hkust-nlp/vrt-baseline

---
dataset_info:
features:
- name: query
dtype: string
- name: response
dtype: string
splits:
- name: train
num_bytes: 475633098
num_examples: 590601
download_size: 104156576
dataset_size: 475633098
configs:
- config_name: default
data_files:
- split: train
path: data/train-*
license: mit
task_categories:
- text-generation
language:
- en
tags:
- synthetic
- mathematics
pretty_name: VRT-Baseline
size_categories:
- 100K<n<1M
---

> [!NOTE]
> This dataset is the **VRT baseline** dataset used to train baseline models `*-VRT` in Table 2 of the paper.
> > Another ablation baseline to DART is vanilla rejection tuning (VRT), where we synthesize a dataset of the same size of 0.59M examples with DeepSeekMath-7B-RL, using vanilla rejection sampling as described in §2.1.

# 🎯 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.

<style>
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<div class="container">
<img src="https://tongyx361.github.io/assets/dart-math/main-results.png" alt="Main results averaged on 2 in-domain and 4 challenging out-of-domain mathematical reasoning benchmarks.">
<img src="https://tongyx361.github.io/assets/dart-math/main-nresp-vs-query.png" alt="Number of responses v.s. query descending in difficulty in DART-Math datasets and similar-sized VRT baseline.">
</div>
<div class="caption">
Figure 1: <strong>Left:</strong> Average accuracy on 6 mathematical benchmarks. We compare with models fine-tuned on the best, public instruction tuning datasets for mathematical problem-solving:
MetaMath <a href="https://openreview.net/forum?id=N8N0hgNDRt">(Yu et al., 2024)</a> with 395K
examples,
MMIQC <a href="https://arxiv.org/abs/2401.09003">(Liu et al., 2024a)</a> with 2.3 million examples,
as well as vanilla rejection tuning (VRT) with 590K examples.
Both <em>DART-Math (Uniform)</em> and <em>DART-Math (Prop2Diff)</em> use 590K training examples.
<strong>Right:</strong> Number of responses for each query descending by difficulty across 3 synthesis strategies.
Queries are from the MATH training split <a href="https://datasets-benchmarks-proceedings.neurips.cc/paper/2021/hash/be83ab3ecd0db773eb2dc1b0a17836a1-Abstract-round2.html">(Hendrycks et al., 2021)</a>.
VRT is the baseline biased towards easy queries, while <em>Uniform</em> and <em>Prop2Diff</em> are proposed in this work to balance and bias towards difficult queries respectively.
Points are slightly shifted and downsampled for clarity.
</div>

`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},
}
```