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Contrastive transfer with a within-class covariance penalty

Statement

Assume the Axiom of Choice and the one-negative latent-class model with $τ < 1$ . Fix either source hinge or logistic loss, $M \geq 1$ , and 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, under the measurability hypotheses of the uniform-generalization result. For each comparator $f \in F$ and every $0 < δ < 1$ , with probability at least $1 - δ$ over the IID triplet sample,

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

where $L_{s u p}$ is the infimum of the average binary-task risk over all linear classifiers, $s (f)$ is the within-class covariance deviation, and

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

is the uniform-deviation quantity for the same sample and loss. The event is simultaneous for all comparators.

Statement references

Latent-class contrastive representation learning: defines the linear-classifier infimum, mean-classifier risk, distinct-class contrastive risk and covariance deviation $s (f)$ .

Class-collision loss is bounded by within-class covariance deviation: supplies the collision decomposition and $L_{u n}^{=} (f) \leq 1 + \sqrt{2} L_{0} s (f)$ .

Contrastive transfer bound with class collisions: supplies the simultaneous empirical-minimizer transfer bound.

Saunshi et al., Theorem 4.5, Lemma 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, a measurable exact empirical minimizer, $M \geq 1$ , and $0 < δ < 1$ .

[A1]

AC enters through the empirical-minimizer transfer theorem's uniform-generalization supplier. The Axiom of Choice, Contrastive transfer bound with class collisions

[F1]

The average risk infimum over all linear weights is at most the risk of the particular class-mean weight: $L_{s u p} (g) \leq L_{s u p}^{μ} (g)$ . Latent-class contrastive representation learning

[F2]

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

[F3]

For $τ > 0$ , $L_{u n} (f) = τ L_{u n}^{=} (f) + (1 - τ) L_{u n}^{\neq} (f)$ and $L_{u n}^{=} (f) \leq 1 + \sqrt{2} L_{0} s (f)$ . If $τ = 0$ , $L_{u n} (f) = L_{u n}^{\neq} (f)$ and $s (f) = 0$ . Class-collision loss is bounded by within-class covariance deviation

Proof

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

1.1F1given

By the definition of $L_{s u p}$ as an infimum over all linear weights, its value for $\hat{f}$ is at most the risk of the particular mean weight, so $L_{s u p} (\hat{f}) \leq L_{s u p}^{μ} (\hat{f})$ .

1.2A1F2F3givenalgebra

On the event in [F2], fix any comparator $f$ . If $τ > 0$ , [F3] gives $L_{u n} (f) - τ = τ (L_{u n}^{=} (f) - 1) + (1 - τ) L_{u n}^{\neq} (f) \leq τ \sqrt{2} L_{0} s (f) + (1 - τ) L_{u n}^{\neq} (f)$ . If $τ = 0$ , the same inequality is equality under the stipulated convention $s (f) = 0$ .

2.1F2step 1.1step 1.2givenalgebra∎

Divide the inequality in step 1.2 by $1 - τ > 0$ and substitute it into [F2]. Together with step 1.1, this gives the displayed bound for every comparator on the same event, whose probability is at least $1 - δ$ .

Depends on

Used by

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

Dependency tree · two levels

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Theorem 4.5, Lemma 4.4 and proof route (standard reference, not scraped)
Nikunj Saunshi et al., Supplementary Appendices A.1 and A.3 (standard reference, not scraped)