{"id":45759,"date":"2025-07-30T01:06:02","date_gmt":"2025-07-29T17:06:02","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/45759.html"},"modified":"2025-07-30T01:06:02","modified_gmt":"2025-07-29T17:06:02","slug":"2025%e5%b9%b4nature%e5%ad%90%e5%88%8a%e6%96%b0%e7%ae%97%e6%b3%95-%e6%a2%af%e5%ba%a6%e4%b8%8b%e9%99%8d%e7%ae%97%e6%b3%95adam-gradient-descent-%e9%99%84matlab%e5%85%8d%e8%b4%b9%e4%bb%a3","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/45759.html","title":{"rendered":"2025\u5e74Nature\u5b50\u520a\u65b0\u7b97\u6cd5\u2014\u2014\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5Adam Gradient Descent-\u9644Matlab\u514d\u8d39\u4ee3\u7801"},"content":{"rendered":"

1\u3001\u7b80\u4ecb<\/h4>\n

\u5728\u8fc7\u53bb\u7684\u51e0\u5e74\u91cc，\u65e0\u6570\u57fa\u4e8e\u7fa4\u4f53\u667a\u80fd\u7684\u5143\u542f\u53d1\u5f0f\u7b97\u6cd5\u88ab\u5f15\u5165\u5e76\u5e7f\u6cdb\u5e94\u7528\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u7b97\u6cd5\u501f\u9274\u4e86\u751f\u7269\u884c\u4e3a，\u4f46\u5b83\u4eec\u76f8\u4f3c\u7684\u542f\u53d1\u5f0f\u8303\u5f0f\u548c\u6a21\u5757\u5316\u8bbe\u8ba1\u5bfc\u81f4\u4e86\u590d\u6742\u4f18\u5316\u95ee\u9898\u4e2d\u7684\u4e0d\u5e73\u8861\u63a2\u7d22\u548c\u5229\u7528\u3002\u5c06\u6570\u5b66\u5c5e\u6027\u4e0e\u968f\u673a\u641c\u7d22\u8fc7\u7a0b\u76f8\u7ed3\u5408\u7684\u5143\u542f\u53d1\u5f0f\u7b97\u6cd5\u53ef\u4ee5\u5e2e\u52a9\u7a81\u7834\u4f20\u7edf\u7684\u8fdb\u5316\u8303\u5f0f，\u589e\u5f3a\u4e2a\u4f53\u4f18\u5316\u3002\u4e3a\u4e86\u8ffd\u6c42\u8fd9\u4e00\u76ee\u6807，\u672c\u7814\u7a76\u5f15\u5165\u4e86\u4e00\u79cd\u57fa\u4e8e\u6570\u5b66\u7684\u521b\u65b0\u5143\u542f\u53d1\u5f0f\u7b97\u6cd5，\u79f0\u4e3a\u4e9a\u5f53\u68af\u5ea6\u4e0b\u964d\u4f18\u5316\u5668（AGDO），\u65e8\u5728\u89e3\u51b3\u6301\u7eed\u4f18\u5316\u548c\u5de5\u7a0b\u6311\u6218\u3002AGDO\u53d7\u5230Adam\u4f18\u5316\u5668\u7684\u542f\u53d1，\u4f7f\u7528\u4e09\u4e2a\u89c4\u5219\u63a2\u7d22\u4e86\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b：\u6e10\u8fdb\u5f0f\u68af\u5ea6\u52a8\u91cf\u79ef\u5206\u3001\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u548c\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u3002\u6e10\u8fdb\u5f0f\u68af\u5ea6\u52a8\u91cf\u79ef\u5206\u548c\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u5f88\u597d\u5730\u5e73\u8861\u4e86\u63a2\u7d22\u548c\u5f00\u53d1，\u800c\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u5219\u7ec6\u5316\u4e86\u5f00\u53d1\u65b9\u9762\u3002AGDO\u7684\u6027\u80fd，\u7ed3\u5408\u51e0\u4e2a\u77e5\u540d\u548c\u65b0\u5f15\u5165\u7684\u5143\u542f\u53d1\u5f0f，\u5728CEC2017\u57fa\u51c6\u4e0a\u8de8\u5404\u4e2a\u7ef4\u5ea6\u548c\u516d\u4e2a\u5b9e\u9645\u5de5\u7a0b\u6311\u6218\u8fdb\u884c\u4e86\u8bc4\u4f30\u3002Wilcoxon\u79e9\u548c\u68c0\u9a8c\u8bc1\u5b9e\u4e86\u5176\u529f\u6548\u3002<\/p>\n

2\u3001\u63d0\u51fa\u7684AGDO\u7b97\u6cd5<\/h4>\n

\u672c\u90e8\u5206\u6df1\u5165\u63a2\u8ba8\u4e86\u4e9a\u5f53\u4f18\u5316\u7b97\u6cd5\u80cc\u540e\u7684\u57fa\u672c\u7406\u8bba\u548c\u539f\u7406\u3002\u6b64\u5916，\u6df1\u5165\u9610\u8ff0\u4e86\u4e9a\u5f53\u68af\u5ea6\u4e0b\u964d\u4f18\u5316\u5668（AGDO）\u7684\u7075\u611f，\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u5176\u4f18\u5316\u8fc7\u7a0b\u548c\u5173\u952e\u673a\u5236\u3002<\/p>\n

2.1 \u52a8\u673a<\/h5>\n

\u968f\u673a\u68af\u5ea6\u4f18\u5316\u5728\u8bb8\u591a\u79d1\u5b66\u548c\u5de5\u7a0b\u5b66\u79d1\u4e2d\u5177\u6709\u91cd\u8981\u7684\u5b9e\u7528\u4ef7\u503c\u3002\u8fd9\u4e9b\u9886\u57df\u7684\u8bb8\u591a\u95ee\u9898\u53ef\u4ee5\u7b80\u5316\u4e3a\u4f18\u5316\u53c2\u6570\u76ee\u6807\u51fd\u6570，\u76ee\u6807\u662f\u6700\u5927\u5316\u6216\u6700\u5c0f\u5316\u7279\u5b9a\u6807\u91cf\u503c79\u3002\u5f53\u76ee\u6807\u51fd\u6570\u53ef\u4ee5\u4e0e\u5176\u53c2\u6570\u76f8\u533a\u522b\u65f6，\u68af\u5ea6\u4e0b\u964d\u6210\u4e3a\u4e00\u79cd\u6709\u6548\u7684\u4f18\u5316\u6280\u672f，\u56e0\u4e3a\u8ba1\u7b97\u6240\u6709\u53c2\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\u5728\u8ba1\u7b97\u4e0a\u7c7b\u4f3c\u4e8e\u8bc4\u4f30\u51fd\u6570\u672c\u8eab80\u3002\u7136\u800c，\u5728\u73b0\u5b9e\u4e2d，\u76ee\u6807\u51fd\u6570\u5f80\u5f80\u662f\u968f\u673a\u7684，\u5177\u6709\u4e0d\u786e\u5b9a\u6027\u548c\u6ce2\u52a8\u6027\u30022014\u5e74，\u4e00\u9636\u65b9\u6cd5Adam\u88ab\u63d0\u51fa\u6765\u5904\u7406\u9ad8\u7ef4\u53c2\u6570\u7a7a\u95f4\u4e2d\u7684\u968f\u673a\u76ee\u6807\u4f18\u5316\u95ee\u9898\u3002Adam\u6280\u672f\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u968f\u673a\u4f18\u5316\u65b9\u6cd5，\u53ea\u9700\u8981\u4e00\u9636\u68af\u5ea6\u548c\u6700\u5c11\u7684\u5185\u5b58\u4f7f\u7528\u3002Adam，\u201c\u81ea\u9002\u5e94\u77e9\u4f30\u8ba1\u201d\u7684\u7f29\u5199，\u901a\u8fc7\u8bc4\u4f30\u68af\u5ea6\u7684\u7b2c\u4e00\u548c\u7b2c\u4e8c\u77e9\u6765\u8ba1\u7b97\u6bcf\u4e2a\u53c2\u6570\u7684\u552f\u4e00\u5b66\u4e60\u7387\u3002Adam\u65b9\u6cd5\u7ed3\u5408\u4e86\u4e24\u79cd\u6d41\u884c\u65b9\u6cd5\u7684\u4f18\u52bf：AdaGrad\u548cRMSProp\u3002AdaGrad\u5728\u7a00\u758f\u68af\u5ea682\u4e0a\u8868\u73b0\u826f\u597d，\u800cRMSProp\u5728\u5728\u7ebf\u548c\u975e\u5e73\u6ed1\u8bbe\u7f6e83\u3002\u5e73\u6ed1\u76ee\u6807，\u5904\u7406\u7a00\u758f\u68af\u5ea6\u7684\u80fd\u529b，\u4ee5\u53ca\u4e00\u79cd\u6b65\u8fdb\u9000\u706b\u5f62\u5f0f\u7684\u81ea\u7136\u6267\u884c\u3002\u8fd9\u4e9b\u7279\u6027\u4f7fAdam\u65b9\u6cd5\u6210\u4e3a\u8bb8\u591a\u5b66\u4e60\u673a\u5668\u548c\u6df1\u5ea6\u5b66\u4e60\u5e94\u7528\u4e2d\u5f3a\u5927\u7684\u4f18\u5316\u5de5\u5177\u3002<\/p>\n

2.2 \u68af\u5ea6\u4e0b\u964d\u6cd5（GDM）<\/h5>\n

\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u662f\u4e00\u79cd\u7528\u4e8e\u5b9a\u4f4d\u76ee\u6807\u51fd\u6570\u5c40\u90e8\u6700\u5c0f\u503c\u7684\u4e3b\u8981\u4f18\u5316\u6280\u672f，\u5b83\u88ab\u5e7f\u6cdb\u7528\u4e8e\u673a\u5668\u5b66\u4e60\u548c\u6df1\u5ea6\u5b66\u4e60\u3002\u68af\u5ea6\u4e0b\u964d\u65b9\u6cd5\u80cc\u540e\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5229\u7528\u76ee\u6807\u51fd\u6570\u7684\u68af\u5ea6\u6570\u636e\u6765\u6307\u5bfc\u4f4d\u7f6e\u7684\u66f4\u65b0\u3002\u7279\u522b\u662f，\u68af\u5ea6\u6307\u793a\u76ee\u6807\u51fd\u6570\u5728\u7ed9\u5b9a\u70b9\u7684\u6700\u9661\u589e\u957f\u8def\u5f84，\u800c\u5411\u68af\u5ea6\u7684\u53cd\u5411\u79fb\u52a8\u5bfc\u81f4\u76ee\u6807\u51fd\u6570\u7684\u6700\u5feb\u4e0b\u964d\u3002\u6211\u4eec\u5728\u68af\u5ea6\u7684\u53cd\u5411\u91cd\u590d\u8c03\u6574\u4ee5\u5b9a\u4f4d\u51fd\u6570\u7684\u5c40\u90e8\u6700\u5c0f\u503c\u3002<\/p>\n

\u4e3a\u4e86\u66f4\u597d\u5730\u7406\u89e3\u68af\u5ea6\u4e0b\u964d\u6cd5(GDM)，\u6211\u4eec\u9700\u8981\u5b9a\u4e49\u9700\u8981\u6700\u5c0f\u5316\u7684\u76ee\u6807\u51fd\u6570f(x)f(x)<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u8fd9\u91ccx=[x1,x2,\u2026,xn]x = [x_1, x_2, \\\\ldots, x_n]<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u4e00\u4e2an\u7ef4\u5411\u91cf，\u76ee\u6807\u662f\u627e\u5230\u80fd\u4f7ff(x)f(x)<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u503c\u6700\u5c0f\u7684xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u3002\u76ee\u6807\u51fd\u6570\u7684\u659c\u7387\u8868\u793a\u4e3a\u4e00\u4e2a\u5411\u91cf（\u89c1\u56fe2）\u3002\u56fe\u4e2d\u7684\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>（\u5373xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>）\u8868\u793a\u51fd\u6570\u5bf9\u6bcf\u4e2a\u53d8\u91cf\u7684\u53d8\u5316\u7387：<\/p>\n

