{"id":1184654,"date":"2026-07-06T15:37:00","date_gmt":"2026-07-06T22:37:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-blog-post&#038;p=1184654"},"modified":"2026-09-01T02:36:16","modified_gmt":"2026-09-01T09:36:16","slug":"%e5%a4%a7%e6%a8%a1%e5%9e%8b%e9%a2%84%e8%ae%ad%e7%bb%83%e4%b8%ad%e7%9a%84%e5%8f%88%e7%a8%b3%e5%8f%88%e5%bf%ab%ef%bc%9asso-%e5%a6%82%e4%bd%95%e7%bb%9f%e4%b8%80%e8%ae%ad%e7%bb%83%e7%a8%b3%e5%ae%9a","status":"publish","type":"msr-blog-post","link":"https:\/\/www.microsoft.com\/en-us\/research\/articles\/%e5%a4%a7%e6%a8%a1%e5%9e%8b%e9%a2%84%e8%ae%ad%e7%bb%83%e4%b8%ad%e7%9a%84%e5%8f%88%e7%a8%b3%e5%8f%88%e5%bf%ab%ef%bc%9asso-%e5%a6%82%e4%bd%95%e7%bb%9f%e4%b8%80%e8%ae%ad%e7%bb%83%e7%a8%b3%e5%ae%9a\/","title":{"rendered":"\u5927\u6a21\u578b\u9884\u8bad\u7ec3\u4e2d\u7684&#8221;\u53c8\u7a33\u53c8\u5feb&#8221;\uff1aSSO \u5982\u4f55\u7edf\u4e00\u8bad\u7ec3\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\uff1f"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u5927\u6a21\u578b\u9884\u8bad\u7ec3\u4e00\u76f4\u662f\u4e00\u4ef6&#8221;\u65e2\u8981\u53c8\u8981&#8221;\u7684\u4e8b\u60c5\uff1a\u65e2\u5e0c\u671b\u8bad\u7ec3\u7a33\u5b9a\u3001\u53ef\u6301\u7eed\u6269\u5c55\uff0c\u53c8\u5e0c\u671b\u8bad\u7ec3\u6548\u7387\u8fbe\u5230\u6781\u81f4\u3002\u8fd9\u4e00\u8fc7\u7a0b\u5c31\u50cf\u201c\u6234\u7740\u9563\u94d0\u8df3\u821e\u201d\uff0c\u65e2\u8981\u6ee1\u8db3\u5404\u79cd\u7ea6\u675f\uff0c\u53c8\u8981\u5c3d\u53ef\u80fd\u53d1\u6325\u6a21\u578b\u6f5c\u529b\uff0c\u65e0\u7591\u662f\u4e00\u95e8\u517c\u5177\u7406\u8bba\u4e0e\u5de5\u7a0b\u4e4b\u7f8e\u7684 AI \u827a\u672f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fd1\u65e5\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u7684\u7814\u7a76\u5458\u4eec\u63d0\u51fa\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u4f18\u5316\u5668Spectral Sphere Optimizer\uff08SSO\uff09\u3002\u8be5\u5de5\u4f5c\u4ece \u03bcP\uff08Maximal Update Parametrization\uff09\u7406\u8bba\u51fa\u53d1\uff0c\u9996\u6b21\u5c06\u8bad\u7ec3\u7a33\u5b9a\u6027\u4e0e\u6700\u901f\u4e0b\u964d\u76ee\u6807\u7edf\u4e00\u5230\u540c\u4e00\u4e2a\u7406\u8bba\u6846\u67b6\u4e2d\uff0c\u63a8\u5bfc\u51fa\u4e86\u517c\u987e\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\u7684\u552f\u4e00\u6570\u5b66\u89e3\u3002\u5728\u5de5\u7a0b\u5b9e\u73b0\u4e0a\uff0cSSO \u901a\u8fc7\u8c31\u7403\u7ea6\u675f\u3001\u5207\u7a7a\u95f4\u66f4\u65b0\u4e0e\u8c31\u7403\u56de\u7f29\u7b49\u8bbe\u8ba1\uff0c\u5728\u65e0\u9700\u4f9d\u8d56 Weight Decay \u7b49\u4f20\u7edf\u7a33\u5b9a\u5316\u6280\u5de7\u7684\u60c5\u51b5\u4e0b\uff0c\u5b9e\u73b0\u4e86\u66f4\u52a0\u7a33\u5b9a\u3001\u9ad8\u6548\u7684\u5927\u6a21\u578b\u9884\u8bad\u7ec3\u3002\u76f8\u5173\u8bba\u6587\u5df2\u88abICML 2026\u4f5c\u4e3aOral\u5de5\u4f5c\u63a5\u6536\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><em>\u8bba\u6587\u4fe1\u606f\u5df2\u6574\u7406\u81f3\u6587\u672b\uff0c\u6b22\u8fce\u70b9\u51fb\u76f8\u5173\u94fe\u63a5\uff0c\u4e86\u89e3\u66f4\u591a\u6280\u672f\u8be6\u60c5\u3002<\/em><\/strong><\/p>\n\n\n\n<h2 id=\"\u4ec0\u4e48\u624d\u662f\u771f\u6b63\u7684\u8bad\u7ec3\u7a33\u5b9a\u6027\" class=\"wp-block-heading\">\u4ec0\u4e48\u624d\u662f\u771f\u6b63\u7684\u8bad\u7ec3\u7a33\u5b9a\u6027\uff1f<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">&#8220;\u5feb&#8221;\u5176\u5b9e\u5f88\u597d\u5b9a\u4e49\u2014\u2014\u8ba9&nbsp;Loss&nbsp;\u4e0b\u964d\u5f97\u6700\u5feb\u5373\u53ef\uff0c\u4e5f\u5c31\u662f\u7ecf\u5178\u7684\u6700\u901f\u4e0b\u964d\uff08steepest descent\uff09\u3002\u771f\u6b63\u56f0\u96be\u7684\u662f\uff0c&#8221;\u7a33&#8221;\u7a76\u7adf\u5e94\u8be5\u5982\u4f55\u5b9a\u4e49\uff1f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7814\u7a76\u5458\u4eec\u8ba4\u4e3a\uff0c\u4ece\u6839\u672c\u4e0a\u8bb2\uff0c\u53ef\u6301\u7eed\u3001\u53ef\u6269\u5c55\u7684\u7a33\u5b9a\u6027\uff0c\u5e94\u5f53\u6765\u81ea\u6a21\u578b\u81ea\u8eab\u7684&#8221;\u5065\u5eb7\u6307\u6807&#8221;\u3002\u5982\u679c\u6a21\u578b\u6240\u6709\u5c42\u7684\u8f93\u5165\u8f93\u51fa\u59cb\u7ec8\u4fdd\u6301\u7edf\u4e00\u7684\u5c3a\u5ea6\uff0c\u5e76\u4e14\u8fd9\u4e00\u5c3a\u5ea6\u4e0e\u5c42\u7684\u5f62\u72b6\u65e0\u5173\uff0c\u90a3\u4e48\u6fc0\u6d3b\u503c\uff08activation\uff09\u624d\u771f\u6b63\u8fbe\u5230\u4e86\u7a33\u5b9a\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e8b\u5b9e\u4e0a\uff0c\u8fd9\u6b63\u662f\u7ecf\u5178 \u03bcP&nbsp;\u7406\u8bba\u6240\u63cf\u8ff0\u7684\u6027\u8d28\u3002\u6ee1\u8db3 \u03bcP\u540e\uff0c\u6a21\u578b\u5373\u4f7f\u6269\u5c55\u5230\u4efb\u610f\u5bbd\u5ea6\uff0c\u4e5f\u80fd\u591f\u4fdd\u6301\u4e00\u81f4\u7684\u8bad\u7ec3\u52a8\u6001\uff0c\u5e76\u5e26\u6765\u8bb8\u591a\u91cd\u8981\u4f18\u52bf\uff0c\u4f8b\u5982\uff1a\u5c0f\u6a21\u578b\u641c\u7d22\u5f97\u5230\u7684\u5b66\u4e60\u7387\u7b49\u8d85\u53c2\u6570\uff0c\u53ef\u4ee5\u76f4\u63a5\u8fc1\u79fb\u81f3\u66f4\u5927\u7684\u6a21\u578b\uff1b\u65e0\u9700\u91cd\u65b0\u8c03\u53c2\u5373\u53ef\u5b8c\u6210\u6a21\u578b\u6269\u5c55\uff1b\u53ef\u4ee5\u7528\u4e8e\u62df\u5408\u89c4\u6a21\u6cd5\u5219\uff08scaling law\uff09\uff0c\u5e76\u9884\u6d4b\u6a21\u578b\u6700\u7ec8\u80fd\u591f\u8fbe\u5230\u7684\u6700\u4f18&nbsp;Loss\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u53e5\u8bdd\u6982\u62ec\uff0c\u672c\u7bc7\u8bba\u6587\u4ece\u03bcP&nbsp;\u7406\u8bba\u4f5c\u4e3a\u7a33\u5b9a\u6027\u6700\u4f73\u5ea6\u91cf\u8fd9\u4e00\u89c6\u89d2\u51fa\u53d1\uff0c\u63a8\u5bfc\u51fa\u4e86\u4f7fLoss&nbsp;\u6700\u901f\u4e0b\u964d\u7684\u4f18\u5316\u5668\uff0c\u800c&nbsp;SSO&nbsp;\u6b63\u662f\u8fd9\u4e00\u95ee\u9898\u552f\u4e00\u7684\u6570\u5b66\u89e3\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"757\" height=\"264\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-1.jpg\" alt=\"chart, line chart\" class=\"wp-image-1184600\" style=\"aspect-ratio:2.8675262607656693;object-fit:cover;width:770px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-1.jpg 757w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-1-300x105.