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InternLM-Math

State-of-the-art bilingual open-sourced Math reasoning LLMs. A solver, prover, verifier, augmentor.

📑 Paper 💻 Github 🤗 Demo 🤗 Checkpoints <img src="./assets/modelscope_logo.png" width="20px" /> ModelScope

News

• [2024.10.22] We release InternLM2.5-Step-Prover policy and critic models with 14k proves found in Lean-Workbook. 🤗Dataset 🤗Policy-model 🤗Critic-model 📑 Paper
• [2024.07.25] We release Lean-Github and InternLM2-Step-Prover with 29K theorems compiled from 100+ Lean 4 repos and a 7B models fine-tuned on Lean-Github and Lean-Workbook with SOTA performance on MiniF2F-test (54.5%), ProofNet (18.1%), and Putnam (5 problems). 🤗Dataset 🤗Model 📑 Paper 📖 README
• [2024.06.06] We release Lean-Workbook with 57K math problems formalized in Lean 4 with 5K searched proof for autoformalization and auto theorem proving. 🤗Dataset 📑 Paper
• [2024.05.24] We release updated version InternLM2-Math-Plus with 4 sizes and state-of-the-art performances including 1.8B, 7B, 20B, and 8x22B. We improve informal math reasoning performance (chain-of-thought and code-interpreter) and formal math reasoning performance (LEAN 4 translation and LEAN 4 theorem proving) significantly.
• [2024.02.10] We add tech reports and citation references.
• [2024.01.31] We add MiniF2F results with evaluation codes!
• [2024.01.29] We add checkpoints from ModelScope. Update results about majority voting and Code Interpreter. Tech report is on the way!
• [2024.01.26] We add checkpoints from OpenXLab, which ease Chinese users to download!

InternLM2-Math-Plus

Checkpoints

Formal Math Reasoning

We evaluate the performance of InternLM2-Math-Plus on formal math reasoning benchmark MiniF2F-test. The evaluation setting is same as Llemma with LEAN 4.

This is how to reproduce our performance on MiniF2F.

Informal Math Reasoning

We evaluate the performance of InternLM2-Math-Plus on informal math reasoning benchmark MATH and GSM8K. InternLM2-Math-Plus-1.8B outperforms MiniCPM-2B in the smallest size setting. InternLM2-Math-Plus-7B outperforms Deepseek-Math-7B-RL which is the state-of-the-art math reasoning open source model. InternLM2-Math-Plus-Mixtral8x22B achieves 68.5 on MATH (with Python) and 91.8 on GSM8K.

For tool-calling inference and evaluation, please see the agent section.

We also evaluate models on MathBench-A. InternLM2-Math-Plus-Mixtral8x22B has comparable performance compared to Claude 3 Opus.

Introduction (For InternLM2-Math)

• 7B and 20B Chinese and English Math LMs with better than ChatGPT performances. InternLM2-Math are continued pretrained from InternLM2-Base with ~100B high quality math-related tokens and SFT with ~2M bilingual math supervised data. We apply minhash and exact number match to decontaminate possible test set leakage.
• Add Lean as a support language for math problem solving and math theorem proving. We are exploring combining Lean 3 with InternLM-Math for verifiable math reasoning. InternLM-Math can generate Lean codes for simple math reasoning tasks like GSM8K or provide possible proof tactics based on Lean states.
• Also can be viewed as a reward model, which supports the Outcome/Process/Lean Reward Model. We supervise InternLM2-Math with various types of reward modeling data, to make InternLM2-Math can also verify chain-of-thought processes. We also add the ability to convert a chain-of-thought process into Lean 3 code.
• A Math LM Augment Helper and Code Interpreter. InternLM2-Math can help augment math reasoning problems and solve them using the code interpreter which makes you generate synthesis data quicker!

Models

InternLM2-Math-Base-7B and InternLM2-Math-Base-20B are pretrained checkpoints. InternLM2-Math-7B and InternLM2-Math-20B are SFT checkpoints.

Model	Model Type	Transformers(HF)	OpenXLab	ModelScope	Release Date
InternLM2-Math-Base-7B	Base	🤗internlm/internlm2-math-base-7b		<img src="./assets/modelscope_logo.png" width="20px" /> internlm2-math-base-7b	2024-01-23
InternLM2-Math-Base-20B	Base	🤗internlm/internlm2-math-base-20b		<img src="./assets/modelscope_logo.png" width="20px" /> internlm2-math-base-20b	2024-01-23
InternLM2-Math-7B	Chat	🤗internlm/internlm2-math-7b		<img src="./assets/modelscope_logo.png" width="20px" /> internlm2-math-7b	2024-01-23
InternLM2-Math-20B	Chat	🤗internlm/internlm2-math-20b		<img src="./assets/modelscope_logo.png" width="20px" /> internlm2-math-20b	2024-01-23

Performance

Pretrain Performance

We evaluate pretrain checkpoints based on greedy decoding with few-shot COT. Details of pretraining will be introduced in the tech report.

We evaluate pretrain checkpoints using few-shot on MiniF2F. Please see eval/pretrain/minif2f for evaluation.

