---
language:
- en
license: other
license_name: deepseek
license_link: https://github.com/deepseek-ai/DeepSeek-Math/blob/main/LICENSE-MODEL
library_name: transformers
tags:
- mathematics
datasets:
- hkust-nlp/dart-math-hard
metrics:
- accuracy
pipeline_tag: text-generation
base_model: deepseek-ai/deepseek-math-7b-base
model-index:
  - name: dart-math-dsmath-7b-prop2diff
    results:
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: hendrycks/competition_math
          name: MATH
          split: test
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 53.6
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: openai/gsm8k
          name: GSM8K
          config: main
          split: test
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 86.8
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: college-math
          name: CollegeMath
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 40.7
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: deepmind-mathematics
          name: DeepMind-Mathematics
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 61.6
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: Hothan/OlympiadBench
          name: OlympiadBench-OE_TO_maths_en_COMP
          config: OE_TO_maths_en_COMP
          split: train
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 21.7
      - task:
          type: text-generation
          name: Mathematical Problem-Solving
        dataset:
          type: TIGER-Lab/TheoremQA
          name: TheoremQA
          split: test
        metrics:
          - type: accuracy
            name: Pass@1 (0-shot CoT)
            value: 32.2
---

# DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving

📝 [Paper@arXiv](https://arxiv.org/abs/2407.13690) | 🤗 [Datasets&Models@HF](https://huggingface.co/collections/hkust-nlp/dart-math-665704599b35de59f8fdf6c1) | 🐱 [Code@GitHub](https://github.com/hkust-nlp/dart-math)

🐦 [Thread@X(Twitter)](https://x.com/tongyx361/status/1811413243350454455) | 🐶 [中文博客@知乎](https://zhuanlan.zhihu.com/p/708371895) | 📊 [Leaderboard@PapersWithCode](https://paperswithcode.com/paper/dart-math-difficulty-aware-rejection-tuning#results) | 📑 [BibTeX](https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#citation)

> [!IMPORTANT]
> 🔥 Excited to find **[our `DART-Math-DSMath-7B` (Prop2Diff)](https://huggingface.co/hkust-nlp/dart-math-dsmath-7b-prop2diff) [comparable](https://github.com/project-numina/aimo-progress-prize/blob/main/report/numina_dataset.pdf) to the AIMO winner [NuminaMath-7B](https://huggingface.co/AI-MO/NuminaMath-7B-CoT)** on CoT,
> but based solely on [MATH](https://huggingface.co/datasets/hkust-nlp/dart-math-pool-math-query-info) & [GSM8K](https://huggingface.co/datasets/hkust-nlp/dart-math-pool-gsm8k-query-info) prompt set, leaving much room to improve!
> Besides, our [`DART` method](https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#dars--difficulty-aware-rejection-sampling) is also fully compatible with [tool-integrated reasoning](https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#tool-integrated-reasoning-reasoning-in-natural-language-interleaved-with-python-code).
> Find more details and join the discussion under this [X thread](https://x.com/tongyx361/status/1815112376649134172)!

## Models: `DART-Math`

`DART-Math` models achieve performance **superior or competitive to previous SOTAs** on 2 in-domain and 4 challenging out-of-domain mathematical reasoning benchmarks, despite using **much smaller datasets** and **no proprietary model like GPT-4**.

