{"id":439,"date":"2020-12-04T08:28:00","date_gmt":"2020-12-04T08:28:00","guid":{"rendered":"https:\/\/tensor.agenthub.uk\/?p=439"},"modified":"2024-05-13T03:22:40","modified_gmt":"2024-05-13T03:22:40","slug":"%e4%b8%ba%e4%bb%80%e4%b9%88%e6%98%af%e6%a2%af%e5%ba%a6%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/tensorzen.blog\/?p=439","title":{"rendered":"\u4e3a\u4ec0\u4e48\u662f\u68af\u5ea6\uff1f"},"content":{"rendered":"\n

\u68af\u5ea6\u4e0b\u964d\u662f\u4e00\u4e2a\u4f18\u5316\u7b97\u6cd5\uff0c\u4e3a\u4ec0\u4e48\u662f\u68af\u5ea6\u5462\uff1f\u4e3a\u4ec0\u4e48\u4e0d\u662f\u786c\u5ea6\uff1f\u957f\u5ea6\uff1f\u96f6\u5ea6\uff1f\u767e\u5ea6\uff1f<\/p>\n\n\n\n

\u4f18\u5316\u95ee\u9898\u5206\u6210\u65e0\u7ea6\u675f\u4f18\u5316\u548c\u6709\u7ea6\u675f\u4f18\u5316\uff0c\u6709\u7ea6\u675f\u4f18\u5316\u6700\u7ec8\u90fd\u8981\u8c03\u6574\u6210\u65e0\u7ea6\u675f\u4f18\u5316\u95ee\u9898\uff0c\u6240\u4ee5\u6211\u4eec\u76f4\u63a5\u770b\u65e0\u7ea6\u675f\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u7684\u6807\u51c6\u5f62\u5f0f\u5982\u4e0b\uff1a <\/p>\n\n\n\n

$$\\min_{x}f(x)$$<\/p>\n\n\n\n

\u4e25\u683c\u7684\u6570\u5b66\u5b9a\u4e49\u91cc\u8981\u6c42$f(x)$\u662f\u4e00\u4e2a\u5e73\u6ed1\u51fd\u6570,\u4e00\u822c\u6765\u8bf4$x$\u662f\u5411\u91cf\uff0c\u6bd4\u5982\u673a\u5668\u5b66\u4e60\u91cc\u9762\u52a8\u4e0d\u52a8\u5c31\u662f\u4e0a\u767e\u7ef4\u7684\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a$x$\u4f7f\u5f97$f(x)$\u6700\u5c0f\uff0c\u5c0f\u5b66\u4e8c\u5e74\u7ea7\u5728\u5b66\u4e60\u5b8c\u5fae\u79ef\u5206\u540e\u6211\u4eec\u77e5\u9053\u6700\u5c0f\u503c\u80af\u5b9a\u51fa\u73b0\u5728$f(x)$\u4e00\u9636\u5bfc\u6570\u7b49\u4e8e0\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u4f7f\u4e00\u9636\u5bfc\u6570\u7b49\u4e8e0\u7684$x$\u53eb\u9a7b\u70b9\uff0c\u90a3\u4f60\u53ef\u80fd\u5c31\u8981\u95ee\u4e86\uff0c\u65e2\u7136\u53ef\u4ee5\u5f97\u5230\u89e3\u6790\u89e3(analytical solution\/closed solution)\uff0c\u4e3a\u4ec0\u4e48\u8fd8\u8981\u7528\u68af\u5ea6\u4e0b\u964d\u6c42\u5462\uff1f\u539f\u56e0\u5927\u6982\u5982\u6b64\uff1a\u5982\u679c$f(x)$\u6781\u5176\u590d\u6742\u7684\u8bdd\uff0c\u8ba1\u7b97\u5b83\u7684\u4e00\u9636\u5bfc\u6570\u975e\u5e38\u56f0\u96be\uff0c\u5176\u6b21\u8fd8\u5f97\u8ba9\u8ba1\u7b97\u673a\u6c42\u89e3$f'(x)=0$\u4e3a\u4e86\u6c42\u89e3\u4e00\u4e2a\u95ee\u9898\uff0c\u5f15\u5165\u53e6\u4e00\u4e2a\u590d\u6742\u95ee\u9898\uff5e\u8fd9\u4e0d\u662f\u89e3\u51b3\u95ee\u9898\u7684\u601d\u8def\uff5e\u6240\u4ee5\u624d\u4f1a\u51fa\u73b0\u4f18\u5316\u7b97\u6cd5\u3002<\/p>\n\n\n\n

