Consider the Ridge LRR is being used to learn prediction function ypred=wT x where w,x ∈ R2 & mean absolute error (MAE) is used to measure the prediction error. A weight of 0.20 is associated with the regularizer. At an intermediate step of training process assume that the parameter w = [-3.00,4.00]T . In the next step for the I/P x=[1.00,2.00]T , the predicted value of y is noted. Let the relation b/w x=[x1,x2] T & the true value of y be ytrue = x1+x2. The value of the overall regularized loss for instance is _____(upto 2 decimal).

The correct answer is 7.00.

To find the overall regularized loss, we need to calculate the prediction error (using Mean Absolute Error, or MAE) and add the regularized penalty (L2 or Ridge regularization).

Step 1: Calculate the predicted value of y
The prediction function is given by:
$$y_{\text{pred}}=w^{T}x$$ Given the parameter vector: $$w={[-3.00,4.00]}^T$$ and the input vector: $$x={[1.00,2.00]}^T$$ We compute the dot product:
$$y_{\text{pred}}=(-3.00\times 1.00)+(4.00\times 2.00)=-3.00+8.00=5.00$$

Step 2: Calculate the true value of y
The relationship for the true value is:
$$y_{\text{true}}=x_1+x_2$$ Using the components of the input vector $x$ where $x_1=1.00$ and $x_2=2.00$:
$$y_{\text{true}}=1.00+2.00=3.00$$

Step 3: Calculate the Prediction Error (MAE)
The absolute error for this instance is:
$$\text{MAE}=|y_{\text{true}}-y_{\text{pred}}|$$ Substituting the values:
$$\text{MAE}=|3.00-5.00|=|-2.00|=2.00$$

Step 4: Calculate the Regularization Term
Ridge regression utilizes L2 regularization, where the penalty is the squared L2 norm of the parameter vector:
$${||w||}_2^2=w_1^2+w_2^2$$ Substituting the weights:
$${||w||}_2^2={(-3.00)}^2+{\left(4.00\right)}^2=9.00+16.00=25.00$$

Step 5: Calculate the Overall Regularized Loss
The overall regularized loss is the sum of the MAE loss and the weighted regularization term:
$$\text{Total Loss}=\text{MAE}+\lambda {||w||}_2^2$$ Given the regularization weight $\lambda =0.20$:
$$\text{Total Loss}=2.00+0.20\times 25.00=2.00+5.00=7.00$$