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Norm and margin generalization for neural networks — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

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

A two-class one-layer spectral-margin calculation

Example

Assume the axiom of choice. Take one layer with input width $1$ , output width $2$ , identity activation with Lipschitz constant $ρ_{1} = 1$ , reference matrix $M_{1} = 0$ , and layer matrix $A_{1} = (1, - 1)^{⊤}$ . Let the input distribution be the point mass at $(x, y) = (1, 1)$ , so a sample of size $n = 1$ always has $X = (1)$ and label $1$ . At margin level $γ = 1$ and confidence $δ = 1 / 10$ , the score vector is $(1, - 1)$ , its margin is $2$ , its empirical margin loss and population error are both zero, and the spectral network complexity is $R_{A} = 2$ . The simultaneous theorem's right-hand side is $8 + 3 \sqrt{Q / 2}$ , where $Q = \log 60 + 2 \log 3 + 2 \log 3 + 2 \log 4 + 2 \log (2 + \sqrt{2}) \approx 13.72,$ so the bound evaluates to approximately $15.86$ . The example records the exact margin and complexity and shows the small-sample bound is valid but loose.

Example references

Spectral network complexity with reference matrices: defines the spectral complexity formula and induced operator norm.

Multiclass margins, ramp loss, and empirical margin loss: defines multiclass margin, empirical margin loss, and population error.

A simultaneous spectral margin bound for multiclass networks: supplies the simultaneous probability bound and its confidence expression.

Facts & Assumptions

Given: The one-layer network, point-mass distribution, sample size, margin level, and confidence parameter in the Example.

[F1]

The spectral norm is the induced Euclidean operator norm, the transpose-difference mixed norm sums its column Euclidean norms, and $R_{A}$ is their prescribed spectral-complexity expression. Spectral network complexity with reference matrices

[F2]

The margin is true-label score minus the largest competing score; empirical margin loss counts examples with margin at most the threshold, and population error is zero-one classification error. Multiclass margins, ramp loss, and empirical margin loss

[F3]

Under the stated iid and choice assumptions, the simultaneous theorem has an additive $8 / n$ term, a leading term proportional to $\log n$ , and confidence term $3 \sqrt{Q / (2 n)}$ . A simultaneous spectral margin bound for multiclass networks

Verification

technique · direct

1.1F1givenalgebra

The only column of $A_{1}$ has Euclidean norm $\sqrt{2}$ , so its induced operator norm is $\sqrt{2}$ . The transpose $A_{1}^{⊤} = (1, - 1)$ has two scalar columns of absolute values $1$ and $1$ , so $∥ A_{1}^{⊤} - M_{1}^{⊤} ∥_{2, 1} = 2$ . Also $∥ X ∥_{F} = 1$ .

2.1F2givenstep 1.1algebra

Applying the network map to $x = 1$ gives scores $(1, - 1)$ ; [F2] therefore gives margin $1 - (- 1) = 2$ , empirical margin loss $1 {2 \leq 1} = 0$ , and zero population error under the point-mass law.

3.1F1F3step 1.1step 2.1algebra∎

Substitution into the spectral-complexity definition yields $R_{A} = \sqrt{2} {((2 / \sqrt{2})^{2 / 3})}^{3 / 2} = 2.$ For $n = 1$ the leading term in [F3] vanishes because $\log 1 = 0$ , and $8 / n = 8$ . Substituting $δ = 1 / 10$ , $γ = 1$ , $∥ X ∥_{F} = 1$ , $L = 1$ , $∥ A_{1}^{⊤} - M_{1}^{⊤} ∥_{2, 1} = 2$ , and $∥ A_{1} ∥_{σ} = \sqrt{2}$ into the theorem's $Q$ gives the displayed expression. The inequality is $0 \leq 8 + 3 \sqrt{Q / 2} \approx 15.86$ .

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

Changing the reference matrix changes complexity, not the network

Example

Use the one-layer network with identity activation, $A_{1} = (1, - 1)^{⊤}$ , and input $x = 1$ . Its scores are $(1, - 1)$ for either choice of reference matrix. With reference $M_{1} = A_{1}$ , its spectral network complexity is $R_{A} = 0$ ; with reference $M_{1} = 0$ , it is $R_{A} = 2$ . The layer operator norm and every prediction stay fixed, while the reference-dependent complexity changes.

Example references

Spectral network complexity with reference matrices: defines the network score map, reference-dependent complexity, and induced operator norm.

Multiclass margins, ramp loss, and empirical margin loss: defines predictions by score maximization and the multiclass margin.

