Distilling LLM Feedback for Lean Theorem Proving

Post-training for reasoning models typically combines supervised fine-tuning with reinforcement learning from verifiable rewards, most commonly with GRPO. However, this algorithm suffers from sparse rewards, limited exploration, and mode collapse. Building upon recent works on self-distillation, we propose Feedback Distillation, a training method where the model is trained to match, at the token level, its own distribution conditioned on privileged feedback produced by a language model. Feedback Distillation is on-policy, offers token-level supervision, and can inject external knowledge. Evaluating our method for Lean4 theorem-proving, we find that Feedback Distillation maintains greater diversity in generated trajectories than GRPO, yielding higher policy entropy and better pass@$k$ scaling. The two methods are complementary: initializing GRPO from a Feedback Distillation checkpoint outperforms either method alone. All in all, our results suggest a promising avenue to improve post-training for complex reasoning.