., Graf, V ., Hwang, J. D., Yang, J., Bras, R. L., Tafjord, O., Wilhelm, C., Soldaini, L., Smith, N. A., Wang, Y ., Dasigi, P., and Hajishirzi, H. Tulu 3: Pushing Fr...

work page Pith review arXiv 2025
[3]

Shao, Z., Wang, P., Zhu, Q., Xu, R., Song, J., Bi, X., Zhang, H., Zhang, M., Li, Y

URLhttp://arxiv.org/abs/2505.21444. Shao, Z., Wang, P., Zhu, Q., Xu, R., Song, J., Bi, X., Zhang, H., Zhang, M., Li, Y . K., Wu, Y ., and Guo, D. DeepSeekMath: Pushing the Limits of Mathemat- ical Reasoning in Open Language Models, 2024. URL http://arxiv.org/abs/2402.03300. Song, F., Yu, B., Li, M., Yu, H., Huang, F., Li, Y ., and Wang, H. Preference Rank...

work page doi:10.18653/v1/2025.emnlp-main.583 2024
[4]

A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models

URLhttp://arxiv.org/abs/2510.08049. 9 Learning from Saturated Data: Signals Beyond Correctness for LLM Training A. Task and Data Details This appendix collects ablation studies, hyperparameter sweeps, and auxiliary analyses that support the main results. A.1. Difficulty Partitioning: Formal Setup Problem settingLet questions x be drawn from some distribut...

work page Pith review arXiv
[5]

Subtract 706 from 381: 381 - 706 = -325
[6]

Subtract 602 from -325: -325 - 602 = -927
[7]

Both responses arrive at the correct answer but differ in verbosity and reasoning style

Add 383 to -927: -927 + 383 = -544 So, the final answer is−544 . Both responses arrive at the correct answer but differ in verbosity and reasoning style. 11 Learning from Saturated Data: Signals Beyond Correctness for LLM Training Table 6.Evaluation configuration. Parameter Value Evaluation datasethannoh/chainsum eval GSM8K(test split) Number of questions...
[8]

**Brief Analysis** of conflicting answers

Show all 41 references

[10]

**Justification** for eliminating incorrect solutions
[11]

**Clear Explanation** for selecting the best approach. An evaluation ends with: **Judgment: [IDX]** *(Where ‘IDX‘ is the index$0 - {max_idx}$of the best solution.)* --- ## EVALUATION A **Problem:** Compute the following step by step: 19 Learning from Saturated Data: Signals Be...
[12]

Let’s solve the problem step by step:
[13]

**Addition:** \\[ 593615 + 204846 = 798461 \\]
[14]

**Subtraction:** \\[ 798461 - 838944 = -40483 \\]
[15]

**Addition:** \\[ -40483 + 816336 = 775853 \\]
[16]

**Subtraction:** \\[ 775853 - 913166 = -137313 \\]
[17]

**Subtraction:** \\[ -137313 - 338746 = -476059 \\] The final answer is: \\boxed{-476059}
[19]

Compute 593615 + 204846
[20]

Subtract 838944 from the result
[21]

Add 816336 to the new result
[22]

Subtract 913166 from the new result
[23]

Subtract 338746 from the new result Now, let’s compute each step:
[24]

593615 + 204846 = 798461
[25]

798461 - 838944 = -40483
[26]

-40483 + 816336 = 775853
[27]

775853 - 913166 = -137313
[28]

### EVALUATION **Initial Screening:** Both solutions arrive at -476059

-137313 - 338746 = -476059 So, the final answer is \\boxed{-476059}. ### EVALUATION **Initial Screening:** Both solutions arrive at -476059. No answer disagreement; both are arithmetically correct. **Analysis:** - Solution 0: Proceeds directly to computation. Each step is labe...
[29]

**Brief Analysis** of any answer disagreements
[30]

**Detailed Evaluation** of mathematically sound solutions
[31]

**Clear Explanation** for selecting the best approach. 21 Learning from Saturated Data: Signals Beyond Correctness for LLM Training An evaluation ends with: **Judgment: [IDX]** *(Where ‘IDX‘ is the index$0 - {max_idx}$of the best solution.)* --- ## EVALUATION A **Problem:** Fo...
[32]

To solve this problem, we need to calculate the amount of money received from recycling the cans and the newspapers separately, and then add those amounts together
[33]

Your family collected 144 cans

**Calculate the money received from recycling cans:** We know that for every 12 cans, you receive$0.50. Your family collected 144 cans. To find out how many sets of 12 cans they have, we divide the total number of cans by 12: \\\\[ \\\\frac{{144 \\\\text{{ cans}}}}{{12 \\\\tex...
[34]

Your family collected 20 kilograms

**Calculate the money received from recycling newspapers:** We know that for every 5 kilograms of newspapers, you receive$1.50. Your family collected 20 kilograms. To find out how many sets of 5 kilograms they have: \\\\[ \\\\frac{{20 \\\\text{{ kg}}}}{{5 \\\\text{{ kg/set}}}}...
[35]

**Add the amounts:** \\\\[ \\\\$6.00 + \\\\$6.00 = \\\\$12.00 \\\\] Therefore, the total amount of money your family would receive is$\\\\boxed{{12}}$
[36]

Let’s solve this step by step:
[37]

- Your family collected 144 cans

Calculate the money earned from recycling cans: - For every 12 cans, you receive$0.50. - Your family collected 144 cans. - To find out how many sets of 12 cans there are in 144 cans, divide 144 by 12: 144 / 12 = 12 sets. - Multiply the number of sets by the amount earned per s...
[38]

