Explain AUC, activations, ensembles, and imbalance

Respond to every sub-question with exact answers. AUC/Ranking: With scores s = [0.10, 0.40, 0.35, 0.80, 0.60] and labels y = [0, 1, 0, 1, 0], calculate the ROC AUC exactly using pairwise positive–negative comparisons, without a library. Next, plot the ROC points and compute the area using the trapezoidal rule; the two results should agree. How would severe class imbalance (1% positives) affect your interpretation of AUC versus Average Precision? Activations: For each case, choose the output-layer activation and loss function, and explain your choice: (a) single-label multi-class with K=7, (b) multi-label with K=7, (c) regression bounded to [0,1], (d) unbounded regression with outliers. Describe the vanishing-gradient problem for sigmoid and tanh, and explain why leaky-ReLU or GELU can be beneficial in hidden layers. MSE vs MAE: Explain how they differ in optimization and robustness, covering gradients, outlier influence, and median versus mean optimality. Ensembles: Compare bagging and boosting with respect to bias/variance trade-offs, and state when each would be preferable for noisy data. Overfitting: Give two concrete, testable diagnostics, including plots or metrics, and two mitigation approaches that do not leak validation information. Overview: This question assesses skills in model evaluation metrics (ROC AUC and Average Precision), handling class imbalance, selecting output activations and loss functions, robustness to outliers (MSE vs MAE), ensemble methods, and overfitting diagnostics in the Machine Learning domain.