Cost Function for Logistic regression

Cost function for logistic

logistic hyphothesis

$H(x)=\frac{1}{1+e^{-x}}$

$cost(W)=\frac{1}{m}\sum{c(H(x),y)}$

$c(H(x),y)\begin{cases} -log(H(x)) & \text{y=1} \\ -log(1-H(x)) & \text{y=0} \end{cases}$

tensorflow에서 구현할때 if문을 달아야한다는 점이 있으므로

$c(H(x),y)=-ylog(H(x))-(1-y)log(1-H(x))$

로 쓸 수 있다.

$y=1\text{일때 } (1-y)\text{가 0이되고 }y=0\text{일때 -y=0이 되며 (1-y)는 1이 됨을 이용함}$

$cost(W)=-\frac{1}{m}\sum ylog(H(x))+(1-y)log(1-H(x))$
$W:=W-\alpha\frac{\alpha}{\alpha W}cost(W)$

#cost function
cost = tf.reduce_mean(-tf.reduce_sum(Y*tf.log(hypothesis) + (1-y)*tf.log(1-hypothesis)))

#Minimize
a = tf.Variable(0.1) # Learning rate, alpha.
optimizer = tf.train.GradientDescentOptimizer(a)
train = optimizer.minimize(cost)

tf.train.GradientDescentOptimizer(a)는 $W:=W-\alpha\frac{\alpha}{\alpha W}cost(W)$와 같다