{"id":5549,"date":"2025-01-16T17:27:16","date_gmt":"2025-01-16T09:27:16","guid":{"rendered":"https:\/\/nullthought.net\/?p=5549"},"modified":"2025-03-03T12:25:31","modified_gmt":"2025-03-03T04:25:31","slug":"rstar-math%ef%bc%9a%e6%9c%89%e6%95%88%e6%8f%90%e5%8d%87%e5%b0%8f%e5%9e%8b%e8%af%ad%e8%a8%80%e6%a8%a1%e5%9e%8b%ef%bc%88small-language-models-slms%ef%bc%89%e5%9c%a8%e6%95%b0%e5%ad%a6%e6%8e%a8%e7%90%86","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5549","title":{"rendered":"rStar-Math\uff1a\u6709\u6548\u63d0\u5347\u5c0f\u578b\u8bed\u8a00\u6a21\u578b\uff08Small Language Models, SLMs\uff09\u5728\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e2d\u7684\u8868\u73b0"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/www.arxiv.org\/abs\/2501.04519\" target=\"_blank\" rel=\"noreferrer noopener\">rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking<\/a><\/strong>\u63d0\u51fa\u4e86rStar-Math\uff0c\u4ee5\u8bc1\u660e\u5c0f\u578b\u8bed\u8a00\u6a21\u578b\uff08Small Language Models\uff0cSLMs\uff09\u80fd\u591f\u4e0eOpenAI o1\u7684\u6570\u5b66\u63a8\u7406\u80fd\u529b\u76f8\u5ab2\u7f8e\uff0c\u751a\u81f3\u8d85\u8d8a\uff0c\u800c\u65e0\u9700\u4f9d\u8d56\u66f4\u9ad8\u7ea7\u6a21\u578b\u7684\u84b8\u998f\u8fc7\u7a0b\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aXinyu Guan, Li Lyna Zhang, Yifei Liu, Ning Shang, Youran Sun, Yi Zhu, Fan Yang, Mao Yang\uff0c\u5747\u6765\u81eaMicrosoft Research Asia\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"351\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-15-1024x351.png\" alt=\"\" class=\"wp-image-5550\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-15-1024x351.png 1024w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-15-300x103.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-15-768x264.png 768w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-15.png 1046w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/www.arxiv.org\/abs\/2501.04519\" target=\"_blank\" rel=\"noreferrer noopener\">rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e00\u3001\u5f15\u8a00<\/h5>\n\n\n\n<p>\u672c\u6587\u63d0\u51fa\u4e86rStar-Math\uff0c\u4e00\u4e2a\u4e13\u6ce8\u4e8e\u63d0\u5347\u5c0f\u578b\u8bed\u8a00\u6a21\u578b\uff08Small Language Models, SLMs\uff09\u5728\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e2d\u8868\u73b0\u7684\u7cfb\u7edf\u3002\u76f8\u6bd4\u4f9d\u8d56\u4e8e\u5927\u578b\u8bed\u8a00\u6a21\u578b\uff08Large