\u2207J(x)=[\u2202f(x)\u2202x1,\u2202f(x)\u2202x2,\u22ef\u2009,\u2202f(x)\u2202xn](1)\\\\nabla J(x) =
\n\\\\begin{bmatrix}
\n\\\\frac{\\\\partial f(x)}{\\\\partial x_1}, & \\\\frac{\\\\partial f(x)}{\\\\partial x_2}, & \\\\cdots, & \\\\frac{\\\\partial f(x)}{\\\\partial x_n}
\n\\\\end{bmatrix} \\\\tag{1}<\/span><\/span>\u2207<\/span>J<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span><\/span>\u2202<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u2207J(x)\\\\nabla J(x)<\/span><\/span>\u2207<\/span>J<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u5f53\u524d\u4f4d\u7f6e\u635f\u5931\u51fd\u6570J(x)J(x)<\/span><\/span>J<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u68af\u5ea6\u3002<\/p>\n

\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe1 GDM\u66f4\u65b0\u65b9\u5411\u53ca\u5176\u539f\u7406
\n\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df，\u5e38\u7528\u7684\u635f\u5931\u51fd\u6570\u662f\u5747\u65b9\u8bef\u5dee(MSE)，\u5b83\u91cf\u5316\u4e86\u9884\u6d4b\u503c\u4e0e\u5b9e\u9645\u7ed3\u679c\u4e4b\u95f4\u7684\u5dee\u5f02\u3002MSE\u662f\u4e00\u4e2a\u51f8\u51fd\u6570，\u53ef\u4ee5\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5(GDM)\u6765\u6700\u5c0f\u5316\u8fd9\u4e2a\u635f\u5931\u51fd\u6570\u4ee5\u627e\u5230\u6700\u4f18\u4f4d\u7f6e\u3002MSE\u5173\u4e8e\u4f4d\u7f6exx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u7684\u68af\u5ea6\u4e3a：<\/p>\n

\u2207J(x)=2m\u2211i=1m(h(x,x(i))\u2212yi)\u22c5\u2207xh(x,x(i))(2)\\\\nabla J(x)=\\\\frac{2}{m}\\\\sum\\\\nolimits_{i=1}^{m}\\\\left(h\\\\left(x,x^{(i)}\\\\right)-y_{i}\\\\right)\\\\cdot\\\\nabla_{x}h\\\\left(x,x^{(i)}\\\\right)\\\\tag{2}<\/span><\/span>\u2207<\/span>J<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>h<\/span><\/span>(<\/span><\/span>x<\/span>,<\/span><\/span>x<\/span><\/span>(<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>\u2207<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>(<\/span><\/span>x<\/span>,<\/span><\/span>x<\/span><\/span>(<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>(<\/span>2<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u91ccmm<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u6837\u672c\u6570\u91cf，yiy_{i}<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7b2cii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u4e2a\u6837\u672c\u7684\u5b9e\u9645\u503c，\u51fd\u6570\u7684\u9884\u6d4b\u503cy^j\\\\hat{y}_{j}<\/span><\/span><\/span>y<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5373y^i=h(x,x(i))\\\\hat{y}_{i}=h\\\\left(x,x^{(i)}\\\\right)<\/span><\/span><\/span>y<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>(<\/span><\/span>x<\/span>,<\/span><\/span>x<\/span><\/span>(<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u6b64\u5916，x(i)x^{(i)}<\/span><\/span>x<\/span><\/span>(<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5bf9\u5e94\u4e8e\u7b2cii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u4e2a\u6837\u672c\u7684\u4f4d\u7f6e\u5411\u91cf\u3002<\/p>\n

\u4e3a\u4e86\u786e\u5b9a\u4ece\u5f53\u524d\u70b9\u79fb\u52a8\u7684\u8ddd\u79bb，\u5c06\u6b65\u957f\u03b1\\\\alpha<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u4e58\u4ee5\u635f\u5931\u51fd\u6570\u7684\u68af\u5ea6：<\/p>\n

\u4e0b\u964d\u8ddd\u79bb=\u03b1\u22c5\u2202\u2202xiJ(x1,x2,\u2026,xn)(3){\\\\rm \u4e0b\u964d\u8ddd\u79bb}=\\\\alpha\\\\cdot\\\\frac{\\\\partial}{\\\\partial x_{i}}J(x_{1},x_{2},\\\\ldots,x_{n})\\\\tag{3}<\/span><\/span>\u4e0b\u964d\u8ddd\u79bb<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>J<\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>(<\/span>3<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u68af\u5ea6\u4e0b\u964d\u7684\u57fa\u672c\u601d\u60f3\u662f\u6cbf\u7740\u76ee\u6807\u51fd\u6570\u68af\u5ea6\u7684\u76f8\u53cd\u65b9\u5411\u79fb\u52a8\u6765\u66f4\u65b0\u4f4d\u7f6e\u3002\u5047\u8bbextx_{t}<\/span><\/span>x<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u7b2ctt<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u6b21\u8fed\u4ee3\u7684\u4f4d\u7f6e\u5411\u91cf，\u5728\u6bcf\u6b21\u8fed\u4ee3\u6b65\u9aa4\u4e2d，\u6839\u636e\u68af\u5ea6\u7684\u8d1f\u65b9\u5411\u66f4\u65b0\u4f4d\u7f6e，\u68af\u5ea6\u4e0b\u964d\u6cd5\u7684\u6700\u7ec8\u66f4\u65b0\u516c\u5f0f\u4e3a：<\/p>\n

Xt+1=xt\u2212\u03b1\u22c5\u2207J(x)(4)X_{t+1}=x_{t}-\\\\alpha\\\\cdot\\\\nabla J(x)\\\\tag{4}<\/span><\/span>X<\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>\u2207<\/span>J<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span>(<\/span>4<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b9e\u9645\u4e0a，\u68af\u5ea6\u4e0b\u964d\u6cd5(GDM)\u9762\u4e34\u7740\u51e0\u4e2a\u95ee\u9898，\u5305\u62ec\u978d\u70b9\u95ee\u9898，\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u4f1a\u963b\u788d\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\u3002\u978d\u70b9\u51fa\u73b0\u5728\u76ee\u6807\u51fd\u6570\u68af\u5ea6\u7b49\u4e8e\u96f6\u4f46\u8be5\u70b9\u4e0d\u662f\u5c40\u90e8\u6700\u5c0f\u503c\u6216\u6700\u5927\u503c\u7684\u4f4d\u7f6e\u3002\u978d\u70b9\u5728\u67d0\u4e9b\u65b9\u5411\u4e0a\u8868\u73b0\u4e3a\u6700\u5c0f\u503c，\u5728\u5176\u4ed6\u65b9\u5411\u4e0a\u8868\u73b0\u4e3a\u6700\u5927\u503c\u6216\u5e73\u5766\u533a\u57df(\u56fe3)\u3002\u978d\u70b9\u7684\u5b58\u5728\u5bfc\u81f4GDM\u505c\u6ede\u4e0d\u524d，\u56e0\u4e3a\u5728\u978d\u70b9\u5904\u68af\u5ea6\u4e3a\u96f6，\u5bfc\u81f4\u4f4d\u7f6e\u66f4\u65b0\u505c\u6b62\u3002\u6b64\u5916，\u5728\u978d\u70b9\u9644\u8fd1，\u68af\u5ea6\u7684\u53d8\u5316\u975e\u5e38\u5c0f，\u5bfc\u81f4\u66f4\u65b0\u6b65\u957f\u6781\u5c0f，\u4f7f\u5f97GDM\u7684\u6536\u655b\u901f\u5ea6\u663e\u8457\u51cf\u6162\u3002
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe2 \u978d\u70b9\u529f\u80fd\u6f14\u793a
\n\u5728GDM\u5c31\u50cf\u6211\u4eec\u5f92\u6b65\u65c5\u884c\u65f6\u91c7\u53d6\u7684\u6b65\u9aa4\u5927\u5c0f，\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u03b1\\\\alpha<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u63a7\u5236\u597d\u6211\u4eec\u6bcf\u4e00\u6b65\u8d70\u7684\u8ddd\u79bb，\u786e\u4fdd\u6211\u4eec\u4e0d\u4f1a\u8d70\u592a\u5927\u7684\u4e00\u6b65，\u5176\u5b9e\u5c31\u662f\u4e0d\u8981\u8d70\u5f97\u592a\u5feb\u800c\u65e9\u65e9\u75b2\u52b3（\u9677\u5165\u5c40\u90e8\u6700\u4f18）\u800c\u9519\u8fc7\u6700\u4f4e\u70b9；\u540c\u65f6，\u6211\u4eec\u8fd8\u9700\u8981\u786e\u4fdd\u6211\u4eec\u4e0d\u8981\u8d70\u5f97\u592a\u6162（\u6536\u655b\u901f\u5ea6\u592a\u6162），\u5bfc\u81f4\u592a\u9633\u5df2\u7ecf\u4e0b\u5c71，\u8fd8\u6ca1\u6709\u8d70\u5230\u5c71\u5e95\u3002<\/p>\n

3 Adam\u68af\u5ea6\u4e0b\u964d\u4f18\u5316\u5668(AGDO)<\/h4>\n

\u63d0\u51fa\u7684AGDO\u7b97\u6cd5\u901a\u8fc7\u4ee5\u4e0b\u65b9\u5f0f\u5b9e\u73b0\u66f4\u9ad8\u6548\u7684\u5168\u5c40\u641c\u7d22\u548c\u4f18\u5316：\u5e94\u7528GDM\u8fdb\u884c\u4f4d\u7f6e\u4fe1\u606f\u641c\u7d22\u548c\u65b9\u5411\u5f15\u5bfc，\u91c7\u7528\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u7b56\u7565\u8fdb\u884c\u63a2\u7d22，\u5e76\u901a\u8fc7\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u548c\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u4f18\u5316\u5f00\u53d1\u8fc7\u7a0b\u3002<\/p>\n

3.1 \u521d\u59cb\u5316<\/h5>\n

\u4f18\u5316\u95ee\u9898\u7684\u80cc\u666f\u662f\u5904\u7406\u65e0\u7ea6\u675f\u5355\u76ee\u6807\u4f18\u5316\u4efb\u52a1，\u65e8\u5728\u8bc6\u522b\u6574\u4e2a\u5b9a\u4e49\u57dfR\\\\mathbb{R}<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span>\u5185\u4f7f\u76ee\u6807\u51fd\u6570\u6700\u5927\u6216\u6700\u5c0f\u7684\u53d8\u91cf\u7ec4\u5408\u3002\u95ee\u9898\u5b9a\u4e49\u5982\u4e0b：<\/p>\n

min\u2061x:RN\u00d7Dmf(x)(5)\\\\min_{x:\\\\mathbb{R}^{N\\\\times Dm}} f(x)\\\\tag{5}<\/span><\/span><\/span>x<\/span>:<\/span>R<\/span><\/span>N<\/span>\u00d7<\/span>D<\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span>(<\/span>5<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2df(x)f(x)<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u8981\u6700\u5c0f\u5316\u7684\u9002\u5e94\u5ea6\u51fd\u6570，x=[xij]\u2208RN\u00d7Dimx=[x_{ij}]\\\\in\\\\mathbb{R}^{N\\\\times Dim}<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>N<\/span>\u00d7<\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5f85\u4f18\u5316\u7684\u53d8\u91cf\u77e9\u9635，xijx_{ij}<\/span><\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u4f18\u5316\u5411\u91cf(\u5373\u7b2cjj<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u7ef4\u4e2d\u7b2cii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u4e2a\u4e2a\u4f53\u5411\u91cf)，DimDim<\/span><\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u5f85\u4f18\u5316\u95ee\u9898\u7684\u7ef4\u5ea6\u3002\u6bcf\u4e2axijx_{ij}<\/span><\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u53d6\u503c\u53d7\u4ee5\u4e0b\u6761\u4ef6\u7ea6\u675f：<\/p>\n

