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[Submitted on 17 Sep 2026] Title:Learning from the Gap Between Pass@K and Pass@1 View a PDF of the paper titled Learning from the Gap Between Pass@K and Pass@1, by Xuan Liu and 2 other authors View PDF HTML (experimental) Abstract:Large language models (LLMs) are increasingly trained with reinforcement learning from verifiable rewards (RLVR). An exact verifier can also support test-time scaling by selecting a passing response from multiple samples, while other deployments use beam search, adaptive sampling, or tools. We study single-sample decoding, where each query receives one response without search, to ask whether search-exposed behavior can be absorbed into the model. Existing verified-response post-training recipes do not generally distinguish problems already solved on the first decode from failures recovered within K samples. Under a fixed budget, this can spend examples repeating behavior the deployed policy already has. We introduce GapFT, which selects training evidence by the source checkpoint's single-sample outcome and fine-tunes on the Pass@K-Pass@1 gap: problems the policy fails on one sample but solves within K samples. We match training examples, processed tokens, and optimizer steps while keeping the objective unchanged. GapFT fills the matched budget with recovered failures and uses an exact decomposition to distinguish corrections of recovered and missed failures from regressions on first-decode successes. On LogiQA 2.0 and ReClor with Llama-3.1-8B, GapFT improves Pass@1 by 14.4 and 13.9 points over the source model, outperforms budget-matched uniform verified RFT at the same learning rate, and matches fine-tuning on the full verified pool using one third of the data. A single decode matches the source model's verifier-selected Pass@4 accuracy. A randomized control attributes gains to covering distinct failures, and our analysis relates available gains to transferable failure support. A three-seed Qwen2.5-7B replication retains positive gains over uniform RFT on both logic tasks. Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Cite as: arXiv:2609.35793 [cs.LG] (or arXiv:2609.35793v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.35793 arXiv-issued DOI via DataCite Submission history From: Xuan Liu [view email] [v1] Thu, 17 Sep 2026 04:34:33 UTC (253 KB) Full-text links: Access Paper: View a PDF of the paper titled Learning from the Gap Between Pass@K and Pass@1, by Xuan Liu and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs stat stat.ML References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)