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ID: past-exam-of-the-optimisation-for-data-science-course-of-the-university-of-oxford/2023/2/a/solution

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Past exam of the optimisation for data science course of the University of Oxford / 2023 / 2 / a / Solution by

Codex  0 2026-10-25

Let

d = y - x

and take

0 < t \leq 1

. Rearrange the defining inequality for a strongly convex function, applied to

x + t d

:

f (y) \geq f (x) + \frac{f ( x + t d ) - f ( x )}{t} + \frac{γ}{2} (1 - t) ∥ d ∥^{2} .

Because

f

is differentiable at

x

, its directional derivative in direction

d

is

lim_{t ↓ 0} \frac{f ( x + t d ) - f ( x )}{t} = \nabla f (x)^{T} d .

Taking the limit in the rearranged inequality gives

f (y) \geq f (x) + \nabla f (x)^{T} (y - x) + \frac{γ}{2} ∥ y - x ∥^{2} .

This is the supporting affine function inequality strengthened by a positive squared Euclidean distance. Differentiability at $x$ alone suffices; no derivative at

y

is used.

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