Training Reasoning Models on Saturated Problems via Failure-Prefix Conditioning
This addresses a bottleneck in improving reasoning abilities of large language models for AI applications, but it is incremental as it builds on existing RLVR methods.
The paper tackles the problem of training reinforcement learning models on saturated reasoning problems by proposing failure-prefix conditioning, which reallocates exploration to expose models to rare incorrect reasoning trajectories, resulting in performance gains matching those from medium-difficulty problems while preserving token efficiency.
Reinforcement Learning with Verifiable Rewards (RLVR) has substantially improved the reasoning abilities of large language models (LLMs), yet training often stalls as problems become saturated. We identify the core challenge as the poor accessibility of informative failures: learning signals exist but are rarely encountered during standard rollouts. To address this, we propose failure-prefix conditioning, a simple and effective method for learning from saturated problems. Rather than starting from the original question, our approach reallocates exploration by conditioning training on prefixes derived from rare incorrect reasoning trajectories, thereby exposing the model to failure-prone states. We observe that failure-prefix conditioning yields performance gains matching those of training on medium-difficulty problems, while preserving token efficiency. Furthermore, we 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 suggest that failure-prefix conditioning offers an effective pathway to extend RLVR training on saturated problems.