--- language: - en license: llama3 library_name: transformers tags: - mathematics datasets: - hkust-nlp/dart-math-hard metrics: - accuracy pipeline_tag: text-generation base_model: meta-llama/Meta-Llama-3-8B model-index: - name: dart-math-llama3-8b-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: 46.6 - 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: 81.1 - task: type: text-generation name: Mathematical Problem-Solving dataset: type: college-math name: CollegeMath metrics: - type: accuracy name: Pass@1 (0-shot CoT) value: 28.8 - 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: 48.0 - 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: 14.5 - 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: 19.4 --- # 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) | 📑 [BibTeX](https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#citation) ## 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}, } ```