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iclr26/3K1y4KbWAx/appendix_text_v3.txt

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+[p. 15 | section: Appendix | type: TableOfContents]
+A LLM Usage 15 B Theoretical Proof 16 B.1 Notion 16 B.2 GRPO as Preference Optimization 16 B.3 Gradient Dynamics and NTK Alignment 18 B.3.1 Change in Log-Probability 18 B.3.2 Main Generalization Result 19 B.4 Unifying Trajectory Divergence and Domain Discrepancy 20 B.5 Main Theorem: Generalization Bound 21 B.6 Main Theorem: Convergence Analysis 23 B.7 Addition Proofs 24 C Discussion and Limitations 26 D Experiment Details 27 D.1 Detailed Setup 27 D.2 System Prompt 28 D.3 Baseline Description 28 E More Experiments 29 E.1 Comparison with More Supervised RLVR Baselines 29 E.2 Extend TraPO to More Models 29 E.3 TraPO Is a Universal Component 30 E.4 Run TraPO on DeepMath 30 E.5 Different Selection Strategies 31 E.6 Stability of TraPO 31 E.7 Training Cost Analysis of TraPO 31 E.8 Different Ways of Utilizing Reliable Passrate Databases 31 F More Related Work 31 G Pseudo Code 32
+
+[p. 15 | section: A LLM USAGE | type: Text]
+In the preparation of this paper, the LLM was used solely for language editing and proofreading to improve clarity and readability.
+
+[p. 16 | section: B THEORETICAL PROOF | type: Text]
+In this section, we provide proofs for the generalization error bound and convergence of the proposed semi-supervised framework TRAPO.
+
+[p. 26 | section: C DISCUSSION AND LIMITATIONS | type: Text]
+First, our results demonstrate that semi-supervised training using 4K labeled data combined with 16K unlabeled data outperforms fully supervised training on 45K labeled data. This encouraging
+
+[p. 27 | section: C DISCUSSION AND LIMITATIONS | type: Text]
+finding aligns with the insight proposed by Li et al. (2025b) in the context of RLVR training: thorough training (i.e., more training epochs) on smaller curated datasets can yield better performance than training with larger datasets for fewer epochs. Our work further extends this observation by showing that unlabeled data, when carefully selected using guidance from labeled data training, can effectively enhance the model's reasoning capabilities, thus amplifying the benefits of semisupervised RLVR.
+
+[p. 27 | section: C DISCUSSION AND LIMITATIONS | type: Text]
+In addition, due to computational constraints, our evaluation is currently limited to models under the 7B parameter scale. Exploring the applicability and scalability of this semi-supervised paradigm to larger language models (e.g., 13B or beyond) remains an important direction for future research, as larger models may benefit even more from effective utilization of unlabeled data.
+
+[p. 27 | section: C DISCUSSION AND LIMITATIONS | type: Text]
+One key observation from our experiments comparing Qwen2.5-Math-7B and LLaMA-3.1-8B is that the more effective supervised training with labeled data is on a model, the better the labeled data guides the selection and utilization of unlabeled data. Conversely, if RLVR training with labeled data yields only marginal gains, its impact on unlabeled data filtering is also limited. Therefore, we recommend applying TRAPO primarily in settings where the labeled data and model are well aligned.
+
+[p. 27 | section: C DISCUSSION AND LIMITATIONS | type: Text]
+Finally, we believe it is a promising direction, similar to active learning, to investigate what types of labeled examples most effectively guide unlabeled data training, and we plan to explore this in future work.
+
+[p. 31 | section: F MORE RELATED WORK | type: Text]
+Semi-supervised Reinforcement Learning. Semi-supervised learning has been widely studied in weakly supervised scenarios (Zhu et al., 2025; Tang et al., 2024; 2025; 2026; Zhou et al., 2024; 2026), where labeled and unlabeled data are combined to improve model performance under limited annotation budgets (Blum & Mitchell, 1998; Chapelle et al., 2009; Subramanya & Bilmes, 2011;
+
+[p. 32 | section: F MORE RELATED WORK | type: TableGroup]
+Table 7: Wall-clock training time (reported as "GPU-hours × GPUs") across data regimes. Unsupervised Supervised Semi-Supervised (TRAPO) ∼7 × 8 ∼13 × 8 ∼25 × 8 ∼39 × 8 ∼26 × 8 ∼38 × 8 ∼55 × 8 ∼11 × 8 ∼57 × 8
+
+[p. 32 | section: F MORE RELATED WORK | type: Text]
+Rasmus et al., 2015; Laine & Aila, 2016; Tarvainen & Valpola, 2017; Berthelot et al., 2019; Xie et al., 2020; Sohn et al., 2020) . In reinforcement learning, early work explored combining rewardbased learning with self-supervised signals or pseudo-rewards derived from environment dynamics or intrinsic motivation (Dud´ık et al., 2011; Finn et al., 2016; Thomas & Brunskill, 2016; Kallus & Uehara, 2020; Zhou et al., 2023) . These methods typically treat supervised and unsupervised signals independently, for instance by summing reward and consistency objectives, or by pre-training on unlabeled data before fine-tuning on labeled trajectories.
+
+[p. 32 | section: F MORE RELATED WORK | type: Text]
+However, such semi-supervised RL approaches are ill-suited for large language model (LLM) training under verifiable rewards (RLVR). In RLVR, the policy is optimized using feedback signals derived from answer verification (e.g., correctness of final outputs), rather than explicit action-level rewards. Unsupervised methods in this space rely on internal consistency, such as low token entropy (Agarwal et al., 2025) , high self-certainty (Zhao et al., 2025) , or majority voting (Zuo et al., 2025) , to construct pseudo-rewards. While these signals can guide exploration, they often reinforce incorrect or degenerate reasoning patterns in the absence of external supervision, leading to model collapse (Zhang et al., 2025c) .
+
+[p. 32 | section: F MORE RELATED WORK | type: Text]
+Our work departs from prior approaches by introducing a guidance mechanism: the labeled data are not merely used to provide an additional reward signal, but to actively steer the selection and utilization of unlabeled samples. Specifically, we observe that reliable reasoning trajectories on unlabeled data exhibit learning dynamics similar to those on labeled data. By measuring trajectory similarity in the reward model space, TRAPO identifies high-quality unlabeled samples whose reasoning patterns are consistent with verified ones. This ensures that unsupervised signals are only leveraged when they align with externally validated behavior, preventing the amplification of spurious patterns.
+
+[p. 32 | section: F MORE RELATED WORK | type: Text]
+This paradigm shift from independent combination to supervised guidance addresses a key limitation of traditional methods. In high dimensional open ended generation tasks such as reasoning with LLMs consistency alone is insufficient for correctness. Without supervision to anchor the learning process models easily overfit to superficial patterns or self reinforced errors. TRAPO resolves this by using minimal labeled data as a "north star" enabling stable and effective learning from large amounts of unlabeled data. As we show empirically this leads to superior performance and data efficiency surpassing both fully supervised baselines trained on orders of magnitude more labels and unsupervised methods that fail to generalize.
+
+[p. 32 | section: G PSEUDO CODE | type: Text]
+We provide the pseudo code 1.

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+      "title": "Appendix",
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+      "title": "A LLM USAGE",
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