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-1-240x84.jpg 240w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><figcaption class=\"wp-element-caption\">\u56fe1\uff1aDense-1.7B\u6a21\u578b\u6fc0\u6d3b\u503c\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u53d8\u5316\u8d8b\u52bf<\/figcaption><\/figure>\n\n\n\n<h2 id=\"\u4e3a\u4ec0\u4e48\u9700\u8981\u8c31\u7403\u7ea6\u675f\" class=\"wp-block-heading\">\u4e3a\u4ec0\u4e48\u9700\u8981\u8c31\u7403\u7ea6\u675f\uff1f<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u90a3\u4e48\uff0c\u5728&nbsp;SSO&nbsp;\u51fa\u73b0\u4e4b\u524d\uff0c\u4eba\u4eec\u662f\u5982\u4f55\u4fdd\u8bc1\u5927\u6a21\u578b\u8bad\u7ec3\u7a33\u5b9a\u6027\u7684\uff1f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u65b9\u6cd5\u5176\u5b9e\u5f88\u591a\u3002\u6700\u5e38\u89c1\u7684\u662f&nbsp;AdamW&nbsp;\u5f15\u5165\u7684Decoupled Weight Decay\uff08\u89e3\u8026\u5f0f\u6743\u91cd\u8870\u51cf\uff09\u3002\u5b83\u901a\u8fc7\u4e0d\u65ad\u5c06\u6743\u91cd\u5411&nbsp;0&nbsp;\u62c9\u8fd1\u6765\u7a33\u5b9a\u8bad\u7ec3\uff0c\u4f46\u8fd9\u4e5f\u4f1a\u5bfc\u81f4\u76f8\u5bf9\u66f4\u65b0\u91cf\uff08\u0394w\/w\uff09\u4e0d\u65ad\u53d8\u5316\uff0c\u5e76\u4e0e\u5b66\u4e60\u7387\u8026\u5408\uff0c\u4ece\u800c\u5e26\u6765\u590d\u6742\u4e14\u96be\u4ee5\u5206\u6790\u7684\u8bad\u7ec3\u52a8\u6001\u3002\u56e0\u6b64\uff0c\u5b66\u672f\u754c\u53c8\u63d0\u51fa\u4e86\u5404\u79cdWeight Decay Scheduler\uff08\u6743\u91cd\u8870\u51cf\u8c03\u5ea6\u7b56\u7565\uff09\uff0c\u5e0c\u671b\u8fdb\u4e00\u6b65\u6539\u5584\u8fd9\u4e00\u95ee\u9898\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u8d28\u4e0a\uff0c&#8221;\u6743\u91cd\u5f80&nbsp;0&nbsp;\u62c9&#8221;\u4e0e\u5b66\u4e60\u7387\u59cb\u7ec8\u5904\u4e8e\u4e00\u79cd\u535a\u5f08\u5173\u7cfb\u3002\u867d\u7136\u7406\u8bba\u4e0a\u80fd\u591f\u7ed9\u51fa\u6743\u91cd\u4e0a\u754c\uff08\u7ea6\u4e3a1\/\u03bb\uff09\uff0c\u4f46\u5b9e\u9645\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u4ecd\u7136\u5bb9\u6613\u51fa\u73b0\u5404\u79cd\u4e0d\u7a33\u5b9a\u73b0\u8c61\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53e6\u4e00\u65b9\u9762\uff0c\u5404\u79cd\u521d\u59cb\u5316\u6280\u5de7\u4e5f\u65e0\u6cd5\u4ece\u6839\u672c\u4e0a\u89e3\u51b3\u95ee\u9898\uff0c\u56e0\u4e3a\u968f\u7740\u8bad\u7ec3\u4e0d\u65ad\u8fdb\u884c\uff0c\u6743\u91cd\u6700\u7ec8\u90fd\u4f1a\u9010\u6e10\u504f\u79bb\u521d\u59cb\u5316\u5c3a\u5ea6\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9664\u6b64\u4e4b\u5916\uff0c\u8fd8\u6709\u5927\u91cf\u4f9d\u8d56\u67b6\u6784\u8bbe\u8ba1\u7684\u7a33\u5b9a\u5316\u6280\u5de7\uff0c\u4f8b\u5982\uff0cQK-Norm&nbsp;\u7528\u4e8e\u7a33\u5b9aAttention Logits\uff0cSandwich Norm&nbsp;\u7528\u4e8e\u540c\u65f6\u7ea6\u675f\u8f93\u5165\u4e0e\u8f93\u51fa\u5c3a\u5ea6\u3002\u8fd9\u4e9b\u65b9\u6cd5\u867d\u7136\u80fd\u591f\u63d0\u5347\u7a33\u5b9a\u6027\uff0c\u5374\u5f80\u5f80\u9700\u8981\u989d\u5916\u8c03\u53c2\uff0c\u751a\u81f3\u53ef\u80fd\u727a\u7272\u6a21\u578b\u80fd\u529b\u4e0a\u9650\uff0c\u5e76\u4e0d\u662f\u4e00\u79cd\u8db3\u591f\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u56e0\u6b64\uff0c\u7814\u7a76\u5458\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u66f4\u52a0\u7edf\u4e00\u7684\u65b9\u6cd5\uff1a\u65e2\u80fd\u591f\u6446\u8131&nbsp;Weight Decay\uff08\u6743\u91cd\u8870\u51cf\uff09\uff0c\u53c8\u80fd\u591f\u5c3d\u53ef\u80fd\u5730\u51cf\u5c11\u989d\u5916\u7684Norm&nbsp;\u8bbe\u8ba1\u3002<\/p>\n\n\n\n<h2 id=\"\u4ece-\u03bcp-\u7406\u8bba\u63a8\u5bfc\u552f\u4e00\u6570\u5b66\u89e3\" class=\"wp-block-heading\">\u4ece\u00a0\u03bcP\u00a0\u7406\u8bba\u63a8\u5bfc\u552f\u4e00\u6570\u5b66\u89e3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u5982\u679c\u5e0c\u671b\u6a21\u578b\u6fc0\u6d3b\u503c\u6ee1\u8db3 \u03bcP&nbsp;\u7406\u8bba\uff0c\u90a3\u4e48\u524d\u5411\u4f20\u64ad\u4e2d\u7684\u6743\u91cdW\uff0c\u4ee5\u53ca\u53cd\u5411\u4f20\u64ad\u4e2d\u7684\u66f4\u65b0\u91cf \u03a6=\u0394W\uff0c\u90fd\u5fc5\u987b\u6ee1\u8db3\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"397\" height=\"67\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-2.jpg\" alt=\"formula\" class=\"wp-image-1184601\" style=\"aspect-ratio:5.925925925925926;object-fit:cover;width:330px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-2.jpg 397w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-2-300x51.