ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Reported double-descent experiments for neural networks and tree ensembles

Example

Belkin, Hsu, Ma, and Mandal report empirical double-descent curves for random Fourier features on a $10^{4}$ -example, ten-class MNIST subset, with interpolation at $10^{4}$ features; for a fully connected two-layer neural network on $4 \cdot 10^{3}$ examples of dimension $784$ and ten classes, with the observed interpolation threshold near $n K$ parameters; and for random forests on a $10^{4}$ -example MNIST subset, with model complexity indexed jointly by maximum tree leaves and number of trees. These are finite experimental observations; they do not assert a universal risk curve or a theorem about arbitrary neural networks, trees, or ensembles.

Example references

Belkin, Hsu, Ma, and Mandal, Figures 2, 4, and 5

Facts & Assumptions

Given: The reported empirical plots and experiment descriptions in the source cited in Example references.

[F1]

Figure 2 describes the random-Fourier-feature MNIST experiment and its interpolation threshold; Figure 4 describes the two-layer neural-network experiment; Figure 5 describes the random-forest experiment. The paper labels these as empirical evidence, not a universal theorem.

Verification

1.1givenF1

The Figure 2 description gives $n = 10^{4}$ , ten classes, and the threshold $N = n = 10^{4}$ ; these are the random-feature quantities stated in the Example.

1.2givenF1

The Figure 4 description gives $n = 4 \cdot 10^{3}$ , input dimension $784$ , ten outputs, and threshold observed at $n K$ parameters. The Figure 5 description gives $n = 10^{4}$ and indexes forest complexity by maximum leaves and tree count.

2.1givenF1∎

The source reports experiments for those specified datasets and model families, so the Example is an empirical report with those qualifications; it does not claim a theorem for models beyond the experiments.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Belkin, Hsu, Ma, and Mandal, Reconciling modern machine learning practice and the bias-variance trade-off (standard reference, not scraped)