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Supervised fine-tuning (SFT) improves in-domain performance but can degrade out-of-domain (OOD) generalization. Prior work suggests that thi… (voir plus)s degradation is related to changes in dominant singular subspaces of pretrained weight matrices. However, directly identifying loss-sensitive directions with Hessian or Fisher information is computationally expensive at LLM scale. In this work, we propose preserving projected rotations in pretrained singular subspaces as an efficient proxy for Fisher-sensitive directions, which we call Rotation-Preserving Supervised Fine-Tuning (RPSFT). RPSFT penalizes changes in the projected top-
Reinforcement learning with verifiable rewards improves reasoning, while the allocation of learning signal shapes which solutions remain acc… (voir plus)essible under repeated sampling. Group-relative objectives assign equal advantages to equally rewarded responses, making aggregate credit proportional to sampled mode frequency. We introduce Exploration-Preserving Policy Optimization (ExPPO), a lightweight advantage-shaping rule that redistributes credit using prompt-relative, length-normalized response surprisal and prompt pass rate. ExPPO combines bounded shaping with shared normalization to preserve verifier polarity and approximately maintain each prompt group's total absolute sequence-advantage mass. Our analysis characterizes response-level credit allocation alongside sampled mode updates, deriving local conditions for gains in entropy and correct-mode discovery. Experiments show improved in-domain and out-of-domain reasoning coverage, higher aggregate response accuracy, and strong coverage at large sampling budgets. A controlled multi-answer evaluation further demonstrates increased correct-mode yield and gains in diversity among verified-correct responses. Code is available at https://github.com/jinhangzhan/ExPPO
The two-stage fine-tuning paradigm of Supervised Fine-Tuning (SFT) followed by Reinforcement Learning (RL) has empirically shown better reas… (voir plus)oning performance than one-stage SFT for the post-training of Large Language Models (LLMs). However, the evolution and mechanism behind the synergy of SFT and RL are still under-explored and inconclusive. In our study, we find the well-known claim "SFT memorizes, RL generalizes" is over-simplified, and discover that: (1) OOD performance peaks at the early stage of SFT and then declines (OOD forgetting), the best SFT checkpoint cannot be captured by training/test loss; (2) the subsequent RL stage does not generate fundamentally better OOD capability, instead it plays an \textbf{OOD restoration} role, recovering the lost reasoning ability during SFT; (3) The recovery ability has boundaries, \ie{} \textbf{if SFT trains for too short or too long, RL cannot recover the lost OOD ability;} (4) To uncover the underlying mechanisms behind the forgetting and restoration process, we employ SVD analysis on parameter matrices, manually edit them, and observe their impacts on model performance. Unlike the common belief that the shift of model capacity mainly results from the changes of singular values, we find that they are actually quite stable throughout fine-tuning. Instead, the OOD behavior strongly correlates with the \textbf{rotation of singular vectors}. Our findings re-identify the roles of SFT and RL in the two-stage fine-tuning and discover the rotation of singular vectors as the key mechanism. %reversing the rotations induced by SFT, which shows recovery from forgetting, whereas imposing the SFT parameter directions onto a RL-tuned model results in performance degradation. Code is available at https://github.com/xiaodanguoguo/RL_Heals_SFT