\u4f20\u7edf\u56de\u5f52\u6a21\u578b\u5982\u7ebf\u6027\u56de\u5f52\uff0c\u5bf9\u4e8e\u6837\u672c$(x,y)$\u662f\u76f4\u63a5\u57fa\u4e8e\u6a21\u578b\uff0c\u901a\u8fc7\u9884\u6d4b\u503c$f(x_i){y}$\u548c\u771f\u5b9e\u503c$y$\u4e4b\u95f4\u7684\u5dee\u522b\u8ba1\u7b97\u635f\u5931\uff0c\u5e76\u4e14\u5f53$f(x_i){y}=y$\u65f6\u635f\u5931\u624d\u4e3a\u96f6\u3002<\/p>\n
\u652f\u6301\u5411\u91cf\u56de\u5f52(support vector regression, SVR)\u5219\u53ef\u4ee5\u5bb9\u5fcd$f(x_i){y}$\u548c$y$\u4e4b\u95f4\u6709\u6700\u591a$\\epsilon$\u7684\u504f\u5dee\uff0c\u5373\u5f53$|f(x_i){y}-y|>\\epsilon$\u7684\u65f6\u5019\u624d\u8ba1\u7b97\u635f\u5931\uff0c\u8fd9\u76f8\u5f53\u4e8e\u4ee5$f(x_i){y}$\u4e3a\u4e2d\u5fc3\uff0c\u6784\u5efa\u4e86\u4e00\u4e2a\u5bbd\u5ea6\u4e3a$2\\epsilon$\u7684\u95f4\u9694\u5e26\uff0c\u5982\u679c\u6837\u672c\u843d\u5165\u95f4\u9694\u5e26\uff0c\u5219\u4ed6\u7684\u5206\u7c7b\u5c31\u662f\u6b63\u786e\u7684\u3002<\/p>\n
\u7ebf\u6027\u53ef\u5206SVM\u76ee\u6807\u51fd\u6570\u4f18\u5316\u95ee\u9898\u4e3a \u652f\u6301\u5411\u91cf\u56de\u5f52\u7531\u4e8e\u6709\u4e00\u4e2a\u95f4\u9694\u5e26\uff0c\u56e0\u6b64\u5b83\u7684\u635f\u5931\u5ea6\u91cf\u51fd\u6570\u4e3a \u7531\u4e8eSVR\u7684\u95f4\u9694\u5e26\u662f\u81ea\u5df1\u5f15\u5165\u7684\uff0c\u6240\u4ee5SVR\u7684\u76ee\u6807\u51fd\u6570\u53d8\u4e3a \u8054\u7cfb\u7ba1\u7406\u5458\u5fae\u4fe1tutu19192010\uff0c\u6ce8\u518c\u8d26\u53f7<\/p>\n<\/div>
\n$$
\n\\begin{align}
\n& \\underbrace{\\min}_{\\omega,b} {\\frac{1}{2}}{||\\omega||}^2 \\
\n& s.t. \\quad y_i(\\omega{x_i}+b)\\geq1, \\quad i=1,2,\\ldots,m
\n\\end{align}
\n$$
\n \u7ebf\u6027SVM\u7531\u4e8e\u5728\u76ee\u6807\u51fd\u6570\u4e2d\u52a0\u5165\u4e86\u677e\u5f1b\u56e0\u5b50$\\xii>0$\uff0c\u76ee\u6807\u51fd\u6570\u4f18\u5316\u95ee\u9898\u4e3a
\n$$
\n\\begin{align}
\n& \\underbrace{\\min}<\/em>{\\omega,b,\\xi} {\\frac{1}{2}}{||\\omega||}^2 + C\\sum_{i=1}^m\\xi_i \\
\n& s.t. \\quad y_i(\\omega{x_i}+b)\\geq1-\\xi_i, \\quad i=1,2,\\ldots,m \\
\n& \\quad\\quad \\xi_i\\geq0, \\quad i=1,2,\\ldots,m
\n\\end{align}