Language Models, LLMs\uff09\u8fdb\u884c\u77e5\u8bc6\u84b8\u998f\u7684\u4f20\u7edf\u65b9\u6cd5\uff0crStar-Math\u5b8c\u5168\u901a\u8fc7\u81ea\u6211\u8fdb\u5316\u7684\u65b9\u5f0f\uff0c\u4e0d\u4f9d\u8d56\u6559\u5e08\u6a21\u578b\uff0c\u4ece\u96f6\u5f00\u59cb\u53d1\u5c55\u51fa\u5f3a\u5927\u7684\u6570\u5b66\u63a8\u7406\u80fd\u529b\u3002\u5176\u6838\u5fc3\u601d\u60f3\u662f\u5229\u7528<strong>\u7cfb\u7edf2\u5f0f\u63a8\u7406\uff08System 2 reasoning\uff09<\/strong>\uff0c\u5373\u4e00\u79cd\u7c7b\u4f3c\u4eba\u7c7b\u6df1\u5ea6\u601d\u8003\u7684\u9010\u6b65\u63a8\u7406\u65b9\u5f0f\uff0c\u4ece\u800c\u8d85\u8d8a\u5f53\u524d\u5927\u6a21\u578b\u5728\u6570\u5b66\u95ee\u9898\u4e0a\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u6765\u8bf4\uff0crStar-Math\u901a\u8fc7\u4ee5\u4e0b\u4e09\u5927\u6280\u672f\u521b\u65b0\uff0c\u5b9e\u73b0\u4e86\u5c0f\u6a21\u578b\u5728\u6570\u5b66\u63a8\u7406\u80fd\u529b\u4e0a\u7684\u91cd\u5927\u7a81\u7834\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u4ee3\u7801\u589e\u5f3a\u7684\u9010\u6b65\u94fe\u5f0f\u601d\u7ef4\u6570\u636e\u5408\u6210\u65b9\u6cd5<\/strong>\uff1a\u901a\u8fc7\u8499\u7279\u5361\u6d1b\u6811\u641c\u7d22\uff08Monte Carlo Tree Search, MCTS\uff09\u751f\u6210\u9010\u6b65\u9a8c\u8bc1\u7684\u63a8\u7406\u8def\u5f84\uff0c\u786e\u4fdd\u4e2d\u95f4\u6bcf\u4e00\u6b65\u63a8\u7406\u5747\u6b63\u786e\u65e0\u8bef\uff0c\u5e76\u901a\u8fc7\u4ee3\u7801\u6267\u884c\u8fdb\u884c\u9a8c\u8bc1\u3002<\/li>\n\n\n\n<li><strong>\u8fc7\u7a0b\u504f\u597d\u6a21\u578b\uff08Process Preference Model, PPM\uff09<\/strong>\uff1a\u6539\u8fdb\u4e86\u5956\u52b1\u6a21\u578b\u8bad\u7ec3\u65b9\u6cd5\uff0c\u4e0d\u518d\u4f9d\u8d56\u4f20\u7edf\u7684\u9010\u6b65\u6253\u5206\u673a\u5236\uff0c\u800c\u662f\u901a\u8fc7\u504f\u597d\u5bf9\uff08preference pairs\uff09\u7684\u65b9\u5f0f\uff0c\u66f4\u9ad8\u6548\u5730\u4f18\u5316\u63a8\u7406\u6b65\u9aa4\u7684\u8bc4\u4ef7\u3002<\/li>\n\n\n\n<li><strong>\u56db\u8f6e\u81ea\u6211\u8fdb\u5316\u7b56\u7565<\/strong>\uff1a\u5728\u65e0\u4efb\u4f55\u9884\u8bad\u7ec3\u6570\u636e\u7684\u60c5\u51b5\u4e0b\uff0c\u901a\u8fc7\u81ea\u6211\u751f\u6210\u3001\u4f18\u5316\u548c\u8fed\u4ee3\uff0c\u9010\u6b65\u63d0\u5347SLM\u7684\u63a8\u7406\u80fd\u529b\uff0c\u89e3\u51b3\u66f4\u590d\u6742\u7684\u6570\u5b66\u95ee\u9898\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u5