A simultaneous spectral margin bound for multiclass networks: uses reference matrices in its explicit complexity terms.

Facts & Assumptions

Given: The one-layer identity-activation network, its matrix, input, and the two stated reference matrices.

[F1]

The score map depends on the layer matrix and activation; the reference matrix enters $R_{A}$ through the transpose-difference mixed norm. Spectral network complexity with reference matrices

[F2]

The predicted label maximizes the score coordinates, and the margin compares the true-label score with the maximum competing score. Multiclass margins, ramp loss, and empirical margin loss

[F3]

The theorem's layer terms depend on both the layer matrix norms and their distances from the chosen reference matrices. A simultaneous spectral margin bound for multiclass networks

Verification

technique · direct

1.1F1F2givenalgebra

Since the matrix and activation are identical in both cases, the network computes $A_{1} x = (1, - 1)$ for either reference. Its operator norm is $\sqrt{2}$ , so the predicted class is class $1$ and the margin for label $1$ is $2$ .

2.1F1F3step 1.1algebra∎

If $M_{1} = A_{1}$ , the transpose-difference norm is zero and the defining formula gives $R_{A} = 0$ . If $M_{1} = 0$ , that norm is $2$ by summing the two scalar column norms of $(1, - 1)$ ; hence $R_{A} = \sqrt{2} (2 / \sqrt{2}) = 2$ . The calculations change the complexity input only; [F3] does not assert that the floored explicit theorem term vanishes when $R_{A} = 0$ .

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

A small empirical margin loss does not determine population error

Example

Assume the axiom of choice and take confidence level $δ = 1 / 2$ . Let the input always be $x = 1$ and let the label distribution assign probability $0.6$ to class $1$ and $0.4$ to class $2$ . Use one identity-activation layer with $A_{1} = (1, 0)^{⊤}$ and reference $M_{1} = 0$ . For a sample of size $n = 1$ , at margin threshold $γ = 1 / 2$ the event that the sampled label is class $1$ has probability $0.6$ , has empirical margin loss zero, while the network's population zero-one error is $0.4$ . The simultaneous theorem still permits this event because its $8 / n$ term alone is $8$ ; the example shows that one zero empirical margin-loss observation does not itself determine population error.

Example references

Spectral network complexity with reference matrices: defines the network map and norm conventions.

Multiclass margins, ramp loss, and empirical margin loss: defines the population zero-one error and empirical margin loss.

Population and empirical risk: defines population expectation and empirical averages.

A simultaneous spectral margin bound for multiclass networks: supplies the finite-sample bound including the additive $8 / n$ term.

Facts & Assumptions

Given: The two-point label distribution, deterministic input, one-layer network, sample size, and margin threshold in the Example.

[F1]

The network uses the zero-bias recurrence with an identity activation, and the operator norm is induced by the Euclidean norm. Spectral network complexity with reference matrices

[F2]

The margin is true score minus the maximum competing score; empirical margin loss counts margins at most $γ$ and population zero-one error is the probability of a wrong predicted label. Multiclass margins, ramp loss, and empirical margin loss

[F3]

The theorem's upper bound includes the additive $8 / n$ correction. A simultaneous spectral margin bound for multiclass networks

Verification

technique · direct

1.1F1F2givenalgebra

The scores at $x = 1$ are $(1, 0)$ . For label $1$ the margin is $1 - 0 = 1$ ; for label $2$ it is $0 - 1 = - 1$ . Thus the network predicts class $1$ on both possible labeled outcomes.

2.1F2step 1.1givenalgebra

By the label probabilities in the Example and [F2], the population zero-one error is $0.4$ . If the sample label is class $1$ , which occurs with probability $0.6$ , its only margin is $1 > 1 / 2$ , so its empirical margin loss is $0$ .

3.1F3step 2.1givenalgebra∎

For $n = 1$ , [F3] contributes $8 / n = 8$ to the theorem's right-hand side, already exceeding the population error $0.4$ ; all other displayed terms are nonnegative. Therefore the stated sample event is consistent with the theorem and demonstrates why empirical margin loss alone is not a population-error guarantee.

Norm and margin generalization for neural networks — Examples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

A two-class one-layer spectral-margin calculation

Example

Example references

Facts & Assumptions

Verification

Changing the reference matrix changes complexity, not the network

Example

Example references

Facts & Assumptions

Verification

A small empirical margin loss does not determine population error

Example

Example references

Facts & Assumptions

Verification

Sources