Projected Distillation: On Learning to Solve Hard Problems from Close, Correct Targets

Shripad V. Deshmukh, Christian Bock, Kangrui Ruan, Abhijeet Awasthi, Vijay Lingam +2

On-policy learning is central to post-training: training on a model's own generations limits distribution shift and helps preserve existing capabilities. On hard problems, however, on-policy RL provides no reward-based learning signal when all attempts fail. On-policy distillation provides next-token supervision from a teacher, but on hard problems, if a rollout goes wrong early, subsequent supervision is conditioned on prefixes that already contain the mistake. Correct behavior can instead come from a gold or teacher solution, but such off-policy targets can differ substantially from the model's generations and, as we show, may erode existing capabilities. We therefore treat on-policyness as a continuum and propose projected distillation, which minimally edits the student's failed attempts into correct targets. The method uses contrastive updates to favor corrections over failed attempts and refreshes these corrections as the student evolves, enabling near-on-policy learning even when all student attempts fail. Theoretically, we prove that the student's learning regret grows with the edit size: it scales linearly with the horizon under minimal correction, as in DAgger, and…