Mathematical reasoning poses a significant challenge for language models due to its complex and structured nature. In this paper, we introduce DeepSeekMath 7B, which continues pre-training DeepSeek-Coder-Base-v1.5 7B with 120B math-related tokens sourced from Common Crawl, together with natural language and code data. DeepSeekMath 7B has achieved an impressive score of 51.7% on the competition-level MATH benchmark without relying on external toolkits and voting techniques, approaching the performance level of Gemini-Ultra and GPT-4. Self-consistency over 64 samples from DeepSeekMath 7B achieves 60.9% on MATH. The mathematical reasoning capability of DeepSeekMath is attributed to two key factors: First, we harness the significant potential of publicly available web data through a meticulously engineered data selection pipeline. Second, we introduce Group Relative Policy Optimization (GRPO), a variant of Proximal Policy Optimization (PPO), that enhances mathematical reasoning abilities while concurrently optimizing the memory usage of PPO.

數(shù)學(xué)推理對(duì)語(yǔ)言模型構(gòu)成了重大挑戰(zhàn)，因?yàn)槠鋸?fù)雜且結(jié)構(gòu)化的特性。在本文中，我們介紹了DeepSeekMath 7B，它在DeepSeek-Coder-Base-v1.5 7B的基礎(chǔ)上進(jìn)行了繼續(xù)預(yù)訓(xùn)練，使用了來(lái)自Common Crawl的120B與數(shù)學(xué)相關(guān)的標(biāo)記，以及自然語(yǔ)言和代碼數(shù)據(jù)。DeepSeekMath 7B在競(jìng)爭(zhēng)級(jí)MATH基準(zhǔn)測(cè)試中取得了51.7%的優(yōu)異成績(jī)，且未依賴外部工具包和投票技術(shù)，接近Gemini-Ultra和GPT-4的性能水平。DeepSeekMath 7B在64個(gè)樣本上的自一致性達(dá)到了60.9%的MATH成績(jī)。DeepSeekMath的數(shù)學(xué)推理能力歸因于兩個(gè)關(guān)鍵因素：首先，我們通過(guò)精心設(shè)計(jì)的數(shù)據(jù)選擇流程，充分利用了公開(kāi)可用的網(wǎng)絡(luò)數(shù)據(jù)的巨大潛力。其次，我們引入了群體相對(duì)策略優(yōu)化（GRPO），這是一種近端策略優(yōu)化（PPO）的變體，旨在增強(qiáng)數(shù)學(xué)推理能力，同時(shí)優(yōu)化PPO的內(nèi)存使用。

trainer = GRPOTrainer(
    model="Qwen/Qwen2.5-3B-Instruct",
    reward_funcs="weqweasdas/RM-Gemma-2B",
    args=training_args,
    train_dataset=dataset,
    peft_cnotallow=LoraConfig(task_type="CAUSAL_LM"),
)

def completion_reward(completions, **kwargs):
    '''獎(jiǎng)勵(lì)函數(shù)，對(duì)較長(zhǎng)的補(bǔ)全給予更高的分?jǐn)?shù)'''
    return [float(len(completion))/100 for completion in completions]

prompts = ["The sky is", "The sun is"]
completions = [" blue.", " in the sky."]
print("completion_reward: ", completion_reward(prompts=prompts, completinotallow=completions))

def format_reward(completions, **kwargs):
    '''格式獎(jiǎng)勵(lì)'''
    pattern = r"<think>.*?</think>\s*<answer>.*?</answer>"
    responses = [completion[0]["content"] for completion in completions]
    matches = [re.match(pattern, response) for response in responses]
    return [0.5if match else0.0for match in matches]

