ExampleConstruction: AI-generatedVerification: 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.

Constant and inverse-time schedules give different stability bounds

Example

For each $r \in {1, 2}$ , let $(X_{r}, Σ_{X_{r}})$ , $(Y_{r}, Σ_{Y_{r}})$ , and $(A_{r}, Σ_{A_{r}})$ be measurable spaces, let $P_{r}$ be a probability measure on $X_{r} \times Y_{r}$ , let $ℓ_{r} : A_{r} \times Y_{r} \to [0, + \infty)$ be measurable, and let $f_{w}^{(r)} : X_{r} \to A_{r}$ be measurable for $w \in R^{d}$ , where $d \geq 1$ . Put $g_{r} (w, (x, y)) = ℓ_{r} (f_{w}^{(r)} (x), y)$ and assume $g_{r}$ and $\nabla_{w} g_{r}$ are jointly measurable. Let $n = 1000$ , $T = 100$ , let $S_{r} = (Z_{1}^{(r)}, \dots, Z_{n}^{(r)})$ be IID with law $P_{r}$ , and define $R_{P_{r}} (f_{w}^{(r)}) = \int g_{r} (w, z) P_{r} (d z)$ and ${\hat{R}}_{S_{r}} (f_{w}^{(r)}) = n^{- 1} \sum_{i = 1}^{n} g_{r} (w, Z_{i}^{(r)})$ . For each regime, let $I_{1}^{(r)}, \dots, I_{T}^{(r)}$ be independent of $S_{r}$ with $P (I_{t}^{(r)} = i) = 1 / n$ for every $t, i$ ; dependence between times is allowed. Start from fixed $w_{0}^{(r)} \in R^{d}$ and use $w_{t}^{(r)} = w_{t - 1}^{(r)} - α_{t}^{(r)} \nabla_{w} g_{r} (w_{t - 1}^{(r)}, Z_{I_{t}^{(r)}}^{(r)})$ . In regime $r = 1$ , assume each $g_{1} (\cdot, z)$ is totally differentiable and convex, has $1$ -Lipschitz gradient, and is $1$ -Lipschitz in values; use $α_{t}^{(1)} = 1$ . In regime $r = 2$ , assume each $g_{2} (\cdot, z)$ is totally differentiable, takes values in $[0, 1]$ , has $1$ -Lipschitz gradient and is $1$ -Lipschitz in values, with no convexity assumption; use $α_{t}^{(2)} = 1 / t$ . Assume the displayed output risk variables are measurable and integrable. The convex theorem's stability and absolute expected-generalization upper bound is $1 / 5$ . The nonconvex theorem's upper bound is $20 \sqrt{2} / 999$ . This comparison concerns only these theorem certificates under their distinct loss hypotheses, not the actual stability or generalization gaps of arbitrary training problems.

Example references

Expected generalization of convex stochastic gradient training supplies the convex bound.

Expected generalization of nonconvex stochastic gradient training supplies the nonconvex bound.

Hardt, Recht, and Singer, arXiv-version Theorems 3.8 and 3.12 (conference Theorems 3.7 and 3.8): full arXiv paper.

Facts & Assumptions

Given: $n = 1000$ , $T = 100$ , $L = β = 1$ , constant convex-regime steps $α_{t} = 1$ , and nonconvex-regime steps $α_{t} = 1 / t$ with $c = 1$ .

[F1]

Expected generalization of convex stochastic gradient training: for admissible convex-regime steps, the bound is $2 L^{2} n^{- 1} \sum_{t} α_{t}$ .

[F2]

Expected generalization of nonconvex stochastic gradient training: for $n \geq 2$ and nonincreasing $α_{t} \leq c / t$ , the bound is $(1 + 1 / (β c)) (2 c L^{2})^{1 / (β c + 1)} T^{β c / (β c + 1)} / (n - 1)$ .

Verification

1.1givenF1algebra

For the convex schedule, $1 \leq 2 / β = 2$ and $\sum_{t = 1}^{100} α_{t} = 100$ . Substitution in [F1] gives $2 \cdot 100 / 1000 = 1 / 5$ .

1.2givenF2algebra

The nonconvex schedule is nonincreasing, satisfies $α_{t} = c / t$ , and $q = β c = 1$ . Substitution in [F2] gives $2 \sqrt{2} \sqrt{100} / 999 = 20 \sqrt{2} / 999$ .

2.1step 1.1step 1.2algebra∎

Both denominators are positive. The comparison $20 \sqrt{2} / 999 < 1 / 5$ is equivalent to $100 \sqrt{2} < 999$ ; squaring these positive quantities reduces it to $20000 < 998001$ . This compares only the stated upper bounds.

Depends on

Used by

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Dependency tree · two levels

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Sources

Moritz Hardt, Benjamin Recht, and Yoram Singer, Train faster, generalize better: Stability of stochastic gradient descent, arXiv-version Theorems 3.8 and 3.12 (conference Theorems 3.7 and 3.8) (standard reference, not scraped)
Nikolai Matni, ESE680 Lecture 16: Algorithmic Stability and Stochastic Gradient Descent, Theorem 3 (standard reference, not scraped)