b9e\u9a8c\u7ed3\u679c\u8868\u660e\uff0crStar-Math\u5728\u591a\u4e2a\u6570\u5b66\u57fa\u51c6\u4e0a\u8fbe\u5230\u4e86\u6216\u8d85\u8d8a\u4e86\u6700\u5148\u8fdb\u6a21\u578b\u7684\u6c34\u5e73\u3002\u4f8b\u5982\uff0c\u5728MATH\u57fa\u51c6\u4e0a\uff0crStar-Math\u5c06Qwen2.5-Math-7B\u6a21\u578b\u7684\u51c6\u786e\u7387\u4ece58.8%\u63d0\u9ad8\u523090.0%\uff1b\u5728AIME\uff08\u7f8e\u56fd\u6570\u5b66\u5965\u6797\u5339\u514b\u9080\u8bf7\u8d5b\uff09\u6d4b\u8bd5\u4e2d\uff0c\u80fd\u591f\u89e3\u51b353.3%\u7684\u95ee\u9898\uff0c\u76f8\u5f53\u4e8e\u8fdb\u5165\u5168\u7f8e\u9ad8\u4e2d\u6570\u5b66\u7ade\u8d5b\u5b66\u751f\u4e2d\u7684\u524d20%\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e8c\u3001\u80cc\u666f\u4e0e\u52a8\u673a<\/h5>\n\n\n\n<p>\u968f\u7740\u5927\u8bed\u8a00\u6a21\u578b\u7684\u5feb\u901f\u53d1\u5c55\uff0c\u5176\u5728\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e2d\u7684\u8868\u73b0\u53d7\u5230\u4e86\u5e7f\u6cdb\u5173\u6ce8\u3002\u7136\u800c\uff0c\u76ee\u524d\u4e3b\u6d41\u65b9\u6cd5\u4ecd\u9762\u4e34\u4ee5\u4e0b\u4e09\u4e2a\u4e3b\u8981\u6311\u6218\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u9ad8\u8d28\u91cf\u6570\u5b66\u63a8\u7406\u6570\u636e\u7a00\u7f3a<\/strong><br>\u6570\u5b66\u63a8\u7406\u9700\u8981\u6a21\u578b\u751f\u6210\u4e0d\u4ec5\u6b63\u786e\u7684\u6700\u7ec8\u7b54\u6848\uff0c\u8fd8\u9700\u8981\u5b8c\u6574\u4e14\u51c6\u786e\u7684\u63a8\u7406\u8def\u5f84\u3002\u7136\u800c\uff0c\u73b0\u6709\u6570\u636e\u96c6\u5927\u591a\u4f9d\u8d56\u4eba\u7c7b\u6807\u6ce8\u6216\u901a\u8fc7\u5927\u578b\u6a21\u578b\uff08\u5982GPT-4\uff09\u8fdb\u884c\u84b8\u998f\u751f\u6210\u3002\u8fd9\u79cd\u65b9\u6cd5\u4e0d\u4ec5\u6210\u672c\u9ad8\u6602\uff0c\u4e14\u751f\u6210\u7684\u6570\u636e\u5f80\u5f80\u53d7\u5230\u6559\u5e08\u6a21\u578b\u80fd\u529b\u7684\u9650\u5236\uff0c\u5c24\u5176\u5728\u590d\u6742\u95ee\u9898\u4e0a\u96be\u4ee5\u53d6\u5f97\u7a81\u7834\u3002<\/li>\n\n\n\n<li><strong>\u4e2d\u95f4\u63a8\u7406\u8def\u5f84\u7684\u9519\u8bef\u7d2f\u79ef\u95ee\u9898<\/strong><br>\u5728\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e2d\uff0c\u6b63\u786e\u7684\u6700\u7ec8\u7b54\u6848\u5e76\u4e0d\u610f\u5473\u7740\u63a8\u7406\u8def\u5f84\u5b8c\u5168\u6b63\u786e\u3002\u4e2d\u95f4\u6b65\u9aa4\u4e2d\u4efb\u4f55\u9519\u8bef\u90fd\u4f1a\u663e\u8457\u964d\u4f4e\u6570\u636e\u8d28\u91cf\uff0c\u5e76\u5f71\u54cd\u6a21\u578b\u7684\u8bad\u7ec3\u6548