= Rademacher symmetrization inequality
{c}
For an <independent and identically distributed random variables>[i.i.d. sample] and an integrable <function class> $\mathcal F$,
$$
\mathbb E\sup_{f\in\mathcal F}
\frac1n\sum_{i=1}^n\bigl(f(Z_i)-\mathbb Ef(Z_i)\bigr)
\leq2\mathcal R_n(\mathcal F).
$$
Introduce an independent ghost sample, replace each expectation by its ghost-sample average using <Jensen inequality>, and then multiply each paired difference by an independent <Rademacher random variable>[Rademacher sign]. Splitting the resulting supremum into its two sample contributions gives the factor two.
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