Linear Regression: closed form (Normal Equation)

The problem, from first principles

Full Linear Regression Training Loop finds the best-fit line iteratively: guess, measure the error, nudge the weights, repeat, for as many epochs as it takes to converge. For plain linear regression specifically, that iteration turns out to be unnecessary: the loss surface (02-mse-loss's mean squared error, as a function of the weights) is a smooth, convex bowl with exactly one minimum, and calculus can solve for that minimum's exact location directly, in one shot, with no learning rate, no epochs, no convergence to wait for at all.

This is the Normal Equation, and understanding it matters for more than just efficiency: it's the moment where "why does gradient descent need a learning rate at all" gets a genuine answer, for THIS specific, simple problem, it doesn't, gradient descent is solving something calculus can solve exactly. Most of the models this curriculum builds (anything with a nonlinearity) don't have this luxury, which is precisely why gradient descent is the general-purpose tool used everywhere else.

From theory to code

Theory derives the exact solution by setting the loss's gradient to zero and solving algebraically, folding the bias into the weight vector by augmenting the input with a constant column of ones.

Implement closed_form_linear_regression(input, target) against that reasoning. The signature and docstring are already in the editor.

Constraints

• Return (weight, bias) in the exact same (1, in_features)/(1,) shapes Full Linear Regression Training Loop returns.
• Use np.linalg.pinv (the pseudoinverse), not np.linalg.inv, for the reason Theory explains.
• Fold the bias into the augmented system (a column of ones appended to input), rather than solving for it separately.

Hints

Open one at a time. Each gives away a little more than the last.