🦣 MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning

University of Waterloo, The Ohio State University, HKUST, University of Edinburg, §01.AI
*Xiang Yue and Wenhu Chen are the leading authors of the project. They contributed equally to this project.
yue.149@osu.edu , wenhuchen@uwaterloo.ca

Abstract

We introduce 🦣MAmmoTH, a series of open-source large language models (LLMs) specifically tailored for general math problem-solving. The MAmmoTH models are trained on MathInstruct, our meticulously curated instruction tuning dataset. MathInstruct is compiled from 13 math datasets with intermediate rationales, six of which have rationales newly curated by us. It boasts the hybrid of chain-of- thought (CoT) and program-of-thought (PoT) rationales, and also ensures exten- sive coverage of diverse fields in math. The hybrid of CoT and PoT can not only unleash the potential of tool use but also allow different thought processes for dif- ferent math problems. As a result, the MAmmoTH series substantially outperform existing open-source models on nine mathematical reasoning datasets across all scales with an average accuracy gain ranging from 12% to 29%. Remarkably, our MAmmoTH-7B model reaches 35% on MATH (a competition-level dataset), which exceeds the best open-source 7B model (WizardMath) by 25%, and the MAmmoTH-34B model achieves 46% accuracy on MATH, even surpassing GPT- 4’s CoT result. Our work underscores the importance of diverse problem coverage and the use of hybrid rationales in developing superior math generalist models.

The hybrid instruction tuning of 🦣MAmmoTH

Figure 1: The hybrid instruction tuning of 🦣MAmmoTH, a series of models trained to solve a diverse set of mathematical problems using CoTs and PoTs.

Our Dataset: MathInstruct

Training Dataset Type Annotation Samples Characteristics Fields
GSM8K CoT Human 7K Grade Schol Exam Pre-Algebra
GSM8K-RFT CoT Llama 28K Llama + Validated Pre-Algebra
AQuA-RAT CoT Human 90K GRE/GMAT Exam Inter-Algebra
MATH CoT Human 7K Math Competition Pre-Algebra, Inter-Algebra, Algebra, Probability, NumTheory, Calculus, Geometry
TheoremQA CoT GPT-4 600 GPT4 + Validated Algebra, Probability, NumTheory, Calculus, Geometry
Camel-Math CoT GPT-4 50K GPT4 (Unvalidated) Algebra, Probability, NumTheory, Calculus, Geometry
College-Math CoT GPT-4 1.8K GPT4 (Unvalidated) Algebra
GSM8K PoT GPT4 14K GPT4 + Validated Pre-Algebra
AQuA-RAT PoT GPT4 9.7K GPT4 + Validated Inter-Algebra
MATH PoT GPT4 7K GPT4 + Validated Pre-Algebra, Inter-Algebra, Algebra, Probability
TheoremQA PoT GPT4 700 GPT4 + Validated Algebra, Probability, NumTheory, Calculus, Geometry
MathQA PoT Human 25K AQuA-RAT Subset Inter-Algebra
NumG PoT Human 13K Lila Annotated Pre-Algebra

Overall Results

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Figure 2: Overall results of 🦣MAmmoTH on the in-domain and out-of-domain datasets.

Overall, we can see that MAmmoTH and MAmmoTH-Coder are able to outperform the SoTA model at different scales. In general, the performance gain for OOD datasets is more significant than IND datasets. These results show us the potential of our models as a mathematical generalist. On several datasets, MAmmoTH-Coder-34B and MAmmoTH-70B are even surpassing closed-source LLMs (see more break down results below).

Where does the gain come from?

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Figure 3: Investigation of the influence of CoT \& PoT hybrid training on the 7B Llama-2 model. Key insights include: 1) The SoTA model, utilizing dataset-specific CoT fine-tuning on GSM and MATH, displays strong performance within its domains but struggles in OOD scenarios; 2) Diverse data sources in MathInstruct enable better math generalist model; 3) Fine-tuning on the PoT subsets generally outperforms fine-tuning on the CoT subsets; 4) Hybrid training yields the best-performing model.

