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--- |
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language: |
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- en |
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license: llama3 |
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library_name: transformers |
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tags: |
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- mathematics |
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datasets: |
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- hkust-nlp/dart-math-hard |
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metrics: |
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- accuracy |
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pipeline_tag: text-generation |
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base_model: meta-llama/Meta-Llama-3-8B |
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model-index: |
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- name: dart-math-llama3-8b-prop2diff |
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results: |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: hendrycks/competition_math |
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name: MATH |
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split: test |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 46.6 |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: openai/gsm8k |
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name: GSM8K |
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config: main |
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split: test |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 81.1 |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: college-math |
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name: CollegeMath |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 28.8 |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: deepmind-mathematics |
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name: DeepMind-Mathematics |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 48.0 |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: Hothan/OlympiadBench |
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name: OlympiadBench-OE_TO_maths_en_COMP |
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config: OE_TO_maths_en_COMP |
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split: train |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 14.5 |
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- task: |
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type: text-generation |
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name: Mathematical Problem-Solving |
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dataset: |
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type: TIGER-Lab/TheoremQA |
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name: TheoremQA |
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split: test |
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metrics: |
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- type: accuracy |
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name: Pass@1 (0-shot CoT) |
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value: 19.4 |
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--- |
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# DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving |
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📝 [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) |
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🐦 [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) |
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## Models: `DART-Math` |
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`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**. |
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| 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 | |
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| :----------------------------------------------------------------------------------------------------- | -----------------------------------------------------------------: | ---------------------------------------------: | -----------------------------------------------------------------------------------------------------------: | -----------------------------------------------------------------------------------------------: | ------------------------------------------------------------------------------------------------------------------: | -----------------------------------------------------------------------------------------: | -------: | |
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| 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) | -- | -- | -- | -- | |
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| Llama-3-70B-MetaMath | 44.9 | 88.0 | 31.9 | 53.2 | 11.6 | 21.9 | 41.9 | |
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| [`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 | |
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| [`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** | |
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| DeepSeekMath-7B-MetaMath | 43.7 | 81.8 | 33.7 | 53.0 | 13.6 | 23.2 | 41.5 | |
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| [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 | |
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| [`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 | |
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| [`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** | |
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| Mistral-7B-MetaMath | 29.8 | 76.5 | 19.3 | 28.0 | 5.9 | 14.0 | 28.9 | |
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| [`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 | |
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| [`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** | |
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| Llama-3-8B-MetaMath | 32.5 | 77.3 | 20.6 | 35.0 | 5.5 | 13.8 | 30.8 | |
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| [`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 | |
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| [`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** | |
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***Abbreviations**: College (CollegeMath), DM (DeepMind Mathematics), Olympiad (OlympiadBench-Math), Theorem (TheoremQA). |
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**Bold** means the best score by SFT on the respective base model here. |
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To reproduce our results, please refer to [the `DART-Math` GitHub repository](https://github.com/hkust-nlp/dart-math).* |
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## Prompt Template |
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All the `DART-Math` models use the [Alpaca](https://github.com/tatsu-lab/stanford_alpaca) prompt template: |
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``` |
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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 |
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``` |
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## Training Dataset |
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We construct our traning datasets by applying **Difficulty-Aware Rejection Sampling** (`DARS`) to the **MATH and GSM8K** training sets. |
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`DARS` tackle **severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries**, in previous datasets. |
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These biases are primarily caused by vanilla rejection sampling, where **the same number of responses is |
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sampled for each query**, yet the likelihood of obtaining correct responses for difficult queries is significantly lower, sometimes even zero. |
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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. |
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## Training Setup |
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We perform standard instruction tuning to several base models including Llama3-8B & Mistral-7B & Llama3-70B as representatives of general models and DeepSeekMath- |
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7B as the representative of math-specialized model |
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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), |
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leading to `DART-Math (Prop2Diff)` & `DART-Math (Uniform)` respectively. |
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For simplicity, we keep most hyper-parameters the same across different models and datasets: |
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- Model max length (of [packed](https://github.com/MeetKai/functionary/tree/main/functionary/train/packing) sequence): 4096 |
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- Batch size: 64 |
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- Warm-up ratio: 0.03 |
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- Learning rate scheduler: cosine |
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- Prompt template: [Alpaca](https://github.com/tatsu-lab/stanford_alpaca) |
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Several other key hyper-parameters are tuned as follow: |
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| Base Model | Max. L.R. | # of Epochs | # of Grad. Acc. Steps | # of A100 GPUs | |
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|:--------------- | ---------:| -----------:| ---------------------:| --------------:| |
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| Mistral-7B | `1e-5` | 3 | 1 | 8 | |
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| Llama3-8B | `5e-5` | 1 | 2 | 8 | |
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| Llama3-70B | `2e-5` | 1 | 1 | 32 | |
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| DeepSeekMath-7B | `5e-5` | 3 | 1 | 8 | |
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- 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). |
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- For **Llama3** models, preliminary experiments indicate that **training for 1 epoch consistently outperforms 3 epochs**. |
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Please refer to [Appendix A.1 of our paper](https://tongyx361.github.io/assets/dart-math/paper-dart-math.pdf) for more details. |
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## Other Details |
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- 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.) |
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## Citation |
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If you find our data, model or code useful for your work, please kindly cite [our paper](https://arxiv.org/abs/2407.13690): |
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```latex |
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@article{tong2024dartmath, |
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title={DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving}, |
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author={Yuxuan Tong and Xiwen Zhang and Rui Wang and Ruidong Wu and Junxian He}, |
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year={2024}, |
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eprint={2407.13690}, |
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archivePrefix={arXiv}, |
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primaryClass={cs.CL}, |
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url={https://arxiv.org/abs/2407.13690}, |
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} |
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``` |
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