Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 31J (Mathematics of Machine Learning)

For a convex function $f$, its subdifferential isFor $f(x)=\gamma |x|$,The defining inequality shows directly that $0\in \partial f(x)$ exactly when $x$ is a global minimizer.

Strict convexity meansfor distinct $x,y$ and $0<\lambda <1$. Two distinct minimizers would make the midpoint have a strictly smaller value, so a minimizer is unique.

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: