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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<!-- Meta tags for social media banners, these should be filled in appropriatly as they are your "business card" -->
<!-- Replace the content tag with appropriate information -->
<meta name="description" content="DESCRIPTION META TAG">
<meta property="og:title" content="MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models" />
<meta property="og:description" content="SOCIAL MEDIA DESCRIPTION TAG TAG" />
<meta property="og:url" content="URL OF THE WEBSITE" />
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<title>MetaMath</title>
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<link href="https://fonts.googleapis.com/css?family=Google+Sans|Noto+Sans|Castoro" rel="stylesheet">
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<script defer src="static/js/fontawesome.all.min.js"></script>
<script src="static/js/bulma-carousel.min.js"></script>
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<script src="static/js/index.js"></script>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body>
<section class="hero">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column has-text-centered">
<h1 class="title is-1 publication-title">MetaMath:<br />Bootstrap Your Own
Mathematical Questions for Large Language Models</h1>
<div class="is-size-5 publication-authors">
<!-- Paper authors -->
<span class="author-block"><a href="https://yulonghui.github.io/" target="_blank">Longhui
Yu</a><sup>1,*</sup>,</span>
<span class="author-block"><a href="https://wayson.tech/" target="_blank">Weisen
Jiang</a><sup>2,3,*</sup>,</span>
<span class="author-block"><a href="https://han-shi.github.io/" target="_blank">Han
Shi</a><sup>4,‡</sup>,</span>
<span class="author-block"><a href="https://lovesykun.cn/about.html"
target="_blank">Jincheng
Yu</a><sup>3,4</sup>,</span>
<span class="author-block"><a href="https://scholar.google.com/citations?user=DFme0joAAAAJ"
target="_blank">Zhengying
Liu</a><sup>4</sup>,</span>
<br />
<span class="author-block"><a href="https://cse.sustech.edu.cn/faculty/~zhangy/"
target="_blank">Yu
Zhang</a><sup>2</sup>,</span>
<span class="author-block"><a href="https://www.cse.ust.hk/~jamesk/" target="_blank">James
T.
Kwok</a><sup>3</sup>,</span>
<span class="author-block"><a href="https://scholar.google.com/citations?user=XboZC1AAAAAJ"
target="_blank">Zhenguo
Li</a><sup>4</sup>,</span>
<span class="author-block"><a href="https://mlg.eng.cam.ac.uk/adrian/"
target="_blank">Adrian
Weller</a><sup>1,5</sup>,</span>
<span class="author-block"><a href="https://wyliu.com/" target="_blank">Weiyang
Liu</a><sup>1,6,‡</sup></span>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup>1</sup>University of Cambridge,</span>
<span class="author-block"><sup>2</sup>Southern University of Science and Technology,</span>
<span class="author-block"><sup>3</sup>Hong Kong University of Science and
Technology,</span>
<span class="author-block"><sup>4</sup>Huawei Noah's Ark Lab,</span>
<span class="author-block"><sup>5</sup>The Alan Turing Institute,</span>
<span class="author-block"><sup>6</sup>Max Planck Institute for Intelligent Systems</span>
<span class="eql-cntrb"><small><br><sup>*</sup>Equal Contribution, <sup>‡</sup>Corresponding
author</small></span>
<span class="eql-cntrb"><small><br><a
href="mailto:longhuiyu98@gmail.com">longhuiyu98@gmail.com</a>, <a
href="mailto:shi.han@huawei.com">shi.han@huawei.com</a>, <a
href="mailto:wl396@cam.ac.uk">wl396@cam.ac.uk</a></small></span>
</div>
<div class="column has-text-centered">
<div class="publication-links">
<span class="link-block">
<a href="https://huggingface.co/meta-math/MetaMath-7B-V1.0" target="_blank"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
🤗
</span>
<span>Models</span>
</a>
</span>
<span class="link-block">
<a href="https://huggingface.co/datasets/meta-math/MetaMathQA" target="_blank"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
🤗
</span>
<span>Dataset</span>
</a>
</span>
<!-- Github link -->
<span class="link-block">
<a href="https://github.com/meta-math/MetaMath" target="_blank"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
<i class="fab fa-github"></i>
</span>
<span>Code</span>
</a>
</span>
<!-- ArXiv abstract Link -->
<span class="link-block">
<a href="https://arxiv.org/abs/2309.12284" target="_blank"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
<i class="ai ai-arxiv"></i>
</span>
<span>arXiv</span>
</a>
</span>
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<!-- Paper abstract -->
<section class="section hero is-light">
<div class="container is-max-desktop">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Abstract</h2>
<div class="content has-text-justified">
<p>
Large language models (LLMs) have pushed the limits of natural language understanding and
exhibited excellent problem-solving ability. Despite the great success, most existing
open-source LLMs (e.g., LLaMA-2) are still far away from satisfactory for solving mathematical problems due to its complex reasoning procedures. To bridge
this gap, we propose MetaMath, a finetuned language model that specializes in
mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by
rewriting the question from multiple perspectives, which results in
a new dataset called MetaMathQA. Then we finetune the LLaMA-2 models on MetaMathQA.
