The <ridge regression> estimator minimizes
$$
\lVert Y-X\beta\rVert_2^2+\lambda\lVert\beta\rVert_2^2.
$$
Its gradient vanishes exactly when $(X^TX+\lambda I)\beta=X^TY$. Since $\lambda>0$ makes this matrix positive definite,
$$
\widehat\beta_\lambda=(X^TX+\lambda I)^{-1}X^TY.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 5 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-205/5/a/solution