\u679c\u3002\u73b0\u6709\u65b9\u6cd5\u5f80\u5f80\u96be\u4ee5\u533a\u5206\u63a8\u7406\u8def\u5f84\u4e2d\u5b58\u5728\u7684\u7ec6\u5fae\u9519\u8bef\u3002<\/li>\n\n\n\n<li><strong>\u5956\u52b1\u6a21\u578b\u7684\u8bad\u7ec3\u74f6\u9888<\/strong><br>\u9ad8\u6548\u7684\u6570\u5b66\u63a8\u7406\u9700\u8981\u4f9d\u8d56\u4e8e\u80fd\u591f\u7cbe\u786e\u8bc4\u4f30\u6bcf\u4e00\u6b65\u63a8\u7406\u7684\u5956\u52b1\u6a21\u578b\u3002\u7136\u800c\uff0c\u76ee\u524d\u7684\u5956\u52b1\u6a21\u578b\u8bad\u7ec3\u4f9d\u8d56\u5927\u91cf\u4eba\u5de5\u6807\u6ce8\uff0c\u96be\u4ee5\u5927\u89c4\u6a21\u6269\u5c55\u3002\u6b64\u5916\uff0c\u81ea\u52a8\u5316\u7684\u6807\u6ce8\u65b9\u6cd5\uff08\u5982\u57fa\u4e8e\u7ed3\u679c\u7684\u6253\u5206\uff09\u5f80\u5f80\u8fc7\u4e8e\u7c97\u7cd9\uff0c\u65e0\u6cd5\u63d0\u4f9b\u8db3\u591f\u7cbe\u7ec6\u7684\u53cd\u9988\u4fe1\u53f7\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u9488\u5bf9\u6b64\uff0crStar-Math\u63d0\u51fa\u4e86\u4e00\u79cd\u5168\u65b0\u7684\u601d\u8def\uff1a\u901a\u8fc7\u5c0f\u578b\u6a21\u578b\u81ea\u6211\u751f\u6210\u9ad8\u8d28\u91cf\u6570\u636e\uff0c\u5e76\u7ed3\u5408\u9010\u6b65\u9a8c\u8bc1\u548c\u504f\u597d\u5efa\u6a21\u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u6a21\u578b\u63a8\u7406\u80fd\u529b\u7684\u81ea\u6211\u8fed\u4ee3\u4f18\u5316\u3002\u8fd9\u4e0d\u4ec5\u6446\u8131\u4e86\u5bf9\u5927\u578b\u6a21\u578b\u7684\u4f9d\u8d56\uff0c\u4e5f\u4e3a\u5c0f\u6a21\u578b\u5728\u590d\u6742\u4efb\u52a1\u4e0a\u7684\u5e94\u7528\u63d0\u4f9b\u4e86\u65b0\u7684\u53ef\u80fd\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e09\u3001\u65b9\u6cd5\u8bba<\/h5>\n\n\n\n<p><strong>rStar-Math<\/strong>\u7684\u65b9\u6cd5\u8bba\u53ef\u4ee5\u5206\u4e3a\u4ee5\u4e0b\u4e09\u4e2a\u5173\u952e\u90e8\u5206\uff1a<\/p>\n\n\n\n<p><strong>1. \u57fa\u4e8eMCTS\u7684\u9010\u6b65\u9a8c\u8bc1\u63a8\u7406\u8def\u5f84\u751f\u6210<\/strong><br>MCTS\u662f\u4e00\u79cd\u9002\u5408\u89e3\u51b3\u9010\u6b65\u51b3\u7b56\u95ee\u9898\u7684\u641c\u7d22\u7b97\u6cd5\uff0c\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u68cb\u7c7b\u6e38\u620f\u4e2d\u7684\u667a\u80fd\u51b3\u7b56\u3002\u672c\u6587\u5c06\u5176\u5f15\u5165\u6570\u5b66\u63a8\u7406\u4efb\u52a1\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u751f\u6210\u9ad8\u8d28\u91cf\u7684\u9010\u6b65\u9a8c\u8bc1\u63a8\u7406\u8def\u5f84\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u9010\u