Experimental results on two popular benchmarks (i.e., GSM8K and MATH) for
mathematical reasoning emonstrate that MetaMath outperforms all open-source LLMs by a
significant margin. Our MetaMath-7B model achieves 66.5% on GSM8K and 19.8% on MATH,
exceeding the state-of-the-art models of the same size by 11.5% and 8.7%. Particularly,
MetaMath-70B achieves an accuracy of 82.3% on GSM8K, slightly better than
GPT-3.5-Turbo. We release the MetaMathQA dataset, the pretrained
MetaMath models with different mode size and the training code for public use.
</p>
</div>
</div>
</div>
</div>
</section>
<!-- End paper abstract -->
<section class="hero">
<div class="hero-body">
<div class="container is-max-desktop has-text-centered">
<h2 class="title is-3">Overview</h2>
<div class="columns is-centered">
<div class="column is-four-fifths">
<div class="item">
<!-- Your image here -->
<img style="width: 100%;" src="static/images/metamath.svg" alt="metamath" />
<h2 class="subtitle">
Figure 1: Overview of the <i>MetaMathQA</i> data and <i>MetaMath</i>.
</h2>
</div>
</div>
</div>
</div>
</div>
</section>
<section class="hero">
<div class="hero-body">
<div class="container is-max-desktop">
<h2 class="title is-3 has-text-centered">MetaMathQA</h2>
<div class="columns is-centered">
<div class="column is-four-fifths">
<div class="item">
<article class="message is-dark">
<div class="message-header">
<p>Answer Augmentation (AnsAug)</p>
</div>
<div class="message-body">
<p>
<b>Generated Answer:</b>
James buys 5 packs of beef that are 4 pounds each,
so he buys a total of 5 * 4 = 20 pounds of beef. The price of beef is $5.50
per pound, so he pays 20 * $5.50 = $110. The answer is: 110
</p>
</div>
</article>
<article class="message is-dark">
<div class="message-header">
<p>Rephrasing Question</p>
</div>
<div class="message-body">
<p>
<b>Question:</b> What is the total amount that James paid when he purchased 5 packs of beef, each weighing 4
pounds, at a price of $5.50 per pound?
</p>
<p>
<b>Rephrase the above question:</b> Each pack of beef weighs 4 pounds, so 5 packs weigh 4 * 5 = 20 pounds in total. The price per
pound of beef is $5.50, so the total cost for 20 pounds is 20 * $5.50 = $110. Therefore, James paid a total of
$110. The answer is: 110.
</p>
</div>
</article>
<article class="message is-dark">
<div class="message-header">
<p>Self-Verification Question</p>
</div>
<div class="message-body">
<p><b>Question:</b> James buys <span style="color: red">x</span> packs of beef that
are
4 pounds each. The price of beef is \$5.50 per pound. He paid 110. <span
style="color: red">What is the value of unknown variable x?</span></p>
<p>
<b>Answer:</b> To solve this problem, we need to determine the value of x, which
represents the number of packs of beef that James bought.
Each pack of beef weighs 4 pounds and costs $5.50 per pound.
The total amount James paid is $110.
