Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 31J / a / Solution

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent Rademacher signs, also independent of the i.i.d. sample $Z_{1:n}$. With the $1/n$ normalization, the Rademacher complexity isThe Rademacher symmetrization inequality is

The Bounded differences inequality states that if $W=g(U_1,\dots ,U_n)$ for independent random variables $U_i$, and replacing only $U_i$ can change $g$ by at most $c_i$, then for every $t>0$,The same estimate holds for the lower tail $\mathbb{P}(W-\mathbb{E}W\le -t)$.