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

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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.

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The diagonal AdaGrad scale is the coordinatewise square root

Statement

For the diagonal AdaGrad choice $H_{t} = diag (\sqrt{(G_{t})_{11}}, \dots, \sqrt{(G_{t})_{d d}})$ with $G_{t} = \sum_{s = 1}^{t} g_{s} g_{s}^{⊤}$ , the $i$ th diagonal entry is

$(H_{t})_{i i} = \sqrt{\sum_{s = 1}^{t} g_{s, i}^{2}}, i = 1, \dots, d .$

Statement references

Gram-matrix AdaGrad with full or singular diagonal metrics: defines the accumulated Gram matrix and the diagonal AdaGrad scale.

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes: defines the coordinate formula for matrix products.

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: identifies the transpose entries used in the outer product.

Facts & Assumptions

Given: The vectors $g_{1}, \dots, g_{t} \in R^{d}$ and the diagonal-scale definition in the statement.

[F1]

Gram-matrix AdaGrad with full or singular diagonal metrics: $G_{t} = \sum_{s = 1}^{t} g_{s} g_{s}^{⊤}$ and $H_{t} = diag (\sqrt{(G_{t})_{11}}, \dots, \sqrt{(G_{t})_{d d}})$ for the diagonal choice.

[F2]

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: $(g_{s} g_{s}^{⊤})_{i i} = g_{s, i} g_{s, i}$ by the matrix-product entry formula and transpose convention.

Proof

1.1givenF1F2algebra

For each $i$ , the definition of matrix multiplication gives $(G_{t})_{i i} = \sum_{s = 1}^{t} (g_{s} g_{s}^{⊤})_{i i} = \sum_{s = 1}^{t} g_{s, i} g_{s, i} = \sum_{s = 1}^{t} g_{s, i}^{2}$ .

2.1step 1.1F1algebra∎

Taking the nonnegative scalar square root of the identity in step 1.1 and using the diagonal-scale definition in [F1] gives $(H_{t})_{i i} = \sqrt{\sum_{s = 1}^{t} g_{s, i}^{2}}$ for every $i$ .

Depends on

Used by

AdaGrad with a singular diagonal metric Example

Dependency tree · two levels

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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)