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MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning
Ke Wang, Houxing Ren, Aojun Zhou, Zimu Lu, Sichun Luo, Weikang Shi, Renrui Zhang, Linqi Song, Mingjie Zhan, Hongsheng Li
TL;DR
Open-source models lacked an effective recipe for combining natural-language reasoning, code generation, and execution feedback for mathematical problem solving. MathCoder addresses this with MathCodeInstruct and customized fine-tuning and inference, achieving state-of-the-art open-source results and outperforming GPT-4 on MATH. The approach remains constrained by GPT-4-based data generation and struggles with theorem proving and complex geometry.
Problem
Open-source models lacked an effective recipe for solving math problems by combining natural-language reasoning, code generation, and execution feedback.
Method
MathCoder constructs the 80k-example MathCodeInstruct dataset and fine-tunes open-source models on solutions interleaving natural language, code, and execution results.
Results
MathCoder achieves state-of-the-art performance among open-source models on GSM8K and MATH, surpasses ChatGPT-3.5 and PaLM-2, and outperforms GPT-4 on MATH.
Takeaways & Limitations
MathCoder provides an open-source family of models capable of code-based mathematical reasoning with strong benchmark performance.
Takeaways & Limitations
MathCoder’s capabilities are constrained by GPT-4’s data-generation capabilities, and the models cannot solve theorem-proving problems or complex geometry reliably.
Abstract
from arXiv · showhide
The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.
1 INTRODUCTION
Open-source models still lag behind closed-source systems on mathematical reasoning, motivating MathCoder’s integration of natural-language reasoning, code generation, and execution feedback through a dedicated dataset and fine-tuning pipeline.
- Motivation: Open-source LLMs lag significantly behind closed-source models on mathematical reasoning, leaving no effective recipe for GPT-4 Code Interpreter-style problem solving.The gap persists in both code generation and natural-language reasoning.
- Dataset: MathCodeInstruct contains 80k math problems with solutions interleaving natural language, executable code, and execution results.The dataset combines GSM8K and MATH problems with newly generated problems produced through problem interpolation prompting.
- Dataset: Problem interpolation generates questions with difficulty levels between GSM8K and MATH, bridging grade-school and challenging high-school mathematics.MathCoder-Initial generates solutions for the newly created problems before they are combined with seed data.
- Training and inference: The customized training and inference pipeline marks text, code, and execution results with special tokens and feeds runtime outputs back into autoregressive reasoning.Jupyter Notebooks execute detected code blocks during inference, after which the model continues predicting from the updated context.
- Results: MathCoder models fine-tuned on MathCodeInstruct show significant improvements across mathematical reasoning benchmarks.The models are produced by fine-tuning open-source Llama-2 and CodeLlama variants.
- Approach: MathCoder integrates natural-language reasoning, code generation, and execution feedback into open-source pretrained language models.The framework is presented as a systematic study targeting enhanced mathematical reasoning.
2 MATHCODER: SPECIALIZING LLAMA FOR MATHEMATICAL REASONING
MathCoder constructs MathCodeInstruct by combining seed solutions, interpolated problems, and LCE-formatted reasoning, then fine-tunes open-source models to generate solutions with executable code.
- 2.1 MATHCODEINSTRUCT DATASET: MathCodeInstruct combines seed data D0 with interpolated data D1 generated through problem interpolation prompting.The seed data comes from GSM8K and MATH solutions, while D1 adds newly generated problems.
- 2.1 MATHCODEINSTRUCT DATASET: Problem interpolation asks GPT-4 to create problems between relatively simple GSM8K examples and difficult MATH examples.GPT-4 evaluates whether generated problems occupy the intended intermediate difficulty range.
- 2.1 MATHCODEINSTRUCT DATASET: 83.2% of generated problems were harder than GSM8K, while 95.6% were easier than MATH.These results indicate that interpolation produced problems with intermediate difficulty.
- 2.2 SUPERVISED FINE-TUNING AND INFERENCE: The combined dataset fine-tunes Llama-2 and CodeLlama into the MathCoder-L and MathCoder-CL model families.The framework uses supervised fine-tuning after constructing D = {D0, D1}.
