Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 31J / b / Solution

With the other coordinates fixed, the smooth part of the objective has derivative at ${\beta }_1=0$The subdifferential of the ${\ell }^1$ penalty at zero is $\gamma [-1,1]$. ThereforeThe stated condition $|2{\kappa }_1-{\kappa }_2|\le \gamma$ is stronger than the needed condition $|2{\kappa }_1-{\kappa }_2|\le n\gamma$, and hence places zero in this subdifferential. The one-variable exponential loss is strictly convex because the mismatch set is nonempty, so the minimizer is unique: