Step 1: Calculate the predicted value $y_{\text{pred}}$
The predicted value is calculated using the linear model formula $y_{\text{pred}} = \mathbf{w}^T \mathbf{x}$.
$\mathbf{w}^T \mathbf{x} = \begin{bmatrix} -3 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = (-3 \cdot 1) + (4 \cdot 2) = -3 + 8 = 5$
So, $y_{\text{pred}} = 5$.
Step 2: Calculate the true value $y_{\text{true}}$
The true value is given by the formula $y_{\text{true}} = x_1 + x_2$.
$y_{\text{true}} = 1 + 2 = 3$
So, $y_{\text{true}} = 3$.
Step 3: Calculate the Mean Absolute Error (MAE) loss
The MAE loss is the absolute difference between the true and predicted values: $\text{MAE} = |y_{\text{true}} - y_{\text{pred}}|$.
$\text{MAE} = |3 - 5| = |-2| = 2$
Step 4: Calculate the $L_2$ regularization term
The $L_2$ regularization term is given by $\lambda \cdot \text{L}_2\text{-norm}^2$, where the $\text{L}_2\text{-norm}^2$ of $\mathbf{w}$ is $\sum w_i^2$.
$\text{L}_2\text{-norm}^2 = (-3)^2 + (4)^2 = 9 + 16 = 25$
The regularization term is $\lambda \cdot 25 = 0.20 \cdot 25 = 5$.
Step 5: Calculate the total regularized loss
The total regularized loss is the sum of the MAE loss and the $L_2$ regularization term.
$\text{Regularized Loss} = \text{MAE} + \text{Regularization Term} = 2 + 5 = 7$
Answer: The regularized loss for this instance is $\mathbf{7}$.