prompts = [
    [{"role": "assistant", "content": "What is the result of (1 + 2) * 4?"}],
    [{"role": "assistant", "content": "What is the result of (3 + 1) * 2?"}],
]
completions = [
    [{"role": "assistant", "content": "<think>The sum of 1 and 2 is 3, which we multiply by 4 to get 12.</think><answer>(1 + 2) * 4 = 12</answer>"}],
    [{"role": "assistant", "content": "The sum of 3 and 1 is 4, which we multiply by 2 to get 8. So (3 + 1) * 2 = 8."}],
]
print("format_reward: ", format_reward(prompts=prompts, completinotallow=completions))

trainer = GRPOTrainer(
    model=model,
    processing_class=tokenizer,
    reward_funcs=[
        ...
        format_reward,
        completion_reward,
    ],
    args=training_args,
    train_dataset=data,
    ...
)

import re
from datasets import load_dataset
from transformers import AutoModelForCausalLM, AutoTokenizer
from peft import LoraConfig
from trl import GRPOConfig, GRPOTrainer

SYSTEM_PROMPT = """
按照如下格式生成：
<think>
...
</think>
<answer>
...
</answer>
"""

def process_data(data):
    return data.map(
        lambda x: {
            "prompt": [
                {"role": "system", "content": SYSTEM_PROMPT},
                {"role": "user", "content": x["question_zh-cn"]},
            ],
            "answer": x["answer_only"],
        }
    )

def extract_answer(text):
    answer = text.split("<answer>")[-1]
    answer = answer.split("</answer>")[0]
    return answer.strip()

def correctness_reward(completions, answer, **kwargs):
    responses = [completion[0]["content"] for completion in completions]
    extracted_responses = [extract_answer(r) for r in responses]
    return [2.0if response == str(ans) else0.0for response, ans in zip(extracted_responses, answer)]

def completion_reward(completions, **kwargs):
    '''獎(jiǎng)勵(lì)函數(shù)，對(duì)較長(zhǎng)的補(bǔ)全給予更高的分?jǐn)?shù)'''
    return [float(len(completion)) / 100for completion in completions]

prompts = ["The sky is", "The sun is"]
completions = [" blue.", " in the sky."]
print("completion_reward: ", completion_reward(prompts=prompts, completinotallow=completions))

def digit_reward(completions, **kwargs):
    responses = [completion[0]["content"] for completion in completions]
    extracted_responses = [extract_answer(r) for r in responses]
    return [0.5if response.isdigit() else0.0for response in extracted_responses]

def format_reward(completions, **kwargs):
    '''格式獎(jiǎng)勵(lì)'''
    pattern = r"<think>.*?</think>\s*<answer>.*?</answer>"
    responses = [completion[0]["content"] for completion in completions]
    matches = [re.match(pattern, response) for response in responses]
    return [0.5if match else0.0for match in matches]

prompts = [
    [{"role": "assistant", "content": "What is the result of (1 + 2) * 4?"}],
    [{"role": "assistant", "content": "What is the result of (3 + 1) * 2?"}],
]
completions = [
    [{"role": "assistant", "content": "<think>The sum of 1 and 2 is 3, which we multiply by 4 to get 12.</think><answer>(1 + 2) * 4 = 12</answer>"}],
    [{"role": "assistant", "content": "The sum of 3 and 1 is 4, which we multiply by 2 to get 8. So (3 + 1) * 2 = 8."}],
]
print("format_reward: ", format_reward(prompts=prompts, completinotallow=completions))

def mark_reward(completions, **kwargs):
    '''標(biāo)記獎(jiǎng)勵(lì)（改善格式獎(jiǎng)勵(lì)稀疏問(wèn)題）'''
    def mark_num(text):
        reward = 0
        if text.count("<think>\n") == 1:
            reward += 0.125

        if text.count("</think>\n") == 1:
            reward += 0.125

        if text.count("<answer>\n") == 1:
            reward += 0.125

        if text.count("</answer>\n") == 1:
            reward += 0.125 * 2

        return reward

    responses = [completion[0]["content"] for completion in completions]
    return [mark_num(response) for response in responses]