6b65\u751f\u6210\u4e0e\u4ee3\u7801\u9a8c\u8bc1<\/strong>\uff1a\u5728\u6bcf\u4e00\u6b65\u63a8\u7406\u4e2d\uff0c\u6a21\u578b\u540c\u65f6\u751f\u6210\u81ea\u7136\u8bed\u8a00\u63cf\u8ff0\u548c\u5bf9\u5e94\u7684Python\u4ee3\u7801\u3002\u751f\u6210\u7684\u4ee3\u7801\u9700\u901a\u8fc7\u6267\u884c\u9a8c\u8bc1\uff0c\u53ea\u6709\u4ee3\u7801\u80fd\u591f\u6b63\u786e\u6267\u884c\u7684\u8def\u5f84\u624d\u88ab\u4fdd\u7559\u3002<\/li>\n\n\n\n<li><strong>Q\u503c\u6807\u6ce8\u4e0e\u8bc4\u5206<\/strong>\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u4e2d\u95f4\u6b65\u9aa4\uff0c\u901a\u8fc7MCTS\u7684\u56de\u6eaf\u673a\u5236\uff0c\u8d4b\u4e88\u5176\u57fa\u4e8e\u6700\u7ec8\u7ed3\u679c\u7684\u8d21\u732e\u503c\uff08Q\u503c\uff09\u3002\u901a\u8fc7\u591a\u6b21\u641c\u7d22\u8fed\u4ee3\uff0c\u786e\u4fddQ\u503c\u6807\u6ce8\u7684\u51c6\u786e\u6027\u548c\u53ef\u9760\u6027\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u90e8\u5206\u65b9\u6cd5\u6709\u6548\u89e3\u51b3\u4e86\u4f20\u7edf\u63a8\u7406\u8def\u5f84\u751f\u6210\u4e2d\u5e38\u89c1\u7684\u9519\u8bef\u7d2f\u79ef\u95ee\u9898\u3002<\/p>\n\n\n\n<p><strong>2. \u8fc7\u7a0b\u504f\u597d\u6a21\u578b\uff08PPM\uff09<\/strong><br>\u4f20\u7edf\u5956\u52b1\u6a21\u578b\u7684\u8bad\u7ec3\u5f80\u5f80\u4f9d\u8d56\u7cbe\u786e\u7684\u9010\u6b65\u8bc4\u5206\uff0c\u8fd9\u4e0d\u4ec5\u6210\u672c\u9ad8\u6602\uff0c\u4e14\u96be\u4ee5\u786e\u4fdd\u4e00\u81f4\u6027\u3002rStar-Math\u901a\u8fc7\u6784\u9020\u504f\u597d\u5bf9\uff0c\u66ff\u4ee3\u4e86\u4f20\u7edf\u7684\u9010\u6b65\u6253\u5206\u673a\u5236\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u504f\u597d\u5bf9\u7684\u6784\u9020<\/strong>\uff1a\u5728\u6bcf\u4e2a\u6b65\u9aa4\u4e2d\uff0c\u4eceMCTS\u751f\u6210\u7684\u8def\u5f84\u4e2d\u9009\u62e9\u4e24\u4e2a\u9ad8\u8bc4\u5206\u7684\u6b63\u6837\u672c\u548c\u4e24\u4e2a\u4f4e\u8bc4\u5206\u7684\u8d1f\u6837\u672c\uff0c\u5f62\u6210\u6b63\u8d1f\u504f\u597d\u5bf9\u3002<\/li>\n\n\n\n<li><strong>\u57fa\u4e8e\u6392\u5e8f\u635f\u5931\u7684\u4f18\u5316<\/strong>\uff1a\u901a\u8fc7\u5bf9\u504f\u597d\u5bf9\u7684\u6392\u5e8f\u635f\u5931\u51fd\u6570\u8fdb\u884c\u4f18\u5316\uff0c\u8bad\u7ec3\u8fc7\u7a0b\u504f\u597d\u6a21\u578b\uff0c\u4f7f\u5176\u80fd\u591f\u66f4\u51c6\u786e\u5730\u9884\u6d4b\u6bcf\u4e00\u6b65\u63a8\u7406\u7684\u5f97\u5206\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u79cd\u65b9\u6cd5\u907f\u514d\u4e86\u4f20\u7edf\