Unbounded directional risk of fixed ridge shrinkage (${\sup }_{{\beta }^0,\Vert x{\Vert }_2=1}(R_R(x)-R_{OLS}(x))=\infty$)

For fixed positive ridge regression penalty, choose a unit eigenvector $v$ of $X^{\mathsf{T}}X$ with eigenvalue $d>0$, and signal ${\beta }^0=Bv$. The directional ridge risk is $({\lambda }^2{B}^2+{\sigma }^{2}d)/(d+\lambda {)}^2$, whereas ordinary least squares has risk ${\sigma }^2/d$. Letting $|B|$ grow makes the ridge squared bias arbitrarily large.