\n$$<\/p>\n\u652f\u6301\u5411\u91cf\u56de\u5f52\u635f\u5931\u5ea6\u91cf\u51fd\u6570<\/h2>\n
\n$$
\nl(f(x_i),y_i) =
\n\\begin{cases}
\n0, & if\\,|f(x_i)-y_i|\\leq\\epsilon \\
\n|f(x_i)-y_i|-\\epsilon, & if\\,|f(x_i)-y_i|>\\epsilon \\
\n\\end{cases}
\n$$<\/p>\n\u652f\u6301\u5411\u91cf\u56de\u5f52\u76ee\u6807\u51fd\u6570\u4f18\u5316\u95ee\u9898<\/h2>\n
\n$$
\n\\underbrace{\\min}{\\omega,b}\\,{\\frac{1}{2}}{||\\omega||}^2 + C\\sum<\/em>{i=1}^ml(f(x_i)-y_i)
\n$$
\n \u5982\u679c\u548c\u7ebf\u6027SVM\u4e00\u6837\u5f15\u5165\u677e\u5f1b\u56e0\u5b50\uff0c\u4f46\u662f\u7531\u4e8e\u6211\u4eec\u7684\u8bef\u5dee\u5ea6\u91cf\u4e2d\u7684$|f(x_i)-y_i|\\leq\\epsilon$\u662f\u7edd\u5bf9\u503c\u5c0f\u4e8e\uff0c\u56e0\u6b64\u8fd9\u4e2a\u4e0d\u7b49\u5f0f\u5176\u5b9e\u662f\u4e24\u4e2a\u4e0d\u7b49\u5f0f\uff0c\u5219SVR\u9700\u8981\u5f15\u5165\u4e24\u4e2a\u677e\u5f1b\u56e0\u5b50$\\xi_i$\u548c$\\hat{\\xii}$\uff0c\u5219SVR\u7684\u4f18\u5316\u95ee\u9898\u5c06\u53d8\u6210
\n$$
\n\\underbrace{\\min}<\/em>{\\omega,b,\\xi_i,\\hat{\\xii}}{\\frac{1}{2}}||w||^2+C\\sum<\/em>{i=1}^m(\\xi_i+\\hat{\\xi_i})
\n$$
\n$$
\n\\begin{align}
\ns.t. & f(x_i)-y_i\\leq\\epsilon+\\xi_i, \\
\n& y_i-f(x_i)\\leq\\epsilon+\\hat{\\xi_i}, \\
\n& \\xi_i\\geq0,\\hat{\\xi_i}\\geq0,\\,i=1,2,\\cdots,m
\n\\end{align}
\n$$
\n \u5bf9SVR\u7684\u4f18\u5316\u95ee\u9898\u5f15\u5165\u62c9\u683c\u6717\u65e5\u4e58\u5b50$\\mu_i\\geq0,\\hat{\\mu_i}\\geq0,\\alpha_i\\geq0,\\hat{\\alphai}\\geq0$\uff0c\u901a\u8fc7\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5\u5373\u53ef\u5f97\u5230\u62c9\u683c\u6717\u65e5\u51fd\u6570
\n$$
\n\\begin{align}
\n& L(w,b,\\alpha,\\hat{\\alpha},\\xi,\\hat{\\xi},\\mu,\\hat{\\mu}) \\
\n& = \\frac{1}{2}||w||^2+C\\sum<\/em>{i=1}^m(\\xi_i+\\hat{\\xii})-\\sum<\/em>{i=1}^m\\mu_i\\xii-\\sum<\/em>{i=1}^m\\hat{\\mu_i}\\hat{\\xii} \\
\n& +\\sum<\/em>{i=1}^m\\alpha_i(f(x_i)-yi-\\epsilon-\\xi)+\\sum<\/em>{i=1}^m\\hat{\\alpha_i}(y_i-f(x_i)-\\epsilon-\\hat{\\xi_i})
\n\\end{align}
\n$$<\/p>\n\u652f\u6301\u5411\u91cf\u56de\u5f52\u76ee\u6807\u51fd\u6570\u5bf9\u5076\u5f62\u5f0f<\/h2>\n
\u67e5\u770b\u66f4\u591a<\/h2>