LBj\u2264xij\u2264UBj\u00a0\u2200i=1,2\u00a0,\u00a0N,\u00a0\u2200j=1,2,\u2026,Dim(6)LB_{j}\\\\leq x_{ij}\\\\leq UB_{j}\\\\ \\\\forall i=1,2\\\\ ,\\\\ N,\\\\ \\\\forall j=1, 2,\\\\ldots,Dim\\\\tag{6}<\/span><\/span>L<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>U<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0<\/span>\u2200<\/span>i<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>2<\/span>\u00a0<\/span>,<\/span>\u00a0<\/span><\/span>N<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>j<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>2<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>D<\/span>im<\/span><\/span><\/span>(<\/span>6<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nLBj=[LB1,LB2,\u2026,LBDim]\u00a0,UBj=[UB1,UB2,\u2026,UBDim]LB_{j}=[LB_{1},LB_{2},\\\\ldots,LB_{Dim}]\\\\ ,UB_{j}=[UB_{1},UB_{2},\\\\ldots,UB_{Dim}]<\/span><\/span>L<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>L<\/span>B<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>L<\/span>B<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>L<\/span>B<\/span><\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span>\u00a0<\/span>,<\/span><\/span>U<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>U<\/span>B<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>U<\/span>B<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>U<\/span>B<\/span><\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f0f(6)\u4e2d，LBjLB_{j}<\/span><\/span>L<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4f18\u5316\u53d8\u91cf\u7b2cjj<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u7ef4\u7684\u4e0b\u754c\u5411\u91cf，UBjUB_{j}<\/span><\/span>U<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u76f8\u5e94\u7684\u4e0a\u754c\u5411\u91cf\u3002\u4e0e\u5176\u4ed6\u5143\u542f\u53d1\u5f0f\u7b97\u6cd5\u7c7b\u4f3c，AGDO\u4ece\u5728\u5019\u9009\u89e3\u8fb9\u754c\u5185\u751f\u6210\u521d\u59cb\u968f\u673a\u79cd\u7fa4\u5f00\u59cb，\u4ee5\u63a2\u7d22\u6700\u4f18\u89e3\u3002\u8003\u8651\u5230\u95ee\u9898\u7684\u7fa4\u4f53\u6027\u8d28，\u79cd\u7fa4\u89c4\u6a21\u4e3aNN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>，\u6bcf\u4e2a\u79cd\u7fa4\u4e2a\u4f53\u7531\u6a21\u7cca\u4f18\u5316\u53d8\u91cf\u6216\u5411\u91cf\u7ec4\u6210\u3002\u56e0\u6b64，\u4f7f\u7528\u5f0f(7)\u751f\u6210\u968f\u673a\u79cd\u7fa4：<\/p>\n

xij=rand(N,Dim)\u00d7(UBj\u2212LBj)+LBj(7)x_{ij}=rand(N,Dim)\\\\times(UB_{j}-LB_{j})+LB_{j}\\\\tag{7}<\/span><\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>N<\/span>,<\/span><\/span>D<\/span>im<\/span>)<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>(<\/span>U<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>L<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>L<\/span>B<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>7<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dxijx_{ij}<\/span><\/span>x<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u79cd\u7fa4\u4e2d\u7b2cii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u4e2a\u4e2a\u4f53\u5728\u7b2cjj<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u7ef4\u7684\u4f4d\u7f6e\u6570\u636e，rand(N,Dim)rand(N,Dim)<\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>N<\/span>,<\/span><\/span>D<\/span>im<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u4e00\u4e2aN\u00d7DimN\\\\times Dim<\/span><\/span>N<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u7684\u77e9\u9635，\u586b\u5145\u67090\u52301\u4e4b\u95f4\u7684\u968f\u673a\u503c\u3002\u5f0f(8)\u63d0\u4f9b\u4e86\u7efc\u5408\u4f18\u5316\u6311\u6218\u7684\u77e9\u9635\u6a21\u578b，\u80fd\u591f\u8bf4\u660e\u79cd\u7fa4\u5728\u4e0d\u540c\u7ef4\u5ea6\u4e0a\u7684\u7ed3\u6784，\u6e05\u6670\u63cf\u8ff0\u4f18\u5316\u8fc7\u7a0b\u4e2d\u79cd\u7fa4\u7684\u591a\u6837\u6027\u548c\u4e2a\u4f53\u7684\u7a7a\u95f4\u5206\u5e03\u3002<\/p>\n

Xn=[x1,1×1,2\u22efx1,Dimx2,1×2,2\u22efx2,Dim\u22ee\u22ee\u22f1\u22eexN,1xN,2\u22efxN,Dim]N\u00d7Dim(8)X_{n}=\\\\begin{bmatrix}x_{1,1}&x_{1,2}&\\\\cdots&x_{1,Dim}\\\\\\\\ x_{2,1}&x_{2,2}&\\\\cdots&x_{2,Dim}\\\\\\\\ \\\\vdots&\\\\vdots&\\\\ddots&\\\\vdots\\\\\\\\ x_{N,1}&x_{N,2}&\\\\cdots&x_{N,Dim}\\\\end{bmatrix}_{N\\\\times Dim}\\\\tag{8}<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span>,<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span>,<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>N<\/span>,<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span>,<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span>,<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>N<\/span>,<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span>,<\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span>,<\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>N<\/span>,<\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span>\u00d7<\/span>D<\/span>im<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>8<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f00\u59cb\u65f6，AGDO\u5c06\u9002\u5e94\u5ea6\u5f97\u5206\u6700\u9ad8\u7684\u4e2a\u4f53\u8bc6\u522b\u4e3a\u6700\u4f18\u4e2a\u4f53，\u4f5c\u4e3a\u68af\u5ea6\u4e0b\u964d\u65b9\u5411\u7684\u8fd1\u4f3c\u6307\u5bfc\u3002\u5177\u4f53\u6765\u8bf4，\u5bf9\u4e8e\u6700\u5c0f\u5316\u95ee\u9898，\u4f18\u52bf\u4e2a\u4f53\u7684\u4f4d\u7f6e\u662f\u5f53\u524d\u627e\u5230\u7684\u5168\u5c40\u6700\u4f18\u89e3\u7684\u8fd1\u4f3c\u503c\u3002\u901a\u8fc7\u9009\u62e9\u9002\u5e94\u5ea6\u503c\u6700\u5c0f\u7684\u4e2a\u4f53\u4f5c\u4e3a\u521d\u59cb\u4f18\u52bf\u4f4d\u7f6e，\u7b97\u6cd5\u80fd\u591f\u5728\u65e9\u671f\u5feb\u901f\u903c\u8fd1\u6700\u4f18\u89e3\u3002\u521d\u59cb\u4f18\u52bf\u4f4d\u7f6eXsuperiorX_{\\\\text{superior}}<\/span><\/span>X<\/span><\/span>superior<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u516c\u5f0f\u5982\u4e0b：<\/p>\n

fbest=min\u2061(f(Xi))(9)f_{best}=\\\\min\\\\left(f\\\\left(X_{i}\\\\right)\\\\right)\\\\tag{9}<\/span><\/span>f<\/span><\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span><\/span>(<\/span>f<\/span><\/span>(<\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span>9<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Xsuperior=X(arg\u2061min\u2061(f(Xi)))(10)X_{\\\\text{superior}}=X\\\\left(\\\\arg\\\\min\\\\left(f\\\\left(X_{i}\\\\right)\\\\right)\\\\right)\\\\tag{10}<\/span><\/span>X<\/span><\/span>superior<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>(<\/span>arg<\/span><\/span><\/span>min<\/span><\/span>(<\/span>f<\/span><\/span>(<\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>)<\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span>10<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dargmin\u8fd0\u7b97\u7b26\u7528\u4e8e\u627e\u5230\u5177\u6709\u6700\u5c0f\u9002\u5e94\u5ea6\u503c\u7684\u4e2a\u4f53\u7684\u7d22\u5f15\u3002<\/p>\n

3.2 \u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408<\/h5>\n

\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u662f\u4e00\u79cd\u4e13\u4e3a\u63a2\u7d22\u9636\u6bb5\u8bbe\u8ba1\u7684\u6280\u672f，\u5176\u7075\u611f\u4e3b\u8981\u6765\u81eaAdam\u4f18\u5316\u5668\u4e2d\u4f7f\u7528\u7684\u68af\u5ea6\u52a8\u91cf（\u4e00\u9636\u77e9\u4f30\u8ba1）\u3002\u8be5\u9636\u6bb5\u57fa\u4e8e\u52a8\u91cf\u8c03\u6574\u7684\u52a8\u6001\u7279\u6027，\u91c7\u7528\u6e10\u8fdb\u5f0f\u52a8\u91cf\u7b56\u7565，\u4ece\u800c\u5f62\u6210\u521b\u65b0\u7684\u4f4d\u7f6e\u66f4\u65b0\u6846\u67b6\u3002\u8be5\u6846\u67b6\u4e3b\u8981\u901a\u8fc7\u7cfb\u7edf\u8c03\u6574\u4f4d\u7f6e\u7cfb\u6570\u6765\u53d1\u73b0\u6f5c\u5728\u66f4\u4f18\u89e3，\u4ee5\u63d0\u9ad8\u7b97\u6cd5\u7684\u5168\u5c40\u641c\u7d22\u6548\u7387\u3002\u4e0e\u56fa\u5b9a\u52a8\u91cf\u7b56\u7565\u4e0d\u540c，\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u901a\u8fc7\u9010\u6b65\u8c03\u6574\u4f4d\u7f6e\u7cfb\u6570\u6765\u9002\u5e94\u4f18\u5316\u8fc7\u7a0b\u7684\u53d8\u5316\u3002\u8fd9\u79cd\u81ea\u9002\u5e94\u7cfb\u7edf\u901a\u8fc7\u7efc\u5408\u8003\u8651\u5f53\u524d\u641c\u7d22\u72b6\u6001\u548c\u5386\u53f2\u4f4d\u7f6e\u6570\u636e，\u5b9e\u73b0\u5b9e\u65f6\u4f4d\u7f6e\u8c03\u6574\u3002\u7ecf\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u540e\u7684\u65b0\u4f4d\u7f6e\u5982\u5f0f(11)\u6240\u793a：<\/p>\n

X1,sumt+1={\u03c9\u2217Xkt+\u03b1t\u2217Xktk=1Xkt+sin\u2061(2\u03c0\u2217dim\u2061\u2217t)\u2217X1,sumt+11<k<2\/dim\u2061(11)X_{1,\\\\text{sum}}^{t+1}=\\\\begin{cases}
\n\\\\omega*X_{k}^{t}+\\\\alpha_{t}*X_{k}^{t} & k=1 \\\\\\\\
\nX_{k}^{t}+\\\\sin(2\\\\pi*\\\\dim*t)*X_{1,\\\\text{sum}}^{t+1} & 1<k<2\/\\\\dim
\n\\\\end{cases}\\\\tag{11}<\/span><\/span>X<\/span><\/span>1<\/span>,<\/span>sum<\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>\u03c9<\/span><\/span>\u2217<\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u03b1<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>sin<\/span>(<\/span>2<\/span>\u03c0<\/span><\/span>\u2217<\/span><\/span>dim<\/span><\/span>\u2217<\/span>t<\/span>)<\/span><\/span>\u2217<\/span><\/span>X<\/span><\/span>1<\/span>,<\/span>sum<\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>k<\/span><\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>1<\/span><\/span><<\/span><\/span>k<\/span><\/span><<\/span><\/span>2\/<\/span><\/span>dim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>11<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dXktX_{k}^{t}<\/span><\/span>X<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7b2ctt<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u6b21\u8fed\u4ee3\u65f6\u641c\u7d22\u4ee3\u7406\u7684\u4f4d\u7f6e，tt<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u8fde\u7eed\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u7684\u6b21\u6570，\u8be5\u53c2\u6570\u6839\u636e\u4f18\u5316\u95ee\u9898\u7684\u7ef4\u5ea6\u4e0d\u65ad\u53d8\u5316\u4ee5\u9002\u5e94\u4e0d\u540c\u573a\u666f\u4e0b\u7684\u95ee\u9898\u590d\u6742\u5ea6\u3002<\/p>\n

