Visa ML Coding Question: Linear Regression Implementation

Train linear regression from scratch using gradient descent and return learned parameters and per-epoch loss history.

You are asked to implement a standard linear model where predictions follow:

y_hat = X @ w + b

and the objective is to minimize the mean squared error (MSE):

L(w, b) = (1/n) * sum_i (y_i - (x_i^T w + b))^2

Your implementation should accept a feature matrix X (n_samples, n_features), target vector y (n_samples,), a learning_rate, and a number of epochs. Use batch gradient descent (vectorized operations recommended) to update parameters. Return a 1-D params array of length n_features + 1 that includes the intercept (bias) — specify in your function docstring whether the intercept is the last or first element (recommendation: place the bias as the last element) — and a loss_history list containing the training MSE after each epoch.

Flow you can expect in an interview:

• Clarify shapes, numeric types, and whether to include an intercept term.
• Derive gradients for w and b and propose a vectorized update rule.
• Implement and run a training loop, computing MSE each epoch and returning params + loss history.

Skill signals to demonstrate: linear algebra/vectorization, gradient derivation, numerical stability (feature scaling), convergence behavior (learning rate choice), and clear function documentation. You may mention analytical (normal equation) comparison as an alternative but implement gradient descent without ML libraries.