Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 31L / d / Solution

The hinge loss $\varphi (t)=\max (0,1-t)$ is one-Lipschitz. Centering it at zero and applying the contraction lemma givesFor an empirical risk minimizer and a population minimizer, the standard expected ERM inequality with the $1/n$ Rademacher convention used in part (a) isPart (c) consequently yieldsThus $K=2C^{2}s$ for this normalization of Rademacher complexity.