In order to better understand what factors contribute to the great gain of 🦣MAmmoTH over existing baselines, we set up a group of control experiments in the Figure 3. We study the following setups:

  1. 🦣MAmmoTH (MathInstruct - CoT): This experiment aims to understand how much our curated CoT data could improve the generalization over the SoTA model WizardMath trained specifically on GSM + MATH. As can be seen, while sacrificing accuracy on GSM + MATH by 3%, our CoT subset fine-tuning improves the overall nine-dataset accuracy from 27% to 32%.
  2. 🦣MAmmoTH (MathInstruct - PoT): This experiment aims to understand the advantage of our PoT subset. As can be observed, our PoT subset fine-tuning can significantly improve the overall accuracy from 27% to 37.5%. This ablation reflects the importance of unlocking the program generation capabilities of our model.
  3. 🦣MAmmoTH (MathInstruct - Hybrid): We further combine CoT and PoT as the hybrid training data to achieve the best overall performance of 45.4%. This combined gain comes from two aspects:
    • The CoT subset can help maintain the generic language-based reasoning skills to handle scenarios where PoT cannot handle well, e.g., the multi-choice questions in AQuA, SAT, and MMLU.
    • The PoT subset can teach the model how to utilize Python APIs to solve complex math problems with high precision, e.g., the MATH problems requiring complex computation.

