  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.
