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Exponential cone programmingLogistic regression exampleReferences

Exponential cone programming

指数锥规划定义为$(y\exp (x/y)\le z,y>0)$，等效于相对熵规划，YALMIP不一定会检测到指数锥，可以使用persepective expoential.

Logistic regression example

LR问题的凸规划方程为 $$\sum \log (1+{\exp }^{-y_i(a^T{x}_i+b)})$$ 可以使用logsumexp直接求解，也可以进行分解后等效转换 $$\{\begin{array}{l}\log (1+e^z)=\log (e^0+e^z)\\ \log (e^0+e^{-y_i(a^T{x}_i+b)})\le t_i\to e^0+e^{-y_i(a^T{x}_i+b)}\le e^{t_i}\end{array}$$ 令 $$\{\begin{array}{l}{e}^{-t_i}\le z_i\\ e^{-y_i(a^T{x}_i+b)-t_i}\le u_i\end{array}$$ 约束转化为 $$z_i+u_i\le 1$$ 目标函数为 $$\min \sum t_i$$

%% 产生两种数据点 N = 50; blues = randn(2,N/2); reds = randn(2,N/2)+2; clf hold on plot(reds(1,:),reds(2,:),'r*'); plot(blues(1,:),blues(2,:),'b*') hold off

使用logsumexp算子求解

%% logsumexp (需要使用mosek 9)进行求解 % blue = 1, red=-1 x = [blues reds]; y = [ones(1, length(blues)) repmat(-1, 1, length(reds))]; a = sdpvar(2, 1); b = sdpvar(1); J = sum(logsumexp([zeros(length(y), 1) (-y.*(a'*x+b))']')); optimize([], J)

统计分类结果

%% count class = sign(value(a'*x+b)); % 进行分类 classB = find(class==1); % 蓝色点 classR = find(class==-1); % 红色点 hold on; plot(x(1, classB), x(2, classB), 'bo') plot(x(1, classR), x(2, classR), 'ro') hold off; nnz(sign(value(a'*x+b))-y) % 计算出非零点的数量，即分类错误点的数量

References

relative entropy optimization and its applications Exponential cone programming