0/1 loss
\[J(\theta) = \sum_i^m L_{0/1}(\theta^Tx)\]where
\[L_{0/1}(f) = \left\{ \begin{array}{ll} 1 & \mbox{if } y\cdot f \lt 0 \\ 0 & \mbox{if } y\cdot f \gt 0 \end{array} \right.\]Hinge loss
\[J(\theta) = max(0, 1 - y\cdot f)\]Logistic loss
\[J(\theta) = -\left[ \sum_{i=1}^m y^{(i)} \log h_\theta(x^{(i)}) + (1-y^{(i)}) \log (1-h_\theta(x^{(i)})) \right]\]where $ h_\theta(x) = \frac{1}{1+\exp(-\theta^\top x)} $ is the sigmoid activation function.
Cross entropy
\(J(\theta) = - \left[ \sum_{i=1}^{m} \sum_{k=0}^{1} 1\left\{y^{(i)} = k\right\} \log P(y^{(i)} = k \vert x^{(i)} ; \theta) \right]\) 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.