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Minimum-norm interpolation and double descent — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

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.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Bias, variance, and signal-to-noise regimes of the isotropic risk curve

Example

For fixed signal norm $r \geq 0$ , noise standard deviation $σ \geq 0$ , and aspect ratio $γ > 0$ , $γ \neq 1$ , the limiting risk has bias and variance components $\begin{array}{ccc} bias & variance \\ 0 < γ < 1 & 0 & \frac{σ^{2} γ}{1 - γ} \\ γ > 1 & r^{2} (1 - \frac{1}{γ}) & \frac{σ^{2}}{γ - 1} . \end{array}$ When $σ > 0$ , the underparameterized branch increases from limit $0$ as $γ ↓ 0$ to $+ \infty$ as $γ ↑ 1$ . On the overparameterized side its variance decreases to $0$ , and its bias increases to $r^{2}$ . When $σ > 0$ , the total overparameterized risk is strictly decreasing if $r \leq σ$ , and if $r > σ$ it has a unique minimum at $γ_{*} = r / (r - σ)$ . If $σ = 0$ , the underparameterized risk is $0$ ; the overparameterized risk is $r^{2} (1 - 1 / γ)$ and is strictly increasing when $r > 0$ . The one-sided risk curves diverge at $γ = 1$ only when $σ > 0$ ; no risk limit at $γ = 1$ is asserted. For $σ > 0$ the global infimum over $γ \neq 1$ is $0$ , approached as $γ ↓ 0$ and not attained; if $r = 0$ , it is also approached as $γ \to \infty$ . For $σ = 0$ , the minimum $0$ is attained throughout $0 < γ < 1$ .

Example references

Separate isotropic ridgeless risk into limiting bias and variance

Isotropic ridgeless least-squares risk on both sides of interpolation

Hastie, Montanari, Rosset, and Tibshirani, §3.2

Facts & Assumptions

Given: $r \geq 0$ , $σ \geq 0$ , and $γ > 0$ with $γ \neq 1$ .

[F1]

The limiting bias and variance components are the two displayed branches (Separate isotropic ridgeless risk into limiting bias and variance).

Verification

1.1givenF1algebra

Summing the two components in [F1] gives the total risk $R (γ) = σ^{2} γ / (1 - γ)$ for $0 < γ < 1$ and $R (γ) = r^{2} (1 - 1 / γ) + σ^{2} / (γ - 1)$ for $γ > 1$ . Differentiating the first expression gives $R^{'} (γ) = σ^{2} / (1 - γ)^{2}$ , so it is strictly increasing for $σ > 0$ and is identically zero for $σ = 0$ .

1.2givenF1algebra

For $γ > 1$ , $R^{'} (γ) = \frac{r^{2}}{γ^{2}} - \frac{σ^{2}}{(γ - 1)^{2}} .$ If $σ > 0$ and $r \leq σ$ , then $r / γ \leq σ / γ < σ / (γ - 1)$ , so $R^{'} < 0$ . If $r > σ > 0$ , the derivative vanishes exactly when $r (γ - 1) = σ γ$ , namely at $γ_{*} = r / (r - σ) > 1$ ; it is negative before this point and positive after it, giving the unique minimum. If $σ = 0$ , the derivative is $r^{2} / γ^{2}$ , which is positive for $r > 0$ and zero for $r = 0$ .

2.1givenF1step 1.1algebra∎

The limits at $γ ↓ 0$ , $γ ↑ 1$ , and $γ \to \infty$ follow directly from the formulas in [F1]. In particular, the singular terms at $γ = 1$ vanish when $σ = 0$ and diverge when $σ > 0$ . For $σ > 0$ every finite-domain risk value is positive while the underparameterized branch tends to zero as $γ ↓ 0$ , proving that its global infimum is $0$ and is not attained. When $σ = 0$ , the underparameterized branch equals $0$ identically.

Minimum-norm interpolation and double descent — Examples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Reported double-descent experiments for neural networks and tree ensembles

Example

Example references

Facts & Assumptions

Verification

Bias, variance, and signal-to-noise regimes of the isotropic risk curve

Example

Example references

Facts & Assumptions

Verification

Sources