\u60c5\u51b51：\u5728\u9884\u63a2\u7d22\u9636\u6bb5，\u8fc7\u4e8e\u9891\u7e41\u7684\u52a8\u91cf\u8c03\u6574\u53ef\u80fd\u5bfc\u81f4\u4e0d\u5fc5\u8981\u7684\u8ba1\u7b97\u5f00\u9500\u548c\u4f18\u5316\u8fc7\u7a0b\u4e2d\u7684\u632f\u8361\u3002\u901a\u8fc7\u5c06\u6b21\u6570kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u8bbe\u4e3a1，\u5e76\u6309\u7167\u5f0f(11)\u524d\u534a\u90e8\u5206\u66f4\u65b0\u4f4d\u7f6e，\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u65b9\u6cd5\u80fd\u5728\u51cf\u5c11\u8ba1\u7b97\u8d44\u6e90\u6d6a\u8d39\u7684\u540c\u65f6\u4fdd\u6301\u8f83\u5feb\u7684\u6536\u655b\u901f\u5ea6\u3002\u8fd9\u79cd\u8c03\u6574\u673a\u5236\u4f7f\u7b97\u6cd5\u5728\u5904\u7406\u7b80\u5355\u95ee\u9898\u65f6\u66f4\u52a0\u9ad8\u6548\u3002\u5f0f(11)\u4e2d\u7684\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u548c\u03b1t\\\\alpha_{t}<\/span><\/span>\u03b1<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5206\u522b\u5982\u5f0f(12)\u548c(13)\u6240\u793a：<\/p>\n

\u03c9=rand()\u2217(1ttotal\u03b2inov\u22122ttotaltinov+12)(12)\\\\omega=rand()*\\\\left(\\\\frac{1}{t_{\\\\text{total}}}\\\\beta_{\\\\text{inov}}-\\\\frac{2}{t_{\\\\text{total}}}t_{\\\\text{inov}}+\\\\frac{1}{2}\\\\right)\\\\tag{12}<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>t<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>inov<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>t<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>inov<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>(<\/span>12<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u03b1t=cos\u2061(1\u2212rand()\u22172\u03c0)(13)\\\\alpha_{t}=\\\\cos(1-rand()*2\\\\pi)\\\\tag{13}<\/span><\/span>\u03b1<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>cos<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span>)<\/span><\/span><\/span>(<\/span>13<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56fe4\u5c55\u793a\u4e86\u03c9\\\\omega<\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u968f\u8fed\u4ee3\u6b21\u6570\u589e\u52a0\u7684\u52a8\u6001\u53d8\u5316\u8fc7\u7a0b\u3002\u521d\u59cb\u65f6\u5728[\u22120.5,0.5][-0.5,0.5]<\/span><\/span>[<\/span>\u2212<\/span>0.5<\/span>,<\/span><\/span>0.5<\/span>]<\/span><\/span><\/span><\/span><\/span>\u8303\u56f4\u5185\u968f\u673a\u8c03\u6574，\u968f\u7740\u8fed\u4ee3\u8fdb\u884c，\u5176\u503c\u5f00\u59cb\u7f13\u6162\u51cf\u5c0f\u3002\u8fd9\u79cd\u9010\u6b65\u51cf\u5c0f\u7684\u8bbe\u8ba1\u65e8\u5728\u8ba9\u7b97\u6cd5\u9010\u6e10\u7f29\u5c0f\u641c\u7d22\u533a\u57df，\u96c6\u4e2d\u5230\u66f4\u4f18\u533a\u57df\u8fdb\u884c\u5f7b\u5e95\u68c0\u67e5\u3002\u975e\u7ebf\u6027\u4e0b\u964d\u8fc7\u7a0b\u4fdd\u8bc1\u4e86\u641c\u7d22\u8303\u56f4\u7684\u9010\u6b65\u6536\u655b，\u540c\u65f6\u4fdd\u7559\u4e00\u5b9a\u63a2\u7d22\u80fd\u529b\u4ee5\u907f\u514d\u9677\u5165\u5c40\u90e8\u6700\u4f18\u3002\u03b1t\\\\alpha_{t}<\/span><\/span>\u03b1<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8ba1\u7b97\u65b9\u5f0f\u901a\u8fc7\u5f15\u5165\u968f\u673a\u89d2\u5ea6\u5e76\u5c06\u5176\u6620\u5c04\u5230[\u22121,1][-1,1]<\/span><\/span>[<\/span>\u2212<\/span>1<\/span>,<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span>\u8303\u56f4，\u4e3a\u4f4d\u7f6e\u66f4\u65b0\u589e\u52a0\u4e86\u968f\u673a\u6027\u5143\u7d20，\u6709\u52a9\u4e8e\u5728\u641c\u7d22\u7a7a\u95f4\u4e2d\u5f15\u5165\u66f4\u591a\u53d8\u5316\u548c\u63a2\u7d22\u3002\u5728\u7b97\u6cd5\u521d\u59cb\u9636\u6bb5，\u589e\u5f3a\u7684\u968f\u673a\u6027\u6709\u5229\u4e8e\u8fdb\u884c\u5168\u9762\u63a2\u7d22\u3002
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe4 \u52a8\u6001w\u8c03\u6574\u8d8b\u52bf
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe5 \u6e10\u8fdb\u5f0f\u68af\u5ea6\u52a8\u91cf\u79ef\u5206\u66f4\u65b0\u8fc7\u7a0b<\/p>\n

3.3 \u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf<\/h5>\n

\u53d7Adam\u4f18\u5316\u5668\u4e2d\u4e8c\u9636\u77e9\u4f30\u8ba1\u7684\u542f\u53d1，\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u5728AGDO\u7684\u5f00\u53d1\u9636\u6bb5\u8d77\u7740\u5173\u952e\u4f5c\u7528\u3002\u901a\u8fc7\u91c7\u7528\u7c7b\u4f3c\u7684\u4e8c\u9636\u77e9\u4f30\u8ba1\u6280\u672f，\u786e\u4fdd\u4f18\u5316\u8fc7\u7a0b\u4e2d\u9ad8\u6548\u5f00\u53d1\u548c\u7a33\u5b9a\u6536\u655b\u3002\u521d\u59cb\u4f4d\u7f6e\u7684\u9009\u62e9\u4e5f\u662f\u8be5\u7cfb\u7edf\u7684\u5173\u952e\u6b65\u9aa4，\u4e0d\u4ec5\u5229\u7528\u539f\u59cb\u521d\u59cb\u5316\u751f\u6210\u7684\u79cd\u7fa4\u4f4d\u7f6e，\u8fd8\u901a\u8fc7"randperm"\u51fd\u6570\u968f\u673a\u751f\u6210\u53e6\u5916N\u4e2a\u521d\u59cb\u4f4d\u7f6e，\u4fdd\u8bc1\u521d\u59cb\u4f4d\u7f6e\u5728\u89e3\u7a7a\u95f4\u4e2d\u5747\u5300\u4e14\u552f\u4e00\u5206\u5e03，\u4e3a\u4f18\u5316\u8fc7\u7a0b\u63d0\u4f9b\u591a\u6837\u5316\u7684\u8d77\u70b9\u3002\u5728\u7b2ct\u6b21\u8fed\u4ee3\u4e2d，\u4f7f\u7528\u968f\u673a\u751f\u6210\u7684\u521d\u59cb\u4f4d\u7f6e\u7d22\u5f15\u6301\u7eed\u4e3a\u7cfb\u7edf\u63d0\u4f9b\u5168\u65b0\u521d\u59cb\u4f4d\u7f6e，\u9009\u62e9\u524d\u4e24\u4e2a\u7d22\u5f15(Xidx1X_{idx1}<\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cXidx2X_{idx2}<\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)\u786e\u4fdd\u4ece\u4e0d\u540c\u8d77\u59cb\u4f4d\u7f6e\u5f00\u59cb\u4f18\u5316，\u4ece\u800c\u589e\u5f3a\u89e3\u7a7a\u95f4\u7684\u63a2\u7d22\u80fd\u529b，\u5177\u4f53\u9009\u62e9\u5982\u5f0f(14)\u6240\u793a：<\/p>\n

Xidx1=Xrp(1)t(14)X_{idx1} = X_{rp(1)}^{t} \\\\tag{14}<\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>r<\/span>p<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>14<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nXidx2=Xrp(2)tX_{idx2} = X_{rp(2)}^{t}<\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>r<\/span>p<\/span>(<\/span>2<\/span>)<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2drp(1)rp(1)<\/span><\/span>r<\/span>p<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u548crp(2)rp(2)<\/span><\/span>r<\/span>p<\/span>(<\/span>2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f"randperm"\u968f\u673a\u751f\u6210\u7684\u524d\u4e24\u4e2a\u6574\u6570\u3002<\/p>\n

\u5728\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u4e2d，\u641c\u7d22\u65b9\u5411\u7684\u786e\u5b9a\u5bf9AGDO\u6027\u80fd\u81f3\u5173\u91cd\u8981\u3002\u65b9\u5411\u6307\u9488\u5728\u6b64\u65b9\u9762\u53d1\u6325\u91cd\u8981\u4f5c\u7528，\u7528\u4e8e\u786e\u5b9a\u4e0b\u4e00\u4e2a\u641c\u7d22\u65b9\u5411，\u5176\u516c\u5f0f\u5982\u4e0b：<\/p>\n

\u03f5=f(Xidx2)\u2212f(Xidx1)\u2223f(Xidx2)\u2212f(Xidx1)\u2223(15)\\\\epsilon = \\\\frac{f(X_{idx2})-f(X_{idx1})}{|f(X_{idx2})-f(X_{idx1})|} \\\\tag{15}<\/span><\/span>\u03f5<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>15<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u03f5\\\\epsilon<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>\u662f\u7cfb\u7edf\u65b9\u5411\u6307\u9488，f(Xidx2)f(X_{idx2})<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u521d\u59cb\u4f4d\u7f6e\u7684\u7cfb\u7edf\u91cd\u65b0\u751f\u6210\u9002\u5e94\u5ea6\u503c，f(Xidx1)f(X_{idx1})<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u521d\u59cb\u5316\u751f\u6210\u7684\u9002\u5e94\u5ea6\u503c\u3002\u65b9\u5411\u6307\u9488\u901a\u8fc7\u5f52\u4e00\u5316\u5c06\u9002\u5e94\u5ea6\u503c\u5dee\u5f02\u8f6c\u6362\u4e3a1\u6216-1，\u4fdd\u8bc1\u65b9\u5411\u9009\u62e9\u4e0d\u53d7\u9002\u5e94\u5ea6\u503c\u5927\u5c0f\u5f71\u54cd，\u4ece\u800c\u5b9e\u73b0\u66f4\u7a33\u5b9a\u548c\u7edf\u4e00\u7684\u66f4\u65b0\u6b65\u957f\u3002\u7cfb\u7edf\u901a\u8fc7\u8ba1\u7b97\u6bcf\u5bf9\u4f4d\u7f6e\u7684\u9002\u5e94\u5ea6\u5dee\u5f02，\u57fa\u4e8e\u5f52\u4e00\u5316\u65b9\u5411\u6307\u793a\u5668\u5b9e\u65f6\u8c03\u6574\u641c\u7d22\u65b9\u5411，\u4f7f\u4f18\u5316\u7b97\u6cd5\u80fd\u51c6\u786e\u9009\u62e9\u66f4\u4f18\u65b9\u5411\u79fb\u52a8，\u6709\u6548\u5f15\u5bfc\u641c\u7d22\u8fc7\u7a0b\u9010\u6b65\u63a5\u8fd1\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8fd9\u79cd\u673a\u5236\u4f7f\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u5728\u590d\u6742\u89e3\u7a7a\u95f4\u4e2d\u4fdd\u6301\u9ad8\u6548\u5f00\u53d1\u548c\u7a33\u5b9a\u6536\u655b\u3002\u7cfb\u7edf\u5c06\u4f4d\u7f6e\u66f4\u65b0\u5206\u4e3a\u5f00\u53d1\u524d\u548c\u5f00\u53d1\u540e\u4e24\u4e2a\u9636\u6bb5，\u80fd\u66f4\u597d\u9002\u5e94\u8fc7\u7a0b\u7684\u4e0d\u540c\u9700\u6c42\u3002<\/p>\n

