Learning from the Gap Between Pass@K and Pass@1

Sampling many responses and keeping one that passes a verifier lets large language models solve problems beyond their single-response ability, but this search must be paid again for every query, while many deployments answer with a single response. Post-training on verified responses can transfer the benefit of search into the model. With a fixed budget, selecting by correctness alone spends slots on problems the model already answers correctly, leaving fewer to correct its failures. To address this imbalance, we propose GapFT, which trains on the gap between Pass@K and Pass@1: problems that the source model fails with one response but solves within K samples. GapFT keeps the objective and training budget fixed and changes only which verified responses enter training; an exact decomposition splits the resulting Pass@1 change into corrected failures and regressions on problems the source model already solved. On LogiQA 2.0 and ReClor with three model families, GapFT is above budget-matched uniform rejection-sampling fine-tuning (RFT) in every setting, with a positive pooled effect, and on Llama-3.1-8B and Mistral-7B it recovers about two thirds to four fifths of the gain of fine-tuning on the entire verified pool with 11-34% of its problems. Further analyses reveal that the gain comes from failures that the first few search samples recover, while failures found only by deeper search displace replay and add no net gain, that filling the same budget with gold-labeled failures search cannot reach lowers accuracy, and that the gain is bounded by how many transferable failures search exposes.