LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Coordinate proximal subproblems are attained uniquely and descend

Statement

For the model and inputs of Randomized proximal coordinate descent with certified inner solves, every selected exact block subproblem has a unique minimizer. If the update is exact, the objective does not increase. If its accepted inner point has certified model gap at most $ϵ_{k}$ , then the new objective is at most the old objective plus $ϵ_{k}$ .

Facts & Assumptions

Given: A feasible iterate $x$ , a selected block $i$ , positive majorizer $M_{i}$ , and proper closed convex block penalty $g_{i}$ .

[F1]

Randomized proximal coordinate descent with certified inner solves: the block model is $q_{i} (u) = ⟨ \nabla_{i} f (x), u - x_{i} ⟩ + (M_{i} / 2) ∥ u - x_{i} ∥_{2}^{2} + g_{i} (u)$ , and an accepted inexact point satisfies $q_{i} (u) - \inf q_{i} \leq ϵ_{k}$ .

[F2]

Existence, resolvent characterization, and firm nonexpansiveness of the proximal map: for proper closed convex $g$ , positive $t$ , and any $v$ , the proximal set ${prox}_{t g} (v)$ is a singleton.

[F3]

Proximal mapping: ${prox}_{t g} (v)$ is the set of global minimizers of $g (u) + ∥ u - v ∥_{2}^{2} / (2 t)$ .

[F4]

Proper closed convex extended-real functions on $R^{n}$ : each $g_{i}$ is proper, closed and convex on its Euclidean block space.

[F5]

Coordinate-block smoothness descent model: changing one block by $h$ raises $f$ by at most $⟨ \nabla_{i} f (x), h ⟩ + (L_{i} / 2) ∥ h ∥_{2}^{2}$ .

Proof

1.1F2F3F4givenalgebra

Complete the square in $q_{i}$ : with $v_{i} = x_{i} - M_{i}^{- 1} \nabla_{i} f (x)$ , its $u$ -dependent terms are $g_{i} (u) + (M_{i} / 2) ∥ u - v_{i} ∥_{2}^{2}$ . Taking $t = 1 / M_{i} > 0$ in [F2]–[F4] proves the exact block minimizer exists and is unique.

2.1F1F5step 1.1algebra∎

For an exact update, $q_{i} (u^{*}) \leq q_{i} (x_{i}) = g_{i} (x_{i})$ . For an accepted inexact update, [F1] gives $q_{i} (u) \leq q_{i} (u^{*}) + ϵ_{k} \leq q_{i} (x_{i}) + ϵ_{k}$ . Apply [F5] with $h = u - x_{i}$ and cancel the unchanged penalties on the other blocks; the resulting objective is at most $F (x)$ in the exact case and at most $F (x) + ϵ_{k}$ in the inexact case.

Depends on

Used by

Cited to discharge well-definedness by Randomized proximal coordinate descent with certified inner solves.

Dependency tree · two levels

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Sources

Stephen J. Wright, Coordinate Descent Algorithms, §3.7 (standard reference, not scraped)
Clément Royer, Regularized, Large-Scale and Distributed Optimization, Lecture 3, §3.1 (standard reference, not scraped)