\u60c5\u51b51：\u5728\u5f00\u53d1\u524d\u9636\u6bb5，\u7cfb\u7edf\u4e3b\u8981\u4efb\u52a1\u662f\u5e7f\u6cdb\u63a2\u7d22\u89e3\u7a7a\u95f4，\u76ee\u6807\u662f\u901a\u8fc7\u591a\u6837\u5316\u641c\u7d22\u7b56\u7565\u53d1\u73b0\u6f5c\u5728\u9ad8\u8d28\u91cf\u89e3\u5e76\u907f\u514d\u9677\u5165\u5c40\u90e8\u6700\u4f18\u3002\u4e3a\u6b64\u7cfb\u7edf\u91c7\u7528\u590d\u6742\u66f4\u65b0\u65b9\u6cd5\u5f15\u5bfc\u641c\u7d22\u65b9\u5411，\u5177\u4f53\u89c1\u5f0f(16)：<\/p>\n

Xi,new1t=Xit+\u03f5\u2217\u03b1\u2217(G\u2212Xidx1t)\u2212\u03b1\u2217(Xidx2t\u2212Xit)(16)X_{i,new1}^{t} = X_{i}^{t} + \\\\epsilon * \\\\alpha * (G – X_{idx1}^{t}) – \\\\alpha * (X_{idx2}^{t} – X_{i}^{t}) \\\\tag{16}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span>1<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03f5<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>(<\/span>G<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>(<\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>(<\/span>16<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u03b1\\\\alpha<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u662f\u7531\u52a8\u6001\u7f29\u653e\u56e0\u5b50(1\u2212\u03b7t)(1-\\\\eta_t)<\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u786e\u5b9a\u7684\u968f\u673a\u6270\u52a8\u5411\u91cf，\u8303\u56f4\u5728[0,(1\u2212\u03b7t)][0,(1-\\\\eta_t)]<\/span><\/span>[<\/span>0<\/span>,<\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)]<\/span><\/span><\/span><\/span><\/span>\u5185，\u03b1\\\\alpha<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u901a\u8fc7\u5f0f(17)\u53d8\u5316\u3002GG<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span>\u901a\u8fc7\u77e9\u4f30\u8ba1\u5bfb\u627e\u6700\u4f18\u4f4d\u7f6e，\u4f30\u8ba1\u539f\u7406\u5982\u5f0f(17)\u6240\u793a：<\/p>\n

\u03b1=(1\u2212\u03b7t)\u2217rand(1,Dim)(17)\\\\alpha = (1-\\\\eta_t) * rand(1,Dim) \\\\tag{17}<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b7<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>1<\/span>,<\/span><\/span>D<\/span>im<\/span>)<\/span><\/span><\/span>(<\/span>17<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5728\u4f4d\u7f6e\u66f4\u65b0\u65b9\u6cd5\u7684\u521d\u59cb\u9636\u6bb5，\u7cfb\u7edf\u6574\u5408\u591a\u79cd\u4f4d\u7f6e\u6570\u636e\u5e7f\u6cdb\u63a2\u7d22\u89e3\u7a7a\u95f4，\u6709\u52a9\u4e8e\u65e9\u671f\u5feb\u901f\u8bc6\u522b\u6709\u5e0c\u671b\u533a\u57df，\u7279\u522b\u662f\u5f53\u89e3\u7a7a\u95f4\u5927\u90e8\u5206\u672a\u88ab\u63a2\u7d22\u65f6\u3002\u901a\u8fc7\u4f7f\u5f53\u524d\u4f4d\u7f6e\u52a8\u6001\u9002\u5e94\u591a\u4e2a\u65b9\u5411，\u7b97\u6cd5\u80fd\u907f\u514d\u56e0\u8fc7\u65e9\u6536\u655b\u800c\u9519\u8fc7\u5176\u4ed6\u53ef\u80fd\u9ad8\u8d28\u91cf\u89e3\u3002\u03b1\\\\alpha<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u7684\u5f15\u5165\u5728\u5f53\u524d\u4f4d\u7f6e\u548c\u591a\u4e2a\u53c2\u8003\u4f4d\u7f6e\u95f4\u53d6\u5f97\u5e73\u8861，\u786e\u4fdd\u7cfb\u7edf\u5728\u5f00\u53d1\u8fc7\u7a0b\u4e2d\u80fd\u627e\u5230\u5408\u7406\u6298\u8877，\u907f\u514d\u8fc7\u5ea6\u4f9d\u8d56\u7279\u5b9a\u4f4d\u7f6e，\u4ece\u800c\u63d0\u9ad8\u4f18\u5316\u7684\u5168\u9762\u6027\u548c\u6548\u7387\u3002<\/p>\n

\u60c5\u51b52：\u8fdb\u5165\u5f00\u53d1\u540e\u9636\u6bb5\u540e，\u7cfb\u7edf\u76ee\u6807\u8f6c\u5411\u7cbe\u7ec6\u8c03\u6574\u4ee5\u4fdd\u8bc1\u89e3\u8d28\u91cf\u548c\u7a33\u5b9a\u6536\u655b\u3002\u8be5\u9636\u6bb5\u7b56\u7565\u805a\u7126\u4e8e\u5feb\u901f\u5f15\u5bfc\u5f53\u524d\u89e3\u63a5\u8fd1\u6700\u4f18\u89e3，\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\u53d8\u5f97\u66f4\u9ad8\u6548\u548c\u96c6\u4e2d，\u5177\u4f53\u66f4\u65b0\u516c\u5f0f\u5982\u5f0f(18)\u6240\u793a：<\/p>\n

Xi,new2t=Xit+\u03b1\u2217(G\u2212Xidx1t)(18)X_{i,new2}^{t} = X_{i}^{t} + \\\\alpha * (G – X_{idx1}^{t}) \\\\tag{18}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span>2<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>(<\/span>G<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>d<\/span>x<\/span>1<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>(<\/span>18<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7cfb\u7edf\u5229\u7528\u7d22\u5f15\u4f4d\u7f6e\u76f4\u63a5\u8c03\u6574\u5230\u6700\u4f18\u89e3\u65b9\u5411，\u8fd9\u79cd\u76f4\u63a5\u9ad8\u6548\u7684\u4fee\u6539\u663e\u8457\u52a0\u901f\u7b97\u6cd5\u6536\u655b，\u7279\u522b\u662f\u5f53\u89e3\u7a7a\u95f4\u63a5\u8fd1\u5168\u5c40\u6700\u4f18\u65f6\u3002\u7531\u4e8e\u66f4\u65b0\u65b9\u6cd5\u76f8\u5bf9\u7b80\u5355，\u964d\u4f4e\u4e86\u8ba1\u7b97\u9700\u6c42，\u4f7f\u7b97\u6cd5\u80fd\u5728\u6700\u7ec8\u4f18\u5316\u9636\u6bb5\u6709\u6548\u5fae\u8c03，\u4ece\u800c\u4fdd\u8bc1\u66f4\u7cbe\u786e\u7684\u6536\u655b\u3002<\/p>\n

\u5728\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u4e2d，\u4f4d\u7f6eGG<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span>\u662f\u5173\u952e\u53c2\u8003\u70b9，\u5728\u4f18\u5316\u8fc7\u7a0b\u4e2d\u8d77\u6838\u5fc3\u4f5c\u7528，\u7531\u77e9\u4f30\u8ba1\u65b9\u6cd5\u786e\u5b9a，\u4e3b\u8981\u7528\u4e8e\u5f15\u5bfc\u548c\u8c03\u6574\u641c\u7d22\u8fc7\u7a0b，\u5177\u4f53\u5b9a\u4e49\u5982\u5f0f(19)：<\/p>\n

G=Xi,bestt\u2212\u03b7\u2217mtvt+\u03f5(19)G = X_{i,best}^{t} – \\\\frac{\\\\eta * m_{t}}{\\\\sqrt{v_{t}} + \\\\epsilon} \\\\tag{19}<\/span><\/span>G<\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b7<\/span><\/span>\u2217<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>19<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dXi,besttX_{i,best}^{t}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7b2ct\u6b65\u6700\u4f18\u89e3\u7684\u5f53\u524d\u4f4d\u7f6e，\u03b7\\\\eta<\/span><\/span>\u03b7<\/span><\/span><\/span><\/span><\/span>\u8bbe\u4e3a0.001\u7684\u5b66\u4e60\u7387，vtv_{t}<\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u8fed\u4ee3\u671f\u95f4\u4e8c\u9636\u77e9\u7684\u504f\u5dee\u6821\u6b63\u4f30\u8ba1，\u03f5\\\\epsilon<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>\u662f\u6781\u5c0f\u5e38\u6570(1e-8)\u907f\u514d\u9664\u4ee5\u96f6，mtm_{t}<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u68af\u5ea6\u7684\u504f\u5dee\u6821\u6b63\u4f30\u8ba1\u3002vtv_{t}<\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cmtm_{t}<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8868\u8fbe\u5f0f\u5982\u4e0b：<\/p>\n

vt=vt1\u2212\u03b22t(20)v_{t} = \\\\frac{v_{t}}{1-\\\\beta_2^t} \\\\tag{20}<\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>20<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nmt=mt1\u2212\u03b21tm_{t} = \\\\frac{m_{t}}{1-\\\\beta_1^t}<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>1<\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f0f(20)\u4e2d，mtm_{t}<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cvtv_{t}<\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5206\u522b\u8868\u793a\u68af\u5ea6\u4e00\u9636\u548c\u4e8c\u9636\u77e9\u4f30\u8ba1，\u03b21\\\\beta_1<\/span><\/span>\u03b2<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c\u03b22\\\\beta_2<\/span><\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u8fd9\u4e9b\u77e9\u4f30\u8ba1\u7684\u6307\u6570\u8870\u51cf\u7387，\u901a\u5e38\u53d6\u503c0.9\u548c0.999\u3002\u5b83\u4eec\u7684\u66f4\u65b0\u65b9\u7a0b\u5982\u4e0b：<\/p>\n

vt=\u03b22\u2217vt\u22121+(1\u2212\u03b22)\u2217gt2(21)v_{t} = \\\\beta_2 * v_{t-1} + (1-\\\\beta_2)*g_{t}^2 \\\\tag{21}<\/span><\/span>v<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>v<\/span><\/span>t<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>g<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>21<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nmt=\u03b21\u2217mt\u22121+(1\u2212\u03b21)\u2217gtm_{t} = \\\\beta_1 * m_{t-1} + (1-\\\\beta_1)*g_{t}<\/span><\/span>m<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>m<\/span><\/span>t<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>g<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dgtg_{t}<\/span><\/span>g<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u7b2ct\u6b21\u8fed\u4ee3\u8ba1\u7b97\u7684\u4f4d\u7f6e\u68af\u5ea6\u4fe1\u606f，\u7531\u5f0f(22)\u786e\u5b9a：<\/p>\n

gt=Xi,bestt\u2212C\u2217P(22)g_{t} = X_{i,best}^{t} – C * P \\\\tag{22}<\/span><\/span>g<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>C<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>P<\/span><\/span><\/span>(<\/span>22<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Xi,besttX_{i,best}^{t}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5f53\u524d\u6700\u4f18\u89e3\u4f4d\u7f6e，\u5373\u5728\u7b2ct\u6b21\u8fed\u4ee3\u4e2d\u627e\u5230\u7684\u6700\u4f18\u89e3\u3002CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>\u548cPP<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span>\u7684\u6570\u5b66\u516c\u5f0f\u5982\u4e0b：<\/p>\n

P=Xi,meant\u2212Xi,new1t(23)P = X_{i,mean}^{t} – X_{i,new1}^{t} \\\\tag{23}<\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>m<\/span>e<\/span>an<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span>1<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>23<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nC=1\u2212rand()C = 1 – rand()<\/span><\/span>C<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dXi,meantX_{i,mean}^{t}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>m<\/span>e<\/span>an<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5f53\u524d\u4f4d\u7f6e\u7684\u5747\u503c\u4fe1\u606f，\u4e0e\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u4e2d\u7684\u66f4\u65b0\u4f4d\u7f6e\u4fe1\u606f\u7ed3\u5408，\u4e3a\u68af\u5ea6\u4fe1\u606f\u63d0\u4f9b\u53c2\u8003\u65b9\u5411\u3002CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>\u662f\u533a\u95f4[0,1]\u5185\u7684\u968f\u673a\u8c03\u6574\u56e0\u5b50，\u786e\u4fdd\u68af\u5ea6\u8c03\u6574\u6b65\u9aa4\u7684\u968f\u673a\u6027\u548c\u591a\u6837\u6027\u3002<\/p>\n