- Your family collected 20 kilograms of newspapers

Calculate the money earned from recycling newspapers: - For every 5 kilograms of newspapers, you receive$1.50. - Your family collected 20 kilograms of newspapers. - To find out how many sets of 5 kilograms there are in 20 kilograms, divide 20 by 5: 20 / 5 = 4 sets. - Multiply ...
[39]

Money earned from cans:$6.00

Add the money earned from both cans and newspapers: - Money earned from cans:$6.00 - Money earned from newspapers:$6.00 - Total money earned:$6.00 +$6.00 =$12.00 Final answer: You would receive$12.00. ### EVALUATION **Initial Screening:** Both solutions arrive at$12.00. No ans...
[40]

Step 1: The baker keeps one-quarter for himself

The total number of cookies baked is 48. Step 1: The baker keeps one-quarter for himself. One-quarter of 48: 48 / 4 = 12 cookies kept. Step 2: Find the remaining cookies. Remaining = 48 - 12 = 36 cookies. Step 3: Divide among 3 friends. Each friend gets: 36 / 3 = 12 cookies. C...
[41]

No disagreement

Cookies the baker keeps: 48 / 4 = 12 Remaining to share: 48 - 12 = 36 Each friend gets: 36 / 3 = 12 $\\\\boxed{{12}}$ ### EVALUATION **Initial Screening:** Both solutions reach 12. No disagreement. **Analysis:** - Solution 0: Correct and systematic, but adds a verification ste...
[42]

Then it goes at 40 mph for 1.5 hours, which is 40 * 1.5 = 60 miles

The car drives at 60 mph for 2 hours, so that’s 60 * 2 = 120 miles. Then it goes at 40 mph for 1.5 hours, which is 40 * 1.5 = 60 miles. Total: 120 + 60 = 180 miles. The answer is$\\\\boxed {{180}}$
[43]

No disagreement

Distance in first segment: 60 mph * 2 hours = 120 miles Distance in second segment: 40 mph * 1.5 hours = 60 miles Total distance: 120 + 60 = 180 miles $\\\\boxed{{180}}$ ### EVALUATION **Initial Screening:** Both solutions reach 180. No disagreement. **Analysis:** - Solution 0...

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

Lost in the Middle: How Language Models Use Long Contexts

URLhttp://arxiv.org/abs/2602.03876. Cobbe, K., Kosaraju, V ., Bavarian, M., Chen, M., Jun, H., Kaiser, L., Plappert, M., Tworek, J., Hilton, J., Nakano, R., Hesse, C., and Schulman, J. Training Verifiers to Solve Math Word Problems, 2021. URL http://arxiv. org/abs/2110.14168. DeepSeek-AI, Guo, D., Yang, D., Zhang, H., Song, J., Zhang, R., Xu, R., Zhu, Q.,...

work page Pith review doi:10.1162/tacl 2021

[2] [2]

Tulu 3: Pushing Frontiers in Open Language Model Post-Training

URL https://github.com/huggingface/ math-verify. Lambert, N., Morrison, J., Pyatkin, V ., Huang, S., Ivison, H., Brahman, F., Miranda, L. J. V ., Liu, A., Dziri, N., Lyu, S., Gu, Y ., Malik, S., Graf, V ., Hwang, J. D., Yang, J., Bras, R. L., Tafjord, O., Wilhelm, C., Soldaini, L., Smith, N. A., Wang, Y ., Dasigi, P., and Hajishirzi, H. Tulu 3: Pushing Fr...

work page Pith review arXiv 2025

[3] [3]

Shao, Z., Wang, P., Zhu, Q., Xu, R., Song, J., Bi, X., Zhang, H., Zhang, M., Li, Y

URLhttp://arxiv.org/abs/2505.21444. Shao, Z., Wang, P., Zhu, Q., Xu, R., Song, J., Bi, X., Zhang, H., Zhang, M., Li, Y . K., Wu, Y ., and Guo, D. DeepSeekMath: Pushing the Limits of Mathemat- ical Reasoning in Open Language Models, 2024. URL http://arxiv.org/abs/2402.03300. Song, F., Yu, B., Li, M., Yu, H., Huang, F., Li, Y ., and Wang, H. Preference Rank...

work page doi:10.18653/v1/2025.emnlp-main.583 2024

[4] [4]

A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models

URLhttp://arxiv.org/abs/2510.08049. 9 Learning from Saturated Data: Signals Beyond Correctness for LLM Training A. Task and Data Details This appendix collects ablation studies, hyperparameter sweeps, and auxiliary analyses that support the main results. A.1. Difficulty Partitioning: Formal Setup Problem settingLet questions x be drawn from some distribut...

work page Pith review arXiv

[5] [5]

Subtract 706 from 381: 381 - 706 = -325

[6] [6]

Subtract 602 from -325: -325 - 602 = -927

[7] [7]

Both responses arrive at the correct answer but differ in verbosity and reasoning style

Add 383 to -927: -927 + 383 = -544 So, the final answer is−544 . Both responses arrive at the correct answer but differ in verbosity and reasoning style. 11 Learning from Saturated Data: Signals Beyond Correctness for LLM Training Table 6.Evaluation configuration. Parameter Value Evaluation datasethannoh/chainsum eval GSM8K(test split) Number of questions...

[8] [8]

**Brief Analysis** of conflicting answers