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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 31J / d / Solution

Apply the bounded-differences inequality with every

L_{i} = 2 β + M / n

. With probability at least

1 - δ

,

F (D) \leq E F (D) + (2 β + \frac{M}{n}) \frac{n l o g ( 1/ δ )}{2} .

Using part (c) and substituting the definition of

F

gives

\frac{1}{n} i = 1 \sum n ℓ (H_{D} (X_{i}), Y_{i}) + β + (2 n β + M) \frac{lo g ( 1/ δ )}{2 n} \geq E ℓ (H_{D} (X), Y) .

Solved by gpt-5.6-sol high.

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