L1 vs L2 Regularization

You are given a labeled dataset of n samples with d real-valued features and a binary target variable. Implement two regularized logistic regression solvers: one using L1 (Lasso) regularization and one using L2 (Ridge) regularization. For each solver you must:

1. Accept a regularization strength λ>0, a convergence tolerance ε>0, and a maximum number of iterations T.
2. Fit the logistic regression weights w∈ℝᵈ and intercept b∈ℝ by minimizing the penalized negative log-likelihood:
   • L1: 1/n Σᵢ log(1+exp(−yᵢ(wᵀxᵢ+b))) + λ‖w‖₁
   • L2: 1/n Σᵢ log(1+exp(−yᵢ(wᵀxᵢ+b))) + λ‖w‖₂²
3. Return the learned weights w, the intercept b, the number of non-zero weights after convergence, and the final training loss.

You may use any standard first-order optimization algorithm (coordinate descent, proximal gradient, ISTA/FISTA, etc.) but you must implement the update rules yourself; do not call high-level sklearn-style solvers. Your implementation must handle the non-smooth L1 term explicitly (e.g., via soft-thresholding) and the smooth L2 term via its gradient.

After fitting both models on the same dataset, compare them by reporting:

• The number of weights that are exactly zero (up to numerical precision) for L1 vs L2.
• The ℓ₂ norm of the weight vector for each model.
• The qualitative difference in the resulting weight vectors when features are highly correlated.