Put $s=x^Tx>0$ and $z=x^TY$. With objective $\lVert Y-x\beta\rVert_2^2+\lambda\beta^2$, <ridge regression> gives
$$
\widehat\beta=\frac z{s+\lambda}.
$$
For the duplicated design, the objective depends on $\beta_1+\beta_2$ through the loss and symmetry makes the minimum-penalty decomposition equal:
$$
\widehat\beta_1=\widehat\beta_2=\frac z{2s+\lambda},
\qquad
\widehat\beta_1+\widehat\beta_2=\frac{2z}{2s+\lambda}.
$$
Duplicating a predictor therefore halves its effective ridge penalty.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-218/1/a/solution