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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 31K / a / i / Solution

Let

ε_{1}, \dots, ε_{n}

be independent Rademacher signs. For the set of evaluation vectors

F (z_{1 : n}) = {(f (z_{1}), \dots, f (z_{n})) : f \in F},

the empirical Rademacher complexity is

R (F (z_{1 : n})) = E_{ε} [sup_{f \in F} \frac{1}{n} \sum_{i = 1}^{n} ε_{i} f (z_{i})] .

For an independent sample

Z_{1}, \dots, Z_{n}

,

R_{n} (F) = E_{Z_{1 : n}} R (F (Z_{1 : n})) .

Solved by gpt-5.6-sol high.

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