\u6b63\u5411\u4f20\u64ad\u662f\u6307\u5bf9\u795e\u7ecf\u7f51\u7edc\u6cbf\u7740\u4ece\u8f93\u5165\u5c42\u5230\u8f93\u51fa\u5c42\u7684\u987a\u5e8f\uff0c\u4f9d\u6b21\u8ba1\u7b97\u5e76\u5b58\u50a8\u6a21\u578b\u7684\u4e2d\u95f4\u53d8\u91cf\uff08\u5305\u62ec\u8f93\u51fa\uff09\u3002\u4e3a\u7b80\u5355\u8d77\u89c1\uff0c\u5047\u8bbe\u8f93\u5165\u662f\u4e00\u4e2a\u7279\u5f81\u4e3a$\\boldsymbol{x} \\in \\mathbb{R}^d$\u7684\u6837\u672c\uff0c\u4e14\u4e0d\u8003\u8651\u504f\u5dee\u9879\uff0c\u90a3\u4e48\u4e2d\u95f4\u53d8\u91cf<\/p>\n\n\n\n
$$\\boldsymbol{z} = \\boldsymbol{W}^{(1)} \\boldsymbol{x},$$<\/p>\n\n\n\n
\u5176\u4e2d$\\boldsymbol{W}^{(1)} \\in \\mathbb{R}^{h \\times d}$\u662f\u9690\u85cf\u5c42\u7684\u6743\u91cd\u53c2\u6570\u3002\u628a\u4e2d\u95f4\u53d8\u91cf$\\boldsymbol{z} \\in \\mathbb{R}^h$\u8f93\u5165\u6309\u5143\u7d20\u8fd0\u7b97\u7684\u6fc0\u6d3b\u51fd\u6570$\\phi$\u540e\uff0c\u5c06\u5f97\u5230\u5411\u91cf\u957f\u5ea6\u4e3a$h$\u7684\u9690\u85cf\u5c42\u53d8\u91cf<\/p>\n\n\n\n
$$\\boldsymbol{h} = \\phi (\\boldsymbol{z}).$$<\/p>\n\n\n\n
\u9690\u85cf\u5c42\u53d8\u91cf$\\boldsymbol{h}$\u4e5f\u662f\u4e00\u4e2a\u4e2d\u95f4\u53d8\u91cf\u3002\u5047\u8bbe\u8f93\u51fa\u5c42\u53c2\u6570\u53ea\u6709\u6743\u91cd$\\boldsymbol{W}^{(2)} \\in \\mathbb{R}^{q \\times h}$\uff0c\u53ef\u4ee5\u5f97\u5230\u5411\u91cf\u957f\u5ea6\u4e3a$q$\u7684\u8f93\u51fa\u5c42\u53d8\u91cf<\/p>\n\n\n\n
$$\\boldsymbol{o} = \\boldsymbol{W}^{(2)} \\boldsymbol{h}.$$<\/p>\n\n\n\n
\u5047\u8bbe\u635f\u5931\u51fd\u6570\u4e3a$\\ell$\uff0c\u4e14\u6837\u672c\u6807\u7b7e\u4e3a$y$\uff0c\u53ef\u4ee5\u8ba1\u7b97\u51fa\u5355\u4e2a\u6570\u636e\u6837\u672c\u7684\u635f\u5931\u9879<\/p>\n\n\n\n
$$L = \\ell(\\boldsymbol{o}, y).$$<\/p>\n\n\n\n
\u6839\u636e$L_2$\u8303\u6570\u6b63\u5219\u5316\u7684\u5b9a\u4e49\uff0c\u7ed9\u5b9a\u8d85\u53c2\u6570$\\lambda$\uff0c\u6b63\u5219\u5316\u9879\u5373<\/p>\n\n\n\n
$$s = \\frac{\\lambda}{2} \\left(|\\boldsymbol{W}^{(1)}|_F^2 + |\\boldsymbol{W}^{(2)}|_F^2\\right),$$<\/p>\n\n\n\n
\u5176\u4e2d\u77e9\u9635\u7684Frobenius\u8303\u6570\u7b49\u4ef7\u4e8e\u5c06\u77e9\u9635\u53d8\u5e73\u4e3a\u5411\u91cf\u540e\u8ba1\u7b97$L_2$\u8303\u6570\u3002\u6700\u7ec8\uff0c\u6a21\u578b\u5728\u7ed9\u5b9a\u7684\u6570\u636e\u6837\u672c\u4e0a\u5e26\u6b63\u5219\u5316\u7684\u635f\u5931\u4e3a<\/p>\n\n\n\n
$$J = L + s.$$<\/p>\n\n\n\n
\u5c06$J$\u79f0\u4e3a\u6709\u5173\u7ed9\u5b9a\u6570\u636e\u6837\u672c\u7684\u76ee\u6807\u51fd\u6570\uff0c\u5e76\u5728\u4ee5\u4e0b\u7684\u8ba8\u8bba\u4e2d\u7b80\u79f0\u76ee\u6807\u51fd\u6570\u3002<\/p>\n\n\n\n
