0/1 loss
where
Hinge loss
Logistic loss
where $ h_\theta(x) = \frac{1}{1+\exp(-\theta^\top x)} $ is the sigmoid activation function.
Cross entropy
where $P(y^{(i)} = k \vert x^{(i)}; \theta) = \frac{\exp(\theta^{(k)\top}x^{(i)})}{\sum_{j=1}^K \exp(\theta^{(j)\top} x^{(i)})}$ is the softmax activation function.