Mathematical reasoning is regarded as a necessary ability for Language Models (LMs). Recent works demonstrate large LMs’ impressive performance in solving math problems. The success is attributed to their Chain-of-Thought (CoT) reasoning abilities, i.e., the ability to decompose complex questions into step-by-step reasoning chains, but such ability seems only to emerge from models with abundant parameters. This work investigates how to incorporate relatively small LMs with the capabilities of multi-step reasoning. We propose to inject such abilities by continually pre-training LMs on a synthetic dataset MsAT which is composed of Multi-step Arithmetic Tasks. Our experiments on four math word problem datasets show the effectiveness of the proposed method in enhancing LMs’ math reasoning abilities.

Fine-tuning a pre-trained language model via the contrastive learning framework with a large amount of unlabeled sentences or labeled sentence pairs is a common way to obtain high-quality sentence representations. Although the contrastive learning framework has shown its superiority on sentence representation learning over previous methods, the potential of such a framework is under-explored so far due to the simple method it used to construct positive pairs. Motivated by this, we propose a method that makes hard positives from the original training examples. A pivotal ingredient of our approach is the use of prefix that attached to a pre-trained language model, which allows for differentiable data augmentation during contrastive learning. Our method can be summarized in two steps: supervised prefix-tuning followed by joint contrastive fine-tuning with unlabeled or labeled examples. Our experiments confirm the effectiveness of our data augmentation approach. The proposed method yields significant improvements over existing methods under both semi-supervised and supervised settings. Our experiments under a low labeled data setting also show that our method is more label-efficient than the state-of-the-art contrastive learning methods.