ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

One two-call SPSA estimate for an expensive biological simulator

Example

Suppose a costly black-box biological simulator reports a scalar mismatch for three positive kinetic rates. At $θ = (0.20, 0.20, 0.20)$ , take $c = 0.02$ and Rademacher perturbation $Δ = (1, - 1, 1)$ . If the two simulator calls return $13.1$ and $12.7$ , the SPSA estimate is $(10, - 10, 10)$ and uses two calls, compared with six coordinatewise central-difference calls.

Facts & Assumptions

Given: The parameter, step, perturbation, two observed simulator outputs, and the SPSA procedure in the Statement.

[F1]

Two-evaluation simultaneous-perturbation gradient estimator: with two values $Y^{\pm}$ , the coordinate output is $(Y^{+} - Y^{-}) / (2 c Δ_{i})$ .

Verification

1.1givenalgebra

The perturbed parameters are $θ + c Δ = (0.22, 0.18, 0.22)$ and $θ - c Δ = (0.18, 0.22, 0.18)$ , so both remain in the positive-rate domain.

1.2givenF1algebra

The common numerator is $13.1 - 12.7 = 0.4$ and $2 c = 0.04$ . By [F1], the three outputs are $0.4 / 0.04 = 10$ , $0.4 / (- 0.04) = - 10$ , and $0.4 / 0.04 = 10$ .

2.1step 1.2givenalgebra∎

The estimate uses the two simulator values for all coordinates, whereas one central-difference pair per coordinate uses $2 d = 6$ calls at $d = 3$ . The finite calculation verifies the estimator output only; it does not certify its bias, variance, or convergence for the biological model.

Depends on

Two-evaluation simultaneous-perturbation gradient estimator

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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