{"url":"/method/dart","slug":"dart","name":"DART","full_name":null,"full_name_withheld":false,"description_markdown":"# šŸŽÆ DART-Math\r\n\r\n> Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving\r\n\r\nšŸ“ [Paper@arXiv](https://arxiv.org/abs/2407.13690)Ā |Ā šŸ¤— [Datasets&Models@HF](https://huggingface.co/collections/hkust-nlp/dart-math-665704599b35de59f8fdf6c1) | šŸ± [Code@GitHub](https://github.com/hkust-nlp/dart-math)\r\n\r\nšŸ¦ [Thread@X(Twitter)](https://x.com/tongyx361/status/1811413243350454455) | šŸ¶ [äø­ę–‡åšå®¢@ēŸ„乎](https://zhuanlan.zhihu.com/p/708371895) | šŸ“Š [Leaderboard@PapersWithCode](https://paperswithcode.com/paper/dart-math-difficulty-aware-rejection-tuning#results) | šŸ“‘ [BibTeX](https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#citation)\r\n\r\n## Datasets: `DART-Math`\r\n\r\n`DART-Math` datasets are the **state-of-the-art** and **data-efficient** **open-source** instruction tuning datasets for mathematical reasoning.\r\n\r\n`DART-Math-Hard` contains \\~585k mathematical QA pair samples constructed by applying `DARS-Prop2Diff` to the query set from MATH and GSK8K training sets, achieves **SOTA** on many challenging mathematical reasoning benchmarks. It introduces a **deliberate bias towards hard queries**, opposite to vanilla rejection sampling.\r\n\r\nPerformance produced by `DART-Math-Hard` is usually but not necessarily **slightly better (\\~1% absolutely)** than `DART-Math-Uniform`, which contains \\~591k samples constructed by applying `DARS-Uniform`.\r\n\r\n### Comparison between Mathematical Instruction Tuning Datasets\r\n\r\nMost of previous datasets are **constructed with ChatGPT**, and many of them are **not open-source**, especially for ones of the best performance.\r\n\r\n| Math SFT Dataset | # of Samples | [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) | Synthesis Agent(s) | Open-Source |\r\n| :--------------------------------------------------------------------------------- | -----------: | -----------------------------------------------------------------: | ---------------------------------------------: | -----------------------------------------------------------------------------------------------------------: | :---------------------- | :-------------------------------------------------------------------------: |\r\n| [WizardMath](https://arxiv.org/abs/2308.09583) | 96k | 32.3 | 80.4 | 23.1 | GPT-4 | āœ— |\r\n| [MetaMathQA](https://arxiv.org/abs/2309.12284) | 395k | 29.8 | 76.5 | 19.3 | GPT-3.5 | [āœ“](https://huggingface.co/datasets/meta-math/MetaMathQA) |\r\n| [MMIQC](https://arxiv.org/abs/2401.09003) | **2294k** | 37.4 | 75.4 | _28.5_ | **GPT-4+GPT-3.5+Human** | [**āœ“**](https://huggingface.co/datasets/Vivacem/MMIQC) |\r\n| [Orca-Math](https://arxiv.org/abs/2402.14830) | 200k | -- | -- | -- | GPT-4 | [āœ“](https://huggingface.co/datasets/microsoft/orca-math-word-problems-200k) |\r\n| [Xwin-Math-V1.1](https://arxiv.org/abs/2403.04706) | **1440k** | _45.5_ | **84.9** | 27.6 | **GPT-4** | **āœ—** |\r\n| [KPMath-Plus](https://arxiv.org/abs/2403.02333) | **1576k** | **46.8** | 82.1 | -ā€“ | **GPT-4** | **āœ—** |\r\n| [MathScaleQA](https://arxiv.org/abs/2403.02884) | 2021k | 35.2 | 74.8 | 21.8 | GPT-3.5+Human | āœ— |\r\n| [`DART-Math-Uniform`](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform) | **591k** | 43.5 | _82.6_ | 26.9 | **DeepSeekMath-7B-RL** | [**āœ“**](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform) |\r\n| [`DART-Math-Hard`](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) | **585k** | _45.5_ | 81.1 | **29.4** | **DeepSeekMath-7B-RL** | [**āœ“**](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) |\r\n\r\nMATH and GSM8K are **in-domain**, while College(Math) is **out-of-domain**. Performance here are of models fine-tuned from [Mistral-7B](https://huggingface.co/mistralai/Mistral-7B-v0.1), except for Xwin-Math-V1.1 based on [Llama2-7B](https://huggingface.co/meta-llama/Llama-2-7b-hf). **Bold**/_Italic_ means the best/second best score here.\r\n\r\n## Dataset Construction: `DARS` - Difficulty-Aware Rejection Sampling\r\n\r\nPrevious works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results.\r\nHowever, our analysis of these datasets reveals **severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries**.\r\n\r\nMotivated by the observation above, we propose to *Difficulty-Aware Rejection Sampling* (`DARS`), to collect more responses for more difficult queries.\r\nSpecifically, we introduce two strategies to increase the number of correct responses for difficult queries:\r\n\r\n1) **Uniform**, which involves sampling responses for each query until **each query accumulates $k_u$ correct\r\nresponses**, where $k_u$ is a preset hyperparameter determined by the desired size of the synthetic dataset;\r\n2) **Prop2Diff**, where we continue sampling responses until the number of correct responses for each\r\nquery is **proportional to its difficulty score**. The most challenging queries will receive $k_p$ responses\r\nand kp is a hyperparameter. This method introduces a deliberate bias in the opposite direction to\r\nvanilla rejection sampling, towards more difficult queries, inspired by previous works\r\nthat demonstrate **difficult samples can be more effective to enhance model capabilities** ([Sorscher et al.,\r\n2022](https://proceedings.neurips.cc/paper_files/paper/2022/hash/7b75da9b61eda40fa35453ee5d077df6-Abstract-Conference.html); [Liu et al., 2024b](https://openreview.net/forum?id=BTKAeLqLMw)).\r\n\r\nSee [Figure 1 (Right)](https://tongyx361.github.io/assets/dart-math/main-nresp-vs-query.png) for examples of `DART-Math-Uniform` by `DARS-Uniform` and `DART-Math-Hard` by `DARS-Prop2Diff`.\r\n\r\n## Citation\r\n\r\nIf you find our data, model or code useful for your work, please kindly cite [our paper](https://arxiv.org/abs/2407.13690):\r\n\r\n```latex\r\n@article{tong2024dartmath,\r\n title={DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving},\r\n author={Yuxuan Tong and Xiwen Zhang and Rui Wang and Ruidong Wu and Junxian He},\r\n year={2024},\r\n eprint={2407.13690},\r\n archivePrefix={arXiv},\r\n primaryClass={cs.CL},\r\n url={https://arxiv.org/abs/2407.13690},\r\n}\r\n```","description_state":"present","introduced_year":null,"introduced_by":{"title":null,"paper":null,"first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":null},"source":{"url":null,"title":null,"url_on_a_paper_host":false},"code_snippet_url":"https://github.com/hkust-nlp/dart-math?tab=readme-ov-file#-data-synthesis","code_snippet_url_on_a_code_host":true,"categories":[{"area":"Natural Language Processing","area_id":"natural-language-processing","collection":"Text Data Augmentation","url":"/methods/category/text-data-augmentation","pwc_aliases":[]}],"n_papers_tagged":32,"archive_num_papers":32,"papers_newest_first":[{"paper":null,"title":"DART: Distilling Autoregressive Reasoning to Silent 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