We can set up the equation as follows:
Number of packs of beef * Weight per pack * Price per pound = Total amount paid;
x * 4 * $5.50 = $110;
Let's simplify and solve for x:
4x * $5.50 = $110;
22x = $110;
To isolate x, we divide both sides of the equation by 22:
22x / 22 = $110 / 22;
x = $5;
The value of x is 5.
</p>
</div>
</article>
<article class="message is-dark">
<div class="message-header">
<p>FOBAR Question</p>
</div>
<div class="message-body">
<p>
<b>Question:</b>
James buys <span style="color: red">x</span> packs of beef that are 4 pounds
each.
The price of beef is $5.50 per pound. How much did he pay?
<span style="color: red">If we know the answer to the above question is 110,
what is
the value of unknown variable x?</span>
</p>
<p>
<b>Answer:</b>
James buys x packs of beef that are 4 pounds each, so he buys a total of 4x
pounds
of beef.
The price of beef is $5.50 per pound, so the total cost of the beef is 5.50 * 4x
=
22x.
We are given that the total cost is $110, so we can write: 22x = 110.
Dividing both sides by 22, we get: x = 5.
The value of x is 5.
</p>
</div>
</article>
</div>
</div>
</div>
</div>
</div>
</section>
<section class="hero">
<div class="hero-body">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column has-text-centered is-fifths-fifths">
<h2 class="title is-3">Comprehensive Results</h2>
<div class="content has-text-justified">
<table>
<thead>
<tr>
<th>Model</th>
<th>#params</th>
<th>GSM8K</th>
<th>MATH</th>
</tr>
<tr>
</tr>
</thead>
<tbody id="tabResults">
<tr class="th">
<td colspan="4" style="text-align: center; font-weight: bold;">
Closed-source Model</td>
</tr>
<tr>
<td>GPT-4</td>
<td>-</td>
<td>92.0</td>
<td>42.5</td>
</tr>
<tr>
<td>GPT-3.5-Turbo</td>
<td>-</td>
<td>80.8</td>
<td>34.1</td>
</tr>
<tr>
<td>PaLM</td>
<td>8B</td>
<td>4.1</td>
<td>1.5</td>
</tr>
<tr>
<td>PaLM</td>
<td>62B</td>
<td>33.0</td>
<td>4.4</td>
</tr>
<tr>
<td>PaLM</td>
<td>540B</td>
<td>56.5</td>
<td>8.8</td>
</tr>
<tr>
<td>PaLM-2</td>
<td>540B</td>
<td>80.7</td>
<td>34.3</td>
</tr>
<tr>
<td>Flan-PaLM 2</td>
<td>540B</td>
<td>84.7</td>
<td>33.2</td>
</tr>
<tr>
<td>Minerva</td>
<td>8B</td>
<td>16.2</td>
<td>14.1</td>
</tr>
<tr>
<td>Minerva</td>
<td>62B</td>
<td>52.4</td>
<td>27.6</td>
</tr>
<tr>
<td>Minerva</td>
<td>540B</td>
<td>58.8</td>
<td>33.6</td>
</tr>
<tr class="th">
<td colspan="4" style="text-align: center; font-weight: bold;">
Open-source models (1-10B)</td>
</tr>
<tr>
<td>LLaMA-1</td>
<td>7B</td>
<td>11.0</td>
<td>2.9</td>
</tr>
<tr>
<td>LLaMA-2</td>
<td>7B</td>
<td>14.6</td>
<td>2.5</td>
</tr>
<tr>
<td>MPT</td>
<td>7B</td>
<td>6.8</td>
<td>3.0</td>
</tr>
<tr>
<td>Falcon</td>
<td>7B</td>
<td>6.8</td>
<td>2.3</td>
</tr>
<tr>
<td>InternLM</td>
<td>7B</td>
<td>31.2</td>
<td>-</td>
</tr>
<tr>
<td>GPT-J</td>
<td>6B</td>
<td>34.9</td>
<td>-</td>
</tr>
<tr>
<td>ChatGLM 2</td>
<td>6B</td>
<td>32.4</td>
