Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 1 / d

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- Solution d

Solution

For

p \in \partial J (v)

, the Bregman divergence is

D_{J}^{p} (u, v) = J (u) - J (v) - ⟨ p, u - v ⟩ .

Put

r = A u_{α, δ} - f^{δ}

and

e = f^{δ} - f

, so

∥ e ∥_{Y} \leq δ

. Comparison with

u^{†}

gives

∥ r ∥_{Y} + α J (u_{α, δ}) \leq δ + α J (u^{†}) .

Using

p^{†} = A^{*} w^{†}

,

D_{J}^{p^{†}} (u_{α, δ}, u^{†}) = J (u_{α, δ}) - J (u^{†}) - ⟨ w^{†}, r + e ⟩ \leq (\frac{1}{α} + ∥ w^{†} ∥_{Y^{*}}) δ + (∥ w^{†} ∥_{Y^{*}} - \frac{1}{α}) ∥ r ∥_{Y} .

For

0 < α < α_{0}

the last coefficient is negative, so

D_{J}^{p^{†}} (u_{α, δ}, u^{†}) \leq C_{α} δ, C_{α} = \frac{1}{α} + ∥ w^{†} ∥_{Y^{*}} .

The estimate holds for any fixed admissible

α

; it does not require

α \to 0

with the noise level.

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