MathBench: Evaluating the Theory and Application Proficiency of LLMs with a Hierarchical Mathematics Benchmark

[📄Paper] [📚LeaderBoard]

☀️Introduction

MathBench is an All in One math dataset for language model evaluation, with:

• A Sophisticated Five-Stage Difficulty Mechanism: Unlike the usual mathematical datasets that can only evaluate a single difficulty level or have a mix of unclear difficulty levels, MathBench provides 3709 questions with a gradient difficulty division by education stages, ranging from basic arithmetic to primary, middle, high school, and college levels, allowing you to get a clear overview of the comprehensive difficulty evaluation results.
• Bilingual Gradient Evaluation: Apart from the basic calculation part which is language-independent, MathBench provides questions in both Chinese and English for the four-stage difficulty datasets from primary to college.
• Implementation of the Robust Circular Evaluation (CE) Method: MathBench use CE as the main evaluation method for questions. Compared to the general Accuracy evaluation method, CE requires the model to answer the same multiple-choice question multiple times, with the order of the options changing each time. The model is considered correct on this question only if all answers are correct. The results of CE can reflect the model's capabilities more realistically, providing more valuable evaluation results.
• Support for Basic Theory Questions: For every stage, MathBench provides questions that cover the basic theory knowledge points of the corresponding stage, to ascertain whether the model has genuinely mastered the fundamental concepts of each stage or merely memorized the answers.

🚀 What's New

• [2024.5.20] MathBench has been accepted by ACL2024 (findings), We also released performance of more models such as GPT-4o, Qwen-Max-0428, Llama3 and DeepSeek-V2-API on MathBench. 🎉🎉🎉
• [2024.3.14] We release the complete version of MathBench, which includes a total of 3,709 problems in both Chinese and English. The dataset covers both applied and theoretical math problems. Each problem is labeled with a three-level taxonomy. 🎉🎉🎉
• [2024.1.26] We release the Application questions of MathBench. 🎉🎉🎉

🌲Dataset Structure

📒Model Performance

We use zero-shot CoT set for multiple-choice questions and few-shot (8) CoT set for all textual questions. The results are shown in the following table, we present the results with common Accuracy and Circular Evaluation (CE) metrics.

Here is the CE result of MathBench.

MathBench-A demonstrates the performance of the model on application problems

Models	Arith	Primary	Middle	High	College	Avg.
Closed-source Models
GPT-3.5-Turbo-0125	72.7	72.3	27.3	18.3	14.3	41.0
GLM4	61.7	80.0	55.7	38.7	20.7	51.3
GPT-4-0125-Preview	76.0	82.3	59.0	41.3	35.3	58.8
Qwen-Max-0428	72.3	86.3	65.0	45.0	27.3	59.2
DeepSeek-V2-API	82.7	89.3	59.0	39.3	29.3	59.9
Claude-3-Opus	85.7	85.0	58.0	42.7	43.7	63.0
GPT-4o-2024-05-13	77.7	87.7	76.3	59.0	54.0	70.9
Open-source Chat Models
~7B
Yi-6B-Chat	35.3	36.3	7.0	3.0	4.3	17.2
ChatGLM3-6B	38.0	41.0	13.7	5.3	1.7	19.9
DeepSeek-7B-Chat	48.3	47.7	8.7	4.3	2.7	22.3
Qwen-7B-Chat	50.7	50.7	22.0	9.3	6.0	27.7
InternLM2-Chat-7B	52.0	66.3	30.0	13.7	8.7	34.1
Llama-3-8B-Instruct	54.7	71.0	25.0	19.0	14.0	36.7
GLM-4-9B-Chat	55.0	69.0	54.0	40.3	20.0	47.7
Yi-1.5-9B-Chat	72.7	80.7	45.3	34.3	22.7	51.1
Qwen2-7B-Instruct	69.7	80.7	53.3	38.3	25.0	53.4
10~34B
Baichuan2-13B-Chat	40.0	44.7	13.7	4.7	1.7	20.9
Yi-34B-Chat	50.7	62.0	23.0	14.7	7.7	31.6
Qwen-14B-Chat	63.7	61.7	39.0	21.0	12.0	39.5
InternLM2-Chat-20B	62.3	72.7	37.7	24.7	13.0	42.1
Yi-1.5-34B-Chat	69.7	82.3	50.0	35.3	23.7	52.2
~70B
DeepSeek-67B-Chat	62.0	72.7	33.3	21.3	12.0	40.3
Qwen-72B-Chat	72.0	71.7	53.7	32.0	19.0	49.7
Llama-3-70B-Instruct	70.3	86.0	53.0	38.7	34.0	56.4
Qwen1.5-110B-Chat	70.3	82.3	64.0	47.3	28.0	58.4
Qwen2-72B-Instruct	76.3	89.0	71.7	51.7	46.3	67.0
Mathematical Models
MammoTH-7B	27.0	24.3	2.7	1.7	0.7	11.3
MammoTH-13B	35.0	43.0	5.0	4.7	5.0	18.5
MammoTH-70B	35.7	60.0	11.0	10.7	6.0	24.7
Metamath-Llemma-7B	51.7	51.0	8.3	8.3	5.0	24.9
InternLM2-Chat-Math-7B	53.7	67.0	41.3	18.3	8.0	37.7
DeepSeek-Math-7B-Instruct	61.0	74.0	30.3	24.7	14.3	40.9
InternLM2-Chat-Math-20B	58.7	70.0	43.7	24.7	12.7	41.9
DeepSeek-Math-7B-RL	68.0	83.3	44.3	33.0	23.0	50.3