\u6c42\u89e3\u7ea6\u675f\u95ee\u9898\u80af\u5b9a\u8131\u79bb\u4e0d\u4e86\u8ba1\u7b97\u673a\u4e16\u754c\u6700\u4f1f\u5927\uff08\u4e5f\u8bb8\uff09\u7684\u601d\u60f3\u2014\u2014\u8fed\u4ee3\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u521d\u59cb$x$\uff0c\u53eb\u5b83$x_0$\uff0c\u901a\u8fc7\u67d0\u79cd\u7b56\u7565\u53ef\u5230\u4e00\u4e2a\u65b0\u7684$x$\uff0c\u53eb\u5b83$x_1$\uff0c\u8981\u6c42$f(x_1) < f(x_0)$\uff0c\u518d\u627e\u4e00\u4e2a\u65b0\u7684$x$\uff0c\u53eb$x_2$\uff0c\u8981\u6c42$f(x_2) < f(x_1)$\uff0c\u4e00\u76f4\u8fd9\u6837\u66f4\u65b0$x$\uff0c\u4e00\u76f4\u5f97\u5230\u66f4\u5c0f\u7684$f(x)$\uff0c\u76f4\u5230\u4f60\u6ee1\u610f\u4e3a\u6b62\uff0c\u5982\u4f55\u4ece$x_0$\u5230$x_1$\u76f4\u5230$x_n$\u5c31\u662f\u5404\u79cd\u4f18\u5316\u7b97\u6cd5\u5e72\u7684\u6d3b\uff0c\u5927\u6982\u6709\u4e24\u79cd\u601d\u8def\uff0c\u4e00\u79cd\u53eb\u7ebf\u6027\u641c\u7d22\uff08Linear Search\uff09\uff0c\u53e6\u4e00\u79cd\u53eb\u4fe1\u4efb\u57df(Trsut Range)\uff0cTrust Range\u662f\u5728\u4e00\u4e2a\u5c40\u90e8\u7a7a\u95f4\u4e2d\u7528\u4e00\u4e2a\u7b80\u5355\u51fd\u6570\u66ff\u6362\u539f\u51fd\u6570\u5f97\u5230\u8ba1\u7b97\u8be5\u7b80\u5355\u51fd\u6570\u7684\u6700\u5c0f\u503c\uff0c\u4f9d\u6b21\u6267\u884c\u8fd9\u4e2a\u8fc7\u7a0b\u3002\u68af\u5ea6\u4e0b\u964d\u5c5e\u4e8e\u7ebf\u6027\u641c\u7d22\uff0c\u4e8e\u662f\u54b1\u4eec\u8fd8\u662f\u4e3b\u8981\u8bf4\u7ebf\u6027\u641c\u7d22\u3002<\/p>\n\n\n\n

\u65e2\u7136\u8981\u6c42$f(x_{k+1}) < f(x_{k})$\uff0c\u53ea\u8981\u4fdd\u8bc1\u6bcf\u6b21\u8fed\u4ee3\u90fd\u80fd\u4fdd\u8bc1\u51fd\u6570\u503c\u53d8\u5c0f\u4e0d\u5c31\u884c\u4e86\uff1f\u5982\u4f55\u505a\u5462\uff1f\u8fd9\u91cc\u9700\u8981\u5f15\u8350\u4e00\u4e2a\u4eba\uff0cTaylor<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u5f53\u7136\u4e0d\u662f\u9709\u9709\uff0c\u662f\u4e0b\u9762\u8fd9\u4f4d\uff1a<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u4ee5\u8fd9\u4f4d\u6cf0\u52d2\u547d\u540d\u7684\u516c\u5f0f\u8fd8\u8bb0\u7684\u5417\uff1f<\/p>\n\n\n\n

