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ID: past-exam-of-the-optimisation-for-data-science-course-of-the-university-of-oxford/2023/1/c/solution

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Past exam of the optimisation for data science course of the University of Oxford / 2023 / 1 / c / Solution by

Codex  0 2026-10-25

g_{j} = G_{1/ L} (x^{j})

,

r_{j} = ∥ x^{j} - x^{*} ∥^{2}

and

δ_{j} = ϕ (x^{j}) - ϕ (x^{*})

. Here

g_{j}

is the proximal gradient mapping,

r_{j}

is a squared Euclidean distance, and

δ_{j} \geq 0

because

x^{*}

is a global minimizer. The update gives

x^{j + 1} = x^{j} - g_{j} / L

.

First use the supplied inequality (1) with

z = x^{j}

:

ϕ (x^{j + 1}) \leq ϕ (x^{j}) - \frac{1}{2 L} ∥ g_{j} ∥^{2} \leq ϕ (x^{j}) .

Thus the objective function values, and hence the errors

δ_{j}

, are nonincreasing. Next use (1) with

z = x^{*}

:

δ_{j + 1} \leq g_{j}^{T} (x^{j} - x^{*}) - \frac{1}{2 L} ∥ g_{j} ∥^{2} .

Expansion of the squared Euclidean norm gives

r_{j + 1} = r_{j} - \frac{2}{L} g_{j}^{T} (x^{j} - x^{*}) + \frac{1}{L ^{2}} ∥ g_{j} ∥^{2},

so the last bound is exactly

δ_{j + 1} \leq L (r_{j} - r_{j + 1}) /2

.

The telescoping sum over

j = 0, \dots, k - 1

therefore gives

\sum_{j = 0}^{k - 1} δ_{j + 1} \leq \frac{L}{2} (r_{0} - r_{k}) \leq \frac{L}{2} r_{0} .

Since each summand is at least

δ_{k}

, we obtain the proximal gradient objective error bound

ϕ (x^{k}) - ϕ (x^{*}) \leq \frac{L}{2 k} ∥ x^{0} - x^{*} ∥^{2}, k \geq 1.

The bound holds for the last iterate. The monotonicity of the objective function values is what converts an averaged estimate into this last-iterate estimate.

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