Answer: (iii) Cross-Entropy (Log Loss / Binary Cross-Entropy)

Explanation:

Logistic regression models $p(y=1|x)= \sigma(w^T x+b)$ where $\sigma(z)=1/(1+e^{-z})$ and maximizes likelihood, equivalent to minimizing:

$J(w)= -\frac1m \sum_{i} [y_i \log \hat{y}_i + (1-y_i)\log(1-\hat{y}_i)]$ — Cross-Entropy / Log Loss

• MSE (i) is for linear regression; using it with sigmoid makes optimization non-convex and slow.
• Euclidean Distance (ii) not a classification loss.
• Hinge Loss (iv) is for SVM: $\max(0,1-y \cdot f(x))$.

Thus (iii) Cross-Entropy is correct. It heavily penalizes confident wrong predictions and yields convex optimization.