\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u4f4d\u7f6e\u66f4\u65b0\u7684\u6700\u7ec8\u516c\u5f0f\u5982\u5f0f(24)\u6240\u793a：<\/p>\n

Xi,newt={Xi,new1trand()>\u03b7Xi,new2t\u5426\u5219(24)X_{i,new}^{t} = \\\\begin{cases}
\nX_{i,new1}^{t} & rand() > \\\\eta \\\\\\\\
\nX_{i,new2}^{t} & \\\\text{\u5426\u5219}
\n\\\\end{cases} \\\\tag{24}<\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span>1<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span>2<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>an<\/span>d<\/span>(<\/span>)<\/span><\/span>><\/span><\/span>\u03b7<\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>24<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4fe1\u8d56\u57df\u7b56\u7565\u786e\u5b9e\u662f\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u4e2d\u7684\u91cd\u8981\u8003\u91cf，\u4e3b\u8981\u7528\u4e8e\u901a\u8fc7\u786e\u4fdd\u6bcf\u6b21\u8fed\u4ee3\u7684\u4f4d\u7f6e\u66f4\u65b0\u4e0d\u4f1a\u8fc7\u5927\u6765\u63a7\u5236\u4f18\u5316\u6b65\u8c03，\u907f\u514d\u53ef\u80fd\u7684\u4e0d\u7a33\u5b9a\u548c\u4f4e\u6548\u641c\u7d22\u3002\u7cfb\u7edf\u901a\u8fc7\u63a5\u53d7\u6216\u62d2\u7edd\u66f4\u65b0\u4f4d\u7f6e\u6765\u8c03\u6574\u4fe1\u8d56\u57df，\u786e\u4fdd\u9ad8\u6548\u63a2\u7d22\u89e3\u7a7a\u95f4\u800c\u4e0d\u4f1a\u8fc7\u5ea6\u504f\u79bb，\u4ece\u800c\u63d0\u9ad8\u4f18\u5316\u7cfb\u7edf\u7684\u6709\u6548\u6027\u548c\u9c81\u68d2\u6027\u3002<\/p>\n

\u4fe1\u8d56\u57df\u7b56\u7565\u5305\u542b\u4e24\u90e8\u5206：\u4f4d\u7f6e\u66f4\u65b0\u5224\u65ad\u548c\u6700\u4f18\u4f4d\u7f6e\u66f4\u65b0\u3002\u9996\u5148\u8003\u8651\u4f4d\u7f6e\u66f4\u65b0\u7684\u6761\u4ef6：<\/p>\n

Xit+1={Xi,newt\u5982\u679c\u00a0f(Xi,newt)<f(Xit)Xit\u5426\u5219(25)X_{i}^{t+1} = \\\\begin{cases}
\nX_{i,new}^{t} & \\\\text{\u5982\u679c } f(X_{i,new}^{t}) < f(X_{i}^{t}) \\\\\\\\
\nX_{i}^{t} & \\\\text{\u5426\u5219}
\n\\\\end{cases} \\\\tag{25}<\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span>,<\/span>n<\/span>e<\/span>w<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>25<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u610f\u5473\u7740\u5f53\u65b0\u4f4d\u7f6e\u7684\u9002\u5e94\u5ea6\u51fd\u6570\u503c\u4f18\u4e8e\u5f53\u524d\u4f4d\u7f6eXitX_{i}^{t}<\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u65f6\u63a5\u53d7\u65b0\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\u8003\u8651\u6700\u4f18\u4f4d\u7f6e\u7684\u66f4\u65b0：<\/p>\n

Xbestt+1={Xit+1\u5982\u679c\u00a0f(Xit+1)<f(Xbestt)Xbestt\u5426\u5219(26)X_{best}^{t+1} = \\\\begin{cases}
\nX_{i}^{t+1} & \\\\text{\u5982\u679c } f(X_{i}^{t+1}) < f(X_{best}^{t}) \\\\\\\\
\nX_{best}^{t} & \\\\text{\u5426\u5219}
\n\\\\end{cases} \\\\tag{26}<\/span><\/span>X<\/span><\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5982\u679c<\/span>\u00a0<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>t<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><<\/span><\/span>f<\/span>(<\/span>X<\/span><\/span>b<\/span>es<\/span>t<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u5426\u5219<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>26<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u6837，\u4fe1\u8d56\u57df\u7b56\u7565\u786e\u4fdd\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u5728\u5f00\u53d1\u8fc7\u7a0b\u4e2d\u80fd\u63a2\u7d22\u65b0\u7684\u66f4\u4f18\u89e3，\u540c\u65f6\u4fdd\u6301\u89e3\u7684\u7a33\u5b9a\u6027\u5e76\u907f\u514d\u8fc7\u5ea6\u504f\u79bb\u3002\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u7684\u66f4\u65b0\u8fc7\u7a0b\u5982\u56fe6\u6240\u793a\u3002
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe6 \u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u66f4\u65b0\u8fc7\u7a0b<\/p>\n

3.4 \u7cfb\u7edf\u4f18\u5316\u7b97\u5b50<\/h5>\n

\u5728\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u7684\u5f00\u53d1\u9636\u6bb5，\u867d\u7136\u7cfb\u7edf\u5df2\u5c55\u73b0\u51fa\u4f18\u5f02\u7684\u4f18\u5316\u641c\u7d22\u80fd\u529b，\u4f46\u5728\u89e3\u7a7a\u95f4\u7684\u6574\u4f53\u5f00\u53d1\u4e0a\u4ecd\u53ef\u80fd\u5b58\u5728\u4e0d\u8db3\u3002\u4e3a\u89e3\u51b3\u8fd9\u4e9b\u95ee\u9898，\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u9636\u6bb5\u5f3a\u8c03\u5168\u5c40\u4f18\u5316\u65b9\u6cd5\u548c\u5f15\u5bfc\u7b56\u7565，\u901a\u8fc7\u5f15\u5165\u5173\u952e\u98de\u884c\u7b56\u7565\u5171\u540c\u589e\u5f3a\u89e3\u7a7a\u95f4\u63a2\u7d22\u80fd\u529b，\u4f7f\u7cfb\u7edf\u80fd\u8fdb\u884c\u66f4\u5e7f\u6cdb\u7684\u641c\u7d22\u5e76\u63d0\u5347\u6574\u4f53\u6027\u80fd\u3002\u5176\u8fd0\u7b97\u516c\u5f0f\u5982\u5f0f(27)\u6240\u793a：<\/p>\n

Xk1,k2,\u20261,km=Xm1,m2+k1exp\u2061(\u03d5)+\u03b4\u2217(Xm1,m2\u2212Xi\u2217+\u03b8)(27)X_{k_1,k_2,\\\\ldots}^{1,k_m} = X_{m_1,m_2} + k_1 \\\\exp(\\\\phi) + \\\\delta * (X_{m_1,m_2} – X_i^* + \\\\theta) \\\\tag{27}<\/span><\/span>X<\/span><\/span>k<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>k<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>\u2026<\/span><\/span><\/span><\/span><\/span>1<\/span>,<\/span>k<\/span><\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>X<\/span><\/span>m<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>m<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>exp<\/span>(<\/span>\u03d5<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>(<\/span>X<\/span><\/span>m<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>m<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span>(<\/span>27<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u03b4f\u00a0rand\u00a0S=[1+e(\u03942)2]\u22121(28)\\\\delta f \\\\text{ rand } S = \\\\left[ 1 + e^{\\\\left( \\\\frac{\\\\Delta}{2} \\\\right)^2} \\\\right]^{-1} \\\\tag{28}<\/span><\/span>\u03b4<\/span>f<\/span>\u00a0rand\u00a0<\/span><\/span>S<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>1<\/span><\/span>+<\/span><\/span>e<\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>28<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dXm1,m2X_{m_1,m_2}<\/span><\/span>X<\/span><\/span>m<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span>m<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u503e\u659c\u8fed\u4ee3\u4e2d\u7684\u6700\u4f18\u64cd\u4f5c\u4f4d\u7f6e，k1exp\u2061(\u03d5)k_1 \\\\exp(\\\\phi)<\/span><\/span>k<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>exp<\/span>(<\/span>\u03d5<\/span>)<\/span><\/span><\/span><\/span><\/span>\u8868\u793aLevy\u98de\u884c\u51fd\u6570，\u901a\u8fc7\u5f0f(29)\u8ba1\u7b97：<\/p>\n

k1exp\u2061(\u03d5)=0.01\u00d71\u2223x\u2223\u00d7Tn(29)k_1 \\\\exp(\\\\phi) = 0.01 \\\\times \\\\frac{1}{|x|} \\\\times \\\\frac{T}{\\\\sqrt{n}} \\\\tag{29}<\/span><\/span>k<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>exp<\/span>(<\/span>\u03d5<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>0.01<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>x<\/span>\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>29<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f0f(28)\u4e2d\u03c3\\\\sigma<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span>\u662f0\u52302\u7684\u968f\u673a\u53d8\u91cf（\u672c\u6587\u53d6\u03c3=15\\\\sigma = 15<\/span><\/span>\u03c3<\/span><\/span>=<\/span><\/span><\/span><\/span>15<\/span><\/span><\/span><\/span><\/span>），r1r_1<\/span><\/span>r<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548cr2r_2<\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f0\u52301\u7684\u4efb\u610f\u6570\u3002rr<\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a：<\/p>\n

r=\u0393(1+\u03c3)\u00d7sin\u2061(\u03c0\u03b82)\u0393(1+\u03c32)\u00d7\u03c3\u00d7\u03c02211\/4(30)r = \\\\frac{\\\\Gamma \\\\left( 1 + \\\\sigma \\\\right) \\\\times \\\\sin\\\\left(\\\\frac{\\\\pi\\\\theta}{2}\\\\right)}{\\\\Gamma \\\\left(\\\\frac{1+\\\\sigma}{2}\\\\right) \\\\times \\\\sigma \\\\times \\\\frac{\\\\pi^{2}}{2}} \\\\quad 1^{1\/4} \\\\tag{30}<\/span><\/span>r<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>+<\/span>\u03c3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>\u03c3<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0393<\/span><\/span>(<\/span>1<\/span><\/span>+<\/span><\/span>\u03c3<\/span>)<\/span><\/span><\/span>\u00d7<\/span><\/span>sin<\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>1\/4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>30<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u4e2d\u03b4\\\\delta<\/span><\/span>\u03b4<\/span><\/span><\/span><\/span><\/span>\u548c\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\u5206\u522b\u5982\u5f0f(31)\u548c(32)\u6240\u793a：<\/p>\n

\u03b4=rand(1+1\u03c4randttest\u22122\u03c4testttest+1)(31)\\\\delta = \\\\text{rand} \\\\left( 1 + \\\\frac{1}{\\\\tau_{\\\\text{rand}}} t_{\\\\text{test}} – \\\\frac{2}{\\\\tau_{\\\\text{test}}} t_{\\\\text{test}} + 1 \\\\right) \\\\tag{31}<\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span>rand<\/span><\/span><\/span>(<\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>rand<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>test<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>test<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>test<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>(<\/span>31<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n\u03b8=2ttest\u03c4test(32)\\\\theta = \\\\frac{2t_{\\\\text{test}}}{\\\\tau_{\\\\text{test}}} \\\\tag{32}<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>test<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>t<\/span><\/span>test<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>32<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u03b8\\\\theta<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u662f[0, 2]\u8303\u56f4\u5185\u7684\u7ebf\u6027\u9012\u589e\u51fd\u6570\u3002<\/p>\n