\n

\u8bbe$n$\u662f\u4e00\u4e2a\u6b63\u6574\u6570\u3002\u5982\u679c\u5b9a\u4e49\u5728\u4e00\u4e2a\u5305\u542b$a$\u7684\u533a\u95f4\u4e0a\u7684\u51fd\u6570$f$\u5728\u70b9$a$\u5904$n+1$\u6b21\u53ef\u5bfc\uff0c\u90a3\u4e48\u5bf9\u4e8e\u8fd9\u4e2a\u533a\u95f4\u4e0a\u4efb\u610f$x$\uff0c\u90fd\u6709\uff1a<\/p>\n\n\n\n

$$f(x) = f(a) + \\frac{f^{‘}(a)}{1!}(x -a ) + \\frac{f^{(2)}(a)}{2!}(x – a)^2 + … + \\frac{f^{(n)}(a)}{n!}(x – a)^n + o[(x-a)^n]$$<\/p>\n\n\n\n

\u5176\u4e2d\u7684\u591a\u9879\u5f0f\u79f0\u4e3a\u51fd\u6570\u5728$a$\u5904\u7684\u6cf0\u52d2\u5c55\u5f00\u5f0f\uff0c\u5269\u4f59\u7684$R_{n}(a)$\u5f0f\u6cf0\u52d2\u516c\u5f0f\u7684\u4f59\u9879\uff0c\u662f$(x-a)^n$\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f.<\/p>\n\u6765\u81ea\u7ef4\u57fa\u767e\u79d1<\/a><\/cite><\/blockquote>\n\n\n\n

\u653e\u5728\u6211\u4eec\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u8fd9\u6837\u89e3\u91ca\uff0c\u65b0\u5f15\u5165\u4e00\u4e2a\u53d8\u91cf$\\Delta x = x_{k+1} – x_{k}$\uff0c\u5982\u679c\u8fd9\u4e2a$\\Delta x$\u8db3\u591f\u5c0f\u7684\u8bdd\uff0c$f(x_{k+1})$\u8fd1\u4f3c\u7b49\u4e8e$f(x_k) + \\nabla f(x_k) \\cdot \\Delta x$\uff0c\u8fd9\u91cc\u628a\u5bfc\u6570\u6362\u6210\u68af\u5ea6\u7b26\u53f7$\\nabla$\u4e00\u822c\u6211\u4eec\u78b0\u5230\u7684\u662f\u9ad8\u7eac\u5411\u91cf\u3002 \u5f53\u7136\u4e5f\u53ef\u4ee5\u8ba1\u7b97\u4e8c\u9636\u5bfc\u6570\uff0c\u8fd9\u6837\u8fd1\u4f3c\u7684\u66f4\u51c6\u786e\uff0c\u4f46\u662f\u4e5f\u53d8\u5f97\u590d\u6742\u4e86\uff0c\u6709\u673a\u4f1a\u518d\u804a\u8fd9\u4e2a\uff0c\u73b0\u5728\u6211\u4eec\u5c31\u7528\u4e00\u9636\u5bfc\u6570\u8fd1\u4f3c<\/p>\n\n\n\n

$$f(x_{k+1}) \\approx f(x_k + \\Delta x) \\sim f(x_k) + \\nabla f(x_k)\\cdot \\Delta x$$<\/p>\n\n\n\n

$\\Delta x$\u662f\u5411\u91cf\uff0c\u662f\u5411\u91cf\u5c31\u6709\u4e24\u4e2a\u90e8\u5206\uff0c\u957f\u5ea6\u548c\u65b9\u5411\uff0c$\\Delta x = \\alpha \\rho_{k}$\uff0c$\\alpha$\u662f\u5927\u4e8e0\u7684\u975e\u5e38\u5c0f\u7684\u5e38\u6570\uff0c\u53eb\u5b83\u6b65\u957f\uff0c$\\rho_k$\u662f\u5355\u4f4d\u5411\u91cf\uff0c\u65b0\u4e00\u8f6e\u7684\u51fd\u6570\u503c\u53ef\u4ee5\u5199\u6210<\/p>\n\n\n\n

