Measuring Mathematical Problem Solving With the MATH Dataset (2103.03874v2)

Abstract: Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

An Analysis of "Measuring Mathematical Problem Solving With the MATH Dataset"

The paper "Measuring Mathematical Problem Solving With the MATH Dataset" presents a comprehensive paper on evaluating the mathematical reasoning abilities of machine learning models. Authored by Dan Hendrycks et al., this work introduces the MATH dataset, a collection of $12,500$ advanced competition mathematics problems, each annotated with detailed step-by-step solutions. Additionally, the authors provide the Auxiliary Mathematics Problems and Solutions (AMPS), a pre-training corpus consisting of millions of auxiliary problems designed to bolster the performance of models on mathematical tasks.

Overview of the MATH Dataset

The MATH dataset is uniquely challenging and curated from high school mathematics competitions, encompassing problems that demand more than straightforward applications of K-12 mathematics. In particular, the problems span a range of subjects including Prealgebra, Algebra, Number Theory, Counting and Probability, Geometry, Intermediate Algebra, and Precalculus. Each problem is tagged with a difficulty level from 1 to 5 and formatted using \LaTeX{} and the Asymptote vector graphics language. This consistent formatting enables precise automatic evaluation by normalization and exact match metrics.

A remarkable aspect of this dataset is its capability to generate detailed solutions that can act as learning scaffolds for models, potentially aiding in generating interpretive step-by-step solutions. Despite these affordances, the paper's results show that the performance of Transformer models, such as GPT-2 and GPT-3, is modest at best, with accuracy figures ranging from $3.0\%$ to $6.9\%. Even the largest GPT-3 models, which typically dominate other text-based tasks, display meager performance improvements over smaller models.</p> <h3 class='paper-heading'>Findings and Implications</h3> <p>One of the paper&#39;s central findings is that simply increasing model size, a strategy that has proven highly effective in many other domains, does not yield proportionate benefits for mathematical reasoning tasks. For example, the paper notes an only $28\%$$relative improvement moving from a GPT-2 model with 0.1 billion parameters to one with 1.5 billion parameters.</p> <p>Interestingly, the paper highlights the discrepancy in model performance across different subjects and difficulty levels. For instance, GPT-2 models exhibit relatively higher accuracy on Prealgebra but struggle significantly with high-difficulty problems. This nuance underscores the need for more sophisticated algorithmic approaches beyond just scaling model size.</p> <p>Moreover, the paper explores the utility of integrating step-by-step solutions in the training process. While generating intermediate solution steps did not directly improve test-time performance, the presence of these solutions during training yielded a$$10\%$$relative accuracy gain. This indicates that while current models can benefit from seeing detailed solutions, their ability to emulate such reasoning is still underdeveloped.</p> <h3 class='paper-heading'>The Role of the AMPS Dataset</h3> <p>The AMPS pre-training dataset, comprising both Khan Academy and Mathematica-generated problems, is pivotal in this research. With over$$5$million problems and$100,000$ step-by-step solutions, AMPS significantly aids in pre-training models on foundational mathematical concepts. The paper shows that pre-training on AMPS allows smaller models to rival the performance of substantially larger models that did not undergo similar pre-training.

Future Directions

The paper posits that achieving significant advancements in mathematical reasoning through machine learning will likely necessitate novel algorithmic innovations. The relatively low performance of large-scale Transformers on the MATH dataset hints at intrinsic challenges distinct from those in natural language processing tasks. Future research will need to address these challenges, potentially exploring areas such as algorithmic reasoning, symbolic manipulation, and process-based learning paradigms.

Conclusion

Hendrycks et al. provide a rigorous and multi-faceted examination of machine learning models' mathematical problem-solving skills through the MATH dataset. Their findings highlight both the strides made and the substantial hurdles that remain. The paper's insights into performance limitations and the role of detailed solution exposure suggest that while pre-training and model scaling are valuable, they are insufficient alone. This work lays a foundation for future research endeavors aimed at devising intelligent systems capable of sophisticated mathematical reasoning, with far-reaching implications for AI development in education and beyond.