Mathematical reasoning is a challenging task for large language models
(LLMs), while the scaling relationship of it with respect to LLM capacity is
under-explored. In this paper, we investigate how the pre-training loss,
supervised data amount, and augmented data amount influence the reasoning
performances of a supervised LLM. We find that pre-training loss is a better
indicator of the model's performance than the model's parameter count. We apply
supervised fine-tuning (SFT) with different amounts of supervised data and
empirically find a log-linear relation between data amount and model
performance, and we find better models improve less with enlarged supervised
datasets. To augment more data samples for improving model performances without
any human effort, we propose to apply Rejection sampling Fine-Tuning (RFT). RFT
uses supervised models to generate and collect correct reasoning paths as
augmented fine-tuning datasets. We find with augmented samples containing more
distinct reasoning paths, RFT improves mathematical reasoning performance more
for LLMs. We also find RFT brings more improvement for less performant LLMs.
Furthermore, we combine rejection samples from multiple models which push
LLaMA-7B to an accuracy of 49.3% and outperforms the supervised fine-tuning
(SFT) accuracy of 35.9% significantly.

探索大型语言模型中数学推理的挑战，研究预训练损失、监督数据量和增强数据量对监督型语言模型推理性能的影响，发现预训练损失是模型性能的更好指标，应用不同数量的监督数据和拒绝抽样微调来改善模型性能，发现增加数据量与模型性能呈对数线性关系，增加更多不同推理路径的数据样本能进一步改善语言模型的数学推理性能。