u9010\u6b65\u8bc4\u5206\u4e2d\u7684\u566a\u58f0\u95ee\u9898\uff0c\u540c\u65f6\u663e\u8457\u63d0\u5347\u4e86\u6a21\u578b\u5bf9\u5173\u952e\u63a8\u7406\u6b65\u9aa4\u7684\u6355\u6349\u80fd\u529b\u3002<\/p>\n\n\n\n<p><strong>3. \u56db\u8f6e\u81ea\u6211\u8fdb\u5316\u7b56\u7565<\/strong><br>rStar-Math\u901a\u8fc7\u56db\u8f6e\u81ea\u6211\u8fed\u4ee3\u7684\u65b9\u5f0f\uff0c\u9010\u6b65\u63d0\u5347SLM\u7684\u63a8\u7406\u80fd\u529b\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7b2c\u4e00\u8f6e\uff1a\u521d\u59cb\u5316\u5f3a\u57fa\u7ebf\u6a21\u578b<\/strong><br>\u4f7f\u7528\u5916\u90e8\u6a21\u578b\uff08\u5982DeepSeek-Coder\uff09\u751f\u6210\u521d\u59cb\u8bad\u7ec3\u6570\u636e\uff0c\u8bad\u7ec3\u7b2c\u4e00\u4ee3SLM\u548c\u521d\u6b65\u7684PPM\u3002<\/li>\n\n\n\n<li><strong>\u7b2c\u4e8c\u8f6e\uff1a\u4f18\u5316\u5956\u52b1\u6a21\u578b<\/strong><br>\u901a\u8fc7\u66f4\u7cbe\u786e\u7684Q\u503c\u6807\u6ce8\uff0c\u751f\u6210\u9ad8\u8d28\u91cf\u63a8\u7406\u8def\u5f84\uff0c\u7528\u4e8e\u8bad\u7ec3\u66f4\u53ef\u9760\u7684PPM\u3002<\/li>\n\n\n\n<li><strong>\u7b2c\u4e09\u8f6e\uff1a\u57fa\u4e8ePPM\u7684\u4f18\u5316\u641c\u7d22<\/strong><br>\u4f7f\u7528PPM\u6539\u8fdbMCTS\u641c\u7d22\u8d28\u91cf\uff0c\u751f\u6210\u66f4\u591a\u8986\u76d6\u590d\u6742\u95ee\u9898\u7684\u6570\u636e\u3002<\/li>\n\n\n\n<li><strong>\u7b2c\u56db\u8f6e\uff1a\u89e3\u51b3\u9ad8\u96be\u5ea6\u95ee\u9898<\/strong><br>\u9488\u5bf9\u672a\u89e3\u51b3\u7684\u590d\u6742\u95ee\u9898\uff0c\u901a\u8fc7\u589e\u52a0\u641c\u7d22\u8fed\u4ee3\u6b21\u6570\u548c\u968f\u673a\u6027\uff0c\u8fdb\u4e00\u6b65\u63d0\u5347\u8986\u76d6\u7387\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u6700\u7ec8\uff0c\u8fd9\u79cd\u9010\u6b65\u8fdb\u5316\u7684\u7b56\u7565\u5c06SLM\u7684\u6570\u5b66\u63a8\u7406\u80fd\u529b\u63d0\u5347\u81f3\u6700\u5148\u8fdb\u6c34\u5e73\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u56db\u3001\u5b9e\u9a8c\u4e0e\u7ed3\u679c\u5206\u6790<\/h5>\n\n\n\n<p>\u5b9e\u9a8c\u5728\u591a\u4e2a\u6570\u5b66\u57fa\u51c6\u4e0a\u9a8c\u8bc1\u4e86rStar-Math\u7684\u6709\u6548\u6027\uff0c\u6db5\u76d6\u57fa\u7840\u6570\u5b66\uff08\u5982MATH\uff09\u3001\u7ade\u8d5b\u6570\u5b66\uff08\u5982AIME\u3001\u5965\u8d5b\uff09\u4ee5\u53ca\u5927\u5b66\u6570\u5b66\u7b49\u591a\u4e2a\u9886\u57df\u3002<\/p>\n\n\n\n<p><strong>1. \u57fa\u51c6\u6d4b\u8bd5\u7ed3\u679c<\/strong><br>rStar-Math\u5728\u591a\u4e2a\u4efb\u52a1\u4e0a\u7684\u8868\u73b0\u663e\u8457\u4f18\u4e8e\u73b0\u6709\u65b9\u6cd5\u3002\u4f8b\u5982\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5728MATH\u57fa\u51c6\u4e0a\uff0cQwen2.5-Math-7B\u6a21\u578b\u7684\u51c6\u786e\u7387\u4ece58.8%\u63d0\u5347\u81f390.0%\uff0c\u8d85\u8fc7\u4e86OpenAI