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

Put $s=x^{T}x>0$ and $z=x^{T}Y$. With objective $\Vert Y-x\beta {\Vert }_2^2+\lambda {\beta }^2$, ridge regression givesFor the duplicated design, the objective depends on ${\beta }_1+{\beta }_2$ through the loss and symmetry makes the minimum-penalty decomposition equal:Duplicating a predictor therefore halves its effective ridge penalty.