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支援向量機(Support Vector Machine, SVM)

感知演算法

線性分類器:f(x;w,b)=w,x+bf({\rm{x}};{\rm{w}},b)=\left \langle {\rm{w}},{\rm{x}} \right \rangle+b,決策:sgn[f(x;w,b)]sgn[{f({\rm{x}};{\rm{w}},b)}] 線性感知機(Perception)演算法: 輸入:訓練資料D={(xi,yi)}i=1ND=\{({\rm{x}}_{i},y_{i})\}_{i=1}^{N},學習步長η>0\eta>0

>0 輸出:感知模型的w,b{\rm{w}},b 在這裡插入圖片描述 證明:yift+1(xi)>yift(xi)y_{i}f^{t+1}({\rm{x}}_{i})>y_{i}f^{t}({\rm{x}}_{i}) yift+1(xi)=yi[wt+1,xi+bt+1]=yi[wt+ηyixi,xi+bt+ηRyi]=yi[wt,xi+bt+ηyixi,xi+ηRyi]=yift(xi)+η(xi,xi+R)>yift(xi)y_{i}f^{t+1}({\rm{x}}_{i})=y_{i}[\left \langle {\rm{w}^{t+1}},{\rm{x}}_{i} \right \rangle+b^{t+1}]\\ =y_{i}[\left \langle {\rm{w}^{t}}+\eta y_{i}{\rm{x}}_{i},{\rm{x}}_{i} \right \rangle+b^{t}+\eta Ry_{i}]\\ =y_{i}[\left \langle {\rm{w}^{t}},{\rm{x}}_{i} \right \rangle+b^{t}+\left \langle \eta y_{i}{\rm{x}}_{i},{\rm{x}}_{i} \right \rangle+\eta Ry_{i}]\\ =y_{i}f^{t}({\rm{x}}_{i})+\eta(\left \langle {\rm{x}}_{i},{\rm{x}}_{i} \right \rangle+R)>y_{i}f^{t}({\rm{x}}_{i})
每次更新都會減少錯誤。