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-2-240x41.jpg 240w\" sizes=\"auto, (max-width: 397px) 100vw, 397px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d\u8c31\u8303\u6570\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"184\" height=\"39\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-3.png\" alt=\"formula\" class=\"wp-image-1184602\" style=\"width:230px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-3.png 184w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-3-180x39.png 180w\" sizes=\"auto, (max-width: 184px) 100vw, 184px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u7814\u7a76\u5458\u4eec\u53d1\u73b0\uff0c\u03bcP&nbsp;\u7406\u8bba\u5b9e\u9645\u4e0a\u5929\u7136\u8981\u6c42\u6743\u91cd\u548c\u66f4\u65b0\u90fd\u4f4d\u4e8e\u8c31\u8303\u6570\u7403\uff08Spectral Sphere\uff09\u7684\u7403\u9762\u4e0a\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fdb\u4e00\u6b65\u89c2\u5bdf\u8fd8\u53d1\u73b0\uff0cMuon&nbsp;\u4f18\u5316\u5668\u7684\u66f4\u65b0\u89c4\u5219\u6070\u597d\u6ee1\u8db3\u4e86 \u03bcP\u7406\u8bba\u4e2d\u7684\u540e\u4e00\u6761\u8981\u6c42\uff0c\u5373\u7ea6\u675f\u66f4\u65b0\u91cf\u7684\u8c31\u8303\u6570\u3002\u8fd9\u4e00\u5de7\u5408\u4e5f\u542f\u53d1\u7814\u7a76\u5458\u4eec\u601d\u8003\uff1a\u5982\u679c\u80fd\u591f\u540c\u65f6\u6ee1\u8db3 \u03bcP&nbsp;\u7406\u8bba\u7684\u5168\u90e8\u8981\u6c42\uff0c\u662f\u5426\u80fd\u591f\u5f97\u5230\u4e00\u4e2a\u66f4\u52a0\u7406\u60f3\u7684\u4f18\u5316\u5668\uff1f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u65e2\u7136\u901a\u8fc7\u8c31\u7403\u7ea6\u675f\u5df2\u7ecf\u4fdd\u8bc1\u4e86\u7a33\u5b9a\u6027\uff0c\u63a5\u4e0b\u6765\u9700\u8981\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u5c31\u662f\u5982\u4f55\u5728\u8fd9\u4e00\u7ea6\u675f\u4e0b\uff0c\u4f7f&nbsp;Loss&nbsp;\u4e0b\u964d\u5f97\u6700\u5feb\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Loss&nbsp;\u662f\u5173\u4e8e\u6743\u91cd&nbsp;W&nbsp;\u7684\u51fd\u6570\u3002\u7814\u7a76\u5458\u4eec\u9996\u5148\u5728\u5f53\u524d\u6743\u91cd\u5904\u5bf9&nbsp;Loss&nbsp;\u505a\u4e00\u9636\u6cf0\u52d2\u5c55\u5f00\uff0c\u4ece\u800c\u5f97\u5230\u6700\u901f\u4e0b\u964d\u5bf9\u5e94\u7684\u66f4\u65b0\u91cf \u03a6\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"494\" height=\"74\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-4.jpg\" alt=\"formula\" class=\"wp-image-1184603\" style=\"aspect-ratio:6.6763257026948715;width:393px;height:auto\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-4.jpg 494w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-4-300x45.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-4-240x36.jpg 240w\" sizes=\"auto, (max-width: 494px) 100vw, 494px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u7ed3\u5408\u524d\u8ff0\u8c31\u7403\u7ea6\u675f\uff0c\u5e76\u5c06\u8c31\u7403\u534a\u5f84\u8bb0\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"215\" height=\"66\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-5.png\" alt=\"formula\" class=\"wp-image-1184604\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u4fbf\u53ef\u4ee5\u5f97\u5230\u517c\u987e&#8221;\u53c8\u7a33\u53c8\u5feb&#8221;\u7684\u4f18\u5316\u76ee\u6807\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"379\" height=\"107\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-6.jpg\" alt=\"formula\" class=\"wp-image-1184605\" style=\"width:320px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-6.jpg 379w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-6-300x85.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-6-240x68.jpg 240w\" sizes=\"auto, (max-width: 379px) 100vw, 379px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fd9\u91cc\uff0c\u66f4\u65b0\u91cf\u7684\u8c31\u8303\u6570\u7ea6\u675f\u5199\u6210&nbsp;1\uff0c\u662f\u56e0\u4e3a\u6b64\u5904\u8003\u8651\u7684\u662f\u5355\u4f4d\u66f4\u65b0\uff0c\u5b9e\u9645\u8bad\u7ec3\u65f6\u534a\u5f84R&nbsp;\u53ef\u4ee5\u76f4\u63a5\u5438\u6536\u5230\u5b66\u4e60\u7387\uff08learning rate\uff09\u4e2d\u3002<\/p>\n\n\n\n<h2 id=\"\u8ba9\u66f4\u65b0\u59cb\u7ec8\u8d34\u7740\u7403\u9762\u4e0b\u964d\" class=\"wp-block-heading\">\u8ba9\u66f4\u65b0\u59cb\u7ec8\u8d34\u7740\u7403\u9762\u4e0b\u964d<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a5\u4e0b\u6765\uff0c\u8981\u5bf9\u4e0a\u8ff0\u4f18\u5316\u76ee\u6807\u7ee7\u7eed\u5316\u7b80\u3002\u7814\u7a76\u5458\u4eec\u9996\u5148\u5c06\u6700\u540e\u4e24\u5f0f\u76f8\u51cf\uff0c\u53ef\u5f97\u5230\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"351\" height=\"34\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-7.png\" alt=\"formula\" class=\"wp-image-1184606\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-7.png 351w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-7-300x29.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-7-240x23.png 240w\" sizes=\"auto, (max-width: 351px) 100vw, 351px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d\uff0c<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"239\" height=\"36\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-8.png\" alt=\"formula\" class=\"wp-image-1184607\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u8868\u793a\u5f53\u524d\u6743\u91cd\u5728\u8c31\u7403\u9762\u4e0a\u7684\u68af\u5ea6\u65b9\u5411\uff0c\u4e5f\u5c31\u662f\u8c31\u7403\u9762\u7684\u6cd5\u5411\u65b9\u5411\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3a\u4e86\u4f7f\u4e0a\u5f0f\u6052\u6210\u7acb\uff0c\u5219\u9700\u8981\u6ee1\u8db3\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"122\" height=\"37\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-9.png\" alt=\"formula\" class=\"wp-image-1184608\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fd9\u610f\u5473\u7740\uff0c\u66f4\u65b0\u65b9\u5411\u5fc5\u987b\u59cb\u7ec8\u4f4d\u4e8e\u8c31\u7403\u7684\u5207\u7a7a\u95f4\uff08tangent