INSPIRE: An Internalize-Then-Improve Approach for Example-Driven Mathematical Reasoning
INSPIRE introduces an "Internalize-Then-Improve" framework to help LLMs develop genuine example-based mathematical reasoning rather than merely memorizing solution patterns Reference-Guided Student Internalization (RGSI) generates high-quality preference candidates under the policy model's own distribution, addressing the difficulty of constructing effective preference pairs A stage-wise rubric preference training strategy decomposes learning into method-oriented and correctness-oriented stages,
Analysis
TL;DR
- INSPIRE introduces an "Internalize-Then-Improve" framework to help LLMs develop genuine example-based mathematical reasoning rather than merely memorizing solution patterns
- Reference-Guided Student Internalization (RGSI) generates high-quality preference candidates under the policy model's own distribution, addressing the difficulty of constructing effective preference pairs
- A stage-wise rubric preference training strategy decomposes learning into method-oriented and correctness-oriented stages, reflecting the progressive nature of capability acquisition
- Experiments across multiple model scales and families show consistent improvements, even surpassing larger open-source models, with no degradation on out-of-distribution benchmarks
Why It Matters
This work addresses a critical gap in LLM mathematical reasoning: the distinction between superficial pattern memorization and genuine conceptual internalization. For AI practitioners, it demonstrates that preference optimization can be structured to first teach reasoning strategies before refining correctness, offering a practical blueprint for developing deeper reasoning capabilities in language models.
Technical Details
- Reference-Guided Student Internalization (RGSI): Generates preference candidates that align with the policy model's own distribution, solving the challenge of producing meaningful preference pairs when the model's example-based reasoning ability is initially limited
- Stage-wise Rubric Preference Training: Decomposes training into two stages—method-oriented (learning to adopt example-based reasoning strategies like constructing counterexamples) and correctness-oriented (applying those strategies accurately)—mirroring progressive capability acquisition
- Example-Based Reasoning Focus: Targets the human-like ability to construct counterexamples and test theorem boundaries, a capability largely absent in current LLMs despite rapid progress in general mathematical reasoning
- Multi-scale Evaluation: Validated across multiple model scales and families, demonstrating consistent improvements and the ability to surpass larger open-source baselines while preserving general mathematical reasoning performance on out-of-distribution benchmarks
Industry Insight
- The two-stage preference training paradigm (internalize then improve) could be generalized beyond mathematics to other domains requiring strategic reasoning, such as scientific hypothesis generation or legal argumentation
- The RGSI approach offers a practical solution to a common RLHF bottleneck: when models lack the capability to generate high-quality preferences, using reference-guided internalization can bootstrap the process without requiring external expert annotation
- The finding that example-based reasoning can be trained without degrading general reasoning ability suggests that specialized capability enhancement need not come at the cost of broad competence, an important consideration for deploying reasoning-optimized models in production
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