\u901a\u5e38\u7ed8\u5236\u8ba1\u7b97\u56fe\u6765\u53ef\u89c6\u5316\u8fd0\u7b97\u7b26\u548c\u53d8\u91cf\u5728\u8ba1\u7b97\u4e2d\u7684\u4f9d\u8d56\u5173\u7cfb\u3002\u4e0b\u56fe\u7ed8\u5236\u4e86\u672c\u8282\u4e2d\u6837\u4f8b\u6a21\u578b\u6b63\u5411\u4f20\u64ad\u7684\u8ba1\u7b97\u56fe\uff0c\u5176\u4e2d\u5de6\u4e0b\u89d2\u662f\u8f93\u5165\uff0c\u53f3\u4e0a\u89d2\u662f\u8f93\u51fa\u3002<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u53cd\u5411\u4f20\u64ad<\/h2>\n\n\n\n\u53cd\u5411\u4f20\u64ad\u6307\u7684\u662f\u8ba1\u7b97\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\u68af\u5ea6\u7684\u65b9\u6cd5\u3002\u603b\u7684\u6765\u8bf4\uff0c\u53cd\u5411\u4f20\u64ad\u4f9d\u636e\u5fae\u79ef\u5206\u4e2d\u7684\u94fe\u5f0f\u6cd5\u5219\uff0c\u6cbf\u7740\u4ece\u8f93\u51fa\u5c42\u5230\u8f93\u5165\u5c42\u7684\u987a\u5e8f\uff0c\u4f9d\u6b21\u8ba1\u7b97\u5e76\u5b58\u50a8\u76ee\u6807\u51fd\u6570\u6709\u5173\u795e\u7ecf\u7f51\u7edc\u5404\u5c42\u7684\u4e2d\u95f4\u53d8\u91cf\u4ee5\u53ca\u53c2\u6570\u7684\u68af\u5ea6\u3002\u5bf9\u8f93\u5165\u6216\u8f93\u51fa$\\mathsf{X}, \\mathsf{Y}, \\mathsf{Z}$\u4e3a\u4efb\u610f\u5f62\u72b6\u5f20\u91cf\u7684\u51fd\u6570$\\mathsf{Y}=f(\\mathsf{X})$\u548c$\\mathsf{Z}=g(\\mathsf{Y})$\uff0c\u901a\u8fc7\u94fe\u5f0f\u6cd5\u5219\uff0c\u6709<\/p>\n\n\n\n$$\\frac{\\partial \\mathsf{Z}}{\\partial \\mathsf{X}} = \\text{prod}\\left(\\frac{\\partial \\mathsf{Z}}{\\partial \\mathsf{Y}}, \\frac{\\partial \\mathsf{Y}}{\\partial \\mathsf{X}}\\right),$$<\/p>\n\n\n\n\u5176\u4e2d$\\text{prod}$\u8fd0\u7b97\u7b26\u5c06\u6839\u636e\u4e24\u4e2a\u8f93\u5165\u7684\u5f62\u72b6\uff0c\u5728\u5fc5\u8981\u7684\u64cd\u4f5c\uff08\u5982\u8f6c\u7f6e\u548c\u4e92\u6362\u8f93\u5165\u4f4d\u7f6e\uff09\u540e\u5bf9\u4e24\u4e2a\u8f93\u5165\u505a\u4e58\u6cd5\u3002<\/p>\n\n\n\n\u56de\u987e\u672c\u8282\u4e2d\u6837\u4f8b\u6a21\u578b\uff0c\u5b83\u7684\u53c2\u6570\u662f$\\boldsymbol{W}^{(1)}$\u548c$\\boldsymbol{W}^{(2)}$\uff0c\u56e0\u6b64\u53cd\u5411\u4f20\u64ad\u7684\u76ee\u6807\u662f\u8ba1\u7b97$\\partial J\/\\partial \\boldsymbol{W}^{(1)}$\u548c$\\partial J\/\\partial \\boldsymbol{W}^{(2)}$\u3002\u5c06\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u4f9d\u6b21\u8ba1\u7b97\u5404\u4e2d\u95f4\u53d8\u91cf\u548c\u53c2\u6570\u7684\u68af\u5ea6\uff0c\u5176\u8ba1\u7b97\u6b21\u5e8f\u4e0e\u524d\u5411\u4f20\u64ad\u4e2d\u76f8\u5e94\u4e2d\u95f4\u53d8\u91cf\u7684\u8ba1\u7b97\u6b21\u5e8f\u76f8\u53cd\u3002\u9996\u5148\uff0c\u5206\u522b\u8ba1\u7b97\u76ee\u6807\u51fd\u6570$J=L+s$\u6709\u5173\u635f\u5931\u9879$L$\u548c\u6b63\u5219\u9879$s$\u7684\u68af\u5ea6<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial L} = 1, \\quad \\frac{\\partial J}{\\partial s} = 1.