--- title: dart-math-hard canonical_url: "https://www.modelscope.cn/datasets/hkust-nlp/dart-math-hard" md_url: "https://www.modelscope.cn/datasets/hkust-nlp/dart-math-hard.md" repository: hkust-nlp/dart-math-hard last_updated: 2025-02-17 license: "Apache License 2.0" storage_size: "313 MB" downloads: 1232 stars: 0 --- # dart-math-hard > dart-math-hard - hkust-nlp 在 ModelScope 开源的数据集。🎯 DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving hkust-nlp/dart-math-hard 是 ModelScope 魔搭社区上的数据集，存储大小 313 MB，采用 Apache License 2.0 许可。 - **Repository**: hkust-nlp/dart-math-hard - **License**: Apache License 2.0 - **Storage size**: 313 MB - **Downloads**: 1232 - **Stars**: 0 - **Last updated**: 2025-02-17 Source: https://www.modelscope.cn/datasets/hkust-nlp/dart-math-hard --- # 🎯 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) trained on [`DART-Math-Hard`](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) [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)! ## Datasets: `DART-Math` `DART-Math` datasets are the **state-of-the-art** and **data-efficient** **open-source** instruction tuning datasets for mathematical reasoning.
Main results averaged on 2 in-domain and 4 challenging out-of-domain mathematical reasoning benchmarks. Number of responses v.s. query descending in difficulty in DART-Math datasets and similar-sized VRT baseline.
Figure 1: Left: Average accuracy on 6 mathematical benchmarks. We compare with models fine-tuned on the best, public instruction tuning datasets for mathematical problem-solving: MetaMath (Yu et al., 2024) with 395K examples, MMIQC (Liu et al., 2024a) with 2.3 million examples, as well as vanilla rejection tuning (VRT) with 590K examples. Both DART-Math (Uniform) and DART-Math (Prop2Diff) use 590K training examples. Right: Number of responses for each query descending by difficulty across 3 synthesis strategies. Queries are from the MATH training split (Hendrycks et al., 2021). VRT is the baseline biased towards easy queries, while Uniform and Prop2Diff are proposed in this work to balance and bias towards difficult queries respectively. Points are slightly shifted and downsampled for clarity.
`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. Performance 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`. ### Comparison between Mathematical Instruction Tuning Datasets Most of previous datasets are **constructed with ChatGPT**, and many of them are **not open-source**, especially for ones of the best performance. | 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 | | :--------------------------------------------------------------------------------- | -----------: | -----------------------------------------------------------------: | ---------------------------------------------: | -----------------------------------------------------------------------------------------------------------: | :---------------------- | :-------------------------------------------------------------------------: | | [WizardMath](https://arxiv.org/abs/2308.09583) | 96k | 32.3 | 80.4 | 23.1 | GPT-4 | ✗ | | [MetaMathQA](https://arxiv.org/abs/2309.12284) | 395k | 29.8 | 76.5 | 19.3 | GPT-3.5 | [✓](https://huggingface.co/datasets/meta-math/MetaMathQA) | | [MMIQC](https://arxiv.org/abs/2401.09003) | **2294k** | 37.4 | 75.4 | _28.5_ | **GPT-4+GPT-3.5+Human** | [**✓**](https://huggingface.co/datasets/Vivacem/MMIQC) | | [Orca-Math](https://arxiv.org/abs/2402.14830) | 200k | -- | -- | -- | GPT-4 | [✓](https://huggingface.co/datasets/microsoft/orca-math-word-problems-200k) | | [Xwin-Math-V1.1](https://arxiv.org/abs/2403.04706) | **1440k** | _45.5_ | **84.9** | 27.6 | **GPT-4** | **✗** | | [KPMath-Plus](https://arxiv.org/abs/2403.02333) | **1576k** | **46.8** | 82.1 | -– | **GPT-4** | **✗** | | [MathScaleQA](https://arxiv.org/abs/2403.02884) | 2021k | 35.2 | 74.8 | 21.8 | GPT-3.5+Human | ✗ | | [`DART-Math-Uniform`](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform) | **591k** | 43.5 | _82.6_ | 26.9 | **DeepSeekMath-7B-RL** | [**✓**](https://huggingface.co/datasets/hkust-nlp/dart-math-uniform) | | [`DART-Math-Hard`](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) | **585k** | _45.5_ | 81.1 | **29.4** | **DeepSeekMath-7B-RL** | [**✓**](https://huggingface.co/datasets/hkust-nlp/dart-math-hard) | MATH 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. ## Dataset Construction: `DARS` - Difficulty-Aware Rejection Sampling Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals **severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries**. Motivated by the observation above, we propose to *Difficulty-Aware Rejection Sampling* (`DARS`), to collect more responses for more difficult queries. Specifically, we introduce two strategies to increase the number of correct responses for difficult queries: 1) **Uniform**, which involves sampling responses for each query until **each query accumulates $k_u$ correct responses**, where $k_u$ is a preset hyperparameter determined by the desired size of the synthetic dataset; 2) **Prop2Diff**, where we continue sampling responses until the number of correct responses for each query is **proportional to its difficulty score**. The most challenging queries will receive $k_p$ responses and kp is a hyperparameter. This method introduces a deliberate bias in the opposite direction to vanilla rejection sampling, towards more difficult queries, inspired by previous works that demonstrate **difficult samples can be more effective to enhance model capabilities** ([Sorscher et al., 2022](https://proceedings.neurips.cc/paper_files/paper/2022/hash/7b75da9b61eda40fa35453ee5d077df6-Abstract-Conference.html); [Liu et al., 2024b](https://openreview.net/forum?id=BTKAeLqLMw)). See [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`. ## 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}, } ```