o1-preview\u3002<\/li>\n\n\n\n<li>\u5728AIME\u6d4b\u8bd5\u4e2d\uff0crStar-Math\u89e3\u51b3\u4e8653.3%\u7684\u95ee\u9898\uff0c\u8868\u73b0\u4f18\u4e8e\u5927\u591a\u6570\u5f00\u6e90\u6a21\u578b\u3002<\/li>\n\n\n\n<li>\u5728\u5965\u8d5b\u7ea7\u522b\u548c\u5927\u5b66\u6570\u5b66\u4efb\u52a1\u4e0a\uff0crStar-Math\u540c\u6837\u5c55\u73b0\u4e86\u5f3a\u5927\u7684\u63a8\u7406\u80fd\u529b\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>2. \u9010\u6b65\u9a8c\u8bc1\u8def\u5f84\u7684\u6709\u6548\u6027<\/strong><br>\u76f8\u6bd4\u4e8e\u4f20\u7edf\u7684\u6570\u636e\u5408\u6210\u65b9\u6cd5\uff08\u5982GPT-4\u84b8\u998f\u6570\u636e\uff09\uff0c\u9010\u6b65\u9a8c\u8bc1\u8def\u5f84\u663e\u8457\u63d0\u5347\u4e86\u6a21\u578b\u7684\u8bad\u7ec3\u6548\u679c\u3002\u5728\u540c\u6837\u7684\u8bad\u7ec3\u6761\u4ef6\u4e0b\uff0c\u57fa\u4e8e\u9a8c\u8bc1\u8def\u5f84\u7684\u8bad\u7ec3\u6570\u636e\u4f7f\u6a21\u578b\u7684\u63a8\u7406\u51c6\u786e\u7387\u63d0\u9ad8\u4e865%\u4ee5\u4e0a\u3002<\/p>\n\n\n\n<p><strong>3. PPM\u7684\u4f5c\u7528<\/strong><br>\u5b9e\u9a8c\u8868\u660e\uff0cPPM\u4e0d\u4ec5\u80fd\u6709\u6548\u63d0\u5347\u590d\u6742\u95ee\u9898\u7684\u63a8\u7406\u7cbe\u5ea6\uff0c\u8fd8\u80fd\u81ea\u52a8\u8bc6\u522b\u5173\u952e\u7684\u5b9a\u7406\u5e94\u7528\u6b65\u9aa4\uff08\u5982\u8d39\u9a6c\u5c0f\u5b9a\u7406\u3001\u52fe\u80a1\u5b9a\u7406\u7b49\uff09\uff0c\u4e3a\u6a21\u578b\u63d0\u4f9b\u660e\u786e\u7684\u63a8\u7406\u6307\u5bfc\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e94\u3001\u5173\u952e\u53d1\u73b0<\/h5>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u81ea\u6211\u53cd\u601d\u80fd\u529b\u7684\u6d8c\u73b0<\/strong><br>\u5b9e\u9a8c\u53d1\u73b0\uff0crStar-Math\u5728\u63a8\u7406\u8fc7\u7a0b\u4e2d\u4f1a\u4e3b\u52a8\u8bc6\u522b\u9519\u8bef\u8def\u5f84\u5e76\u8c03\u6574\u7b56\u7565\uff0c\u8fd9\u79cd\u81ea\u6211\u53cd\u601d\u80fd\u529b\u6e90\u4e8eSystem 