space\uff09\u5185\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\uff0c\u5c31\u662f\u66f4\u65b0\u59cb\u7ec8\u8d34\u7740\u7403\u9762\u4e0b\u964d\uff0c\u800c\u4e0d\u662f\u76f4\u63a5\u5411\u7403\u5fc3\u6536\u7f29\u3002\u8fd9\u6837\u65e2\u80fd\u591f\u4fdd\u6301\u6743\u91cd\u4e00\u76f4\u4f4d\u4e8e\u8c31\u7403\u9762\u4e0a\uff0c\u53c8\u4e0d\u4f1a\u635f\u5931\u68af\u5ea6\u5728\u6cd5\u5411\u65b9\u5411\u4e0a\u7684\u6709\u6548\u4fe1\u606f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e8e\u662f\uff0c\u6574\u4e2a\u95ee\u9898\u4fbf\u8f6c\u5316\u4e3a\u4e00\u4e2a\u7ecf\u5178\u7684\u7ebf\u6027\u4f18\u5316\u95ee\u9898\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"566\" height=\"54\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-10.png\" alt=\"formula\" class=\"wp-image-1184609\" style=\"width:459px;height:auto\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-10.png 566w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-10-300x29.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-10-240x23.png 240w\" sizes=\"auto, (max-width: 566px) 100vw, 566px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u7814\u7a76\u5458\u4eec\u8fdb\u4e00\u6b65\u91c7\u7528\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5\u8fdb\u884c\u6c42\u89e3\uff0c\u6784\u9020\u62c9\u683c\u6717\u65e5\u51fd\u6570\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"52\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/\u51fd\u6570.png\" alt=\"formula\" class=\"wp-image-1184625\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/\u51fd\u6570.png 250w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/\u51fd\u6570-240x50.png 240w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u7ea6\u675f\u6761\u4ef6\u4e3a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/mmbiz.qpic.cn\/sz_mmbiz_png\/FkDx3ORhNPSfib0MPR8TpMPOXaYjoDxPdONkj1g4ianPe43oEhTwr1DIbfCAdLl6BVHIcbsBr5JwgLyMKQglqm6G5B4Oo3GgicRZTeJ2a195NY\/640?wx_fmt=png&from=appmsg&tp=wxpic&wxfrom=5&wx_lazy=1#imgIndex=18\" alt=\"formula\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/mmbiz.qpic.cn\/mmbiz_jpg\/FkDx3ORhNPRUmjr3s5chBP0zKKeAZnnYa71Igicjf4d3n7wBiazllvOabiaTP3Dh91wf4W6ltG8v1tGUfabcWMTPvrCDdRODHic8y2RojTR0QIQ\/640?wx_fmt=jpeg&from=appmsg&tp=wxpic&wxfrom=5&wx_lazy=1#imgIndex=19\" alt=\"solution\" style=\"width:670px\"\/><figcaption class=\"wp-element-caption\">\u56fe2\uff1aSSO \u4e2d\u62c9\u683c\u6717\u65e5\u51fd\u6570&nbsp;h(\u03bb)&nbsp;\u7684\u7406\u8bba\u6027\u8d28\u3002\u7531\u4e8e&nbsp;h(\u03bb)&nbsp;\u5355\u8c03\u4e0d\u51cf\u4e14\u96f6\u70b9\u4f4d\u4e8e\u6709\u754c\u533a\u95f4\u5185\uff0c\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u4e8c\u5206\u641c\u7d22\u9ad8\u6548\u6c42\u89e3\u552f\u4e00\u7684\u62c9\u683c\u6717\u65e5\u4e58\u5b50&nbsp;\u03bb\u22c6&nbsp;\u3002<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u6700\u7ec8\u4fbf\u53ef\u5f97\u5230\u6700\u901f\u4e0b\u964d\u5bf9\u5e94\u7684\u6700\u4f18\u66f4\u65b0<img loading=\"lazy\" decoding=\"async\" width=\"55\" height=\"41\" class=\"wp-image-1184626\" style=\"width: 55px;\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/\u5c0f\u7b26\u53f71.png\" alt=\"image\">\uff0c\u5176\u4e2d\u62c9\u683c\u6717\u65e5\u4e58\u5b50&nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"38\" class=\"wp-image-1184628\" style=\"width: 31px;\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/\u5c0f\u7b26\u53f72.jpg\" alt=\"image\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6ee1\u8db3<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"510\" height=\"78\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-13.png\" alt=\"formula\" class=\"wp-image-1184629\" style=\"width:381px;height:auto\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-13.png 510w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-13-300x46.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-13-240x37.png 240w\" sizes=\"auto, (max-width: 510px) 100vw, 510px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e0e&nbsp;Muon&nbsp;\u5bf9\u6bd4\uff0c\u53ef\u4ee5\u66f4\u76f4\u89c2\u5730\u7406\u89e3\u8fd9\u4e00\u7ed3\u679c\u3002Muon&nbsp;\u7684\u66f4\u65b0\u89c4\u5219\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"101\" height=\"32\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-14.png\" alt=\"muon\" class=\"wp-image-1184630\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u800c&nbsp;SSO&nbsp;\u5219\u5f15\u5165\u4e86\u4e00\u4e2a\u5fae\u6270\u9879\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"190\" height=\"36\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-15.png\" alt=\"formula\" class=\"wp-image-1184631\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f7f\u66f4\u65b0\u65b9\u5411\u80fd\u591f\u4e25\u683c\u6295\u5f71\u5230\u8c31\u7403\u5207\u7a7a\u95f4\uff0c\u4ece\u800c\u5728\u6ee1\u8db3 \u03bcP&nbsp;\u7406\u8bba\u7ea6\u675f\u7684\u540c\u65f6\uff0c\u5b9e\u73b0\u771f\u6b63\u610f\u4e49\u4e0a\u7684\u6700\u901f\u4e0b\u964d\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9664\u4e86\u4e0a\u8ff0\u4ee3\u6570\u63a8\u5bfc\u4e4b\u5916\uff0c\u672c\u7bc7\u8bba\u6587\u8fd8\u7ed9\u51fa\u4e86\u5bf9\u5e94\u7684\u51e0\u4f55\u89e3\u91ca\uff0c\u4f7f\u6574\u4e2a\u4f18\u5316\u8fc7\u7a0b\u5177\u6709\u66f4\u52a0\u76f4\u89c2\u7684\u7269\u7406\u610f\u4e49\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"533\" height=\"307\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-16.jpg\" alt=\"diagram\" class=\"wp-image-1184632\" style=\"width:560px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-16.jpg 533w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-16-300x173.