$$<\/p>\n\n\n\n\u5176\u6b21\uff0c\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\u8ba1\u7b97\u76ee\u6807\u51fd\u6570\u6709\u5173\u8f93\u51fa\u5c42\u53d8\u91cf\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{o} \\in \\mathbb{R}^q$\uff1a<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial \\boldsymbol{o}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial L}, \\frac{\\partial L}{\\partial \\boldsymbol{o}}\\right)= \\frac{\\partial L}{\\partial \\boldsymbol{o}}.$$<\/p>\n\n\n\n\u63a5\u4e0b\u6765\uff0c\u8ba1\u7b97\u6b63\u5219\u9879\u6709\u5173\u4e24\u4e2a\u53c2\u6570\u7684\u68af\u5ea6\uff1a<\/p>\n\n\n\n$$\\frac{\\partial s}{\\partial \\boldsymbol{W}^{(1)}} = \\lambda \\boldsymbol{W}^{(1)},\\quad\\frac{\\partial s}{\\partial \\boldsymbol{W}^{(2)}} = \\lambda \\boldsymbol{W}^{(2)}.$$<\/p>\n\n\n\n\u8ba1\u7b97\u6700\u9760\u8fd1\u8f93\u51fa\u5c42\u7684\u6a21\u578b\u53c2\u6570\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(2)} \\in \\mathbb{R}^{q \\times h}$\u3002\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\uff0c\u5f97\u5230<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial \\boldsymbol{W}^{(2)}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{o}}, \\frac{\\partial \\boldsymbol{o}}{\\partial \\boldsymbol{W}^{(2)}}\\right) + \\text{prod}\\left(\\frac{\\partial J}{\\partial s}, \\frac{\\partial s}{\\partial \\boldsymbol{W}^{(2)}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{o}} \\boldsymbol{h}^\\top + \\lambda \\boldsymbol{W}^{(2)}.$$<\/p>\n\n\n\n\u6cbf\u7740\u8f93\u51fa\u5c42\u5411\u9690\u85cf\u5c42\u7ee7\u7eed\u53cd\u5411\u4f20\u64ad\uff0c\u9690\u85cf\u5c42\u53d8\u91cf\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{h} \\in \\mathbb{R}^h$\u53ef\u4ee5\u8fd9\u6837\u8ba1\u7b97\uff1a<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial \\boldsymbol{h}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{o}}, \\frac{\\partial \\boldsymbol{o}}{\\partial \\boldsymbol{h}}\\right)= {\\boldsymbol{W}^{(2)}}^\\top \\frac{\\partial J}{\\partial \\boldsymbol{o}}.$$<\/p>\n\n\n\n\u7531\u4e8e\u6fc0\u6d3b\u51fd\u6570$\\phi$\u662f\u6309\u5143\u7d20\u8fd0\u7b97\u7684\uff0c\u4e2d\u95f4\u53d8\u91cf$\\boldsymbol{z}$\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{z} \\in \\mathbb{R}^h$\u7684\u8ba1\u7b97\u9700\u8981\u4f7f\u7528\u6309\u5143\u7d20\u4e58\u6cd5\u7b26$\\odot$\uff1a<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial \\boldsymbol{z}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{h}}, \\frac{\\partial \\boldsymbol{h}}{\\partial \\boldsymbol{z}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{h}} \\odot \\phi'\\left(\\boldsymbol{z}\\right).