| Model                                                                                                  | [MATH](https://huggingface.co/datasets/hendrycks/competition_math) | [GSM8K](https://huggingface.co/datasets/gsm8k) | [College](https://github.com/hkust-nlp/dart-math/tree/main/data/eval-dsets/mwpbench/college-math-test.jsonl) | [DM](https://github.com/hkust-nlp/dart-math/tree/main/data/eval-dsets/deepmind-mathematics.json) | [Olympiad](https://github.com/hkust-nlp/dart-math/tree/main/data/eval-dsets/olympiadbench/OE_TO_maths_en_COMP.json) | [Theorem](https://github.com/hkust-nlp/dart-math/tree/main/data/eval-dsets/theoremqa.json) |      AVG |
| :----------------------------------------------------------------------------------------------------- | -----------------------------------------------------------------: | ---------------------------------------------: | -----------------------------------------------------------------------------------------------------------: | -----------------------------------------------------------------------------------------------: | ------------------------------------------------------------------------------------------------------------------: | -----------------------------------------------------------------------------------------: | -------: |
| GPT-4 (0314)                                                                                           |                           [52.6](https://arxiv.org/abs/2403.04706) |       [94.7](https://arxiv.org/abs/2403.04706) |                                                                     [24.4](https://arxiv.org/abs/2403.02884) |                                                                                               -- |                                                                                                                  -- |                                                                                         -- |       -- |
| Llama-3-70B-MetaMath                                                                                   |                                                               44.9 |                                           88.0 |                                                                                                         31.9 |                                                                                             53.2 |                                                                                                                11.6 |                                                                                       21.9 |     41.9 |
| [`DART-Math-Llama-3-70B` (Uniform)](https://huggingface.co/hkust-nlp/dart-math-llama3-70b-uniform)     |                                                               54.9 |                                       **90.4** |                                                                                                     **38.5** |                                                                                         **64.1** |                                                                                                                19.1 |                                                                                       27.4 |     49.1 |
| [`DART-Math-Llama-3-70B` (Prop2Diff)](https://huggingface.co/hkust-nlp/dart-math-llama3-70b-prop2diff) |                                                           **56.1** |                                           89.6 |                                                                                                         37.9 |                                                                                         **64.1** |                                                                                                            **20.0** |                                                                                   **28.2** | **49.3** |
| DeepSeekMath-7B-MetaMath                                                                               |                                                               43.7 |                                           81.8 |                                                                                                         33.7 |                                                                                             53.0 |                                                                                                                13.6 |                                                                                       23.2 |     41.5 |
| [DeepSeekMath-7B-RL](https://huggingface.co/deepseek-ai/deepseek-math-7b-rl)                           |                                                               53.1 |                                           88.4 |                                                                                                         41.3 |                                                                                             58.3 |                                                                                                                18.7 |                                                                                       35.9 |     49.3 |
| [`DART-Math-DSMath-7B` (Uniform)](https://huggingface.co/hkust-nlp/dart-math-dsmath-7b-uniform)        |                                                               52.9 |                                       **88.2** |                                                                                                         40.1 |                                                                                             60.2 |                                                                                                                21.3 |                                                                                   **32.5** |     49.2 |
| [`DART-Math-DSMath-7B` (Prop2Diff)](https://huggingface.co/hkust-nlp/dart-math-dsmath-7b-prop2diff)    |                                                           **53.6** |                                           86.8 |                                                                                                     **40.7** |                                                                                         **61.6** |                                                                                                            **21.7** |                                                                                       32.2 | **49.4** |
| Mistral-7B-MetaMath                                                                                    |                                                               29.8 |                                           76.5 |                                                                                                         19.3 |                                                                                             28.0 |                                                                                                                 5.9 |                                                                                       14.0 |     28.9 |
| [`DART-Math-Mistral-7B` (Uniform)](https://huggingface.co/hkust-nlp/dart-math-mistral-7b-uniform)      |                                                               43.5 |                                       **82.6** |                                                                                                         26.9 |                                                                                             42.0 |                                                                                                                13.2 |                                                                                       16.4 |     27.4 |
| [`DART-Math-Mistral-7B` (Prop2Diff)](https://huggingface.co/hkust-nlp/dart-math-mistral-7b-prop2diff)  |                                                           **45.5** |                                           81.1 |                                                                                                     **29.4** |                                                                                         **45.1** |                                                                                                            **14.7** |                                                                                   **17.0** | **38.8** |
| Llama-3-8B-MetaMath                                                                                    |                                                               32.5 |                                           77.3 |                                                                                                         20.6 |                                                                                             35.0 |                                                                                                                 5.5 |                                                                                       13.8 |     30.8 |
| [`DART-Math-Llama-3-8B` (Uniform)](https://huggingface.co/hkust-nlp/dart-math-llama3-8b-uniform)       |                                                               45.3 |                                       **82.5** |                                                                                                         27.1 |                                                                                         **48.2** |                                                                                                                13.6 |                                                                                       15.4 |     38.7 |
| [`DART-Math-Llama-3-8B` (Prop2Diff)](https://huggingface.co/hkust-nlp/dart-math-llama3-8b-prop2diff)   |                                                           **46.6** |                                           81.1 |                                                                                                     **28.8** |                                                                                             48.0 |                                                                                                            **14.5** |                                                                                   **19.4** | **39.7** |