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-16-240x138.jpg 240w\" sizes=\"auto, (max-width: 533px) 100vw, 533px\" \/><figcaption class=\"wp-element-caption\">\u56fe3\uff1a\u6700\u901f\u4e0b\u964d\u66f4\u65b0\u65b9\u5411\u7684\u51e0\u4f55\u7ed3\u6784<\/figcaption><\/figure>\n\n\n\n<h2 id=\"\u5982\u4f55\u8ba9sso-\u9ad8\u6548\u8fd0\u884c\" class=\"wp-block-heading\">\u5982\u4f55\u8ba9SSO\u00a0\u9ad8\u6548\u8fd0\u884c<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fdb\u4e00\u6b65\u89c2\u5bdf\u8fd8\u80fd\u53d1\u73b0\uff0c\u62c9\u683c\u6717\u65e5\u51fd\u6570&nbsp;h(\u03bb)&nbsp;\u5355\u8c03\u9012\u589e\uff0c\u5e76\u4e14\u5176\u96f6\u70b9\u6709\u754c\uff0c\u4f46\u76ee\u524d\u5c1a\u672a\u627e\u5230\u89e3\u6790\u89e3\uff0c\u56e0\u6b64\u5b9e\u9645\u6c42\u89e3\u4ecd\u9700\u8981\u501f\u52a9\u6570\u503c\u8fed\u4ee3\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u5c31\u662f\u4e8c\u5206\u641c\u7d22\uff0c\u800c&nbsp;SSO&nbsp;\u7684\u5b9e\u73b0\u4e5f\u6b63\u662f\u91c7\u7528\u4e86\u8fd9\u4e00\u7b56\u7565\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7814\u7a76\u5458\u4eec\u8fd8\u89c2\u5bdf\u5230\uff0c\u03bb \u7684\u5f62\u5f0f\u4f3c\u4e4e\u53ef\u4ee5\u5199\u6210\u4e00\u4e2a\u4ec5\u4f9d\u8d56\u68af\u5ea6&nbsp;G&nbsp;\u7684\u5747\u503c\uff08mean\uff09\u548c\u6807\u51c6\u5dee\uff08std\uff09\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\uff0c\u800c\u4e0e\u5f53\u524d\u6743\u91cd&nbsp;W&nbsp;\u65e0\u5173\u3002\u4e0d\u8fc7\uff0c\u76ee\u524d\u8fd9\u4e00\u7ed3\u8bba\u5c1a\u672a\u5b8c\u6210\u4e25\u683c\u8bc1\u660e\uff0c\u56e0\u6b64\u8bba\u6587\u53ea\u7ed9\u51fa\u4e86\u4e00\u4e2a\u8f83\u5bbd\u677e\u7684\u7406\u8bba\u754c\uff0c\u4e5f\u671f\u5f85\u672a\u6765\u8fdb\u4e00\u6b65\u5b8c\u5584\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"785\" height=\"267\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-17.jpg\" alt=\"chart, line chart\" class=\"wp-image-1184633\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-17.jpg 785w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-17-300x102.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-17-768x261.jpg 768w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-17-240x82.jpg 240w\" sizes=\"auto, (max-width: 785px) 100vw, 785px\" \/><figcaption class=\"wp-element-caption\">\u56fe4\uff1a\u968f\u673a\u77e9\u9635\u6761\u4ef6\u4e0bh(\u03bb)=\u27e8\u0398,msign(G+\u03bb\u0398)\u27e9\u7684\u7ecf\u9a8c\u66f2\u7ebf<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u81f3\u6b64\uff0cSSO&nbsp;\u7684\u7406\u8bba\u63a8\u5bfc\u5df2\u7ecf\u57fa\u672c\u5b8c\u6210\u3002\u5176\u672c\u8d28\u4e0a\u5c31\u662f\u4e00\u4e2a\u6ee1\u8db3\u53cc\u8c31\u7403\u7ea6\u675f\u7684\u6700\u901f\u4e0b\u964d\u95ee\u9898\u3002\u501f\u52a9\u62c9\u683c\u6717\u65e5\u51fd\u6570\u5355\u8c03\u4e14\u96f6\u70b9\u6709\u754c\u8fd9\u4e00\u5173\u952e\u89c2\u5bdf\uff0c\u901a\u8fc7\u4e8c\u5206\u641c\u7d22\u5373\u53ef\u6c42\u5f97\u62c9\u683c\u6717\u65e5\u4e58\u5b50\uff0c\u8fdb\u800c\u5f97\u5230\u5b9e\u9645\u66f4\u65b0\u65b9\u5411\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e0d\u8fc7\uff0c\u5728\u5b9e\u9645\u5de5\u7a0b\u8bad\u7ec3\u4e2d\uff0c\u53ea\u4f9d\u8d56\u4e00\u9636\u5207\u7a7a\u95f4\u66f4\u65b0\uff0c\u7ecf\u8fc7\u957f\u65f6\u95f4\u8bad\u7ec3\u540e\u4ecd\u53ef\u80fd\u4e0d\u65ad\u7d2f\u79ef\u6570\u503c\u8bef\u5dee\uff0c\u4f7f\u6743\u91cd\u7684\u8c31\u8303\u6570\u9010\u6e10\u504f\u79bb\u76ee\u6807\u8c31\u7403\u3002\u56e0\u6b64\uff0cSSO\u5f15\u5165\u4e86\u4e00\u6b65\u7b80\u5355\u800c\u6709\u6548\u7684\u8c31\u7403\u56de\u7f29\uff08Spectral Retraction\uff09\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"175\" height=\"62\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4.png\" alt=\"formula\" class=\"wp-image-1184634\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u5373\u5bf9\u6574\u4e2a\u6743\u91cd\u77e9\u9635\u8fdb\u884c\u7edf\u4e00\u7f29\u653e\uff0c\u4f7f\u5176\u6700\u5927\u5947\u5f02\u503c\u59cb\u7ec8\u4fdd\u6301\u4e3a\u76ee\u6807\u534a\u5f84&nbsp;R\uff0c\u4ee5\u6d88\u9664\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u4e0d\u65ad\u79ef\u7d2f\u7684\u8bef\u5dee\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u76f8\u6bd4\u8ba1\u7b97&nbsp;msign\uff0c\u5229\u7528\u5e42\u8fed\u4ee3\u6cd5\uff08Power Iteration\uff09\u6c42\u53d6\u6700\u5927\u5947\u5f02\u503c\u7684\u4ee3\u4ef7\u8981\u4f4e\u5f97\u591a\u3002\u7814\u7a76\u5458\u4eec\u89c2\u5bdf\u53d1\u73b0\uff0c\u6a21\u578b\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u8c31\u8303\u6570\u53d8\u5316\u901a\u5e38\u8f83\u4e3a\u5e73\u7f13\uff0c\u5bf9\u5e94\u7684\u5de6\u53f3\u5947\u5f02\u5411\u91cfu\u3001v&nbsp;\u4e5f\u5177\u6709\u8f83\u597d\u7684\u8fde\u7eed\u6027\u3002\u56e0\u6b64\uff0c\u7814\u7a76\u5458\u4eec\u5c06\u4e0a\u4e00\u8f6e\u8ba1\u7b97\u5f97\u5230\u7684\u5947\u5f02\u5411\u91cf\u7f13\u5b58\u4e0b\u6765\uff0c\u4f5c\u4e3a\u4e0b\u4e00\u8f6ePower