\u56fe2\u5c55\u793a\u4e86\u4e0d\u540c\u8fed\u4ee3\u6b21\u6570\u4e0b\u7684\u884c\u4e3a\u53d8\u5316\u3002\u6839\u636e\u56fe\u793a，\u03b4\\\\delta<\/span><\/span>\u03b4<\/span><\/span><\/span><\/span><\/span>\u5728[0,1]\u533a\u95f4\u5185\u7ecf\u5386\u63a5\u8fd10\u7684\u975e\u7ebf\u6027\u4e0b\u964d\u8fc7\u7a0b，\u5176\u4e2d\u5305\u542b\u968f\u673a\u6270\u52a8\u3002\u8fd9\u79cd\u5173\u952e\u6ce2\u52a8\u673a\u5236\u4f7f\u7b97\u5b50\u80fd\u66f4\u4e13\u6ce8\u4e8e\u5e7f\u6cdb\u641c\u7d22\u7b56\u7565，\u4ece\u800c\u63a2\u7d22\u66f4\u5927\u7684\u89e3\u7a7a\u95f4；\u800c\u5728\u65b0\u65b9\u6848\u4e2d，\u4e0b\u964d\u8fc7\u7a0b\u5141\u8bb8\u9010\u6b65\u8fc7\u6e21\u5230\u5c40\u90e8\u533a\u57df，\u540c\u65f6\u4fdd\u6301\u89e3\u7a7a\u95f4\u63a2\u7d22\u3002\u52a8\u6001\u53d8\u5316\u4e0e\u968f\u673a\u6270\u52a8\u7684\u7ed3\u5408\u5e2e\u52a9\u7b97\u5b50\u4f18\u5316\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u7684\u63a2\u7d22\u8fc7\u7a0b，\u907f\u514d\u8fc7\u65e9\u9677\u5165\u5c40\u90e8\u6700\u4f18，\u5e76\u786e\u4fdd\u5168\u5c40\u641c\u7d22\u540e\u7684\u7cbe\u786e\u6536\u655b\u3002
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\u56fe2 \u03b4\\\\delta<\/span><\/span>\u03b4<\/span><\/span><\/span><\/span><\/span>\u53d8\u5316\u8d8b\u52bf
\n\u7cfb\u7edf\u4f18\u5316\u7b97\u5b50\u65e8\u5728\u663e\u8457\u63d0\u5347\u89e3\u7a7a\u95f4\u5f00\u53d1\u548c\u4f18\u5316\u6548\u7387\u3002\u901a\u8fc7\u5f15\u5165Levy\u98de\u884c，\u5f25\u8865\u4e86\u5f00\u53d1\u9636\u6bb5\u7684\u63a2\u7d22\u4e0d\u8db3\u3002\u901a\u8fc7\u52a8\u6001\u8c03\u6574\u641c\u7d22\u7b56\u7565\u5e76\u7075\u6d3b\u5207\u6362\u5168\u5c40\u4e0e\u5c40\u90e8\u641c\u7d22，\u8be5\u9636\u6bb5\u786e\u4fdd\u4f18\u5316\u8fc7\u7a0b\u5728\u76f8\u5bf9\u4f18\u5316\u89e3\u7a7a\u95f4\u65f6\u80fd\u9010\u6b65\u7cbe\u786e\u6536\u655b\u81f3\u6700\u4f18\u89e3\u3002\u7b97\u5b50\u589e\u5f3a\u4e86\u7cfb\u7edf\u7684\u5168\u5c40\u641c\u7d22\u529f\u80fd\u5e76\u4f18\u5316\u5c40\u90e8\u8c03\u6574，\u5927\u5e45\u63d0\u5347\u6574\u4f53\u641c\u7d22\u6548\u7387\u548c\u7cbe\u5ea6，\u4fdd\u8bc1\u89e3\u7a7a\u95f4\u7684\u5168\u9762\u8986\u76d6\u548c\u6700\u4f18\u89e3\u7684\u6709\u6548\u8bc6\u522b\u3002<\/p>\n

3.5 AGDO\u7b97\u6cd5\u6267\u884c\u6d41\u7a0b<\/h5>\n

\u672c\u90e8\u5206\u6df1\u5165\u63a2\u8ba8AGDO\u7684\u63a2\u7d22\u4e0e\u5f00\u53d1\u9636\u6bb5\u3002\u9996\u5148，\u7b97\u6cd5\u7cbe\u5fc3\u751f\u6210\u591a\u7ec4\u96c6\u5408\u4ee5\u542f\u52a8\u4f18\u5316\u8fc7\u7a0b\u3002\u968f\u540e\u901a\u8fc7\u6e10\u8fdb\u68af\u5ea6\u52a8\u91cf\u6574\u5408\u6280\u672f\u9010\u6b65\u4fee\u6b63\u5b9a\u4f4d\u96be\u70b9，\u8bc6\u522b\u9ad8\u8d28\u91cf\u89e3，\u4ece\u800c\u5b8c\u6210\u63a2\u7d22\u9636\u6bb5\u3002\u5728\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u6846\u67b6\u5185，\u91c7\u7528\u65b0\u9896\u7684\u8d77\u59cb\u4f4d\u7f6e\u4fdd\u8bc1\u4f18\u5316\u4ece\u72ec\u7279\u70b9\u5f00\u59cb，\u63d0\u5347\u7b97\u6cd5\u5168\u5c40\u641c\u7d22\u6548\u7387\u3002\u6b64\u5916\u5f15\u5165\u8d77\u59cb\u533a\u57df\u65b9\u6cd5\u786e\u4fdd\u89e3\u7684\u7a33\u5b9a\u6027，\u9632\u6b62\u52a8\u6001\u68af\u5ea6\u4ea4\u4e92\u7cfb\u7edf\u51fa\u73b0\u663e\u8457\u504f\u5dee\u3002\u6700\u7ec8，\u7cfb\u7edf\u589e\u5f3a\u7b97\u5b50（\u672f\u8bedAGDO\u7684\u5f00\u53d1\u9636\u6bb5）\u901a\u8fc7\u6574\u5408Levy\u98de\u884c，\u63d0\u5347\u7b97\u6cd5\u5728\u5c40\u90e8\u641c\u7d22\u4e2d\u7684\u7cbe\u5ea6\u548c\u591a\u6837\u6027\u3002\u7b97\u6cd5\u8be6\u7ec6\u4f2a\u4ee3\u7801\u89c1\u7b97\u6cd53.1，\u6d41\u7a0b\u56fe\u89c1\u56fe3\u3002\u8fd9\u4e9b\u7ec6\u8282\u4e0d\u4ec5\u8bf4\u660eAGDO\u7b97\u6cd5\u7684\u8bbe\u8ba1\u539f\u7406，\u66f4\u7a81\u51fa\u5176\u5728\u63a2\u7d22\u4e0e\u5f00\u53d1\u9636\u6bb5\u7684\u72ec\u7279\u5b9e\u73b0\u548c\u4f18\u5316\u6280\u672f\u3002
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
\n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

4 \u5b8c\u6574\u4ee3\u7801<\/h4>\n

function<\/span> [<\/span>fbest,<\/span> pbest,<\/span> cg]<\/span> =<\/span> AGDO<\/span>(<\/span>N,<\/span>T,<\/span> lb,<\/span> ub,<\/span> dim,<\/span>fobj)<\/span>
\n VarSize =<\/span> [<\/span>1<\/span> dim]<\/span>;<\/span>% Dimension of the problem<\/span>
\n beta1 =<\/span> 0.9<\/span>;<\/span> % Exponential decay rates of biased first original moment estimates <\/span>
\n beta2 =<\/span> 0.999<\/span>;<\/span> % Exponential decay rates of biased second original moment estimates<\/span>
\n epsilon =<\/span> 1e-8<\/span>;<\/span>% Tiny constant for avoiding division by zero<\/span>
\n m =<\/span> zeros<\/span>(<\/span>N,<\/span> dim)<\/span>;<\/span>% Initial estimate of the deviation of the gradient<\/span>
\n v =<\/span> zeros<\/span>(<\/span>N,<\/span> dim)<\/span>;<\/span>% Initial estimate of the deviation of the second-order moments<\/span>
\n lr =<\/span> 0.001<\/span>;<\/span> % Learning rate<\/span>
\n alpha =<\/span> cos<\/span>(<\/span>(<\/span>1<\/span>–<\/span>rand<\/span>(<\/span>N,<\/span>1<\/span>)<\/span>)<\/span>*<\/span>2<\/span>*<\/span>pi<\/span>)<\/span>;<\/span> % Eq.(13)<\/span><\/p>\n

fit =<\/span> zeros<\/span>(<\/span>N,<\/span> 1<\/span>)<\/span>;<\/span>% Fitness of each individual<\/span>
\n bestpos =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span> dim)<\/span>;<\/span>% Best position<\/span>
\n fbest =<\/span> inf<\/span>;<\/span>%Best score (initially infinite)<\/span>
\n po =<\/span> initialization<\/span>(<\/span>N,<\/span> dim,<\/span> ub,<\/span> lb)<\/span>;<\/span>% Initialize population<\/span>
\n %% Record the initial optimal solution and fitness<\/span>
\n for<\/span> i<\/span> =<\/span> 1<\/span>:<\/span>N
\n fit<\/span>(<\/span>i<\/span>)<\/span> =<\/span> fobj<\/span>(<\/span>po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span>)<\/span>;<\/span>
\n if<\/span> fit<\/span>(<\/span>i<\/span>)<\/span> <=<\/span> fbest
\n bestpos =<\/span> po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span>;<\/span>
\n fbest =<\/span> fit<\/span>(<\/span>i<\/span>)<\/span>;<\/span>
\n end<\/span>
\n end<\/span>
\n cg =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span> T)<\/span>;<\/span>% Store convergence information<\/span>
\n %% Main optimization loop<\/span>
\n for<\/span> t =<\/span> 1<\/span>:<\/span>T
\n w =<\/span> rand<\/span>(<\/span>)<\/span>*<\/span>(<\/span>(<\/span>1<\/span>\/<\/span>T^<\/span>2<\/span>)<\/span>*<\/span>t^<\/span>2<\/span>–<\/span>2<\/span>\/<\/span>T*<\/span>t+<\/span>1<\/span>\/<\/span>2<\/span>)<\/span>;<\/span> % Eq.(12)<\/span>
\n a =<\/span> (<\/span>1<\/span> –<\/span> t\/<\/span>T)<\/span> *<\/span> rand<\/span>(<\/span>VarSize)<\/span>;<\/span> % Eq.(17)<\/span>
\n xi =<\/span> rand<\/span>(<\/span>)<\/span>*<\/span>(<\/span>(<\/span>1<\/span>\/<\/span>T^<\/span>2<\/span>)<\/span>*<\/span>t^<\/span>2<\/span>–<\/span>2<\/span>\/<\/span>T*<\/span>t+<\/span>1<\/span>)<\/span>;<\/span> % Eq.(31)<\/span>
\n for<\/span> i<\/span> =<\/span> 1<\/span>:<\/span>N
\n for<\/span> k =<\/span> 1<\/span>:<\/span>dim\/<\/span>2<\/span>
\n % Eq.(11)<\/span>
\n if<\/span> k ==<\/span> 1<\/span>
\n npo_line =<\/span> w .*<\/span> po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span> +<\/span> alpha<\/span>(<\/span>i<\/span>)<\/span> .*<\/span> po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span>;<\/span>
\n else<\/span>
\n npo_line =<\/span> po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span> +<\/span> (<\/span>sin<\/span>(<\/span>2<\/span>*<\/span>pi<\/span>*<\/span>dim*<\/span>t)<\/span> )<\/span> .*<\/span> npo_line;<\/span>
\n end<\/span>
\n A1 =<\/span> randperm<\/span>(<\/span>N)<\/span>;<\/span>
\n A1<\/span>(<\/span>A1 ==<\/span> i<\/span>)<\/span> =<\/span> [<\/span>]<\/span>;<\/span>
\n % Eq.(14)<\/span>
\n a1 =<\/span> A1<\/span>(<\/span>1<\/span>)<\/span>;<\/span>
\n a2 =<\/span> A1<\/span>(<\/span>2<\/span>)<\/span>;<\/span>
\n % Eq.(15)<\/span>
\n zeta =<\/span> (<\/span>(<\/span>fit<\/span>(<\/span>a1)<\/span> –<\/span> fit<\/span>(<\/span>i<\/span>)<\/span>)<\/span> \/<\/span> abs<\/span>(<\/span>fit<\/span>(<\/span>a1)<\/span> –<\/span> fit<\/span>(<\/span>i<\/span>)<\/span>)<\/span>)<\/span>;<\/span>
\n po_mean =<\/span> mean<\/span>(<\/span>po)<\/span>;<\/span>
\n P =<\/span> (<\/span>po_mean –<\/span> npo_line)<\/span>;<\/span>% Eq.(23) (first-order)<\/span>
\n f =<\/span> gtdt<\/span>(<\/span>bestpos,<\/span> P,<\/span> lr,<\/span> t,<\/span> beta1,<\/span> beta2,<\/span> epsilon,<\/span> m,<\/span> v)<\/span>;<\/span>
\n npo_1a =<\/span> npo_line +<\/span> zeta *<\/span>a.*<\/span>(<\/span>f –<\/span> po<\/span>(<\/span>a1,<\/span> :<\/span>)<\/span>)<\/span> –<\/span> a.*<\/span>(<\/span>npo_line –<\/span> po<\/span>(<\/span>a2,<\/span> :<\/span>)<\/span>)<\/span>;<\/span>% Eq.(16)<\/span>
\n npo_1b =<\/span> po<\/span>(<\/span>a1,<\/span> :<\/span>)<\/span> +<\/span> a .*<\/span> (<\/span>f –<\/span> po<\/span>(<\/span>a2,<\/span> :<\/span>)<\/span>)<\/span>;<\/span>% Eq.(18)<\/span>
\n for<\/span> j<\/span> =<\/span> 1<\/span>:<\/span>dim
\n if<\/span> rand\/<\/span>k ><\/span> rand
\n npo_line<\/span>(<\/span>1<\/span>,<\/span> j<\/span>)<\/span> =<\/span> npo_1b<\/span>(<\/span>1<\/span>,<\/span> j<\/span>)<\/span>;<\/span>
\n else<\/span>
\n npo_line<\/span>(<\/span>1<\/span>,<\/span> j<\/span>)<\/span> =<\/span> npo_1a<\/span>(<\/span>1<\/span>,<\/span> j<\/span>)<\/span>;<\/span>
\n end<\/span>
\n end<\/span><\/p>\n

