# Learning from Saturated Data: Signals Beyond Correctness for LLM Training

arXiv: https://arxiv.org/abs/2606.01436 · HTML: https://mlsanity.com/paper/2606.01436/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 31 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 23 of 23)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment](https://mlsanity.com/paper/2304.06767/) (2023)
- [RRHF: Rank Responses to Align Language Models with Human Feedback without tears](https://mlsanity.com/paper/2304.05302/) (2023)
- [Rubrics as Rewards: Reinforcement Learning Beyond Verifiable Domains](https://mlsanity.com/paper/2507.17746/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [Unsupervised Quality Estimation for Neural Machine Translation](https://mlsanity.com/paper/2005.10608/) (2020)
- [Meta-Rewarding Language Models: Self-Improving Alignment with LLM-as-a-Meta-Judge](https://mlsanity.com/paper/2407.19594/) (2024)
- [On Hallucination and Predictive Uncertainty in Conditional Language Generation](https://mlsanity.com/paper/2103.15025/) (2021)
- [The Unreasonable Effectiveness of Entropy Minimization in LLM Reasoning](https://mlsanity.com/paper/2505.15134/) (2025)
- [Easy-to-Hard Generalization: Scalable Alignment Beyond Human Supervision](https://mlsanity.com/paper/2403.09472/) (2024)
- [Can Large Reasoning Models Self-Train?](https://mlsanity.com/paper/2505.21444/) (2025)
- [The Unreasonable Effectiveness of Easy Training Data for Hard Tasks](https://mlsanity.com/paper/2401.06751/) (2024)
- [A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models](https://mlsanity.com/paper/2510.08049/) (2025)
- [SPELL: Self-Play Reinforcement Learning for Evolving Long-Context Language Models](https://mlsanity.com/paper/2509.23863/) (2025)
- [The Delta Learning Hypothesis: Preference Tuning on Weak Data can Yield Strong Gains](https://mlsanity.com/paper/2507.06187/) (2025)
- [IPO: Your Language Model is Secretly a Preference Classifier](https://mlsanity.com/paper/2502.16182/) (2025)
- [On the Entropy Calibration of Language Models](https://mlsanity.com/paper/2511.11966/) (2025)

## Abstract (arXiv)

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.