<td>-</td>
</tr>
<tr>
<td>Qwen</td>
<td>7B</td>
<td>51.6</td>
<td>-</td>
</tr>
<tr>
<td>Baichuan-2</td>
<td>7B</td>
<td>24.5</td>
<td>5.6</td>
</tr>
<tr>
<td>SFT</td>
<td>7B</td>
<td>41.6</td>
<td>-</td>
</tr>
<tr>
<td>RFT</td>
<td>7B</td>
<td>50.3</td>
<td>-</td>
</tr>
<tr>
<td>WizardMath</td>
<td>7B</td>
<td>54.9</td>
<td>10.7</td>
</tr>
<tr>
<td>MetaMath (<b>ours</b>)</td>
<td>7B</td>
<td><b>66.5</b></td>
<td><b>19.8</b></td>
</tr>
<tr class="th">
<td colspan="4" style="text-align: center; font-weight: bold;">
Open-source models (11-50B)</td>
</tr>
<tr>
<td>LLaMA-1</td>
<td>13B</td>
<td>17.8</td>
<td>3.9</td>
</tr>
<tr>
<td>LLaMA-1</td>
<td>33B</td>
<td>35.6</td>
<td>7.1</td>
</tr>
<tr>
<td>LLaMA-2</td>
<td>13B</td>
<td>28.7</td>
<td>3.9</td>
</tr>
<tr>
<td>LLaMA-2</td>
<td>34B</td>
<td>42.2</td>
<td>6.2</td>
</tr>
<tr>
<td>MPT</td>
<td>30B</td>
<td>15.2</td>
<td>3.1</td>
</tr>
<tr>
<td>Falcon</td>
<td>40B</td>
<td>19.6</td>
<td>2.5</td>
</tr>
<tr>
<td>GAL</td>
<td>30B</td>
<td>-</td>
<td>12.7</td>
</tr>
<tr>
<td>Vicuna</td>
<td>13B</td>
<td>27.6</td>
<td>-</td>
</tr>
<tr>
<td>Baichuan-2</td>
<td>13B</td>
<td>52.8</td>
<td>10.1</td>
</tr>
<tr>
<td>SFT</td>
<td>13B</td>
<td>50.0</td>
<td>-</td>
</tr>
<tr>
<td>RFT</td>
<td>13B</td>
<td>54.8</td>
<td>-</td>
</tr>
<tr>
<td>WizardMath</td>
<td>13B</td>
<td>63.9</td>
<td>14.0</td>
</tr>
<tr>
<td>MetaMath (<b>ours</b>)</td>
<td>13B</td>
<td><b>72.3</b></td>
<td><b>22.4</b></td>
</tr>
<tr class="th">
<td colspan="4" style="text-align: center; font-weight: bold;">
Open-source models (50-70B)</td>
</tr>
<tr>
<td>LLaMA-1</td>
<td>65B</td>
<td>50.9</td>
<td>10.6</td>
</tr>
<tr>
<td>LLaMA-2</td>
<td>70B</td>
<td>56.8</td>
<td>13.5</td>
</tr>
<tr>
<td>RFT</td>
<td>70B</td>
<td>64.8</td>
<td>-</td>
</tr>
<tr>
<td>WizardMath</td>
<td>70B</td>
<td>81.6</td>
<td>22.7</td>
</tr>
<tr>
<td>MetaMath (<b>ours</b>) <sup>‡</sup></td>
<td>70B</td>
<td><b>82.3</b></td>
<td><b>26.6</b></td>
</tr>
</div>
</tbody>
</table>
</div>
<h2 class="subtitle">Table 1: Comparison of testing accuracy to existing LLMs on GSM8K and MATH.
<sup>‡</sup>Due to the computing resource limitation, we finetune MetaMath-70B using QLoRA.
</h2>
</div>
</div>
</div>
</div>
</section>
<!--BibTex citation -->
<section class="section" id="BibTeX">
<div class="container is-max-desktop content">
<h2 class="title">BibTeX</h2>
<pre><code>@article{yu2023metamath,
title={MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models},
author={Yu, Longhui and Jiang, Weisen and Shi, Han and Yu, Jincheng and Liu, Zhengying and Zhang, Yu and Kwok, James T and Li, Zhenguo and Weller, Adrian and Liu, Weiyang},
journal={arXiv preprint arXiv:2309.12284},
year={2023}
}</code></pre>
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</section>
<!--End BibTex citation -->
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