- 2.2 SUPERVISED FINE-TUNING AND INFERENCE: MathCodeInstruct solutions interleave reasoning language, mathematical code, and execution results using special tokens and real-time notebook execution.Training computes loss on reasoning language and code, while inference executes generated code and feeds results back into the solution process.
3 EXPERIMENTS
The experiments evaluate MathCoder across in-domain and out-of-domain mathematics benchmarks, model backbones, subjects, and ablations. Results show strong open-source performance, benefits from CodeLlama and problem interpolation, and sensitivity to how execution results are used during training and inference.
- Main Results: MathCoder achieves state-of-the-art results across the five evaluated datasets, although GPT-4 Code Interpreter retains a substantial performance lead over the open-source models.The evaluation covers GSM8K and MATH as in-domain datasets and SVAMP, Mathematics, and SimulEq as out-of-domain datasets.
- Main Results: MathCoder-L-7B outperforms WizardMath-70B on three of five datasets and gains 4.5% in average score.The authors attribute this advantage to training on solutions interleaving natural language, code, and execution results rather than CoT data alone.
- Main Results: MathCoder-CL improves accuracy over corresponding MathCoder-L models by 4.1% at 7B and 3.0% at 13B.These comparisons use CodeLlama and Llama-2 as the respective base models.
- Subject and Level Analysis: MathCoder performs better on algebra and prealgebra but struggles on geometry, especially at higher difficulty levels.The subject-level analysis suggests code is more useful for computationally intensive questions.
- Ablation Study: Problem interpolation significantly improves performance across all five datasets by generating problems with intermediate difficulty levels and increasing problem-set diversity.The interpolation experiment uses CodeLlama-34B as the base model.
- Ablation Study: Ignoring execution results in the supervised fine-tuning loss raises average accuracy from 69.1% to 70.2% across four of five datasets.This setting lets the model focus on natural language and code rather than learning to estimate execution results.
4 RELATED WORK
Related work improves mathematical reasoning through instruction tuning, chain-of-thought reasoning, code-based solving, and distillation. MathCoder differs by combining high-quality math solutions with code and execution-oriented training for open-source models.
- Instruction Tuning: Instruction tuning aligns language models with useful objectives and human preferences, while MathCoder focuses it on high-quality synthetic math solutions.Unlike related math instruction-tuning work, MathCoder uses code to solve the problems.
- Mathematical Reasoning: Mathematical reasoning research commonly uses benchmark datasets and Chain-of-Thought prompting to improve multistep problem solving.Another line of work encourages program-based solving, which motivates MathCoder’s code-integrated approach.
- Distillation: MathCoder can be viewed as self-distillation because its solutions are generated using the model’s own outputs rather than transferred only from a larger teacher.This connects the method to prior work showing that self-distillation can improve model performance.
5 CONCLUSION AND LIMITATION
MathCoder is an open-source model that combines math instruction data with customized supervised fine-tuning for code-based mathematical reasoning. It reports strong benchmark performance, while remaining limited by GPT-4-generated data and difficulty with theorem proving and complex geometry.
- 5 CONCLUSION AND LIMITATION: MathCoder uses instruction-following data construction and customized supervised fine-tuning to support mathematical reasoning with code.The training loss is applied only to natural language and code.
- 5 CONCLUSION AND LIMITATION: 83.9% on GSM8K and 45.2% on MATH are reported as state-of-the-art scores among open-source LLMs.
- 5 CONCLUSION AND LIMITATION: MathCoder outperforms ChatGPT-3.5 and PaLM-2 on GSM8K and MATH, and GPT-4 on MATH.
- 5 CONCLUSION AND LIMITATION: GPT-4-generated training data constrains MathCoder’s capabilities, including an inability to solve theorem-proving problems.
- 5 CONCLUSION AND LIMITATION: As a uni-modal model series, MathCoder still faces challenges with complex geometry problems.
