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

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A negative sample collides with the positive class

Example

Let the latent classes be $a, b$ with equal probabilities, let each class contain one deterministic input, and define a scalar representation by $f (a) = 0$ and $f (b) = 1$ . Use one independent marginal negative and the hinge loss. The probability that the negative class equals the positive-pair class is $τ = 1 / 2$ . Conditional on such a collision, the contrastive score is always zero and the conditional loss is $1$ . Conditional on distinct classes, the ordered pair $(a, b)$ has loss $1$ and $(b, a)$ has loss $0$ , so $L_{u n}^{\neq} (f) = 1 / 2$ and the full contrastive risk is $L_{u n} (f) = 3 / 4$ .

Example references

Latent-class contrastive representation learning: defines the positive-pair and independent-negative sampling laws.

Convexity, Lipschitz bounds, and oscillation of the source contrastive losses: defines the hinge loss used in the calculation.

Saunshi et al., Section 4.2 and Equation (8): decomposes the objective by whether the negative class collides with the positive class.

Facts & Assumptions

Given: Two equally likely classes $a, b$ , deterministic inputs, $f (a) = 0$ , $f (b) = 1$ , one independent marginal negative, and hinge loss.

[F1]

The positive and negative classes are independent with law $(1 / 2, 1 / 2)$ , so equality has probability $1 / 2$ and inequality has probability $1 / 2$ . Latent-class contrastive representation learning

[F2]

The hinge contrastive loss is $(1 + t)_{+}$ and equals $1$ at score $t = 0$ . Convexity, Lipschitz bounds, and oscillation of the source contrastive losses

Proof

Given: The finite sampling model in the Example.

1.1F1F2givenalgebra

If the negative class equals the positive class, the anchor, positive sample and negative sample all have the same representation. Thus $q_{f} = f (x) (f (x^{-}) - f (x^{+})) = 0$ and [F2] gives collision loss $1$ .

1.2F1F2givenalgebra

If the ordered classes are $(a, b)$ , the anchor representation is $0$ , hence $q_{f} = 0$ and loss $1$ . If they are $(b, a)$ , then $q_{f} = 1 (0 - 1) = - 1$ , hence the hinge loss is $0$ . The two ordered cases are equiprobable given inequality, so $L_{u n}^{\neq} (f) = (1 + 0) / 2 = 1 / 2$ .

2.1F1step 1.1step 1.2givenalgebra∎

The collision decomposition weights the two conditional losses by $τ = 1 / 2$ and $1 - τ = 1 / 2$ . Therefore $L_{u n} (f) = (1 / 2) \cdot 1 + (1 / 2) \cdot (1 / 2) = 3 / 4$ , with a positive contribution from negative-class collisions.

Depends on

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Section 4.2, Equation (8) (standard reference, not scraped)