ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Bartlett, Foster, and Telgarsky, Spectrally-normalized margin bounds for neural networks, Theorem 1.1 (standard reference, not scraped)