Nudging the Boundaries of LLM Reasoning

Current online reinforcement learning (RL) algorithms like GRPO share a key limitation in LLM reasoning: they cannot learn from problems that are "unsolvable" to the model. In other words, they can only improve performance on problems where the model is capable of exploring the correct answer. If a problem is too difficult -- such that even hundreds of attempts never produce a correct solution -- the model cannot learn from it. Consequently, the model's "upper limit" remains unchanged after RL training, even though the likelihood of solving easier, solvable problems may increase. These hard, unsolvable samples -- though potentially rich in learning signal -- cannot contribute to training, as no rollouts yield rewards and thus no gradients are produced. To unlock learning from these hard samples, we propose NuRL, a "nudging" method that aims to push the upper bound of LLM reasoning using self-generated hints, i.e., abstract cues that help reduce the problem difficulty for the model. Given a question and its gold answer, the model generates a Chain-of-Thought (CoT) and then produces a hint containing the core knowledge needed to solve the problem. During online RL training, we generate G rollouts from the base policy and use the pass rate to decide whether the hint should be injected. For hard samples with a 0% pass rate, we inject the offline-generated hint and regenerate a new batch of trajectories. This yields two benefits: (1) the hint boosts pass rates (from 0% to non-zero), thereby introducing training signals for previously unsolvable samples, and (2) the hints are self-generated (conditioned on the gold answer), avoiding distributional shift and do not rely on external models. Compared to standard GRPO, NuRL achieves consistent improvements across six diverse benchmarks and three models, while remaining complementary to test-time scaling. Notably, NuRL can raise the model's upper limit, whereas GRPO leaves pass@1024 unchanged from the base model. Furthermore, we present a systematic study of what makes an effective hint and when hints are most useful. Interestingly, the best hints are abstract and high-level -- as revealing gold answers actually hurt performance -- and are most beneficial when applied necessarily and after GRPO has converged.