 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/31l/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a fixed sample

x_{1 : n}

and independent Rademacher signs

σ_{i}

R (H (x_{1 : n})) = E_{σ} sup_{h \in H} \frac{1}{n} \sum_{i = 1}^{n} σ_{i} h (x_{i}) .

R_{n} (H) = E_{X_{1 : n}} R (H (X_{1 : n})) .

The contraction lemma says that if each

ψ_{i}

L

-Lipschitz and

ψ_{i} (0) = 0

, then

E_{σ} sup_{h \in H} \frac{1}{n} \sum_{i} σ_{i} ψ_{i} (h (x_{i})) \leq L R (H (x_{1 : n})) .

Subtracting constants handles maps not vanishing at zero.

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