matcaviar/internlm-math

InternLM-Math

InternLM-Math ^HOT

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

📑 Paper 💻 Github 🤗 Demo 🤗 Checkpoints ModelScope

News

• [2024.02.10] We add tech reports and citation reference.
• [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 Intepreter. Tech report is on the way!
• [2024.01.26] We add checkpoints from OpenXLab, which ease Chinese users to download!

Introduction

• 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		internlm2-math-base-7b	2024-01-23
InternLM2-Math-Base-20B	Base	🤗internlm/internlm2-math-base-20b		internlm2-math-base-20b	2024-01-23
InternLM2-Math-7B	Chat	🤗internlm/internlm2-math-7b		internlm2-math-7b	2024-01-23
InternLM2-Math-20B	Chat	🤗internlm/internlm2-math-20b		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.

Benchmark	GSM8K MAJ@1	GSM8K MAJ@100	MATH MAJ@1	MATH MAJ@256
Llama2-7B	14.6	-	2.5	-
Llemma-7B	36.4	54.0	18.0	33.5
InternLM2-Base-7B	36.5	-	8.6	-
InternLM2-Math-Base-7B	49.2	75.7	21.5	35.6
Minerva-8B	16.2	28.4	14.1	25.4
InternLM2-Base-20B	54.6	-	13.7	-
InternLM2-Math-Base-20B	63.7	84.8	27.3	46.2
Llemma-34B	51.5	69.3	25.0	43.1
Minerva-62B	52.4	68.5	27.6	43.4
Minerva-540B	58.8	78.5	33.6	50.3

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

Benchmark	MiniF2F-test
ReProver	26.5
LLMStep	27.9
Code-Llama-7B	26.2
Code-Llama-34B	25.8
Llemma-7B	26.2
Llemma-34B	25.8
InternLM2-Math-7B-Base	30.3
InternLM2-Math-20B-Base	29.5

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.

Model	Model Type	GSM8K	MATH	Hungary
Qwen-7B-Chat	Genearl	51.7	11.6	-
DeepSeek-7B-Chat	General	63.0	15.8	28.5
InternLM2-Chat-7B	General	70.7	23.0	-
ChatGLM3-6B	General	53.8	20.4	32
MetaMath-Mistral-7B	Mathematics	77.7	28.2	29
MetaMath-Llemma-7B	Mathematics	69.2	30.0	-
InternLM2-Math-7B	Mathematics	78.1	34.6	55
InternLM2-Chat-20B	General	79.6	31.9	-
MetaMath-Llemma-34B	Mathematics	75.8	34.8	-
InternLM2-Math-20B	Mathematics	82.6	37.7	66
Qwen-72B	General	78.9	35.2	52
DeepSeek-67B	General	84.1	32.6	58
ChatGPT (GPT-3.5)	General	80.8	34.1	41
GPT4 (First version)	General	92.0	42.5	68

Code Intepreter Performance

All performance is based on interacting with Python.

Model	GSM8K	MATH
DeepSeek-Coder-Instruct-7B	62.8	28.6
DeepSeek-Coder-Instruct-1.5-7B	72.6	34.1
ToRA-7B	72.6	44.6
MathCODER-CL-7B	67.8	30.2
InternLM2-Chat-7B	77.9	45.1
InternLM2-Math-7B	79.4	50.9
ToRA-13B	75.8	48.1
MathCODER-CL-13B	74.1	35.9
InternLM2-Chat-20B	84.5	51.2
InternLM2-Math-20B	80.7	54.3
MathCODER-CL-34B	81.7	45.2
ToRA-70B	84.3	49.7
GPT-4 Code Interpreter *	97.0	69.7

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:

Description	Query
Solving question via chain-of-thought	{Question}
Solving question via Lean 3	{Question}\nSolve this via Lean 3
Outcome reward model	Given a question and an answer, check is it correct?\nQuestion:{Question}\nAnswer:{COT}
Process reward model	Given a question and an answer, check correctness of each step.\nQuestion:{Question}\nAnswer:{COT}
Reward model	Given a question and two answers, which one is better? \nQuestion:{Question}\nAnswer 1:{COT}\nAnswer 2:{COT}
Convert chain-of-thought to Lean 3	Convert this answer into Lean3. Question:{Question}\nAnswer:{COT}
Convert Lean 3 to chain-of-thought	Convert this lean 3 code into a natural language problem with answers:\n{LEAN Code}
Translate question and chain-of-thought answer to a proof statement	Convert this question and answer into a proof format.\nQuestion:{Question}\nAnswer:{COT}
Translate proof problem to Lean 3	Convert this natural langauge statement into a Lean 3 theorem statement:{Theorem}
Translate Lean 3 to proof problem	Convert this Lean 3 theorem statement into natural language:{STATEMENT}
Suggest a tactic based on Lean state	Given the Lean 3 tactic state, suggest a next tactic:\n{LEAN State}
Rephrase Problem	Describe this problem in another way. {Question}
Augment Problem	Please augment a new problem based on: {Question}
Augment a harder Problem	Increase the complexity of the problem: {Question}
Change specific numbers	Change specific numbers: {Question}
Introduce fractions or percentages	Introduce fractions or percentages: {Question}
Code Interpreter	lagent
In-context Learning	Question:{Question}\nAnswer:{COT}\n...Question:{Question}\nAnswer:{COT}

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}
}