npo_line =<\/span> max<\/span>(<\/span>npo_line,<\/span> lb)<\/span>;<\/span>
\n npo_line =<\/span> min<\/span>(<\/span>npo_line,<\/span> ub)<\/span>;<\/span><\/p>\n

newfit =<\/span> fobj<\/span>(<\/span>npo_line)<\/span>;<\/span>
\n if<\/span> newfit <<\/span> fit<\/span>(<\/span>i<\/span>)<\/span>
\n po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span> =<\/span> npo_line;<\/span>
\n fit<\/span>(<\/span>i<\/span>)<\/span> =<\/span> newfit;<\/span>
\n if<\/span> fit<\/span>(<\/span>i<\/span>)<\/span> <=<\/span> fbest
\n bestpos =<\/span> po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span>;<\/span>
\n fbest =<\/span> fit<\/span>(<\/span>i<\/span>)<\/span>;<\/span>
\n end<\/span>
\n end<\/span>
\n a =<\/span> rand<\/span>(<\/span>VarSize)<\/span>;<\/span>% Eq.(17)<\/span>
\n end<\/span>
\n end<\/span>
\n if<\/span> rand <=<\/span> 1<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>exp<\/span>(<\/span>–<\/span>(<\/span>18<\/span>*<\/span>(<\/span>t\/<\/span>T)<\/span>–<\/span>12<\/span>)<\/span>)<\/span>)<\/span>
\n Step_length=<\/span>levy<\/span>(<\/span>N,<\/span> dim,<\/span> 1.2<\/span>)<\/span>;<\/span>
\n Elite=<\/span>repmat<\/span>(<\/span>bestpos,<\/span> N,<\/span> 1<\/span>)<\/span>;<\/span>
\n for<\/span> i<\/span>=<\/span>1<\/span>:<\/span>N
\n for<\/span> j<\/span>=<\/span>1<\/span>:<\/span>dim
\n npo_2<\/span>(<\/span>i<\/span>,<\/span>j<\/span>)<\/span>=<\/span>Elite<\/span>(<\/span>i<\/span>,<\/span>j<\/span>)<\/span>+<\/span>Step_length<\/span>(<\/span>i<\/span>,<\/span>j<\/span>)<\/span>*<\/span>xi*<\/span>(<\/span>Elite<\/span>(<\/span>i<\/span>,<\/span>j<\/span>)<\/span>–<\/span>po<\/span>(<\/span>i<\/span>,<\/span>j<\/span>)<\/span>*<\/span>(<\/span>2<\/span>*<\/span>t\/<\/span>T)<\/span>)<\/span>;<\/span> % Eq.(27)<\/span>
\n end<\/span>
\n end<\/span>
\n po =<\/span> npo_2;<\/span>
\n po =<\/span> max<\/span>(<\/span>po,<\/span> lb)<\/span>;<\/span>
\n po =<\/span> min<\/span>(<\/span>po,<\/span> ub)<\/span>;<\/span>
\n end<\/span>
\n %% Evaluation<\/span>
\n for<\/span> i<\/span>=<\/span>1<\/span>:<\/span>N
\n newfit<\/span>(<\/span>1<\/span>,<\/span> i<\/span>)<\/span>=<\/span>fobj<\/span>(<\/span>po<\/span>(<\/span>i<\/span>,<\/span> :<\/span>)<\/span>)<\/span>;<\/span>
\n end<\/span>
\n [<\/span>fbestN,<\/span> sorted_indexes]<\/span> =<\/span> sort<\/span>(<\/span>newfit)<\/span>;<\/span>
\n po =<\/span> po<\/span>(<\/span>sorted_indexes<\/span>(<\/span>1<\/span>:<\/span>N)<\/span>,<\/span> :<\/span>)<\/span>;<\/span>
\n if<\/span> fbestN<\/span>(<\/span>1<\/span>)<\/span><<\/span>fbest
\n bestpos =<\/span> po<\/span>(<\/span>1<\/span>,<\/span>:<\/span>)<\/span>;<\/span>
\n fbest=<\/span>fbestN<\/span>(<\/span>1<\/span>)<\/span>;<\/span>
\n end<\/span>
\n %% Record convergence curve <\/span>
\n cg<\/span>(<\/span>t)<\/span> =<\/span> fbest;<\/span>
\n pbest =<\/span> bestpos;<\/span>
\n end<\/span><\/p>\n

end<\/span>
\n% Initialization function<\/span>
\nfunction<\/span> X =<\/span> initialization<\/span>(<\/span>N,<\/span> Dim,<\/span> UB,<\/span> LB)<\/span>
\n B_no =<\/span> size<\/span>(<\/span>UB,<\/span> 2<\/span>)<\/span>;<\/span> % number of boundaries<\/span><\/p>\n

if<\/span> B_no ==<\/span> 1<\/span>
\n X =<\/span> rand<\/span>(<\/span>N,<\/span> Dim)<\/span> .*<\/span> (<\/span>UB –<\/span> LB)<\/span> +<\/span> LB;<\/span>
\n end<\/span>
\n % If each variable has a different lb and ub<\/span>
\n if<\/span> B_no ><\/span> 1<\/span>
\n X =<\/span> zeros<\/span>(<\/span>N,<\/span> Dim)<\/span>;<\/span>
\n for<\/span> i<\/span> =<\/span> 1<\/span>:<\/span>Dim
\n Ub_i =<\/span> UB<\/span>(<\/span>i<\/span>)<\/span>;<\/span>
\n Lb_i =<\/span> LB<\/span>(<\/span>i<\/span>)<\/span>;<\/span>
\n X<\/span>(<\/span>:<\/span>,<\/span> i<\/span>)<\/span> =<\/span> rand<\/span>(<\/span>N,<\/span> 1<\/span>)<\/span> .*<\/span> (<\/span>Ub_i –<\/span> Lb_i)<\/span> +<\/span> Lb_i;<\/span>
\n end<\/span>
\n end<\/span>
\nend<\/span><\/p>\n

% Enhanced Gradient Descent with Adam and Momentum<\/span>
\nfunction<\/span> f =<\/span> gtdt<\/span>(<\/span>X,<\/span> P,<\/span> lr,<\/span> t,<\/span> beta1,<\/span> beta2,<\/span> epsilon,<\/span> m,<\/span> v)<\/span>
\n C =<\/span> 1<\/span> –<\/span> rand;<\/span>
\n grad =<\/span> X –<\/span> C .*<\/span> P;<\/span>
\n %Eq.(21)<\/span>
\n m =<\/span> beta1 *<\/span> m +<\/span> (<\/span>1<\/span> –<\/span> beta1)<\/span> *<\/span> grad;<\/span>% Updating biased first moment estimates<\/span>
\n v =<\/span> beta2 *<\/span> v +<\/span> (<\/span>1<\/span> –<\/span> beta2)<\/span> *<\/span> (<\/span>grad .^<\/span> 2<\/span>)<\/span>;<\/span>% Updating biased second original moment estimates<\/span>
\n %Eq.(20) <\/span>
\n m_hat =<\/span> m \/<\/span> (<\/span>1<\/span> –<\/span> beta1 ^<\/span> t)<\/span>;<\/span>% Calculation of bias-corrected first moment estimates <\/span>
\n v_hat =<\/span> v \/<\/span> (<\/span>1<\/span> –<\/span> beta2 ^<\/span> t)<\/span>;<\/span>% Calculate bias-corrected second original moment estimates<\/span>
\n %Eq.(19)<\/span>
\n f =<\/span> X –<\/span> lr *<\/span> m_hat .\/<\/span> (<\/span>sqrt<\/span>(<\/span>v_hat)<\/span> +<\/span> epsilon)<\/span>;<\/span>
\nend<\/span>
\nfunction<\/span> [<\/span>z]<\/span> =<\/span> levy<\/span>(<\/span>n,<\/span>m,<\/span>beta)<\/span>
\n % beta is set to 1.5 in this paper<\/span>
\n num =<\/span> gamma<\/span>(<\/span>1<\/span>+<\/span>beta)<\/span>*<\/span>sin<\/span>(<\/span>pi<\/span>*<\/span>beta\/<\/span>2<\/span>)<\/span>;<\/span>
\n den =<\/span> gamma<\/span>(<\/span>(<\/span>1<\/span>+<\/span>beta)<\/span>\/<\/span>2<\/span>)<\/span>*<\/span>beta*<\/span>2<\/span>^<\/span>(<\/span>(<\/span>beta–<\/span>1<\/span>)<\/span>\/<\/span>2<\/span>)<\/span>;<\/span>
\n sigma_u =<\/span> (<\/span>num\/<\/span>den)<\/span>^<\/span>(<\/span>1<\/span>\/<\/span>beta)<\/span>;<\/span> % Eq.(30)<\/span>
\n u =<\/span> random<\/span>(<\/span>'Normal'<\/span>,<\/span>0<\/span>,<\/span>sigma_u,<\/span>n,<\/span>m)<\/span>;<\/span>
\n v =<\/span> random<\/span>(<\/span>'Normal'<\/span>,<\/span>0<\/span>,<\/span>1<\/span>,<\/span>n,<\/span>m)<\/span>;<\/span>
\n z =<\/span>u.\/<\/span>(<\/span>abs<\/span>(<\/span>v)<\/span>.^<\/span>(<\/span>1<\/span>\/<\/span>beta)<\/span>)<\/span>;<\/span> % Eq.(29)<\/span>
\nend<\/span><\/p>\n

Xia, Y., Ji, Y. Application of a novel metaheuristic algorithm inspired by Adam gradient descent in distributed permutation flow shop scheduling problem and continuous engineering problems. Sci Rep 15, 21692 (2025). https:\/\/doi.org\/10.1038\/s41598-025-01678-9<\/p>\n","protected":false},"excerpt":{"rendered":"

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