$$f(x_k) + \\alpha \\rho_{k}^{T} \\nabla f(x_k)$$<\/p>\n\n\n\n

\u65e2\u7136\u8981\u6c42\u5b83\u5c0f\u4e8e$f(x_k)$\uff0c\u53ea\u80fd\u52a0\u53f7\u53f3\u8fb9\u7684$\\\\alpha \\rho_{k}^{T} \\nabla f(x_k)$\u90e8\u5206\u5c0f\u4e8e0\u4e86\uff0c\u53cd\u6b63\u662f\u8d70\u4e00\u6b65\uff0c\u90a3\u7d22\u6027\u8ba9\u8fd9\u4e00\u6b65\u5c3d\u53ef\u80fd\u7684\u5c0f\u5427\uff0c\u6700\u5c0f\u5316\u8fd9\u4e00\u90e8\u5206\u5427\uff0c$\\alpha$\u662f\u5e38\u6570\uff0c\u6240\u4ee5\u65b0\u4e00\u8f6e\u7684\u76ee\u6807\u5c31\u662f<\/p>\n\n\n\n

$$\\min \\rho_{k}^{T}\\nabla f(x_k)$$<\/p>\n\n\n\n

\u5570\u55e6\u4e00\u53e5\uff0c\u6700\u5c0f\u5316$\\rho_{k}^{T}\\nabla f(x_k)$\u5c31\u662f\u6700\u5c0f\u5316$f(x_k) + \\alpha \\rho_{k}^{T} \\nabla f(x_k)$\uff0c\u4e5f\u5c31\u662f\u6700\u5c0f\u5316$f(x_k + \\Delta x)$\uff0c\u4e5f\u5c31\u662f\u6700\u5c0f\u5316$f(x_{k+1})$\uff0c\u5373\u4e0b\u4e00\u6b65\u518d\u6b65\u957f$\\alpha$\u56fa\u5b9a\u7684\u60c5\u51b5\u4e0b\u5c3d\u53ef\u80fd\u7684\u8ba9$f(x_{k+1})$\u5c0f\uff0c\u6bcf\u4e00\u6b21\u8fed\u4ee3\u90fd\u5c3d\u53ef\u80fd\u7684\u5c0f\uff0c\u76f4\u5230\u5230\u8fbe\u67d0\u4e2a$x$\u7a33\u5b9a\u4e0b\u6765\u3002\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u601d\u8def\u975e\u5e38\u8d2a\u5a6a\uff0c\u4e8e\u662f\u5f88\u5bb9\u6613\u6389\u8fdb\u53e5\u6b65\u6700\u5c0f\u503c\u65e0\u6cd5\u81ea\u62d4\u3002<\/p>\n\n\n\n

$\\min \\rho_{k}^{T}\\nabla f(x_k)$\u600e\u4e48\u89e3\uff1f \u4e24\u4e2a\u5411\u91cf\u505a\u4e58\u6cd5\uff0c\u5047\u5982\u5411\u91cf\u7684\u5939\u811a\u662f$\\theta$<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u6839\u636e\u5411\u91cf\u4e58\u6cd5\u7684\u5b9a\u4e49$\\rho_{k}^{T} \\nabla f(x_k) = ||\\rho_{k}|| \\cdot ||\\nabla f(x_k)|| \\cos\\theta$,\u5176\u4e2d$ ||\\rho_{k}|| \\cdot ||\\nabla f(x_k)|| $\u662f\u5927\u4e8e0\u7684\uff0c$\\cos \\theta$\u6700\u5c0f\u662f-1,\u4e8e\u662f$\\rho_{k}^{T} \\nabla f(x_k)$\u6700\u5c0f\u503c\u662f$ -||\\rho_{k}|| \\cdot ||\\nabla f(x_k)|| $,\u6b64\u65f6$\\theta = 180^\\circ$\uff0c\u50cf\u8fd9\u6837<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