$$<\/p>\n\n\n\n\u6700\u7ec8\uff0c\u53ef\u4ee5\u5f97\u5230\u6700\u9760\u8fd1\u8f93\u5165\u5c42\u7684\u6a21\u578b\u53c2\u6570\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(1)} \\in \\mathbb{R}^{h \\times d}$\u3002\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\uff0c\u5f97\u5230<\/p>\n\n\n\n$$\\frac{\\partial J}{\\partial \\boldsymbol{W}^{(1)}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{z}}, \\frac{\\partial \\boldsymbol{z}}{\\partial \\boldsymbol{W}^{(1)}}\\right) + \\text{prod}\\left(\\frac{\\partial J}{\\partial s}, \\frac{\\partial s}{\\partial \\boldsymbol{W}^{(1)}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{z}} \\boldsymbol{x}^\\top + \\lambda \\boldsymbol{W}^{(1)}.$$<\/p>\n\n\n\n\u8bad\u7ec3\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b<\/h2>\n\n\n\n\u5728\u8bad\u7ec3\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u65f6\uff0c\u6b63\u5411\u4f20\u64ad\u548c\u53cd\u5411\u4f20\u64ad\u4e4b\u95f4\u76f8\u4e92\u4f9d\u8d56\u3002<\/p>\n\n\n\n\u4e00\u65b9\u9762\uff0c\u6b63\u5411\u4f20\u64ad\u7684\u8ba1\u7b97\u53ef\u80fd\u4f9d\u8d56\u4e8e\u6a21\u578b\u53c2\u6570\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u6a21\u578b\u53c2\u6570\u662f\u5728\u53cd\u5411\u4f20\u64ad\u7684\u68af\u5ea6\u8ba1\u7b97\u540e\u901a\u8fc7\u4f18\u5316\u7b97\u6cd5\u8fed\u4ee3\u7684\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u6b63\u5219\u5316\u9879$s = (\\lambda\/2) \\left(|\\boldsymbol{W}^{(1)}|_F^2 + |\\boldsymbol{W}^{(2)}|_F^2\\right)$\u4f9d\u8d56\u6a21\u578b\u53c2\u6570$\\boldsymbol{W}^{(1)}$\u548c$\\boldsymbol{W}^{(2)}$\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u5f53\u524d\u503c\u662f\u4f18\u5316\u7b97\u6cd5\u6700\u8fd1\u4e00\u6b21\u6839\u636e\u53cd\u5411\u4f20\u64ad\u7b97\u51fa\u68af\u5ea6\u540e\u8fed\u4ee3\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n\u53e6\u4e00\u65b9\u9762\uff0c\u53cd\u5411\u4f20\u64ad\u7684\u68af\u5ea6\u8ba1\u7b97\u53ef\u80fd\u4f9d\u8d56\u4e8e\u5404\u53d8\u91cf\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u53d8\u91cf\u7684\u5f53\u524d\u503c\u662f\u901a\u8fc7\u6b63\u5411\u4f20\u64ad\u8ba1\u7b97\u5f97\u5230\u7684\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u53c2\u6570\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(2)} = (\\partial J \/ \\partial \\boldsymbol{o}) \\boldsymbol{h}^\\top + \\lambda \\boldsymbol{W}^{(2)}$\u7684\u8ba1\u7b97\u9700\u8981\u4f9d\u8d56\u9690\u85cf\u5c42\u53d8\u91cf\u7684\u5f53\u524d\u503c$\\boldsymbol{h}$\u3002\u8fd9\u4e2a\u5f53\u524d\u503c\u662f\u901a\u8fc7\u4ece\u8f93\u5165\u5c42\u5230\u8f93\u51fa\u5c42\u7684\u6b63\u5411\u4f20\u64ad\u8ba1\u7b97\u5e76\u5b58\u50a8\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n\u56e0\u6b64\uff0c\u5728\u6a21\u578b\u53c2\u6570\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u4ea4\u66ff\u5730\u8fdb\u884c\u6b63\u5411\u4f20\u64ad\u548c\u53cd\u5411\u4f20\u64ad\uff0c\u5e76\u6839\u636e\u53cd\u5411\u4f20\u64ad\u8ba1\u7b97\u7684\u68af\u5ea6\u8fed\u4ee3\u6a21\u578b\u53c2\u6570\u3002\u65e2\u7136\u5728\u53cd\u5411\u4f20\u64ad\u4e2d\u4f7f\u7528\u4e86\u6b63\u5411\u4f20\u64ad\u4e2d\u8ba1\u7b97\u5f97\u5230\u7684\u4e2d\u95f4\u53d8\u91cf\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u90a3\u4e48\u8fd9\u4e2a\u590d\u7528\u4e5f\u5bfc\u81f4\u6b63\u5411\u4f20\u64ad\u7ed3\u675f\u540e\u4e0d\u80fd\u7acb\u5373\u91ca\u653e\u4e2d\u95f4\u53d8\u91cf\u5185\u5b58\u3002\u8fd9\u4e5f\u662f\u8bad\u7ec3\u8981\u6bd4\u9884\u6d4b\u5360\u7528\u66f4\u591a\u5185\u5b58\u7684\u4e00\u4e2a\u91cd\u8981\u539f\u56e0\u3002\u53e6\u5916\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8fd9\u4e9b\u4e2d\u95f4\u53d8\u91cf\u7684\u4e2a\u6570\u5927\u4f53\u4e0a\u4e0e\u7f51\u7edc\u5c42\u6570\u7ebf\u6027\u76f8\u5173\uff0c\u6bcf\u4e2a\u53d8\u91cf\u7684\u5927\u5c0f\u8ddf\u6279\u91cf\u5927\u5c0f\u548c\u8f93\u5165\u4e2a\u6570\u4e5f\u662f\u7ebf\u6027\u76f8\u5173\u7684\uff0c\u5b83\u4eec\u662f\u5bfc\u81f4\u8f83\u6df1\u7684\u795e\u7ecf\u7f51\u7edc\u4f7f\u7528\u8f83\u5927\u6279\u91cf\u8bad\u7ec3\u65f6\u66f4\u5bb9\u6613\u8d85\u5185\u5b58\u7684\u4e3b\u8981\u539f\u56e0\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"\u6b63\u5411\u4f20\u64ad \u6b63\u5411\u4f20\u64ad\u662f\u6307\u5bf9\u795e\u7ecf\u7f51\u7edc\u6cbf\u7740\u4ece\u8f93\u5165\u5c42\u5230\u8f93\u51fa\u5c42\u7684\u987a\u5e8f\uff0c\u4f9d\u6b21\u8ba1\u7b97\u5e76\u5b58\u50a8\u6a21\u578b\u7684\u4e2d\u95f4\u53d8\u91cf\uff08\u5305\u62ec\u8f93\u51fa\uff09\u3002\u4e3a\u7b80\u5355\u8d77\u89c1\uff0c\u5047\u8bbe\u8f93\u5165\u662f\u4e00\u4e2a\u7279 …<\/p>","protected":false},"author":1,"featured_media":1698,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[46,3],"tags":[45,44,12,22],"class_list":["post-1697","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-46","category-3","tag-45","tag-44","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1697","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=1697"}],"version-history":[{"count":1,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1697\/revisions"}],"predecessor-version":[{"id":1708,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/1697\/revisions\/1708"