A DATASET EXAMPLES
The dataset examples illustrate MathCodeInstruct creation through solutions containing different numbers of LCE blocks.
- A DATASET EXAMPLES: Fig. 6 presents an example containing one LCE block.
- A DATASET EXAMPLES: Fig. 7 presents an example containing three LCE blocks.
- A DATASET EXAMPLES: Together, the examples show MathCodeInstruct creation processes with one or three LCE blocks.
B EXAMPLES OF DIFFICULTY COMPARISON
The difficulty comparisons cover interpolation problems that are harder than, tied with, or easier than their source problems, with the first two cases quantified explicitly.
- B EXAMPLES OF DIFFICULTY COMPARISON: 83.2% of cases show newly generated interpolation problems as more difficult than the original GSM8K problems.
- B EXAMPLES OF DIFFICULTY COMPARISON: 95.6% of cases show original MATH problems as more difficult than newly generated interpolation problems.
- B EXAMPLES OF DIFFICULTY COMPARISON: 15.3% of all problems are cases where a newly generated interpolation problem ties with its original GSM8K problem.
- B EXAMPLES OF DIFFICULTY COMPARISON: Less than 3% of cases show an original GSM8K problem as slightly more difficult than its newly generated interpolation problem.
C CREATING PROBLEMS USING ONLY GSM8K
Problems generated using only GSM8K are slightly more complicated than the originals, whereas incorporating MATH examples produces more challenging interpolation problems that bridge the datasets’ difficulty gap.
- C CREATING PROBLEMS USING ONLY GSM8K: Problems generated from GSM8K training examples are more difficult than the original problems but remain similar to raw GSM8K problems.They may only change variable names or scale values.
- C CREATING PROBLEMS USING ONLY GSM8K: GSM8K-only generated problems are only slightly more complicated than the raw problems.
- C CREATING PROBLEMS USING ONLY GSM8K: Including MATH problems in interpolation provides GPT-4 a route to generate more challenging problems.
- C CREATING PROBLEMS USING ONLY GSM8K: Interpolation problems combining GSM8K and MATH examples can narrow the difficulty gap between the two datasets.
D MORE EXPERIMENT RESULTS
MathCoder-L and MathCoder-CL are compared across MATH difficulty levels and subjects, with the gains from MathCoder-L to MathCoder-CL concentrated in calculation-intensive areas.
- Improved accuracy from MathCoder-L to MathCoder-CL comes primarily from algebra, number theory, and counting and probability.These subjects require precise calculations.
E CASE STUDY WITH COT, POT AND LCE
The case studies compare LCE solutions with CoT and PoT on mathematical problems, while illustrating GPT-4-based problem interpolation and complexity evaluation. LCE succeeds where CoT and PoT fail in number theory, algebra, and geometry examples.
- LCE versus CoT and PoT: LCE succeeded on number-theory and algebra problems where both CoT and PoT failed.The comparison uses example solutions shown in Fig. 19 and Fig. 20.
- LCE versus CoT and PoT: In geometry, CoT found intermediate coordinates but failed to calculate the area, whereas PoT failed to interpret the problem conditions.The LCE solutions used the conditions correctly and avoided calculation errors in the cited examples.
- Problem interpolation: GPT-4’s interpolation prompt asks for a new problem harder than an easy example but simpler than a much more difficult example.This prompt is illustrated with paired examples and newly generated problems.
- Problem interpolation: MathCodeInstruct examples are generated by combining randomly selected GSM8K and MATH training problems, prompting GPT-4 to interpolate a new problem, and generating an LCE-style solution with MathCoder-Initial.Figures 6 and 7 depict this three-stage construction process.
- Complexity evaluation: GPT-4 evaluates whether generated MathCodeInstruct problems are more difficult than their source problems, tied, or slightly easier.The examples include ties and an uncommon case where a GSM8K source problem is slightly more difficult than the interpolation.
- GSM8K-only generation: GPT-4 also generates new problems from GSM8K examples and evaluates their complexity relative to the originals.Figures 13–15 show this generation-and-evaluation setup.