MathBench-T demonstrates the performance of the model on theoretical problems

Models	Primary	Middle	High	College	Avg.
Closed-source Models
GPT-3.5-Turbo-0125	70.1	56.7	47.3	52.5	56.7
GLM4	88.6	79.5	63.7	60.6	73.1
GPT-4-0125-Preview	87.2	81.0	72.0	73.3	78.4
Claude-3-Opus	86.0	79.0	72.6	77.4	78.7
DeepSeek-V2-API	88.9	83.7	70.3	76.3	79.8
Qwen-Max-0428	90.4	83.2	73.4	74.8	80.4
GPT-4o-2024-05-13	92.2	88.3	82.0	85.6	87.0
Open-source Chat Models
~7B
DeepSeek-7B-Chat	33.3	26.0	14.4	13.6	21.8
ChatGLM3-6B	41.6	32.4	20.2	12.0	26.6
Yi-6B-Chat	48.0	33.5	21.8	23.9	31.8
Qwen-7B-Chat	53.1	43.5	32.9	31.2	40.2
GLM-4-9B-Chat	85.0	78.0	65.8	71.4	75.0
Llama-3-8B-Instruct	60.2	51.3	43.5	53.6	52.1
InternLM2-Chat-7B	67.3	55.8	45.4	42.7	52.8
Yi-1.5-9B-Chat	81.6	74.0	62.7	69.8	72.0
Qwen2-7B-Instruct	89.4	82.9	71.2	70.1	78.4
10~34B
Baichuan2-13B-Chat	45.4	36.9	24.1	21.0	31.9
InternLM2-Chat-20B	64.5	56.2	49.9	43.2	53.4
Yi-34B-Chat	70.9	57.0	43.6	46.8	54.6
Qwen-14B-Chat	71.6	64.0	49.7	49.4	58.7
Yi-1.5-34B-Chat	86.3	80.8	69.3	73.2	77.4
~70B
DeepSeek-67B-Chat	78.1	65.7	55.6	64.6	66.0
Llama-3-70B-Instruct	71.4	64.3	62.1	71.2	67.2
Qwen-72B-Chat	90.9	80.9	67.1	69.8	77.2
Qwen-1.5-110B-Chat	93.4	85.0	76.5	81.5	84.1
Qwen2-72B-Instruct	93.4	89.9	84.4	87.7	88.8
Mathematical Models
MammoTH-7B	11.6	9.1	8.4	6.3	8.8
MammoTH-13B	27.5	18.6	15.0	17.1	19.5
MetaMath-Llemma-7B	36.6	33.5	28.8	25.9	31.2
MammoTH-70B	58.1	47.1	39.3	44.6	47.3
InternLM2-Chat-Math-7B	65.6	60.2	51.7	46.5	56.0
DeepSeek-Math-7B-Instruct	73.3	58.4	49.3	50.3	57.8
InternLM2-Chat-Math-20B	73.2	70.5	60.6	53.0	64.3
DeepSeek-Math-7B-RL	79.6	72.0	61.3	68.7	70.4

🔊Average Application Scores with Stages

Models exhibit similar performances in Arithmetic and Primary stages, while demonstrating a clear performance decline from Primary to College stages.

Bilingual Performance

📊Model Size vs. Average Score

The comparison chart of model parameter size versus performance on MathBench for selected representative models, with models from the same series connected by lines of the same color. The horizontal red dotted line represents the score of GPT-4-0125-Preview.

📈Bilingual Performance

Below is a comparison of the bilingual results of numerous Chat models on MathBench, sorted in ascending order of language average scores.

🖋Inference MathBench with OpenCompass

OpenCompass is a toolkit for evaluating the performance of large language models (LLMs). There are steps for inference MathBench with OpenCompass:

1. Install OpenCompass

conda create --name opencompass python=3.10 pytorch torchvision pytorch-cuda -c nvidia -c pytorch -y
conda activate opencompass
git clone https://github.com/open-compass/opencompass opencompass
cd opencompass
pip install -e .

2. Prepare the dataset, you can download the data from release file

# Download dataset from release file and copy to data/ folder
mkdir data
cp -rf mathbench_v1 ./data/ 

3. Inference MathBench

# Inference MathBench with hf_llama2_7b_chat model
python run.py --models hf_llama2_7b_chat --datasets mathbench_gen

You can also evaluate HuggingFace models via command line.

python run.py --datasets mathbench_gen \
--hf-path meta-llama/Llama-2-7b-chat-hf \  # HuggingFace model path
--model-kwargs device_map='auto' \  # Arguments for model construction
--tokenizer-kwargs padding_side='left' truncation='left' use_fast=False \  # Arguments for tokenizer construction
--max-seq-len 2048 \  # Maximum sequence length the model can accept
--batch-size 8 \  # Batch size
--no-batch-padding \  # Don't enable batch padding, infer through for loop to avoid performance loss
--num-gpus 1  # Number of minimum required GPUs
--summarizer summarizers.groups.mathbench_v1_2024 # Summarizer for MathBench-T and MathBench-A

If you want to see bilingual results for MathBench-A&T, replace summarizers.groups.mathbench_v1_2024 with summarizers.groups.mathbench_v1_2024_lang. To access detailed results for each sub-dataset, use summarizers.mathbench_v1.

You can use the -r command to reuse predictions and display different results when changing the summarizer.

Citation and Tech Report

If you use MathBench in your research, please cite the following paper:

@misc{liu2024mathbench,
      title={MathBench: Evaluating the Theory and Application Proficiency of LLMs with a Hierarchical Mathematics Benchmark}, 
      author={Hongwei Liu and Zilong Zheng and Yuxuan Qiao and Haodong Duan and Zhiwei Fei and Fengzhe Zhou and Wenwei Zhang and Songyang Zhang and Dahua Lin and Kai Chen},
      year={2024},
      eprint={2405.12209},
      archivePrefix={arXiv},
      primaryClass={cs.CL}
}