Iteration&nbsp;\u7684\u521d\u59cb\u5316\uff0c\u4ec5\u9700\u5c11\u91cf\u8fed\u4ee3\u5373\u53ef\u8fbe\u5230\u51e0\u4e4e\u76f8\u540c\u7684\u8ba1\u7b97\u7cbe\u5ea6\uff0c\u4ece\u800c\u663e\u8457\u964d\u4f4e\u4e86\u989d\u5916\u7684\u8ba1\u7b97\u5f00\u9500\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53e6\u4e00\u65b9\u9762\uff0c\u7814\u7a76\u5458\u4eec\u4e5f\u5c1d\u8bd5\u8fc7&nbsp;mclip&nbsp;\u65b9\u6848\uff0c\u5373\u4ec5\u5bf9\u6700\u5927\u5947\u5f02\u503c\u8fdb\u884c\u88c1\u526a\uff0c\u800c\u4fdd\u6301\u5176\u4ed6\u5947\u5f02\u503c\u4e0d\u53d8\u3002\u4ece\u7406\u8bba\u4e0a\u770b\uff0c\u8fd9\u662f\u4e00\u79cd\u66f4\u52a0\u7cbe\u7ec6\u7684\u8c31\u7ea6\u675f\u65b9\u5f0f\u3002\u4f46\u5728\u5b9e\u9645\u5b9e\u9a8c\u4e2d\uff0c\u8be5\u65b9\u6848\u4ecd\u7136\u9047\u5230\u4e86\u8f83\u96be\u514b\u670d\u7684\u6570\u503c\u7cbe\u5ea6\u95ee\u9898\uff0c\u540c\u65f6\u9700\u8981\u5f15\u5165\u66f4\u591a\u7684msign&nbsp;\u8ba1\u7b97\uff0c\u6574\u4f53\u6536\u76ca\u5e76\u4e0d\u660e\u663e\uff0c\u56e0\u6b64\u6700\u7ec8\u6ca1\u6709\u91c7\u7528\u3002<\/p>\n\n\n\n<h2 id=\"\u7528\u8c31\u7403\u66ff\u4ee3\u6743\u91cd\u8870\u51cf\" class=\"wp-block-heading\">\u7528\u8c31\u7403\u66ff\u4ee3\u6743\u91cd\u8870\u51cf<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u8c31\u7403\u56de\u7f29\u8fd8\u6709\u4e00\u4e2a\u91cd\u8981\u6536\u76ca\uff1a\u5b83\u57fa\u672c\u6d88\u9664\u4e86\u9690\u85cf\u5c42\u4f7f\u7528\u6743\u91cd\u8870\u51cf\u7684\u5fc5\u8981\u6027\u3002\u65e2\u7136\u8c31\u7403\u534a\u5f84&nbsp;R&nbsp;\u5df2\u7ecf\u5b9a\u4e49\u4e86\u6743\u91cd\u7684\u6700\u4f73\u5c3a\u5ea6\uff0c\u5e76\u4e14\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u59cb\u7ec8\u4fdd\u6301\u5728\u8fd9\u4e00\u5c3a\u5ea6\u4e0a\uff0c\u90a3\u4e48\u6743\u91cd\u8870\u51cf\u7684\u4f5c\u7528\u5b9e\u9645\u4e0a\u5df2\u7ecf\u88ab\u66ff\u4ee3\u3002\u56e0\u6b64\uff0cSSO\u6210\u529f\u79fb\u9664\u4e86\u9690\u85cf\u5c42\u6743\u91cd\u8870\u51cf\u8fd9\u4e00\u5386\u53f2\u9057\u7559\u8d85\u53c2\u6570\uff0c\u5927\u5e45\u964d\u4f4e\u4e86\u8c03\u53c2\u590d\u6742\u5ea6\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e0e\u6b64\u540c\u65f6\uff0c\u7814\u7a76\u5458\u4eec\u4e5f\u83b7\u5f97\u4e86\u5bf9\u6bcf\u4e2a&nbsp;Attention \/ FFN&nbsp;\u6a21\u5757\u8f93\u5165\u8f93\u51fa\u5c3a\u5ea6\u7684\u5b8c\u5168\u63a7\u5236\u80fd\u529b\u3002\u901a\u8fc7\u8c03\u8282\u53c2\u6570c\uff0c\u5373\u53ef\u6574\u4f53\u63a7\u5236\u5404\u6a21\u5757\u8f93\u51fa\u5c3a\u5ea6\uff0c\u4f7f\u6fc0\u6d3b\u503crms\u5728\u6574\u4e2a\u7f51\u7edc\u4e2d\u4fdd\u6301\u4e00\u81f4\uff0c\u540c\u65f6\u6709\u6548\u6291\u5236\u5f02\u5e38\u503c\uff0c\u8bad\u7ec3\u8fc7\u7a0b\u8868\u73b0\u5f97\u66f4\u52a0\u5065\u5eb7\u3001\u7a33\u5b9a\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"763\" height=\"217\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-18.jpg\" alt=\"graphical user interface, line chart\" class=\"wp-image-1184635\" style=\"width:770px\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-18.jpg 763w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-18-300x85.jpg 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2026\/08\/4-18-240x68.jpg 240w\" sizes=\"auto, (max-width: 763px) 100vw, 763px\" \/><figcaption class=\"wp-element-caption\">\u56fe5\uff1a\u534a\u5f84\u7f29\u653e\u5bf9\u4f18\u5316\u8fc7\u7a0b\u4e0e\u6fc0\u6d3b\u503c\u5c3a\u5ea6\u7684\u5f71\u54cd<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fdb\u4e00\u6b65\u6765\u770b\uff0cc&nbsp;\u8fd8\u53ef\u4ee5\u7406\u89e3\u4e3aAttention\/FFN&nbsp;\u5206\u652f\u4e0e\u6b8b\u5dee\u6d41\u4e4b\u95f4\u7684\u6df7\u5408\u6bd4\u4f8b\uff0c\u5b83\u540c\u65f6\u51b3\u5b9a\u4e86\u6a21\u578b\u6700\u4f18\u521d\u59cb\u5316\u5c3a\u5ea6\u4ee5\u53ca\u6700\u4f18\u6fc0\u6d3b\u503c\u5c3a\u5ea6\uff0c\u4e5f\u662f\u672a\u6765\u503c\u5f97\u6df1\u5165\u7814\u7a76\u7684\u91cd\u8981\u65b9\u5411\u3002<\/p>\n\n\n\n<h2 id=\"sso-\u517c\u987e\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\u7684\u65b0\u4f18\u5316\u6846\u67b6\" class=\"wp-block-heading\">SSO\uff1a\u517c\u987e\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\u7684\u65b0\u4f18\u5316\u6846\u67b6<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u81f3\u6b64\uff0cSSO&nbsp;\u7684\u6574\u4f53\u6d41\u7a0b\u5df2\u7ecf\u5341\u5206\u6e05\u6670\uff1a\u8ba1\u7b97\u6700\u5927\u5947\u5f02\u503c \u2192 \u56de\u7f29\u81f3\u8c31\u7403\u9762 \u2192 \u5728\u8c31\u7403\u5207\u7a7a\u95f4\u5185\u8ba1\u7b97\u66f4\u65b0 \u2192 \u5b8c\u6210\u53c2\u6570\u66f4\u65b0\uff0c\u5982\u6b64\u5faa\u73af\uff0c\u5373\u6784\u6210\u4e86\u5b8c\u6574\u7684&nbsp;SSO&nbsp;\u4f18\u5316\u6d41\u7a0b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5b9e\u9a8c\u7ed3\u679c\u4e5f\u9a8c\u8bc1\u4e86\u8fd9\u4e00\u8bbe\u8ba1\u3002\u4ece\u6a21\u578b\u7684\u201c\u5185\u79d1\u201d\u6307\u6807\u6765\u770b\uff0cSSO&nbsp;\u80fd\u591f\u4fdd\u6301\u51e0\u4e4e\u7406\u60f3\u7684\u6fc0\u6d3b\u503crms\u66f2\u7ebf\uff0c\u5e76\u663e\u8457\u6291\u5236\u5f02\u5e38\u503c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0cAdamW&nbsp;\u7684\u5f02\u5e38\u503c\u5f80\u5f80\u4f1a\u589e\u957f\u5230\u767e\u500d\u4ee5\u4e0a\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ece\u6700\u7ec8\u8bad\u7ec3\u6548\u679c\u6765\u770b\uff0cSSO&nbsp;\u5728\u4e0d\u540c\u6a21\u578b\u89c4\u6a21\u4e0b\u5747\u53d6\u5f97\u4e86\u6bd4AdamW&nbsp;\u548c&nbsp;Muon&nbsp;\u66f4\u4f4e\u7684&nbsp;Loss\uff0c\u540c\u65f6\u4fdd\u6301\u4e86\u4f18\u79c0\u7684\u5b66\u4e60\u7387\u8f6c\u5316\u80fd\u529b\uff0c\u5b9e\u73b0\u4e86\u8bad\u7ec3\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\u7684\u7edf\u4e00\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Controlled