[toc]
常见的损失函数
y_i表示实际值，f_i表示预测值
0-1损失函数
L(y_i, f_i) = \left\{\begin{matrix} 1, ~y_i = f_i\\
0, ~y_i \neq f_i\end{matrix}\right.
等价形式：
L(y_i, f_i) = \frac{1}{2}(1 - sign(y_i\cdot f_i)), ~y_i\in\{\pm1\}
Perceptron感知损失函数(感知机)
L(y_i, f_i) = \left\{\begin{matrix} 1, ~|y_i - f_i| > t\\
0, ~|y_i - f_i| \leq t\end{matrix}\right.
等价形式：
L(y_i, f_i) = max\{0,~-(y_i\cdot f_i)\}, ~y_i\in\{\pm1\}
证明
因为当y_i = {-1, +1}时，|y_i - f_i| = {0, +2}，第一个式子等价于
L(y_i, f_i) = \left\{\begin{matrix} 1, ~|y_i - f_i| = 2~/~y_i\cdot f_i = -1\\
0, ~|y_i - f_i| = 0~/~y_i\cdot f_i = 1\end{matrix}\right.
又等价于
L(y_i, f_i) = max\{0,~-(y_i\cdot f_i)\}, ~y_i\in\{\pm1\}
Hinge损失函数(SVM)
L(y_i, f_i) = max\{0,~1 - y_i\cdot f_i\},~ y \in \{\pm1\}
Loss损失函数(Logistic回归)
L(y_i, f_i) = -\left(y_i\log f_i + (1-y_i)\log{(1-f_i)}\right),~y_i\in \{0,1\}
其中
f(x) = 1/\exp(-w^T\cdot x)
等价于
L(y_i, f_i) = log(1 + \exp(y_i\cdot f_i)),~ y_i \in \{\pm1\}
证明
因为当y_i = {0, +1}时，第一个式子等价于
L(y_i,f_i) = \left\{\begin{matrix} log(1+\exp(-w^T\cdot x),~y_i = 1\\
log(1+\exp(w^T\cdot x),~y_i=0\end{matrix}\right.
等价于，当y_i = {-1, +1}时
L(y_i,f_i) = \left\{\begin{matrix} log(1+\exp(-w^T\cdot x),~y_i = 1\\
log(1+\exp(w^T\cdot x),~y_i=-1\end{matrix}\right.
等价于
L(y_i, f_i) = log(1 + \exp(y_i\cdot f_i)),~ y_i \in \{\pm1\}
指数损失函数(Adaboost)
L(y_i,f_i)=\exp(-y_i\cdot f_i), y_i\in \{\pm1\}
几个损失函数的图像
image
回归损失函数
Square损失函数
L(y_i,f_i)=(y_i - f_i)^2
Absolute损失函数
L(y_i,f_i)=|y_i-f_i|
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