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/1698"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=1697"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=1697"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=1697"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u53cd\u5411\u4f20\u64ad\u6307\u7684\u662f\u8ba1\u7b97\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\u68af\u5ea6\u7684\u65b9\u6cd5\u3002\u603b\u7684\u6765\u8bf4\uff0c\u53cd\u5411\u4f20\u64ad\u4f9d\u636e\u5fae\u79ef\u5206\u4e2d\u7684\u94fe\u5f0f\u6cd5\u5219\uff0c\u6cbf\u7740\u4ece\u8f93\u51fa\u5c42\u5230\u8f93\u5165\u5c42\u7684\u987a\u5e8f\uff0c\u4f9d\u6b21\u8ba1\u7b97\u5e76\u5b58\u50a8\u76ee\u6807\u51fd\u6570\u6709\u5173\u795e\u7ecf\u7f51\u7edc\u5404\u5c42\u7684\u4e2d\u95f4\u53d8\u91cf\u4ee5\u53ca\u53c2\u6570\u7684\u68af\u5ea6\u3002\u5bf9\u8f93\u5165\u6216\u8f93\u51fa$\\mathsf{X}, \\mathsf{Y}, \\mathsf{Z}$\u4e3a\u4efb\u610f\u5f62\u72b6\u5f20\u91cf\u7684\u51fd\u6570$\\mathsf{Y}=f(\\mathsf{X})$\u548c$\\mathsf{Z}=g(\\mathsf{Y})$\uff0c\u901a\u8fc7\u94fe\u5f0f\u6cd5\u5219\uff0c\u6709<\/p>\n\n\n\n
$$\\frac{\\partial \\mathsf{Z}}{\\partial \\mathsf{X}} = \\text{prod}\\left(\\frac{\\partial \\mathsf{Z}}{\\partial \\mathsf{Y}}, \\frac{\\partial \\mathsf{Y}}{\\partial \\mathsf{X}}\\right),$$<\/p>\n\n\n\n
\u5176\u4e2d$\\text{prod}$\u8fd0\u7b97\u7b26\u5c06\u6839\u636e\u4e24\u4e2a\u8f93\u5165\u7684\u5f62\u72b6\uff0c\u5728\u5fc5\u8981\u7684\u64cd\u4f5c\uff08\u5982\u8f6c\u7f6e\u548c\u4e92\u6362\u8f93\u5165\u4f4d\u7f6e\uff09\u540e\u5bf9\u4e24\u4e2a\u8f93\u5165\u505a\u4e58\u6cd5\u3002<\/p>\n\n\n\n
\u56de\u987e\u672c\u8282\u4e2d\u6837\u4f8b\u6a21\u578b\uff0c\u5b83\u7684\u53c2\u6570\u662f$\\boldsymbol{W}^{(1)}$\u548c$\\boldsymbol{W}^{(2)}$\uff0c\u56e0\u6b64\u53cd\u5411\u4f20\u64ad\u7684\u76ee\u6807\u662f\u8ba1\u7b97$\\partial J\/\\partial \\boldsymbol{W}^{(1)}$\u548c$\\partial J\/\\partial \\boldsymbol{W}^{(2)}$\u3002\u5c06\u5e94\u7528\u94fe\u5f0f\u6cd5\u5219\u4f9d\u6b21\u8ba1\u7b97\u5404\u4e2d\u95f4\u53d8\u91cf\u548c\u53c2\u6570\u7684\u68af\u5ea6\uff0c\u5176\u8ba1\u7b97\u6b21\u5e8f\u4e0e\u524d\u5411\u4f20\u64ad\u4e2d\u76f8\u5e94\u4e2d\u95f4\u53d8\u91cf\u7684\u8ba1\u7b97\u6b21\u5e8f\u76f8\u53cd\u3002\u9996\u5148\uff0c\u5206\u522b\u8ba1\u7b97\u76ee\u6807\u51fd\u6570$J=L+s$\u6709\u5173\u635f\u5931\u9879$L$\u548c\u6b63\u5219\u9879$s$\u7684\u68af\u5ea6<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial L} = 1, \\quad \\frac{\\partial J}{\\partial s} = 1.$$<\/p>\n\n\n\n
\u5176\u6b21\uff0c\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\u8ba1\u7b97\u76ee\u6807\u51fd\u6570\u6709\u5173\u8f93\u51fa\u5c42\u53d8\u91cf\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{o} \\in \\mathbb{R}^q$\uff1a<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial \\boldsymbol{o}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial L}, \\frac{\\partial L}{\\partial \\boldsymbol{o}}\\right)= \\frac{\\partial L}{\\partial \\boldsymbol{o}}.$$<\/p>\n\n\n\n
\u63a5\u4e0b\u6765\uff0c\u8ba1\u7b97\u6b63\u5219\u9879\u6709\u5173\u4e24\u4e2a\u53c2\u6570\u7684\u68af\u5ea6\uff1a<\/p>\n\n\n\n
$$\\frac{\\partial s}{\\partial \\boldsymbol{W}^{(1)}} = \\lambda \\boldsymbol{W}^{(1)},\\quad\\frac{\\partial s}{\\partial \\boldsymbol{W}^{(2)}} = \\lambda \\boldsymbol{W}^{(2)}.$$<\/p>\n\n\n\n
\u8ba1\u7b97\u6700\u9760\u8fd1\u8f93\u51fa\u5c42\u7684\u6a21\u578b\u53c2\u6570\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(2)} \\in \\mathbb{R}^{q \\times h}$\u3002\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\uff0c\u5f97\u5230<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial \\boldsymbol{W}^{(2)}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{o}}, \\frac{\\partial \\boldsymbol{o}}{\\partial \\boldsymbol{W}^{(2)}}\\right) + \\text{prod}\\left(\\frac{\\partial J}{\\partial s}, \\frac{\\partial s}{\\partial \\boldsymbol{W}^{(2)}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{o}} \\boldsymbol{h}^\\top + \\lambda \\boldsymbol{W}^{(2)}.$$<\/p>\n\n\n\n
\u6cbf\u7740\u8f93\u51fa\u5c42\u5411\u9690\u85cf\u5c42\u7ee7\u7eed\u53cd\u5411\u4f20\u64ad\uff0c\u9690\u85cf\u5c42\u53d8\u91cf\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{h} \\in \\mathbb{R}^h$\u53ef\u4ee5\u8fd9\u6837\u8ba1\u7b97\uff1a<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial \\boldsymbol{h}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{o}}, \\frac{\\partial \\boldsymbol{o}}{\\partial \\boldsymbol{h}}\\right)= {\\boldsymbol{W}^{(2)}}^\\top \\frac{\\partial J}{\\partial \\boldsymbol{o}}.$$<\/p>\n\n\n\n
\u7531\u4e8e\u6fc0\u6d3b\u51fd\u6570$\\phi$\u662f\u6309\u5143\u7d20\u8fd0\u7b97\u7684\uff0c\u4e2d\u95f4\u53d8\u91cf$\\boldsymbol{z}$\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{z} \\in \\mathbb{R}^h$\u7684\u8ba1\u7b97\u9700\u8981\u4f7f\u7528\u6309\u5143\u7d20\u4e58\u6cd5\u7b26$\\odot$\uff1a<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial \\boldsymbol{z}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{h}}, \\frac{\\partial \\boldsymbol{h}}{\\partial \\boldsymbol{z}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{h}} \\odot \\phi'\\left(\\boldsymbol{z}\\right).$$<\/p>\n\n\n\n
