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Contrastive transfer for an approximate empirical minimizer

Statement

Assume the Axiom of Choice and the one-negative latent-class model with $τ < 1$ . Fix either source hinge or logistic loss, a sample size $M \geq 1$ , and a tolerance $η \geq 0$ . Suppose the risks and suprema in the uniform-generalization statement are measurable and let $\hat{f} \in F$ be a measurable selector satisfying ${\hat{L}}_{u n} (\hat{f}) \leq {\hat{L}}_{u n} (f) + η$ for every $f \in F$ almost surely. For every $0 < δ < 1$ , with probability at least $1 - δ$ ,

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

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. The result assumes such a measurable approximate selector and does not assert its existence.

Statement references

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

Population contrastive excess of an approximate empirical minimizer: supplies the approximate empirical-to-population contrastive-risk bound.

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

Class-collision loss is bounded by within-class covariance deviation: supplies the collision decomposition and collision-loss lower bound.

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

Facts & Assumptions

Given: AC, the one-negative model with $τ < 1$ , either source loss, an approximate selector with tolerance $η \geq 0$ , $M \geq 1$ , and $0 < δ < 1$ .

[A1]

AC enters through the bounded-differences component of the uniform-generalization theorem. The Axiom of Choice, Population contrastive excess of an approximate empirical minimizer

[F1]

With probability at least $1 - δ$ , simultaneously for every comparator $f \in F$ , $L_{u n} (\hat{f}) \leq L_{u n} (f) + 2 Γ_{M} (S, δ) + η$ . 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 and lower bound give $L_{u n}^{\neq} (g) \leq (L_{u n} (g) - τ) / (1 - τ)$ for all $0 \leq τ < 1$ , with the zero-collision case interpreted as equality. Class-collision loss is bounded by within-class covariance deviation

Proof

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

1.1A1F1given

On the simultaneous event in [F1], $L_{u n} (\hat{f}) \leq L_{u n} (f) + 2 Γ_{M} (S, δ) + η$ for every comparator $f$ .

1.2F3givenalgebra

The collision decomposition [F3] applies to $\hat{f}$ because $τ < 1$ , and yields $L_{u n}^{\neq} (\hat{f}) \leq (L_{u n} (\hat{f}) - τ) / (1 - τ)$ .

2.1F1F2step 1.1step 1.2givenalgebra∎

The Jensen comparison [F2], followed by steps 1.1 and 1.2, gives $L_{s u p}^{μ} (\hat{f}) \leq L_{u n}^{\neq} (\hat{f}) \leq \frac{L_{u n} (f) - τ + 2 Γ_{M} (S, δ) + η}{1 - τ}$ simultaneously for every comparator. The event has probability at least $1 - δ$ by [F1].

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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)
Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Chapter 3 empirical-risk comparison (standard reference, not scraped)