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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 30J / c / Solution

The exponential loss is

ϕ (z) = e^{- z}

. For a score

f : X \to R

, its exponential classification risk and empirical risk are

R_{ϕ} (f) = E [e^{- Y f (X)}], R_{ϕ} (f) = \frac{1}{n} \sum_{i = 1}^{n} e^{- Y_{i} f (X_{i})} .

For the set of evaluation vectors

B (x_{1 : n}) = {(h (x_{1}), \dots, h (x_{n})) : h \in B},

its empirical Rademacher complexity is

R (B (x_{1 : n})) = \frac{1}{n} E_{σ} h \in B sup i = 1 \sum n σ_{i} h (x_{i}),

where the

σ_{i}

are independent uniform random signs.

Solved by gpt-5.6-sol high.

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