\u6700\u7ec8\uff0c\u53ef\u4ee5\u5f97\u5230\u6700\u9760\u8fd1\u8f93\u5165\u5c42\u7684\u6a21\u578b\u53c2\u6570\u7684\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(1)} \\in \\mathbb{R}^{h \\times d}$\u3002\u4f9d\u636e\u94fe\u5f0f\u6cd5\u5219\uff0c\u5f97\u5230<\/p>\n\n\n\n
$$\\frac{\\partial J}{\\partial \\boldsymbol{W}^{(1)}}= \\text{prod}\\left(\\frac{\\partial J}{\\partial \\boldsymbol{z}}, \\frac{\\partial \\boldsymbol{z}}{\\partial \\boldsymbol{W}^{(1)}}\\right) + \\text{prod}\\left(\\frac{\\partial J}{\\partial s}, \\frac{\\partial s}{\\partial \\boldsymbol{W}^{(1)}}\\right)= \\frac{\\partial J}{\\partial \\boldsymbol{z}} \\boldsymbol{x}^\\top + \\lambda \\boldsymbol{W}^{(1)}.$$<\/p>\n\n\n\n
\u5728\u8bad\u7ec3\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u65f6\uff0c\u6b63\u5411\u4f20\u64ad\u548c\u53cd\u5411\u4f20\u64ad\u4e4b\u95f4\u76f8\u4e92\u4f9d\u8d56\u3002<\/p>\n\n\n\n
\u4e00\u65b9\u9762\uff0c\u6b63\u5411\u4f20\u64ad\u7684\u8ba1\u7b97\u53ef\u80fd\u4f9d\u8d56\u4e8e\u6a21\u578b\u53c2\u6570\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u6a21\u578b\u53c2\u6570\u662f\u5728\u53cd\u5411\u4f20\u64ad\u7684\u68af\u5ea6\u8ba1\u7b97\u540e\u901a\u8fc7\u4f18\u5316\u7b97\u6cd5\u8fed\u4ee3\u7684\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u6b63\u5219\u5316\u9879$s = (\\lambda\/2) \\left(|\\boldsymbol{W}^{(1)}|_F^2 + |\\boldsymbol{W}^{(2)}|_F^2\\right)$\u4f9d\u8d56\u6a21\u578b\u53c2\u6570$\\boldsymbol{W}^{(1)}$\u548c$\\boldsymbol{W}^{(2)}$\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u5f53\u524d\u503c\u662f\u4f18\u5316\u7b97\u6cd5\u6700\u8fd1\u4e00\u6b21\u6839\u636e\u53cd\u5411\u4f20\u64ad\u7b97\u51fa\u68af\u5ea6\u540e\u8fed\u4ee3\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n
\u53e6\u4e00\u65b9\u9762\uff0c\u53cd\u5411\u4f20\u64ad\u7684\u68af\u5ea6\u8ba1\u7b97\u53ef\u80fd\u4f9d\u8d56\u4e8e\u5404\u53d8\u91cf\u7684\u5f53\u524d\u503c\uff0c\u800c\u8fd9\u4e9b\u53d8\u91cf\u7684\u5f53\u524d\u503c\u662f\u901a\u8fc7\u6b63\u5411\u4f20\u64ad\u8ba1\u7b97\u5f97\u5230\u7684\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u53c2\u6570\u68af\u5ea6$\\partial J\/\\partial \\boldsymbol{W}^{(2)} = (\\partial J \/ \\partial \\boldsymbol{o}) \\boldsymbol{h}^\\top + \\lambda \\boldsymbol{W}^{(2)}$\u7684\u8ba1\u7b97\u9700\u8981\u4f9d\u8d56\u9690\u85cf\u5c42\u53d8\u91cf\u7684\u5f53\u524d\u503c$\\boldsymbol{h}$\u3002\u8fd9\u4e2a\u5f53\u524d\u503c\u662f\u901a\u8fc7\u4ece\u8f93\u5165\u5c42\u5230\u8f93\u51fa\u5c42\u7684\u6b63\u5411\u4f20\u64ad\u8ba1\u7b97\u5e76\u5b58\u50a8\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n
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