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For a fixed sample $z_{1:n}$, its Empirical Rademacher complexity isIts sample expected value is ${\mathcal{R}}_n(\mathcal{F})$.

Because every $f\in \mathcal{F}$ takes values in $\{0,1\}$, replacing one sample point changes $\hat{\mathcal{R}}$ by at most $1/n$. PutThe lower-tail form of the Bounded differences inequality givesoutside an event of probability at most $\delta /3$.

For the excess-loss supremum $G$ from part (b), replacing one observation changes $G$ by at most $2/n$. A second application of the same inequality givesoutside another event of probability at most $\delta /3$. On the intersection of these two events, symmetrization and the union bound yieldThis event has probability at least $1-2\delta /3\ge 1-\delta$, and $4s=2\sqrt{2\log (3/\delta )/n}$. Thus

For a zero-one-valued loss class $\mathcal{F}$, the excess misclassification risk of an empirical risk minimizer $\hat{h}$ over a population minimizer $h^{\ast }$ satisfieswith probability at least $1-\delta$. The proof combines Rademacher symmetrization with the Bounded differences inequality for the supremum of the empirical excess-loss process.

The observable version isIt follows by another application of the Bounded differences inequality to the Empirical Rademacher complexity.