逻辑回归Loss function推导

2018-09-27 本文已影响0人 DeepMine

逻辑回归

$\hat{y} = \sigma(z)$

$\sigma(z) = \frac{1}{{1+e^{-z}}}$

$z = w^Tx+b$

$\hat{y} = p(y=1|x)$

神经网络表示

预测概率

if $y = 1$ : $p(y|x)=\hat{y}$

if $y = 0$ : $p(y|x)=1-\hat{y}$

$=>$
$p(y|x) = \hat{y}^{y}.(1-\hat{y})^{(1-y)}$

$log(p(y|x))=log(\hat{y}^{y}.(1-\hat{y})^{(1-y)}) =y.log\hat{y}+(1-y).log(1-\hat{y})$

$L(y, \hat{y}) = - log(p(y|x))$

最大似然

$P=\prod_{i=1}^N p(y^{(i)}|x^{(i)})$

$LP = \sum_{k=1}^N log(p(y^{(i)}|x^{(i)}))$

$LP = -\sum_{k=1}^N L(y^{(i)},\hat{y}^{(i)})$

求导

$\frac{dL} {d\hat{y}} = -\frac{y} {\hat{y}} +\frac{(1-y)}{(1-\hat{y})}$

$\frac{d\hat{y}} {dz} = \hat{y} . (1-\hat{y})$

$\frac{dL} {dz} = \frac{dL} {d\hat{y}} . \frac{d\hat{y}} {dz} =-y(1-\hat{y}) + (1-y)\hat{y} = -y + \hat{y}$

$\frac{d\hat{z}} {dw} = x$

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