55.55. Lasso Regression (L1)

Implement **Lasso Regression** (Linear Regression with L1 regularization) using gradient descent.

Loss Function:

Loss = (1/n) * sum((y_pred - y)^2) + alpha * sum(|w|)

Gradient Descent Update Rules:

dw = (2/n) * (X^T @ (y_pred - y)) + alpha * sign(w)

db = (2/n) * sum(y_pred - y)

w = w - learning_rate * dw

b = b - learning_rate * db

Where sign(w) returns -1 for negative values, +1 for positive values, and 0 for zero.

Note: The L1 penalty **only applies to weights**, not the bias term.

Your class should have:

Example:

X = [[1, 2, 0], [2, 3, 0], [3, 4, 0], [4, 5, 0]]

y = [5, 8, 11, 14]

model = LassoRegression(alpha=0.1, learning_rate=0.01)

model.fit(X, y, n_iterations=1000)

# L1 tends to push irrelevant feature weights to exactly 0 (sparsity)

**Explanation:** L1 regularization encourages sparsity -- it can drive some weights to exactly zero, effectively performing feature selection. This is useful when you suspect only a subset of features are truly important.

Constraints: