Integrating Arithmetic Learning Improves Mathematical Reasoning in Smaller Models

While large models pre-trained on high-quality data exhibit excellent performance across various reasoning tasks, including mathematical reasoning (e.g. GSM8k, MultiArith), specializing smaller models to excel at mathematical reasoning remains a challenging problem. Common approaches to address this challenge include knowledge distillation, where smaller student models learn from large pre-trained teacher models, and data augmentation, such as rephrasing questions. Despite these efforts, smaller models struggle with arithmetic computations, leading to errors in mathematical reasoning. In this work, we focus on leveraging a programmatically generated arithmetic dataset to enhance the reasoning capabilities of smaller models. We investigate two key approaches to incorporate this dataset -- (1) intermediate fine-tuning, where a model is fine-tuned on the arithmetic dataset before being trained on a reasoning dataset, and (2) integrating the arithmetic dataset into the instruction-tuning mixture, allowing the model to learn arithmetic skills alongside general instruction-following abilities. Our experiments on multiple reasoning benchmarks demonstrate that incorporating an arithmetic dataset, whether through targeted fine-tuning or within the instruction-tuning mixture, enhances the models' arithmetic capabilities, which in turn improves their mathematical reasoning performance.

E-Gen: Leveraging E-Graphs to Improve Continuous Representations of Symbolic Expressions

As vector representations have been pivotal in advancing natural language processing (NLP), some prior research has concentrated on creating embedding techniques for mathematical expressions by leveraging mathematically equivalent expressions. While effective, these methods are limited by the training data. In this work, we propose augmenting prior algorithms with larger synthetic dataset, using a novel e-graph-based generation scheme. This new mathematical dataset generation scheme, E-Gen, improves upon prior dataset-generation schemes that are limited in size and operator types. We use this dataset to compare embedding models trained with two methods: (1) training the model to generate mathematically equivalent expressions, and (2) training the model using contrastive learning to group mathematically equivalent expressions explicitly. We evaluate the embeddings generated by these methods against prior work on both in-distribution and out-of-distribution language processing tasks. Finally, we compare the performance of our embedding scheme against state-of-the-art large language models and demonstrate that embedding-based language processing methods perform better than LLMs on several tasks, demonstrating the necessity of optimizing embedding methods for the mathematical data modality.

Highlighting Named Entities in Input for Auto-Formulation of Optimization Problems

Operations research deals with modeling and solving real-world problems as mathematical optimization problems. While solving mathematical systems is accomplished by analytical software, formulating a problem as a set of mathematical operations has been typically done manually by domain experts. However, recent machine learning models have shown promise in converting textual problem descriptions to corresponding mathematical formulations. In this paper, we present an approach that converts linear programming word problems into meaning representations that are structured and can be used by optimization solvers. Our approach uses the named entity-based enrichment to augment the input and achieves state-of-the-art accuracy, winning the second task of the NL4Opt competition (https://nl4opt.github.io).