$\\rho_k$\u548c$\\nabla f(x_k)$\u7684\u65b9\u5411\u662f\u76f8\u53cd\u7684\uff0c\u5373$\\rho_k = -\\nabla f(x_k)$\uff0c\u6709\u6ca1\u6709\u4e00\u79cd\u4f3c\u66fe\u76f8\u8bc6\u7684\u611f\u89c9\uff0c\u6ca1\u9519\uff0c\u68af\u5ea6\u4e0b\u964d\u516c\u5f0f\u7ec8\u4e8e\u51fa\u73b0\u4e86\uff0c\u65b0\u4e00\u8f6e\u53ef\u4ee5\u4f7f$f(x_{k+1})$\u6700\u5c0f\u5316\u7684\u884c\u8fdb\u65b9\u5411\u662f\u8d1f\u68af\u5ea6\u65b9\u5411\u3002<\/p>\n\n\n\n

\u5982\u679c\u6bcf\u6b21\u6211\u4eec\u90fd\u9009\u62e9\u8d1f\u68af\u5ea6\u65b9\u5411\u66f4\u65b0$x$\uff0c\u518d\u4fdd\u8bc1$f(x_k +\\Delta x) < f(x_k)$\u7684\u540c\u65f6\uff0c\u8fd8\u80fd\u4fdd\u8bc1$f(x_k + \\Delta x)$\u4e00\u76f4\u662f\u6700\u5c0f\u7684\uff0c\u6bcf\u6b21\u8fed\u4ee3\u90fd\u4f1a\u66f4\u65b0$x$\uff0c\u6240\u4ee5\u4e0b\u6b21\u53c8\u53ef\u4ee5\u8ba1\u7b97$\\min f(x_k + \\Delta x)$\uff0c\u8fd9\u6837\u4e00\u81f4\u8fed\u4ee3\u4e0b\u53bb\uff0c\u603b\u4f1a\u627e\u5230\u4e00\u4e2a$x$\u662f\u7684$f(x)$\u6700\u5c0f\uff0c\u5f53\u7136\u6709\u53ef\u80fd\u662f\u5c40\u90e8\u6700\u5c0f\u3002<\/p>\n\n\n\n

PS\uff1a\u65e2\u7136\u4fdd\u8bc1\u6bcf\u6b21\u8fed\u4ee3$f(x_{k+1}) < f(x_k)$\uff0c\u4e5f\u5c31\u662f$\\rho_{k}^{T} \\nabla f(x_k)$\u5c0f\u4e8e0\u5c31\u884c\u5427\uff0c\u6240\u4ee5$\\rho_k$\u5e76\u4e0d\u4e00\u5b9a\u975e\u8981\u9009\u62e9\u8d1f\u68af\u5ea6\u65b9\u5411\uff0c\u4e5f\u80fd\u662f$f(x)$\u9010\u6e10\u53d8\u5c0f<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u53ea\u8981$\\alpha$\u8db3\u591f\u5c0f\uff0c$\\rho_k$\u53ef\u4ee5\u9009\u62e9\u84dd\u8272\u533a\u57df\u7684\u4efb\u610f\u65b9\u5411\uff0c\u90fd\u53ef\u4ee5\u4fdd\u8bc1\u65b0\u4e00\u8f6e\u7684\u8fed\u4ee3$f(x_{k+1})$\u5c0f\u4e8e$f(x_k)$\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e3a\u4ec0\u4e48\u662f\u68af\u5ea6\u5462\uff1f\u4e3a\u4ec0\u4e48\u4e0d\u662f\u786c\u5ea6\uff1f\u957f\u5ea6\uff1f\u96f6\u5ea6\uff1f\u767e\u5ea6\uff1f\u662f\u4e0d\u662f\u53ea\u6709\u68af\u5ea6\u53ef\u4ee5\u4f7f\u6211\u4eec\u8fbe\u5230\u6700\u4f18\u89e3\uff1f<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16,4],"tags":[],"class_list":["post-439","post","type-post","status-publish","format-standard","hentry","category-base","category-machine-learning"],"_links":{"self":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/439","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=439"}],"version-history":[{"count":25,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/439\/revisions"}],"predecessor-version":[{"id":661,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/439\/revisions\/661"}],"wp:attachment":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=439"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=439"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}