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 Incoming links: Massart finite-class lemma

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 31K / b / ii / Solution Created 2026-09-24 Updated 2026-09-25

Condition on the sample. The set of vectors

(D_{i} h (X_{i}))_{i = 1}^{n}, qq u a d h \in H,

has at most

s (H, n)

distinct members, each with Euclidean norm at most

M n

. The Massart finite-class lemma, the Sauer-Shelah growth bound, and

V C (H) \leq p

give

R (F) \leq M \frac{2 l o g s ( H , n )}{n} \leq M \frac{2 p l o g ( n + 1 )}{n} .

Taking expectations and substituting this into part (i) yields

E Q (h) \leq Q (h^{*}) + 2 M \frac{2 p lo g ( n + 1 )}{n} .

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