SFT Peformance

All performance is based on greedy decoding with COT. We notice that the performance of Hungary has a big variance between our different checkpoints, while other performance is very stable. This may be due to the problem amount about Hungary.

Code Intepreter Performance

All performance is based on interacting with Python.

Eval

You can effortlessly evaluate InternLM2-Math across a diverse array of mathematical datasets, such as Math and GSM8K, using OpenCompass with a single command. To get started, simply execute the following in your terminal after installing OpenCompass:

python run.py --models hf_internlm2_chat_math_7b --datasets gsm8k_gen math_gen_736506

Alternatively, for a streamlined experience, you can utilize a predefined configuration file. To do this, run the command below, making sure to adjust the arguments according to your requirements:

python run.py config/eval_internlm_math_chat.py

Inference

LMDeploy

We suggest using LMDeploy(>=0.2.1) for inference.

from lmdeploy import pipeline, TurbomindEngineConfig, ChatTemplateConfig

backend_config = TurbomindEngineConfig(model_name='internlm2-chat-7b', tp=1, cache_max_entry_count=0.3)
chat_template = ChatTemplateConfig(model_name='internlm2-chat-7b', system='', eosys='', meta_instruction='')
pipe = pipeline(model_path='internlm/internlm2-math-7b', chat_template_config=chat_template, backend_config=backend_config)

problem = '1+1='
result = pipe([problem], request_output_len=1024, top_k=1)

Huggingface

import torch
from transformers import AutoTokenizer, AutoModelForCausalLM
tokenizer = AutoTokenizer.from_pretrained("internlm/internlm2-math-7b", trust_remote_code=True)
# Set `torch_dtype=torch.float16` to load model in float16, otherwise it will be loaded as float32 and might cause OOM Error.
model = AutoModelForCausalLM.from_pretrained("internlm/internlm2-math-7b", trust_remote_code=True, torch_dtype=torch.float16).cuda()
model = model.eval()
response, history = model.chat(tokenizer, "1+1=", history=[], meta_instruction="")
print(response)

Special usages

We list some instructions used in our SFT. You can use them to help you. You can use the other ways to prompt the model, but the following are recommended. InternLM2-Math may combine the following abilities but it is not guaranteed.

Translate proof problem to Lean:

Using Lean 3 to solve GSM8K problem:

Generate problem based on Lean 3 code:

Play 24 point game:

Augment a harder math problem:

Fine-tune and others

Please refer to InternLM.

Known issues

Our model is still under development and will be upgraded. There are some possible issues of InternLM-Math. If you find performances of some abilities are not great, welcome to open an issue.

• Jump the calculating step.
• Perform badly at Chinese fill-in-the-bank problems and English choice problems due to SFT data composition.
• Tend to generate Code Interpreter when facing Chinese problems due to SFT data composition.
• The reward model mode can be better leveraged with assigned token probabilities.
• Code switch due to SFT data composition.
• Some abilities of Lean can only be adapted to GSM8K-like problems (e.g. Convert chain-of-thought to Lean 3), and performance related to Lean is not guaranteed.

Citation and Tech Report

@misc{ying2024internlmmath,
      title={InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning}, 
      author={Huaiyuan Ying and Shuo Zhang and Linyang Li and Zhejian Zhou and Yunfan Shao and Zhaoye Fei and Yichuan Ma and Jiawei Hong and Kuikun Liu and Ziyi Wang and Yudong Wang and Zijian Wu and Shuaibin Li and Fengzhe Zhou and Hongwei Liu and Songyang Zhang and Wenwei Zhang and Hang Yan and Xipeng Qiu and Jiayu Wang and Kai Chen and Dahua Lin},
      year={2024},
      eprint={2402.06332},
      archivePrefix={arXiv},
      primaryClass={cs.CL}
}
@misc{ying2024lean,
      title={Lean Workbook: A large-scale Lean problem set formalized from natural language math problems}, 
      author={Huaiyuan Ying and Zijian Wu and Yihan Geng and Jiayu Wang and Dahua Lin and Kai Chen},
      year={2024},
      eprint={2406.03847},
      archivePrefix={arXiv},
      primaryClass={cs.CL}
}
@misc{wu2024leangithubcompilinggithublean,
      title={LEAN-GitHub: Compiling GitHub LEAN repositories for a versatile LEAN prover}, 
      author={Zijian Wu and Jiayu Wang and Dahua Lin and Kai Chen},
      year={2024},
      eprint={2407.17227},
      archivePrefix={arXiv},
      primaryClass={cs.AI},
      url={https://arxiv.org/abs/2407.17227}, 
}
@misc{wu2024internlm25stepproveradvancingautomatedtheorem,
      title={InternLM2.5-StepProver: Advancing Automated Theorem Proving via Expert Iteration on Large-Scale LEAN Problems}, 
      author={Zijian Wu and Suozhi Huang and Zhejian Zhou and Huaiyuan Ying and Jiayu Wang and Dahua Lin and Kai Chen},
      year={2024},
      eprint={2410.15700},
      archivePrefix={arXiv},
      primaryClass={cs.AI},
      url={https://arxiv.org/abs/2410.15700}, 
}