ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

Training risk and independent test risk are different quantities

Example

Use the degree-one clamped B-spline basis $p (x) = (1 - x, x)$ on $[0, 1]$ , with a single fixed training input $x_{1} = 1 / 2$ and response $y_{1} = 1$ . The coefficient vector $\hat{β} = (0, 2)^{T}$ is a least-squares minimizer and gives the fitted function $\hat{g} (x) = 2 x$ . Let the target be $g_{0} (x) = 1$ and the independent test input be uniform on $[0, 1]$ .

Facts & Assumptions

Given: The one-row design matrix $P = (1 / 2, 1 / 2)$ , response $Y = (1)$ , and test law $X^{'} \sim Unif [0, 1]$ .

[F1]

The fitted training vector is the orthogonal projection onto the design column space; any least-squares coefficient representation has that same training fit (Series least squares with a spline basis).

[F2]

The risk bound in this page concerns expected empirical training-design loss, not independent test loss (Expected empirical training risk for spline series least squares).

[F3]

The clamped degree-one B-splines form a basis of the degree-one spline space, and the recurrence defines their values (B-spline spaces and knot conventions, B-spline basis properties and polynomial reproduction).

[A1]

The training input and response are $1 / 2$ and $1$ , the chosen least-squares coefficient is $(0, 2)^{T}$ , and the independent test input is uniform on $[0, 1]$ .

Verification

Given: $x_{1} = 1 / 2$ , $y_{1} = 1$ , $\hat{β} = (0, 2)^{T}$ , $g_{0} \equiv 1$ , and independent $X^{'} \sim Unif [0, 1]$ .

1.1A1F1F3givenalgebra

On the clamped knot vector $(0, 0, 1, 1)$ , the only nonzero degree-zero spline on $(0, 1)$ is $B_{2, 0} = 1$ ; the recurrence in [F3] gives $B_{1, 1} = 1 - x$ and $B_{2, 1} = x$ , so $p (1 / 2) = (1 / 2, 1 / 2)$ . Therefore $p (1 / 2)^{T} \hat{β} = (1 / 2, 1 / 2) (0, 2)^{T} = 1$ , the training residual and empirical training loss are zero, and the fitted vector equals the projection in [F1]. The chosen coefficient vector is one admissible representation of it.

2.1A1F2step 1.1algebra∎

The independent test squared error has expectation $E {\hat{g} (X^{'}) - g_{0} (X^{'})}^{2} = \int_{0}^{1} (2 x - 1)^{2} d x = \frac{4}{3} - 2 + 1 = \frac{1}{3} > 0.$ Thus zero training loss does not imply zero independent test loss, consistent with the scope of [F2].

Depends on

Used by

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Dependency tree · two levels

19 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

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OCW 14.385, Fall 2007, Proposition 3, PDF pp. 10–11 (standard reference, not scraped)