One Gradient Descent Step for Linear Regression

Difficulty: Medium · Concept: Linear Regression — Gradient Descent

For single-feature linear regression $\hat{y} = wx + b$, one step of gradient descent updates both parameters using the gradients derived in Linear Regression, Section 3:

$\frac{\partial J}{\partial w} = \frac{2}{n} \sum_{i=1}^{n} (\hat{y}^{(i)} - y^{(i)}) x^{(i)} \qquad \frac{\partial J}{\partial b} = \frac{2}{n} \sum_{i=1}^{n} (\hat{y}^{(i)} - y^{(i)})$

$w \leftarrow w - \alpha \frac{\partial J}{\partial w} \qquad b \leftarrow b - \alpha \frac{\partial J}{\partial b}$

Your task: implement gradient_descent_step(x, y, w, b, alpha) — given the current w, b, and learning rate alpha, compute one full-batch gradient step over the dataset (x, y) and return the updated (w, b) as a tuple. Don't mutate the inputs in place; return new values.

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