LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A singular AdaGrad metric still has a projection minimizer

Statement

Let $d \geq 1$ , let $K \subseteq R^{d}$ be nonempty, bounded, closed, and convex, let $H$ be a fixed real positive-semidefinite matrix, and let $y \in R^{d}$ . The function $x \mapsto (x - y)^{⊤} H (x - y)$ attains a minimum on $K$ . This conclusion allows $H$ to be singular and makes no uniqueness claim.

Statement references

Gram-matrix AdaGrad with full or singular diagonal metrics: the AdaGrad update uses this quadratic objective with $H = H_{t}$ and $y = y_{t + 1}$ .

Facts & Assumptions

Given: A nonempty bounded closed convex $K \subseteq R^{d}$ , a fixed positive-semidefinite matrix $H$ , and $y \in R^{d}$ .

[F1]

Gram-matrix AdaGrad with full or singular diagonal metrics: the metric projection objective is $Q (x) = (x - y)^{⊤} H (x - y)$ , with $H$ positive semidefinite; its minimizer may be nonunique.

[F2]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $∥ v ∥_{2}^{2} = \sum_{i = 1}^{d} v_{i}^{2}$ , so each coordinate satisfies $∣ v_{i} ∣ \leq ∥ v ∥_{2}$ .

[F3]

Upper and lower semicontinuity on subsets of $R^{n}$ : every continuous real-valued function on a Euclidean subset is lower semicontinuous there.

[F4]

Semicontinuous extreme value theorem on compact Euclidean sets: a real-valued lower semicontinuous function on a nonempty compact Euclidean set attains its minimum.

[F5]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: in finite-dimensional Euclidean space, a set is compact if it is closed and bounded.

Proof

1.1givenF5

By [F5], the nonempty closed bounded set $K$ is compact.

1.2givenF1F2algebrachoose

Since $K$ is bounded, choose $B < \infty$ with $∥ x ∥_{2} \leq B$ for every $x \in K$ , and put $M = B + ∥ y ∥_{2}$ . For $x, z \in K$ , expanding the finite matrix products and factoring each difference gives $Q (x) - Q (z) = \sum_{i, j = 1}^{d} H_{i j} ((x_{i} - z_{i}) (x_{j} - y_{j}) + (z_{i} - y_{i}) (x_{j} - z_{j}))$ . Thus $∣ Q (x) - Q (z) ∣ \leq 2 M (\sum_{i, j} ∣ H_{i j} ∣) ∥ x - z ∥_{2}$ , by [F2]; hence $Q$ is continuous on $K$ .

2.1step 1.1step 1.2F3F4∎

By [F3], $Q$ is lower semicontinuous on $K$ . Apply [F4] to the nonempty compact set in step 1.1 to obtain a minimizer. No step requires $H$ to be invertible, and the argument does not distinguish or select a unique minimizer.

Depends on

Used by

AdaGrad with a singular diagonal metric Example

Cited to discharge well-definedness by Gram-matrix AdaGrad with full or singular diagonal metrics.

Dependency tree · two levels

56 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Elad Hazan, Introduction to Online Convex Optimization, second edition (standard reference, not scraped)
John Duchi, Elad Hazan, and Yoram Singer, Adaptive Subgradient Methods for Online Learning and Stochastic Optimization (standard reference, not scraped)