ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

AMSGrad retains an earlier second-moment maximum

Example

On the one-dimensional domain $F = [- 1, 1]$ , run uncorrected AMSGrad from $x_{1} = 0$ with positive step sizes $α_{1}, α_{2}$ , any first-moment weights in $[0, 1)$ , $β_{2} = 1 / 2$ , and losses $ℓ_{1} (x) = 2 x$ , $ℓ_{2} (x) = 0$ , using their gradients $g_{1} = 2$ , $g_{2} = 0$ . The raw second moment decreases from $v_{1} = 2$ to $v_{2} = 1$ , while the historical maximum remains ${\hat{v}}_{1} = {\hat{v}}_{2} = 2$ .

Example references

Uncorrected AMSGrad with a coordinatewise historical maximum: defines the raw second-moment recurrence and its coordinatewise historical maximum.

Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length: $[- 1, 1]$ is the closed interval used as the domain.

RMSProp, Adam, and AMSGrad moment updates are well-defined: guarantees finite nonnegative moments and well-defined updates on the stated compact domain.

Reddi, Kale, and Kumar, Algorithm 2

Facts & Assumptions

Given: $F = [- 1, 1]$ , $x_{1} = 0$ , $β_{2} = 1 / 2$ , and gradients $g_{1} = 2$ , $g_{2} = 0$ .

[F1]

Uncorrected AMSGrad with a coordinatewise historical maximum: $v_{0} = {\hat{v}}_{0} = 0$ , $v_{t} = β_{2} v_{t - 1} + (1 - β_{2}) g_{t}^{2}$ , and ${\hat{v}}_{t} = \max {{\hat{v}}_{t - 1}, v_{t}}$ .

[F2]

RMSProp, Adam, and AMSGrad moment updates are well-defined: the recurrence and weighted minimizations are well-defined for finite gradients on the nonempty compact convex domain.

[F3]

Convex and strictly convex functions on Euclidean convex sets and A convex subset of $R^{m}$ contains every line segment between two of its points: the interval is convex and the linear losses are convex.

[F4]

Subgradients and the subdifferential of a convex function: $g \in \partial ℓ (x)$ when $ℓ (y) \geq ℓ (x) + g (y - x)$ for all $y$ in the domain.

[F5]

Ordered field and Sign rules for products and monotonicity of multiplication: nonnegative coefficients preserve the interval bounds under weighted sums.

[F6]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: a nonempty closed bounded subset of $R$ is compact.

Verification

Given: The initialization and two gradients in the Example.

1.1F1F2F3F4F5F6givenalgebra

The interval $[- 1, 1]$ is nonempty, closed, and bounded, so [F6] makes it compact. For $x, y \in [- 1, 1]$ and $θ \in [0, 1]$ , [F5] gives $- 1 \leq (1 - θ) x + θ y \leq 1$ , so the interval is convex. For either displayed linear loss, $ℓ ((1 - θ) x + θ y) = (1 - θ) ℓ (x) + θ ℓ (y)$ ; moreover, if $ℓ (x) = a x$ , its gradient $g = a$ satisfies $ℓ (y) = ℓ (x) + g (y - x)$ , so it is a subgradient by [F4]. The first update gives $v_{1} = (1 / 2) 0 + (1 / 2) 2^{2} = 2$ and ${\hat{v}}_{1} = \max {0, 2} = 2$ .

2.1F1F3F4step 1.1givenalgebra∎

The second update gives $v_{2} = (1 / 2) 2 + (1 / 2) 0^{2} = 1$ , so $v_{2} < v_{1}$ ; nevertheless ${\hat{v}}_{2} = \max {2, 1} = 2 = {\hat{v}}_{1}$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar, On the Convergence of Adam and Beyond (standard reference, not scraped)