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Sub-Gaussian contrastive transfer with a mean-classifier margin loss

Statement

Assume the Axiom of Choice and the one-negative latent-class model with $τ < 1$ . Use the source hinge loss for both contrastive training and the binary mean-classifier task. Fix $M \geq 1$ , a measurable exact empirical minimizer $\hat{f} \in F$ satisfying ${\hat{L}}_{u n} (\hat{f}) \leq {\hat{L}}_{u n} (g)$ for every $g \in F$ almost surely, $0 < δ < 1$ , and $0 < ϵ \leq 1$ . Let $f \in F$ be any comparator whose centered class-conditional representation vectors are sub-Gaussian in every direction with common proxy $σ^{2} \geq 0$ . Define

$γ : = {\begin{cases} 1 + 2 R σ \sqrt{2 \ln R + \ln (3 / ϵ)}, & R \geq 1, \\ 1 + 2 R σ \sqrt{\ln (3 / ϵ)}, & 0 < R < 1. \end{cases}$

Under the risk and supremum measurability hypotheses, with probability at least $1 - δ$ ,

$L_{s u p} (\hat{f}) \leq L_{s u p}^{μ} (\hat{f}) \leq γ L_{γ, s u p}^{μ} (f) + ϵ + \frac{\sqrt{2} L_{0} τ s (f) + 2 Γ_{M} (S, δ)}{1 - τ} .$

The uniform-deviation quantity is

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

where $L_{γ, s u p}^{μ} (f)$ is the average task risk of the class-mean weight $w_{μ}$ using the scaled hinge loss $(1 - Y w_{μ}^{⊤} f (X) / γ)_{+}$ . The logistic transfer theorems remain separate; the source margin comparison does not supply their logistic analogue.

Statement references

Class-conditional sub-Gaussian concentration yields a contrastive margin bound: supplies the exact piecewise margin parameter and the hinge comparison $L_{u n}^{\neq} (f) \leq γ L_{γ, s u p}^{μ} (f) + ϵ$ .

Contrastive transfer with a within-class covariance penalty: supplies the empirical-minimizer transfer bound with a within-class covariance term.

Saunshi et al., Corollary 5.1.1 and Lemma 5.1, main paper §5.2.

Saunshi et al., complete proof of Lemma 5.1, Supplementary Appendix A.2.

Facts & Assumptions

Given: AC, the one-negative model with $τ < 1$ , hinge loss, a measurable exact empirical minimizer, comparator $f$ with the stated classwise centered sub-Gaussian proxy, $M \geq 1$ , $0 < δ < 1$ , and $0 < ϵ \leq 1$ .

[A1]

AC enters through the uniform-generalization supplier used by the within-class transfer theorem. The Axiom of Choice, Contrastive transfer with a within-class covariance penalty

[F1]

With probability at least $1 - δ$ , simultaneously for every comparator $f$ , $L_{s u p} (\hat{f}) \leq L_{s u p}^{μ} (\hat{f}) \leq L_{u n}^{\neq} (f) + [\sqrt{2} L_{0} τ s (f) + 2 Γ_{M} (S, δ)] / (1 - τ)$ . Contrastive transfer with a within-class covariance penalty

[F2]

Under the classwise centered sub-Gaussian assumption, the hinge distinct-class risk satisfies $L_{u n}^{\neq} (f) \leq γ L_{γ, s u p}^{μ} (f) + ϵ$ for the specified $γ$ . Class-conditional sub-Gaussian concentration yields a contrastive margin bound

Proof

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

1.1A1F1given

On the event in [F1], the comparator $f$ from the Statement satisfies its transfer inequality.

2.1F1F2step 1.1givenalgebra∎

Apply [F2] to this comparator and substitute its bound for $L_{u n}^{\neq} (f)$ in [F1]. The result is $L_{s u p} (\hat{f}) \leq L_{s u p}^{μ} (\hat{f}) \leq γ L_{γ, s u p}^{μ} (f) + ϵ + \frac{\sqrt{2} L_{0} τ s (f) + 2 Γ_{M} (S, δ)}{1 - τ}$ . The event from [F1] has probability at least $1 - δ$ , proving the corollary.

Depends on

Used by

The hinge margin comparison does not extend to logistic contrastive loss Example

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Corollary 5.1.1 and Lemma 5.1 (standard reference, not scraped)
Nikunj Saunshi et al., Supplementary Appendix A.2, complete proof of Lemma 5.1 (standard reference, not scraped)