if __name__ == "__main__":
    model_name = "Qwen/Qwen2.5-3B-Instruct"
    model = AutoModelForCausalLM.from_pretrained(model_name, cache_dir="./model")
    model.cuda()
    tokenizer = AutoTokenizer.from_pretrained(model_name)
    ds = load_dataset("swulling/gsm8k_chinese", cache_dir="./dataset")
    data = process_data(ds["train"])
    output_dir = "output"
    training_args = GRPOConfig(
        output_dir=output_dir,
        learning_rate=5e-6,
        adam_beta1=0.9,
        adam_beta2=0.99,
        weight_decay=0.1,
        warmup_ratio=0.1,
        lr_scheduler_type="cosine",
        logging_steps=1,
        bf16=True,
        per_device_train_batch_size=1,
        gradient_accumulation_steps=4,
        num_generatinotallow=16,
        max_prompt_length=256,
        max_completion_length=200,
        num_train_epochs=1,
        save_steps=100,
        max_grad_norm=0.1,
        log_on_each_node=False,
        use_vllm=False,
        report_to="tensorboard",
    )

    trainer = GRPOTrainer(
        model=model,
        processing_class=tokenizer,
        reward_funcs=[
            mark_reward,
            format_reward,
            digit_reward,
            completion_reward,
            correctness_reward,
        ],
        args=training_args,
        train_dataset=data,
        peft_cnotallow=LoraConfig(task_type="CAUSAL_LM"),
    )
    trainer.train()
    trainer.save_model(output_dir)

用戶: 
使用給定的數(shù)字 [44, 19, 35]，創(chuàng)建一個(gè)等于 98 的方程。你可以使用基本算術(shù)運(yùn)算（+、-、*、/）一次或多次，但每個(gè)數(shù)字只能使用一次。在 <think> </think> 標(biāo)簽中展示你的思考過(guò)程，并在 <answer> </answer> 標(biāo)簽中返回最終方程，例如 <answer> (1 + 2) / 3 </answer>。在 <think> 標(biāo)簽中逐步思考。

機(jī)器人:  
<span class="think">首先我們需要找到如何將給定的數(shù)字組合起來(lái)得到 98。我們可以嘗試不同的組合，比如加法和減法。注意到 98 可以分解為 90 和 8 的和，因此我們可以考慮用 44 和 35 來(lái)構(gòu)造這兩個(gè)數(shù)。</span>
<span class="think">觀察到 44 和 35 的差是 9，如果我們將 35 減去這個(gè)差，就可以得到 8。即 35 - (44 - 35) = 8。</span>
<span class="think">現(xiàn)在我們有了 8 和 90，可以將它們相加得到 98。所以最終的方程是 35 + (44 - 35)。</span>
<answer> 35 + (44 - 35) </answer>

用戶: 
使用給定的數(shù)字 [44, 19, 35]，創(chuàng)建一個(gè)等于 98 的方程。你可以使用基本算術(shù)運(yùn)算（+、-、*、/）一次或多次，但每個(gè)數(shù)字只能使用一次。在 <think> </think> 標(biāo)簽中展示你的思考過(guò)程，并在 <answer> </answer> 標(biāo)簽中返回最終方程，例如 <answer> (1 + 2) / 3 </answer>。在 <think> 標(biāo)簽中逐步思考。

機(jī)器人:  
<think>
首先，我需要使用數(shù)字44、19和35，通過(guò)基本算術(shù)運(yùn)算（+、-、*、/）來(lái)構(gòu)造一個(gè)等于98的方程，且每個(gè)數(shù)字只能使用一次。

我先考慮將較大的數(shù)字進(jìn)行加法或乘法運(yùn)算，以達(dá)到接近98的目標(biāo)。例如，44加19等于63，再加上35可以得到98。具體計(jì)算如下：

44 + 19 = 63  
63 + 35 = 98  

因此，最終的方程可以表示為44 + 19 + 35 = 98。
</think>

<answer> 44 + 19 + 35 = 98 </answer>