ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

A nonattained risk infimum

Example

Let $X = {x_{0}}$ and $Y = {0}$ with discrete sigma-algebras, let $A = R$ with its Borel sigma-algebra, and let $P$ put mass one on $(x_{0}, 0)$ . For each positive integer $n$ , let $f_{n} (x_{0}) = 1 / n$ and set $F = {f_{n} : n \geq 1}$ . Use squared loss $ℓ (a, y) = (a - y)^{2}$ . For every sample size $m \geq 1$ , the only sample has ${\hat{R}}_{S} (f_{n}) = R_{P} (f_{n}) = 1 / n^{2}$ . The empirical and population infima over $F$ are both $0$ and are not attained. For tolerance $η = 1 / 4$ , the constant selector $A_{1 / 4} (S, x) = 1 / 2$ supplies the approximate-ERM procedure with output $f_{2}$ ; for tolerance $0$ , no exact ERM exists.

Facts & Assumptions

Given: The singleton instance and label spaces, point-mass law, predictor class and squared loss stated in the example.

[F1]

Population and empirical risks are the joint-law integral and finite sample average of the loss. Population and empirical risk

[F2]

An exact ERM attains the empirical infimum; if a finite infimum is not attained, no exact minimizer exists. Empirical risk minimizers

[F3]

The approximate-ERM procedure takes a supplied jointly measurable selector satisfying the samplewise risk condition. Approximate empirical risk minimization with a supplied selector

[F4]

The real line is a complete ordered field, with the usual embedded positive integers. Dedekind-complete ordered real field

[F5]

For every $ε > 0$ in a complete ordered field, some positive integer $n$ satisfies $1 / n < ε$ . For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$

[F6]

The inverse of a positive field element is positive. Inverses of positives are positive, and reciprocation reverses order

[F7]

Every canonical positive integer is positive in the ordered real field. Canonical naturals are positive and strictly increasing

Verification

1.1givenF1F4F6F7algebra

For every sample $S$ and every $n \geq 1$ , the only observed pair is $(x_{0}, 0)$ , so [F1] gives ${\hat{R}}_{S} (f_{n}) = R_{P} (f_{n}) = (1 / n)^{2} = 1 / n^{2}$ . By [F4] and [F7], $n > 0$ , so [F6] gives $1 / n > 0$ and each listed risk is positive.

2.1givenF4F5F6F7step 1.1algebra

Let $ε > 0$ and set $δ = \min {ε, 1 / 2} > 0$ . By [F5] choose $n \geq 1$ with $0 < 1 / n < δ \leq 1 / 2$ . Then $1 / n^{2} < 1 / n < ε$ . Thus $0$ is a lower bound and no positive number is a lower bound; the infimum is $0$ . Step 1.1 shows it is never attained, so no exact ERM exists.

3.1F1F2F3givenstep 1.1step 2.1algebra∎

For each fixed $m$ , the sample space contains only the all- $(x_{0}, 0)$ sample. The constant evaluation map $A_{1 / 4} (S, x) = 1 / 2$ is jointly measurable and its slice is $f_{2}$ , whose risk is $1 / 4 = 0 + 1 / 4$ by steps 1.1 and 2.1. It is therefore an admissible supplied selector for [F3], which returns $f_{2}$ ; tolerance $0$ has no output satisfying exact minimization by step 2.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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

Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms, Chapter 3 §3.2.2 (standard reference, not scraped)