Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 4 / c / i

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- Solution i

Solution

For convex

f

, the subgradient inequality gives

f (x) - f (x_{1}, \dots, z, \dots, x_{n}) \leq \partial_{i} f (x) (x_{i} - z) .

Taking the positive supremum over

z \in [0, 1]

and summing squares shows

V^{+} (x) \leq \sum_{i} (\partial_{i} f (x))^{2} \leq 1.

The modified logarithmic Sobolev inequality from part a with

v = 1

therefore gives

P (Z - E Z \geq t) \leq e^{- t^{2} /2} .

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