Write vectorized pseudocode (Python-like) for batch gradient descent to fit a linear regression model with intercept using mean squared error. Clearly explain each step and variable. Your pseudocode should expose inputs X ∈ R^{m×n} (assume an added intercept column), y ∈ R^{m}, learning_rate α, max_iters, tolerance, and should output θ ∈ R^{n} and cost_history. Derive the update rule from J(θ) = (1/(2m)) ||Xθ − y||^2, show that ∇J(θ) = (1/m) X^T (Xθ − y), and implement θ ← θ − α ∇J(θ). Then: - Stopping criteria: implement at least two (max iters, relative cost decrease or ||∇J||_2 below tolerance) and describe when each is preferable. - Learning rate: discuss constant vs time-decayed schedules; how to pick α to avoid divergence; how feature scaling/standardization affects convergence rate and the condition number of X^T X. - Regularization: modify J(θ) to include L2 with λ (J(θ) = (1/(2m))||Xθ − y||^2 + (λ/(2m))||θ_{1:}||^2 where the intercept is excluded). Write the new gradient and discuss λ selection.