Question 5: You are analyzing a dataset of N = 100 protein-protein in…

You are analyzing a dataset of $N = 100$ protein-protein interaction (PPI) samples. Each interaction is represented by 3 numerical features: $X = [X^{(1)}, X^{(2)}, X^{(3)}]$, where $X^{(1)}, X^{(2)}, X^{(3)} \in \mathbb{R}$. The binary target variable $y$ represents whether an interaction occurs ($y = 1$) or not ($y = 0$).

You aim to train a logistic regression model with L2 regularization (ridge regression) to predict interactions based on these features. The regularized cost function, $J(\theta)$, is defined as:

$J(\theta) = J'(\theta) + \lambda \sum_{i=1}^{3} \theta_i^2$

where, $J'(\theta)$ is the cross entropy loss as seen in lecture videos.

$J(\theta)$ penalizes large weights to avoid overfitting using the following settings:

• Regularization strength $\lambda = 0.1$
• Initial parameters: $\theta_0 = 0.5,\ \theta_1 = -0.5,\ \theta_2 = 0.3,\ \theta_3 = -0.2$
• Learning rate: $\alpha = 0.01$

Compute the regularized cost function $J(\theta)$, for a single protein interaction sample $x_{uv} = [2, 3, 1]$ with target $y_{uv} = 1$.

Note: Use $\lambda/2m$ as the constant in regularization term of the cost function, where $m$ is the batch size.