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• task specific and empirical
  Multinomial Logistic Loss and Cross Entropy Loss
  $J(\theta)=-{1\over m}[\sum^m_{i=1}y^{(i)}\log{h_{\theta}}(x^{(i)}) + (1-y^{(i)})\log(1-h_{\theta}(x^{(i)}))]$

for general case, it should be

$J(\theta)=-[\sum^m_{i=1}y^{(i)}\sum^K_{k=1}1{y^{(i)=k}}\log P(y^{(i)}=k|x^{(i)};\theta)]$