Comprehensive Results: Break Down

Model GSM8K MATH AQuA NumG SVAMP Mathematics SimulEq SAT MMLU
Closed-source Model
GPT-4 - Unknown 92 42.5 72.6 - - 97 - - 95 - -
Code-Interpreter - Unknown 97 69.7 - - - - - - - - -
PaLM-2 - Unknown 80.7 34.3 64.1 - - - - - - - -
Claude-2 - Unknown 85.2 32.5 60.9 - - - - - - - -
Codex (PoT) - No 71.6 36.8 54.1 - - 85.2 - - 68 - -
7B Parameter Model
Llama-1 - No 10.7 2.9 22.6 24.7 15.5 24.5 6.2 4.6 22.7 30.6 17.7
Llama-2 - No 14.6 2.5 30.3 29.9 19.3 34.5 6 5 26.8 29.8 20.4
Galactica-6.7B GAL GAL-Instruct 10.2 2.2 25.6 25.8 15.9 25.6 4.6 4.2 17.5 28 16
Code-Llama (PoT) - No 25.2 14.2 24 26.8 22.3 49.4 21.7 3.5 28.6 26.9 26
AQuA-SFT Llama-2 AQuA 11.2 3.6 35.6 12.2 15.6 - - - - - -
Llama-1 RFT Llama-1 GSM8K 46.5 5.2 18.8 21.1 22.9 21.1 5.1 11 12.5 21.7 14.3
WizardMath Llama-2 GSM8K+MATH 54.9 10.7 26.3 36.1 32 36.1 9.3 12.8 25.4 31.1 28.6
MAmmoTH Llama-2 MathInstruct 53.6 31.5 44.5 61.2 61.2 67.7 46.3 41.2 42.7 42.6 48.1
MAmmoTHc Code-Llama MathInstruct 59.4 33.4 47.2 66.4 51.6 71.4 55.4 45.9 40.5 48.3 52.3
$\Delta$ +5 +21 +12 +30 +20 +22 +34 +33 +14 +17 +24
13-15B Parameter Model
Llama-1 - No 17.8 3.9 26 24.8 18.1 34.7 6.9 5.4 27.7 30.7 21
Llama-2 - No 28.7 3.9 25.1 8.8 16.6 35.1 11.5 5.8 32.7 34.4 23.9
Code-Llama (PoT) - No 36.1 18.1 28.7 29.2 28 60 21.3 3.8 25.9 27.7 27.7
CodeT5+ (PoT) - No 12.5 2.4 20.5 19.4 13.7 - - - - - -
CodeGen+ (PoT) - No 12.7 3.4 24.5 22.5 15.7 - - - - - -
Vicuna-1.5 Llama-2 No 28.4 5.8 24.8 36.9 23.9 55.7 10 6.6 34 34.1 28.1
Llama-1 RFT Llama-1 GSM8K 52.1 5.1 16.1 24.5 24.4 46.5 6.7 10.1 13.2 21.6 19.6
Orca-Platypus Llama-2 Platypus 38.4 3 18.9 35.3 23.9 56.8 12.6 7.9 29.5 41.6 29.7
Platypus Llama-2 Platypus 25.7 2.5 33.4 42.3 25.9 55.4 11.4 7.4 36.8 35.5 29.3
WizardMath Llama-2 GSM8K+MATH 63.9 14 21.2 40.8 34.9 51.9 14.1 14.9 24.5 32.1 27.5
MAmmoTH Llama-2 MathInstruct 62.0 34.2 51.6 68.7 54.1 72.4 49.2 43.2 46.8 47.6 51.8
MAmmoTHc Code-Llama MathInstruct 64.7 36.3 46.9 66.8 53.7 73.7 61.5 47.1 48.6 48.3 55.8
$\Delta$ +1 +20 +18 +26 +19 +14 +40 +33 +12 +7 +26
30-34B Parameter Model
Llama-1 - No 35.6 7.1 33.4 28.4 26.1 48.8 12.8 11.2 33.4 39 29
Code-Llama (PoT) - No 44 25 25.2 29.3 30.8 69.1 34.5 6.8 26.8 21.6 31.7
Llama-1 RFT Llama-1 GSM8K 56.5 7.4 18.5 24.3 26.6 55.4 7.6 12.8 20.4 37.9 26.8
Galactica-30B GAL GAL-Instruct 41.7 12.7 28.7 34.7 29.4 41.6 11.8 13.2 37.7 37.9 28.4
Platypus Llama-1 Platypus 37.8 10.1 27.9 40.5 29.1 51.7 13.8 13.6 38.6 41 31.7
Tulu Llama-2 Tulu 51 10.8 25.5 43.4 32.6 59 10.7 10.3 31.3 39.8 30.2
MAmmoTHc Code-Llama MathInstruct 72.7 43.6 54.7 71.6 60.7 84.3 65.4 51.8 60.9 53.8 63.2
$\Delta$ +16 +21 +21 +28 +28 +15 +31 +38 +22 +13 +32
65-70B Parameter Model
Llama-1 - No 50.9 10.6 35 50.2 36.6 55.3 14.2 15.2 37.4 44.1 33.2
Llama-2 - No 56.8 13.5 40.9 50.4 40.4 63.8 20.5 14 51.3 47.1 39.3
Llama-2-Chat Llama-2 No 54.9 18.6 37 51.6 40.5 71.5 19.2 21.7 44.1 46.9 40.6
Guanaco Llama-2 No 59.2 4.1 45.2 53.5 40.5 66.8 17.8 20.2 50 47.3 40.4
WizardMath Llama-2 GSM8K+MATH 81.6 22.7 20 48.9 43.3 71.8 17.1 37.9 13.2 27.4 33.4
Platypus Llama-2 Platypus 70.6 18.6 51.2 55.4 48.9 51.8 26.3 21.7 55.9 52.5 41.6
MAmmoTH Llama-2 MathInstruct 76.9 41.8 65.0 74.4 64.5 82.4 55.6 51.4 66.4 56.7 62.5
$\Delta$ -5 +19 +14 +19 +16 +11 +29 +14 +11 +4 +21

Reference

Please kindly cite our paper if you use our code, data, models or results:

@article{yue2023mammoth,
  title={MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning},
  author={Xiang Yue, Xingwei Qu, Ge Zhang, Yao Fu, Wenhao Huang, Huan Sun, Yu Su, Wenhu Chen},
  journal={arXiv preprint arXiv:2309.05653},
  year={2023}
}