In math reasoning with large language models (LLMs), fine-tuning data augmentation by query evolution and diverse reasoning paths is empirically verified effective, profoundly narrowing the gap between open-sourced LLMs and cutting-edge proprietary LLMs. In this paper, we conduct an investigation for such data augmentation in math reasoning and are intended to answer: (1) What strategies of data augmentation are more effective; (2) What is the scaling relationship between the amount of augmented data and model performance; and (3) Can data augmentation incentivize generalization to out-of-domain mathematical reasoning tasks? To this end, we create two new dataset AugGSM8K and AugMATH, by complicating and diversifying the queries and sampling multiple reasoning paths from GSM8K and MATH. We obtained a series of LLMs called MuggleMath by fine-tuning LLaMA models on AugGSM8K and AugMATH. MuggleMath substantially achieves new state-of-the-art on GSM8K and MATH. A log-linear relationship and a segmented log-linear are presented between MuggleMath's performance and the amount of augmented data on GSM8K and MATH, respectively. We also find that it is weak in out-of-domain math reasoning generalization from AugGSM8K to MATH and from AugMATH to GSM8K, which suggests that augmenting queries that cover a broader range of subjects is more beneficial for generalization. We release our codes and augmented data in https://github.com/OFA-Sys/gsm8k-ScRel.

翻译：在大语言模型（LLMs）的数学推理中，通过查询演化和多样化推理路径进行微调数据增强的经验性验证表明其效果显著，极大地缩小了开源LLMs与尖端专有LLMs之间的性能差距。本文针对数学推理中的数据增强展开研究，旨在回答以下问题：（1）哪些数据增强策略更为有效；（2）增强数据量与模型性能之间存在何种缩放关系；（3）数据增强能否促进模型在领域外数学推理任务上的泛化能力？为此，我们通过复杂化和多样化GSM8K与MATH数据集的查询，并采样多推理路径，构建了两个新数据集AugGSM8K和AugMATH。基于AugGSM8K和AugMATH对LLaMA模型进行微调，我们得到了一系列名为MuggleMath的LLMs。MuggleMath在GSM8K和MATH上显著达到了新的最优性能。实验揭示了MuggleMath在GSM8K和MATH上的性能与增强数据量之间分别存在对数线性关系和分段对数线性关系。同时，我们发现从AugGSM8K到MATH以及从AugMATH到GSM8K的领域外数学推理泛化能力较弱，这表明覆盖更广泛主题的查询增强更有助于提升泛化性能。我们在https://github.com/OFA-Sys/gsm8k-ScRel 发布了相关代码和增强数据。