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Jensen comparison with the average mean-classifier loss

Statement

Assume the one-negative latent-class model has $τ < 1$ . Let $C_{+}, C_{-}$ be independent with law $ρ$ , conditioned on $C_{+} \neq C_{-}$ , let $X, X^{+}$ be independent with law $D_{C_{+}}$ , and let $X^{-}$ be independent with law $D_{C_{-}}$ . For every measurable bounded representation $f$ and either source hinge or logistic loss, the average binary-task mean-classifier risk satisfies

$L_{s u p}^{μ} (f) \leq L_{u n}^{\neq} (f) .$

Here each binary task uses a uniform label from its two classes, and $L_{u n}^{\neq}$ is the contrastive loss conditional on different positive and negative classes.

Statement references

Latent-class contrastive representation learning: defines the latent-class sampling laws, binary-task risk, mean classifier, and distinct-class contrastive loss.

Convexity, Lipschitz bounds, and oscillation of the source contrastive losses: establishes convexity of both source losses.

Saunshi et al., Lemma 4.3 and complete proof: proves the same one-negative Jensen comparison for uniformly labeled binary tasks.

Facts & Assumptions

Given: The sampling model in the Statement with $τ < 1$ , one representation $f \in F$ , and either source loss $ℓ$ .

[F1]

Conditional on $C_{+} = c_{+}$ and $C_{-} = c_{-}$ , the positive pair is sampled independently from $D_{c_{+}}$ and the negative example independently from $D_{c_{-}}$ . Latent-class contrastive representation learning

[F2]

The loss $ℓ$ is convex, so for each integrable real random variable $Z$ , $ℓ (E Z) \leq E ℓ (Z)$ . Convexity, Lipschitz bounds, and oscillation of the source contrastive losses, Jensen's inequality for expectation

[F3]

The mean-classifier margin for a positive-class input is $f (x)^{⊤} (μ_{c_{+}} - μ_{c_{-}})$ , and the contrastive score is $f (x)^{⊤} (f (x^{-}) - f (x^{+}))$ . Latent-class contrastive representation learning

[F4]

The ordered pair law of $(C_{+}, C_{-})$ conditioned on inequality is invariant under swapping its coordinates; the two draws have joint law $ρ \otimes ρ$ . Independent random elements have product joint law

[F5]

The integral of integrable functions is linear, so conditional expectations of the bounded vector coordinates subtract to the difference of their means. The Lebesgue integral is linear on $L^{1} (μ)$

Proof

Given: The hypotheses in the Statement and Facts & Assumptions.

1.1F1F2F3F5given

Fix distinct classes $c_{+}, c_{-}$ and an anchor $x \in X$ . Conditional on these classes and $x$ , the random score $Z = f (x)^{⊤} (f (X^{-}) - f (X^{+}))$ is integrable and has mean $E [Z] = f (x)^{⊤} (μ_{c_{-}} - μ_{c_{+}})$ by independence and [F5] applied to the bounded coordinates. Jensen's inequality [F2] gives $E [ℓ (Z)] \geq ℓ (f (x)^{⊤} (μ_{c_{-}} - μ_{c_{+}}))$ . Integrating over $x \sim D_{c_{+}}$ and then over distinct ordered class pairs shows that $L_{u n}^{\neq} (f)$ is at least the average mean-classifier loss on the positive label.

2.1F3F4step 1.1algebra∎

By [F4], swapping $(c_{+}, c_{-})$ preserves the conditional ordered-pair law. Therefore the average mean-classifier loss on the negative label equals its average on the positive label: swapping the classes turns its margin $f (x)^{⊤} (μ_{c_{+}} - μ_{c_{-}})$ into the positive-label margin from step 1.1. Since the task label is uniform, the average of the two label risks is exactly the positive-label average. Thus $L_{s u p}^{μ} (f) \leq L_{u n}^{\neq} (f)$ .

Depends on

Used by

Contrastive transfer for an approximate empirical minimizer Corollary
Contrastive transfer bound with class collisions Theorem

Dependency tree · two levels

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Lemma 4.3 and proof (standard reference, not scraped)