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Contrastive transfer bound with class collisions

Statement

Assume the Axiom of Choice and the one-negative latent-class model with $τ < 1$ . Fix either source hinge or logistic loss and a sample size $M \geq 1$ . Suppose all risks and suprema in the uniform-generalization statement are measurable and that a measurable exact empirical minimizer $\hat{f} \in F$ exists, meaning ${\hat{L}}_{u n} (\hat{f}) \leq {\hat{L}}_{u n} (g)$ for every $g \in F$ almost surely. For every $0 < δ < 1$ , with probability at least $1 - δ$ over the IID triplet sample,

$\forall f \in F, L_{s u p}^{μ} (\hat{f}) \leq \frac{L_{u n} (f) - τ + 2 Γ_{M} (S, δ)}{1 - τ},$

where

$Γ_{M} (S, δ) : = \frac{2 \sqrt{18} L_{0} R R_{S} (F)}{M} + 3 B_{ℓ, R} \sqrt{\frac{\ln (4 / δ)}{2 M}}$

is the uniform-deviation quantity for the same sample and loss. This is a guarantee for the average binary task with a uniform label and the class-mean linear classifier.

Statement references

Latent-class contrastive representation learning: defines the one-negative objective, the class-mean classifier risk and the collision probability.

Population contrastive excess of an approximate empirical minimizer: bounds the population contrastive risk of the empirical minimizer against every comparator.

Jensen comparison with the average mean-classifier loss: compares mean-classifier risk with distinct-class contrastive risk for either source loss.

Class-collision loss is bounded by within-class covariance deviation: decomposes total contrastive risk and lower-bounds the collision-conditioned loss by one.

Saunshi et al., Theorem 4.1, Lemmas 4.3–4.4 and proof route.

Saunshi et al., Supplementary Appendices A.1 and A.3.

Facts & Assumptions

Given: AC, the one-negative model with $τ < 1$ , either source loss, the measurable exact empirical minimizer, $M \geq 1$ , and $0 < δ < 1$ .

[A1]

AC enters through the uniform-generalization theorem's bounded-differences step and is carried by the empirical-minimizer excess lemma. The Axiom of Choice, Population contrastive excess of an approximate empirical minimizer

[F1]

With probability at least $1 - δ$ , simultaneously for every comparator $f \in F$ , $L_{u n} (\hat{f}) \leq L_{u n} (f) + 2 Γ_{M} (S, δ)$ . Population contrastive excess of an approximate empirical minimizer

[F2]

For every bounded representation $g$ and either source loss, $L_{s u p}^{μ} (g) \leq L_{u n}^{\neq} (g)$ . Jensen comparison with the average mean-classifier loss

[F3]

The collision decomposition is $L_{u n} (g) = τ L_{u n}^{=} (g) + (1 - τ) L_{u n}^{\neq} (g)$ and $L_{u n}^{=} (g) \geq 1$ when $τ > 0$ ; when $τ = 0$ , $L_{u n} (g) = L_{u n}^{\neq} (g)$ . Class-collision loss is bounded by within-class covariance deviation

Proof

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

1.1A1F1given

On the event in [F1], the empirical-minimizer excess bound holds simultaneously for all comparators $f$ .

1.2F3givenalgebra

By [F3], if $τ > 0$ then $L_{u n}^{\neq} (\hat{f}) = \frac{L_{u n} (\hat{f}) - τ L_{u n}^{=} (\hat{f})}{1 - τ} \leq \frac{L_{u n} (\hat{f}) - τ}{1 - τ}$ . If $τ = 0$ , the same inequality is equality because $L_{u n} = L_{u n}^{\neq}$ . Since $1 - τ > 0$ , these divisions are valid.

2.1F1F2step 1.1step 1.2givenalgebra∎

Apply [F2] to $\hat{f}$ and then use step 1.2 and [F1]. For every comparator on the event in step 1.1, $L_{s u p}^{μ} (\hat{f}) \leq L_{u n}^{\neq} (\hat{f}) \leq \frac{L_{u n} (\hat{f}) - τ}{1 - τ} \leq \frac{L_{u n} (f) - τ + 2 Γ_{M} (S, δ)}{1 - τ}$ . The event has probability at least $1 - δ$ by [F1], proving the claim.

Depends on

Used by

Contrastive transfer with a within-class covariance penalty Theorem

Dependency tree · two levels

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Theorem 4.1, Lemmas 4.3–4.4 and proof route (standard reference, not scraped)
Nikunj Saunshi et al., Supplementary Appendices A.1 and A.3, complete collision and generalization arguments (standard reference, not scraped)