2\u63a8\u7406\u7684\u9010\u6b65\u9a8c\u8bc1\u673a\u5236\u3002<\/li>\n\n\n\n<li><strong>\u5b9a\u7406\u5e94\u7528\u7684\u504f\u597d\u8bc6\u522b<\/strong><br>PPM\u80fd\u591f\u8bc6\u522b\u51fa\u63a8\u7406\u8fc7\u7a0b\u4e2d\u5173\u952e\u7684\u5b9a\u7406\u5e94\u7528\u6b65\u9aa4\uff0c\u4e3a\u590d\u6742\u95ee\u9898\u7684\u89e3\u51b3\u63d0\u4f9b\u6307\u5bfc\u3002<\/li>\n\n\n\n<li><strong>\u5c0f\u6a21\u578b\u7684\u6f5c\u529b<\/strong><br>\u901a\u8fc7\u81ea\u6211\u751f\u6210\u548c\u4f18\u5316\uff0c\u5c0f\u6a21\u578b\u53ef\u4ee5\u5728\u4e0d\u4f9d\u8d56\u5927\u6a21\u578b\u84b8\u998f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u751a\u81f3\u8d85\u8d8a\u5927\u6a21\u578b\u7684\u6027\u80fd\u3002<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">\u516d\u3001\u603b\u7ed3\u4e0e\u5c55\u671b<\/h5>\n\n\n\n<p>rStar-Math\u901a\u8fc7\u7cfb\u7edf2\u5f0f\u63a8\u7406\u65b9\u6cd5\uff0c\u5c55\u73b0\u4e86\u5c0f\u6a21\u578b\u5728\u6570\u5b66\u63a8\u7406\u4efb\u52a1\u4e2d\u7684\u5de8\u5927\u6f5c\u529b\u3002\u672a\u6765\u5de5\u4f5c\u53ef\u8fdb\u4e00\u6b65\u63a2\u7d22\u66f4\u591a\u6837\u5316\u7684\u6570\u636e\u96c6\u548c\u5176\u4ed6\u9886\u57df\uff08\u5982\u4ee3\u7801\u63a8\u7406\u3001\u901a\u7528\u903b\u8f91\u63a8\u7406\uff09\u7684\u5e94\u7528\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587rStar-Math: Small LLMs Can Master Math Reasoning with [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8,36],"tags":[39,96,79,48,97],"class_list":["post-5549","post","type-post","status-publish","format-standard","hentry","category-tech","category-36","tag-ai","tag-deepseek","tag-microsoft","tag-48","tag-distillation"],"rttpg_featured_image_url":null,"rttpg_author":{"display_name":"NullThought","author_link":"https:\/\/nullthought.net\/?author=1"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/nullthought.net\/?cat=8\" rel=\"category\">Tech<\/a> <a href=\"https:\/\/nullthought.net\/?cat=36\" rel=\"category\">\u79d1\u5b66<\/a>","rttpg_excerpt":"\u8bba\u6587rStar-Math: Small LLMs Can Master Math Reasoning with&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5549","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5549"}],"version-history":[{"count":1,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5549\/revisions"}],"predecessor-version":[{"id":5551,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5549\/revisions\/5551"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5549"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5549"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5549"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}