***Abbreviations**: College (CollegeMath), DM (DeepMind Mathematics), Olympiad (OlympiadBench-Math), Theorem (TheoremQA).
**Bold** means the best score by SFT on the respective base model here.
To reproduce our results, please refer to [the `DART-Math` GitHub repository](https://github.com/hkust-nlp/dart-math).*

## Prompt Template

All the `DART-Math` models use the [Alpaca](https://github.com/tatsu-lab/stanford_alpaca) prompt template:

```

Below is an instruction that describes a task. Write a response that appropriately completes the request.\n\n###Instruction:\n{query}\n\n### Response:\n

```

## Training Dataset

We construct our traning datasets by applying **Difficulty-Aware Rejection Sampling** (`DARS`) to the **MATH and GSM8K** training sets.

`DARS` tackle **severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries**, in previous datasets.

These biases are primarily caused by vanilla rejection sampling, where **the same number of responses is
sampled for each query**, yet the likelihood of obtaining correct responses for difficult queries is significantly lower, sometimes even zero.

Please refer to [`DART-Math-Hard`](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) / [`DART-Math-Uniform`](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform) for more details.

## Training Setup

We perform standard instruction tuning to several base models including Llama3-8B & Mistral-7B & Llama3-70B as representatives of general models and DeepSeekMath-
7B as the representative of math-specialized model
on our synthetic datasets [`DART-Math-Hard`](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) & [`DART-Math-Uniform`](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform),
leading to `DART-Math (Prop2Diff)` & `DART-Math (Uniform)` respectively.

For simplicity, we keep most hyper-parameters the same across different models and datasets:

- Model max length (of [packed](https://github.com/MeetKai/functionary/tree/main/functionary/train/packing) sequence): 4096
- Batch size: 64
- Warm-up ratio: 0.03
- Learning rate scheduler: cosine
- Prompt template: [Alpaca](https://github.com/tatsu-lab/stanford_alpaca)

Several other key hyper-parameters are tuned as follow:

| Base Model      | Max. L.R. | # of Epochs | # of Grad. Acc. Steps | # of A100 GPUs |
|:--------------- | ---------:| -----------:| ---------------------:| --------------:|
| Mistral-7B      |    `1e-5` |           3 |                     1 |              8 |
| Llama3-8B       |    `5e-5` |           1 |                     2 |              8 |
| Llama3-70B      |    `2e-5` |           1 |                     1 |             32 |
| DeepSeekMath-7B |    `5e-5` |           3 |                     1 |              8 |

- For **maximum learning rate**, we determine the values by **searching** through `1e-6,5e-6,1e-5,2e-5,5e-5,1e-4` according to the MATH performance after training on MMIQC for 1 epoch, except for Llama3-70B that is so expensive to search for that we derive from Llama3-8B’s learning rate in analogy to the relationship of (per-training) learning rates between [Llama2-7B](https://huggingface.co/meta-llama/Llama-2-7b-hf) and [Llama2-70B](https://huggingface.co/meta-llama/Llama-2-70b-hf) (\~2:1).
- For **Llama3** models, preliminary experiments indicate that **training for 1 epoch consistently outperforms 3 epochs**.

Please refer to [Appendix A.1 of our paper](https://tongyx361.github.io/assets/dart-math/paper-dart-math.pdf) for more details.

## Other Details

- For Mistral-7B-based models, we disable `sliding_window` by default following [the newest Mistral-7B-Instruct](https://huggingface.co/mistralai/Mistral-7B-Instruct-v0.3/blob/main/config.json) (Flash Attention 2 does not support `sliding_window` and XFormer backend in vLLM has throughput \~10% lower in our experiments.)

## Citation

If you find our data, model or code useful for your work, please kindly cite [our paper](https://arxiv.org/abs/2407.13690):

```latex
@article{tong2024dartmath,
  title={DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving},
  author={Yuxuan Tong and Xiwen Zhang and Rui Wang and Ruidong Wu and Junxian He},
  year={2024},
  eprint={2407.13690},
  archivePrefix={arXiv},
  primaryClass={cs.CL},
  url={https://arxiv.org/abs/2407.13690},
}
```
