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[Submitted on 8 Jul 2026] Title:When Does External Guidance Help LLM Reasoning? A Bias-Variance Theory of Guidance-Augmented GRPO View a PDF of the paper titled When Does External Guidance Help LLM Reasoning? A Bias-Variance Theory of Guidance-Augmented GRPO, by Sofia Torres and 3 other authors View PDF HTML (experimental) Abstract:Reinforcement learning with verifiable rewards (RLVR) has become the dominant paradigm for eliciting multi-step reasoning in large language models, and a recent wave of methods (LUFFY, ExPO, PAPO, TAPO) further augments RL with \emph{external guidance} - expert traces, self-explanations, or retrieved thought patterns. Although each method reports empirical gains, none provides convergence rates, bias bounds, or an optimal weighting rule for the guidance signal. We close this gap with \emph{Guidance-Augmented GRPO} (GA-GRPO), a unified theoretical framework that casts external guidance as a stochastic guidance operator G re-writing the question distribution, and analyses the resulting policy-gradient estimator as a biased on-policy estimator whose bias is bounded by the total-variation guidance divergence delta\_G between the guidance-augmented sampling distribution and the policy's own distribution. The framework subsumes vanilla GRPO, LUFFY, ExPO, PAPO, and TAPO as special cases obtained by particular choices of G. Under smoothness and bounded-divergence assumptions we prove that GA-GRPO converges at rate O(1/sqrt(T)) to an O(delta sqrt(T))-neighbourhood of the GRPO stationary point, derive the closed-form MSE-optimal guidance weight lambda-star(T, delta, sigma\_0 squared) = sigma\_0 squared / (sigma\_0 squared + R\_max squared delta squared T), and prove a matching minimax lower bound showing the Omega(delta squared T) bias term is unavoidable. Experiments on Qwen2.5-Math-7B-Base across nine math and OOD benchmarks confirm that optimal-weight GA-GRPO matches or surpasses TAPO, LUFFY, ExPO, and vanilla GRPO while requiring 31\% fewer GPU-hours, and eight analysis experiments validate each theoretical prediction. Subjects: Machine Learning (cs.LG); Computation and Language (cs.CL) Cite as: arXiv:2610.06861 [cs.LG] (or arXiv:2610.06861v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.06861 arXiv-issued DOI via DataCite Submission history From: Carter Adams [view email] [v1] Wed, 8 Jul 2026 14:38:06 UTC (187 KB) Full-text links: Access Paper: View a PDF of the paper titled When Does External Guidance Help LLM Reasoning? A Bias-Variance Theory of Guidance-Augmented GRPO, by Sofia Torres and 3 other authors View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs cs.CL 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?)