ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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A polynomial activation confines univariate outputs to bounded degree

Example

Assume Countable Choice. Let $q \in N_{0}$ and use the polynomial activation $σ (t) = t^{q}$ (with $t^{0} = 1$ ). Every finite single-hidden-layer output restricted to $[0, 1]$ is a polynomial of degree at most $q$ , and the target $f (t) = t^{q + 1}$ stays at uniform distance at least $(q + 1)! / [2^{q + 2} (q + 1)^{q + 1}]$ from every such output.

Example references

Leshno et al., proof of Theorem 1, polynomial-activation obstruction

Facts & Assumptions

Given: $q \in N_{0}$ , $σ (t) = t^{q}$ , and the target $t^{q + 1}$ on $[0, 1]$ .

[F1]

Polynomial activation functions satisfy the hypotheses of the bounded-degree obstruction with their exact polynomial representative (Polynomial activations have a bounded-degree obstruction).

[F2]

A single-hidden-layer output is a finite affine combination of activation units (Single-hidden-layer network realization).

[A1]

Countable Choice is the assumption of the cited obstruction lemma (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

1.1givenF2algebra

By [F2], each unit on a univariate input is $(w t + b)^{q}$ , a polynomial of degree at most $q$ ; finite weighted sums and an output bias preserve that degree bound.

2.1givenF1A1∎

Apply [F1] with the polynomial $p (t) = t^{q}$ to obtain the stated positive lower bound for approximation of $t^{q + 1}$ .

Depends on

Used by

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Dependency tree · two levels

12 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

Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, proof of Theorem 1, Step 1 (standard reference, not scraped)