Polynomial Regression

Implement polynomial regression by first constructing polynomial features, then applying a linear model.

Given scalar inputs x and a degree d, construct features: $\phi(x) = [1, x, x^2, ..., x^d]$

Then predict: $\hat{y} = \phi(x) \cdot w$

And compute MSE loss: $L = \frac{1}{N} \sum (y - \hat{y})^2$

Input:

• x: input tensor of shape (N,)
• w: weight vector of shape (d+1,) for polynomial of degree d
• y: target values of shape (N,)

Output: A dict with “prediction” (shape (N,)) and “loss” (scalar).