LLM Training on Spectral Sphere (ICML 2026 Oral)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bba\u6587\u94fe\u63a5\uff1a<a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"https:\/\/arxiv.org\/abs\/2601.08393\">https:\/\/arxiv.org\/abs\/2601.08393<span class=\"sr-only\"> (opens in new tab)<\/span><\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9879\u76ee\u9875\u9762\uff1a<a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"https:\/\/github.com\/Unakar\/Spectral-Sphere-Optimizer\">https:\/\/github.com\/Unakar\/Spectral-Sphere-Optimizer<span class=\"sr-only\"> (opens in new tab)<\/span><\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Megatron\u4ee3\u7801\uff1a<a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"https:\/\/github.com\/Unakar\/Megatron-LM\/tree\/SSO_main\">https:\/\/github.com\/Unakar\/Megatron-LM\/tree\/SSO_main<span class=\"sr-only\"> (opens in new tab)<\/span><\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5927\u6a21\u578b\u9884\u8bad\u7ec3\u4e00\u76f4\u662f\u4e00\u4ef6&#8221;\u65e2\u8981\u53c8\u8981&#8221;\u7684\u4e8b\u60c5\uff1a\u65e2\u5e0c\u671b\u8bad\u7ec3\u7a33\u5b9a\u3001\u53ef\u6301\u7eed\u6269\u5c55\uff0c\u53c8\u5e0c\u671b\u8bad\u7ec3\u6548\u7387\u8fbe\u5230\u6781\u81f4\u3002\u8fd9\u4e00\u8fc7\u7a0b\u5c31\u50cf\u201c\u6234\u7740\u9563\u94d0\u8df3\u821e\u201d\uff0c\u65e2\u8981\u6ee1\u8db3\u5404\u79cd\u7ea6\u675f\uff0c\u53c8\u8981\u5c3d\u53ef\u80fd\u53d1\u6325\u6a21\u578b\u6f5c\u529b\uff0c\u65e0\u7591\u662f\u4e00\u95e8\u517c\u5177\u7406\u8bba\u4e0e\u5de5\u7a0b\u4e4b\u7f8e\u7684 AI \u827a\u672f\u3002 \u8fd1\u65e5\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u7684\u7814\u7a76\u5458\u4eec\u63d0\u51fa\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u4f18\u5316\u5668Spectral Sphere Optimizer\uff08SSO\uff09\u3002\u8be5\u5de5\u4f5c\u4ece \u03bcP\uff08Maximal Update Parametrization\uff09\u7406\u8bba\u51fa\u53d1\uff0c\u9996\u6b21\u5c06\u8bad\u7ec3\u7a33\u5b9a\u6027\u4e0e\u6700\u901f\u4e0b\u964d\u76ee\u6807\u7edf\u4e00\u5230\u540c\u4e00\u4e2a\u7406\u8bba\u6846\u67b6\u4e2d\uff0c\u63a8\u5bfc\u51fa\u4e86\u517c\u987e\u7a33\u5b9a\u6027\u4e0e\u8bad\u7ec3\u6548\u7387\u7684\u552f\u4e00\u6570\u5b66\u89e3\u3002\u5728\u5de5\u7a0b\u5b9e\u73b0\u4e0a\uff0cSSO \u901a\u8fc7\u8c31\u7403\u7ea6\u675f\u3001\u5207\u7a7a\u95f4\u66f4\u65b0\u4e0e\u8c31\u7403\u56de\u7f29\u7b49\u8bbe\u8ba1\uff0c\u5728\u65e0\u9700\u4f9d\u8d56 Weight Decay \u7b49\u4f20\u7edf\u7a33\u5b9a\u5316\u6280\u5de7\u7684\u60c5\u51b5\u4e0b\uff0c\u5b9e\u73b0\u4e86\u66f4\u52a0\u7a33\u5b9a\u3001\u9ad8\u6548\u7684\u5927\u6a21\u578b\u9884\u8bad\u7ec3\u3002\u76f8\u5173\u8bba\u6587\u5df2\u88abICML 2026\u4f5c\u4e3aOral\u5de5\u4f5c\u63a5\u6536\u3002 \u8bba\u6587\u4fe1\u606f\u5df2\u6574\u7406\u81f3\u6587\u672b\uff0c\u6b22\u8fce\u70b9\u51fb\u76f8\u5173\u94fe\u63a5\uff0c\u4e86\u89e3\u66f4\u591a\u6280\u672f\u8be6\u60c5\u3002 &#8220;\u5feb&#8221;\u5176\u5b9e\u5f88\u597d\u5b9a\u4e49\u2014\u2014\u8ba9&nbsp;Loss&nbsp;\u4e0b\u964d\u5f97\u6700\u5feb\u5373\u53ef\uff0c\u4e5f\u5c31\u662f\u7ecf\u5178\u7684\u6700\u901f\u4e0b\u964d\uff08steepest descent\uff09\u3002\u771f\u6b63\u56f0\u96be\u7684\u662f\uff0c&#8221;\u7a33&#8221;\u7a76\u7adf\u5e94\u8be5\u5982\u4f55\u5b9a\u4e49\uff1f \u7814\u7a76\u5458\u4eec\u8ba4\u4e3a\uff0c\u4ece\u6839\u672c\u4e0a\u8bb2\uff0c\u53ef\u6301\u7eed\u3001\u53ef\u6269\u5c55\u7684\u7a33\u5b9a\u6027\uff0c\u5e94\u5f53\u6765\u81ea\u6a21\u578b\u81ea\u8eab\u7684&#8221;\u5065\u5eb7\u6307\u6807&#8221;\u3002\u5982\u679c\u6a21\u578b\u6240\u6709\u5c42\u7684\u8f93\u5165\u8f93\u51fa\u59cb\u7ec8\u4fdd\u6301\u7edf\u4e00\u7684\u5c3a\u5ea6\uff0c\u5e76\u4e14\u8fd9\u4e00\u5c3a\u5ea6\u4e0e\u5c42\u7684\u5f62\u72b6\u65e0\u5173\uff0c\u90a3\u4e48\u6fc0\u6d3b\u503c\uff08activation\uff09\u624d\u771f\u6b63\u8fbe\u5230\u4e86\u7a33\u5b9a\u3002 \u4e8b\u5b9e\u4e0a\uff0c\u8fd9\u6b63\u662f\u7ecf\u5178 \u03bcP&nbsp;\u7406\u8bba\u6240\u63cf\u8ff0\u7684\u6027\u8d28\u3002\u6ee1\u8db3 \u03bcP\u540e\uff0c\u6a21\u578b\u5373\u4f7f\u6269\u5c55\u5230\u4efb\u610f\u5bbd\u5ea6\uff0c\u4e5f\u80fd\u591f\u4fdd\u6301\u4e00\u81f4\u7684\u8bad\u7ec3\u52a8\u6001\uff0c\u5e76\u5e26\u6765\u8bb8\u591a\u91cd\u8981\u4f18\u52bf\uff0c\u4f8b\u5982\uff1a\u5c0f\u6a21\u578b\u641c\u7d22\u5f97\u5230\u7684\u5b66\u4e60\u7387\u7b49\u8d85\u53c2\u6570\uff0c\u53ef\u4ee5\u76f4\u63a5\u8fc1\u79fb\u81f3\u66f4\u5927\u7684\u6a21\u578b\uff1b\u65e0\u9700\u91cd\u65b0\u8c03\u53c2\u5373\u53ef\u5b8c\u6210\u6a21\u578b\u6269\u5c55\uff1b\u53ef\u4ee5\u7528\u4e8e\u62df\u5408\u89c4\u6a21\u6cd5\u5219\uff08scaling law\uff09\uff0c\u5e76\u9884\u6d4b\u6a21\u578b\u6700\u7ec8\u80fd\u591f\u8fbe\u5230\u7684\u6700\u4f18&nbsp;Loss\u3002 \u4e00\u53e5\u8bdd\u6982\u62ec\uff0c\u672c\u7bc7\u8bba\u6587\u4ece\u03bcP&nbsp;\u7406\u8bba\u4f5c\u4e3a\u7a33\u5b9a\u6027\u6700\u4f73\u5ea6\u91cf\u8fd9\u4e00\u89c6\u89d2\u51fa\u53d1\uff0c\u63a8\u5bfc\u51fa\u4e86\u4f7fLoss&nbsp;\u6700\u901f\u4e0b\u964d\u7684\u4f18\u5316\u5668\uff0c\u800c&nbsp;SSO&nbsp;\u6b63\u662f\u8fd9\u4e00\u95ee\u9898\u552f\u4e00\u7684\u6570\u5b66\u89e3\u3002 \u90a3\u4e48\uff0c\u5728&nbsp;SSO&nbsp;\u51fa\u73b0\u4e4b\u524d\uff0c\u4eba\u4eec\u662f\u5982\u4f55\u4fdd\u8bc1\u5927\u6a21\u578b\u8bad\u7ec3\u7a33\u5b9a\u6027\u7684\uff1f \u65b9\u6cd5\u5176\u5b9e\u5f88\u591a\u3002\u6700\u5e38\u89c1\u7684\u662f&nbsp;AdamW&nbsp;\u5f15\u5165\u7684Decoupled Weight Decay\uff08\u89e3\u8026\u5f0f\u6743\u91cd\u8870\u51cf\uff09\u3002\u5b83\u901a\u8fc7\u4e0d\u65ad\u5c06\u6743\u91cd\u5411&nbsp;0&nbsp;\u62c9\u8fd1\u6765\u7a33\u5b9a\u8bad\u7ec3\uff0c\u4f46\u8fd9\u4e5f\u4f1a\u5bfc\u81f4\u76f8\u5bf9\u66f4\u65b0\u91cf\uff08\u0394w\/w\uff09\u4e0d\u65ad\u53d8\u5316\uff0c\u5e76\u4e0e\u5b66\u4e60\u7387\u8026\u5408\uff0c\u4ece\u800c\u5e26\u6765\u590d\u6742\u4e14\u96be\u4ee5\u5206\u6790\u7684\u8bad\u7ec3\u52a8\u6001\u3002\u56e0\u6b64\uff0c\u5b66\u672f\u754c\u53c8\u63d0\u51fa\u4e86\u5404\u79cdWeight Decay Scheduler\uff08\u6743\u91cd\u8870\u51cf\u8c03\u5ea6\u7b56\u7565\uff09\uff0c\u5e0c\u671b\u8fdb\u4e00\u6b65\u6539\u5584\u8fd9\u4e00\u95ee\u9898\u3002 \u672c\u8d28\u4e0a\uff0c&#8221;\u6743\u91cd\u5f80&nbsp;0&nbsp;\u62c9&#8221;\u4e0e\u5b66\u4e60\u7387\u59cb\u7ec8\u5904\u4e8e\u4e00\u79cd\u535a\u5f08\u5173\u7cfb\u3002\u867d\u7136\u7406\u8bba\u4e0a\u80fd\u591f\u7ed9\u51fa\u6743\u91cd\u4e0a\u754c\uff08\u7ea6\u4e3a1\/\u03bb\uff09\uff0c\u4f46\u5b9e\u9645\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u4ecd\u7136\u5bb9\u6613\u51fa\u73b0\u5404\u79cd\u4e0d\u7a33\u5b9a\u73b0\u8c61\u3002 \u53e6\u4e00\u65b9\u9762\uff0c\u5404\u79cd\u521d\u59cb\u5316\u6280\u5de7\u4e5f\u65e0\u6cd5\u4ece\u6839\u672c\u4e0a\u89e3\u51b3\u95ee\u9898\uff0c\u56e0\u4e3a\u968f\u7740\u8bad\u7ec3\u4e0d\u65ad\u8fdb\u884c\uff0c\u6743\u91cd\u6700\u7ec8\u90fd\u4f1a\u9010\u6e10\u504f\u79bb\u521d\u59cb\u5316\u5c3a\u5ea6\u3002 \u9664\u6b64\u4e4b\u5916\uff0c\u8fd8\u6709\u5927\u91cf\u4f9d\u8d56\u67b6\u6784\u8bbe\u8ba1\u7684\u7a33\u5b9a\u5316\u6280\u5de7\uff0c\u4f8b\u5982\uff0cQK-Norm&nbsp;\u7528\u4e8e\u7a33\u5b9aAttention Logits\uff0cSandwich Norm&nbsp;\u7528\u4e8e\u540c\u65f6\u7ea6\u675f\u8f93\u5165\u4e0e\u8f93\u51fa\u5c3a\u5ea6\u3002\u8fd9\u4e9b\u65b9\u6cd5\u867d\u7136\u80fd\u591f\u63d0\u5347\u7a33\u5b9a\u6027\uff0c\u5374\u5f80\u5f80\u9700\u8981\u989d\u5916\u8c03\u53c2\uff0c\u751a\u81f3\u53ef\u80fd\u727a\u7272\u6a21\u578b\u80fd\u529b\u4e0a\u9650\uff0c\u5e76\u4e0d\u662f\u4e00\u79cd\u8db3\u591f\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002 \u56e0\u6b64\uff0c\u7814\u7a76\u5458\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u66f4\u52a0\u7edf\u4e00\u7684\u65b9\u6cd5\uff1a\u65e2\u80fd\u591f\u6446\u8131&nbsp;Weight Decay\uff08\u6743\u91cd\u8870\u51cf\uff09\uff0c\u53c8\u80fd\u591f\u5c3d\u53ef\u80fd\u5730\u51cf\u5c11\u989d\u5916\u7684Norm&nbsp;\u8bbe\u8ba1\u3002 \u5982\u679c\u5e0c\u671b\u6a21\u578b\u6fc0\u6d3b\u503c\u6ee1\u8db3 \u03bcP&nbsp;\u7406\u8bba\uff0c\u90a3\u4e48\u524d\u5411\u4f20\u64ad\u4e2d\u7684\u6743\u91cdW\uff0c\u4ee5\u53ca\u53cd\u5411\u4f20\u64ad\u4e2d\u7684\u66f4\u65b0\u91cf \u03a6=\u0394W\uff0c\u90fd\u5fc5\u987b\u6ee1\u8db3\uff1a \u5176\u4e2d\u8c31\u8303\u6570\u5b9a\u4e49\u4e3a\uff1a \u7814\u7a76\u5458\u4eec\u53d1\u73b0\uff0c\u03bcP&nbsp;\u7406\u8bba\u5b9e\u9645\u4e0a\u5929\u7136\u8981\u6c42\u6743\u91cd\u548c\u66f4\u65b0\u90fd\u4f4d\u4e8e\u8c31\u8303\u6570\u7403\uff08Spectral Sphere\uff09\u7684\u7403\u9762\u4e0a\u3002 \u8fdb\u4e00\u6b65\u89c2\u5bdf\u8fd8\u53d1\u73b0\uff0cMuon&nbsp;\u4f18\u5316\u5668\u7684\u66f4\u65b0\u89c4\u5219\u6070\u597d\u6ee1\u8db3\u4e86 \u03bcP\u7406\u8bba\u4e2d\u7684\u540e\u4e00\u6761\u8981\u6c42\uff0c\u5373\u7ea6\u675f\u66f4\u65b0\u91cf\u7684\u8c31\u8303\u6570\u3002\u8fd9\u4e00\u5de7\u5408\u4e5f\u542f\u53d1\u7814\u7a76\u5458\u4eec\u601d\u8003\uff1a\u5982\u679c\u80fd\u591f\u540c\u65f6\u6ee1\u8db3 \u03bcP&nbsp;\u7406\u8bba\u7684\u5168\u90e8\u8981\u6c42\uff0c\u662f\u5426\u80fd\u591f\u5f97\u5230\u4e00\u4e2a\u66f4\u52a0\u7406\u60f3\u7684\u4f18\u5316\u5668\uff1f \u65e2\u7136\u901a\u8fc7\u8c31\u7403\u7ea6\u675f\u5df2\u7ecf\u4fdd\u8bc1\u4e86\u7a33\u5b9a\u6027\uff0c\u63a5\u4e0b\u6765\u9700\u8981\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u5c31\u662f\u5982\u4f55\u5728\u8fd9\u4e00\u7ea6\u675f\u4e0b\uff0c\u4f7f&nbsp;Loss&nbsp;\u4e0b\u964d\u5f97\u6700\u5feb\u3002 Loss&nbsp;\u662f\u5173\u4e8e\u6743\u91cd&nbsp;W&nbsp;\u7684\u51fd\u6570\u3002\u7814\u7a76\u5458\u4eec\u9996\u5148\u5728\u5f53\u524d\u6743\u91cd\u5904\u5bf9&nbsp;Loss&nbsp;\u505a\u4e00\u9636\u6cf0\u52d2\u5c55\u5f00\uff0c\u4ece\u800c\u5f97\u5230\u6700\u901f\u4e0b\u964d\u5bf9\u5e94\u7684\u66f4\u65b0\u91cf \u03a6\uff1a \u7ed3\u5408\u524d\u8ff0\u8c31\u7403\u7ea6\u675f\uff0c\u5e76\u5c06\u8c31\u7403\u534a\u5f84\u8bb0\u4e3a\uff1a \u4fbf\u53ef\u4ee5\u5f97\u5230\u517c\u987e&#8221;\u53c8\u7a33\u53c8\u5feb&#8221;\u7684\u4f18\u5316\u76ee\u6807\uff1a \u8fd9\u91cc\uff0c\u66f4\u65b0\u91cf\u7684\u8c31\u8303\u6570\u7ea6\u675f\u5199\u6210&nbsp;1\uff0c\u662f\u56e0\u4e3a\u6b64\u5904\u8003\u8651\u7684\u662f\u5355\u4f4d\u66f4\u65b0\uff0c\u5b9e\u9645\u8bad\u7ec3\u65f6\u534a\u5f84R&nbsp;\u53ef\u4ee5\u76f4\u63a5\u5438\u6536\u5230\u5b66\u4e60\u7387\uff08learning rate\uff09\u4e2d\u3002 \u63a5\u4e0b\u6765\uff0c\u8981\u5bf9\u4e0a\u8ff0\u4f18\u5316\u76ee\u6807\u7ee7\u7eed\u5316\u7b80\u3002\u7814\u7a76\u5458\u4eec\u9996\u5148\u5c06\u6700\u540e\u4e24\u5f0f\u76f8\u51cf\uff0c\u53ef\u5f97\u5230\uff1a \u5176\u4e2d\uff0c \u8868\u793a\u5f53\u524d\u6743\u91cd\u5728\u8c31\u7403\u9762\u4e0a\u7684\u68af\u5ea6\u65b9\u5411\uff0c\u4e5f\u5c31\u662f\u8c31\u7403\u9762\u7684\u6cd5\u5411\u65b9\u5411\u3002 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