Self-Supervised Contrastive Learning is Approximately Supervised Contrastive Learning

TL;DR

This paper demonstrates that self-supervised contrastive learning implicitly approximates a supervised variant, with theoretical analysis showing the gap diminishes as class count increases, and validates findings through empirical experiments.

Contribution

It establishes a theoretical connection between self-supervised and supervised contrastive learning, characterizes the structure of learned representations, and provides bounds on few-shot learning performance.

Findings

The gap between CL and NSCL losses decays as the number of classes increases.

CL and NSCL losses are highly correlated.

Minimizing CL loss implicitly minimizes NSCL loss, supporting effective few-shot learning.

Abstract

Despite its empirical success, the theoretical foundations of self-supervised contrastive learning (CL) are not yet fully established. In this work, we address this gap by showing that standard CL objectives implicitly approximate a supervised variant we call the negatives-only supervised contrastive loss (NSCL), which excludes same-class contrasts. We prove that the gap between the CL and NSCL losses vanishes as the number of semantic classes increases, under a bound that is both label-agnostic and architecture-independent. We characterize the geometric structure of the global minimizers of the NSCL loss: the learned representations exhibit augmentation collapse, within-class collapse, and class centers that form a simplex equiangular tight frame. We further introduce a new bound on the few-shot error of linear-probing. This bound depends on two measures of feature variability--within-class dispersion and variation along the line between class centers. We show that directional variation dominates the bound and that the within-class dispersion's effect diminishes as the number of labeled samples increases. These properties enable CL and NSCL-trained representations to support accurate few-shot label recovery using simple linear probes. Finally, we empirically validate our theoretical findings: the gap between CL and NSCL losses decays at a rate of $\mathcal{O}(\frac{1}{\#\text{classes}})$; the two losses are highly correlated; minimizing the CL loss implicitly brings the NSCL loss close to the value achieved by direct minimization; and the proposed few-shot error bound provides a tight estimate of probing performance in practice. The code and project page of the paper are available at [\href{https://github.com/DLFundamentals/understanding-ssl}{code}, \href{https://dlfundamentals.github.io/ssl-is-approximately-sl/}{project page}].