Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 210 / 3 / c

Decompose $h={\Pi }_{1}h+D^{†}Dh$ using the incidence pseudoinverse decomposition. The Cauchy-Schwarz inequality controls the constant component, and the duality of the ${\ell }_{\infty }$ and ${\ell }_1$ norms controls the remaining component:On the stipulated event, the second term contributes at most $\lambda \Vert Dh{\Vert }_1$ after multiplication by $2/n$. The triangle inequality gives $\Vert Dh{\Vert }_1\le \Vert D\hat{\theta }{\Vert }_1+\Vert D{\theta }^{\ast }{\Vert }_1$, so the negative penalty in the basic inequality for a penalized least-squares estimator cancels. We obtainon an event of probability at least $1-\delta$.