# Training Reasoning Models on Saturated Problems via Failure-Prefix Conditioning

arXiv: https://arxiv.org/abs/2601.20829 · HTML: https://mlsanity.com/paper/2601.20829/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 46 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 25 of 25)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Absolute Zero: Reinforced Self-play Reasoning with Zero Data](https://mlsanity.com/paper/2505.03335/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [Will we run out of data? Limits of LLM scaling based on human-generated data](https://mlsanity.com/paper/2211.04325/) (2022)
- [ProRL: Prolonged Reinforcement Learning Expands Reasoning Boundaries in Large Language Models](https://mlsanity.com/paper/2505.24864/) (2025)
- [The Surprising Effectiveness of Negative Reinforcement in LLM Reasoning](https://mlsanity.com/paper/2506.01347/) (2025)
- [Olmo 3](https://mlsanity.com/paper/2512.13961/) (2025)
- [Competitive Programming with Large Reasoning Models](https://mlsanity.com/paper/2502.06807/) (2025)
- [Self-Evolving Curriculum for LLM Reasoning](https://mlsanity.com/paper/2505.14970/) (2025)
- [SPIRAL: Self-Play on Zero-Sum Games Incentivizes Reasoning via Multi-Agent Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2506.24119/) (2025)
- [Efficient Reinforcement Finetuning via Adaptive Curriculum Learning](https://mlsanity.com/paper/2504.05520/) (2025)
- [BREAD: Branched Rollouts from Expert Anchors Bridge SFT & RL for Reasoning](https://mlsanity.com/paper/2506.17211/) (2025)
- [e3: Learning to Explore Enables Extrapolation of Test-Time Compute for LLMs](https://mlsanity.com/paper/2506.09026/) (2025)
- [ExGRPO: Learning to Reason from Experience](https://mlsanity.com/paper/2510.02245/) (2025)
- [RLVE: Scaling Up Reinforcement Learning for Language Models with Adaptive Verifiable Environments](https://mlsanity.com/paper/2511.07317/) (2025)
- [Prompt Curriculum Learning for Efficient LLM Post-Training](https://mlsanity.com/paper/2510.01135/) (2025)
- [POPE: Learning to Reason on Hard Problems via Privileged On-Policy Exploration](https://mlsanity.com/paper/2601.18779/) (2026)
- [BroRL: Scaling Reinforcement Learning via Broadened Exploration](https://mlsanity.com/paper/2510.01180/) (2025)
- [DisCO: Reinforcing Large Reasoning Models with Discriminative Constrained Optimization](https://mlsanity.com/paper/2505.12366/) (2025)
- [Reinforcement Learning vs. Distillation: Understanding Accuracy and Capability in LLM Reasoning](https://mlsanity.com/paper/2505.14216/) (2025)
- [On the Power of Context-Enhanced Learning in LLMs](https://mlsanity.com/paper/2503.01821/) (2025)

## Built on by (most cited first, 3 of 3)

- [Where Rollouts Begin: Low-Load, High-Leverage First-Token Diversification for RLVR](https://mlsanity.com/paper/2605.28295/) (2026)
- [Addressing Performance Saturation for LLM RL via Precise Entropy Curve Control](https://mlsanity.com/paper/2604.26326/) (2026)
- [From Trajectories to Prefixes: Reusing Teacher Trajectories via Replayed Prefixes and Online Continuation](https://mlsanity.com/paper/2607.19395/) (2026)

## Abstract (arXiv)

As Reinforcement Learning with Verifiable Rewards (RLVR) substantially improves the reasoning abilities of large language models (LLMs), a new bottleneck emerges: more training problems become saturated, that is, the LLM answers the questions correctly for nearly every rollout. On such problems, rewards provide little useful learning signal. While collecting harder problems is a natural response, it is costly and increasingly difficult. We propose failure-prefix conditioning, a simple method that unlocks the remaining signal in saturated problems by shifting exploration toward failure-prone reasoning states. By conditioning on prefixes of rare incorrect trajectories, the method improves the model's ability to recover from misleading early reasoning. We observe that failure-prefix conditioning consistently improves performance where standard RLVR stalls, and achieves gains comparable to training on newly collected medium-difficulty problems. We further analyze the model's robustness, finding that our method reduces performance degradation under misleading failure prefixes, albeit with a mild trade-off in adherence to correct early reasoning. Finally, we demonstrate that an iterative approach, which refreshes failure prefixes during training, unlocks additional gains after performance plateaus. Overall, our results show that saturated problems still contain valuable learning signal, and that failure-prefix conditioning provides an effective way to unlock it.