An Empirical Study of Data Ability Boundary in LLMs' Math Reasoning

TL;DR

This work addresses how to optimally expand math reasoning in open-source LLMs by defining a Minimal Optimal Set (MOS) of reasoning paths and using Mix of Minimal Optimal Sets (MMOS) to cumulatively enhance multiple abilities. It demonstrates that diversity and correctness of reasoning paths, coupled with deduplication and clustering filters, reveal an ability boundary where further path additions yield diminishing returns. By expanding data across related datasets (e.g., combining GSM8K with MATH and TAL-SCQ), the authors show cumulative gains and develop an Auto Problem Generator to probe numerical robustness and educational use. The proposed MMOS framework achieves strong performance at lower data-construction costs, reshaping how supervised data strategies should be designed for math reasoning in LLMs.

Abstract

Large language models (LLMs) are displaying emergent abilities for math reasoning tasks,and there is a growing attention on enhancing the ability of open-source LLMs through supervised fine-tuning (SFT).In this paper, we aim to explore a general data strategy for supervised data to help optimize and expand math reasoning ability.Firstly, we determine the ability boundary of reasoning paths augmentation by identifying these paths' minimal optimal set.Secondly, we validate that different abilities of the model can be cumulatively enhanced by Mix of Minimal Optimal Sets of corresponding types of data, while our models MMOS achieve SOTA performance on series base models under much lower construction costs.Besides, we point out GSM-HARD is not really hard and today's LLMs no longer lack numerical robustness.Also, we provide an Auto Problem Generator for robustness testing and educational applications.Our code and data are publicly available at https://github.com/cyzhh/MMOS.

Figures (5)

• Figure 1: Conceptual figure of the ability boundary
• Figure 2: Visualization of query embedding distribution through t-SNE across six distinct datasets.
• Figure 3: Comparison of test set accuracy on GSM8K, S&A and MATH for models after SFT on Code LLaMA 7B using series subsets of $D^{k}_{u400}$ and $E^{k}_{u400}$ with different data amount.
• Figure 4: Comparison of test set accuracy on GSM8K, S&A and MATH for models after SFT on Code LLaMA 7B using series subsets of $D^{k}_{G+M}$ and $D^{k}_{M}$ with different MATH data amount.
• Figure 5: The relationships of k & N.