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A Statistical Theory of Contrastive Learning via Approximate Sufficient Statistics

Licong Lin, Song Mei

TL;DR

This work develops a theoretical framework for data augmentation-based contrastive learning by extending the notion of approximate sufficient statistics to general ${\mathrm{f}}$-divergences, connecting the minimization of contrastive losses (e.g., InfoNCE) to encoders that are near-sufficient for downstream tasks. It provides risk bounds for downstream regression and classification that decompose into a sufficiency term and augmentation-induced error, and extends the analysis from KL to broader f-divergences, including ${\chi^2}$-sufficiency. Concrete examples in linear regression and topic classification illustrate broad applicability and guide practical adaptation of learned encoders. The results imply that encoders trained via contrastive losses can be effectively reused with limited labeled data, provided the encoders maintain low sufficiency loss and the data augmentations preserve downstream-relevant information. The framework also opens avenues for analyzing alternative contrastive losses and extending to other self-supervised paradigms beyond SimCLR.

Abstract

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of \emph{approximate sufficient statistics}, which we extend beyond its original definition in \cite{oko2025statistical} for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

A Statistical Theory of Contrastive Learning via Approximate Sufficient Statistics

TL;DR

This work develops a theoretical framework for data augmentation-based contrastive learning by extending the notion of approximate sufficient statistics to general -divergences, connecting the minimization of contrastive losses (e.g., InfoNCE) to encoders that are near-sufficient for downstream tasks. It provides risk bounds for downstream regression and classification that decompose into a sufficiency term and augmentation-induced error, and extends the analysis from KL to broader f-divergences, including -sufficiency. Concrete examples in linear regression and topic classification illustrate broad applicability and guide practical adaptation of learned encoders. The results imply that encoders trained via contrastive losses can be effectively reused with limited labeled data, provided the encoders maintain low sufficiency loss and the data augmentations preserve downstream-relevant information. The framework also opens avenues for analyzing alternative contrastive losses and extending to other self-supervised paradigms beyond SimCLR.

Abstract

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of \emph{approximate sufficient statistics}, which we extend beyond its original definition in \cite{oko2025statistical} for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

Paper Structure

This paper contains 57 sections, 16 theorems, 154 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Self-supervised learning and contrastive learning.
  4. Choices of the loss function.
  5. Theoretical understanding of contrastive learning.
  6. Approximate sufficient statistics
  7. Statistical properties of contrastive learning
  8. Setup and the ERM estimator
  9. Why does the SimCLR loss work?
  10. Using the encoder for downstream tasks
  11. Adaptation to downstream regression tasks.
  12. Adaptation to downstream classification tasks.
  13. General ${\mathrm{f}}$-contrastive learning
  14. Finding encoders with low ${\mathrm{f}}$-sufficiency
  15. Implications of low ${\mathrm{f}}$-Sufficiency
  16. ...and 42 more sections

Key Result

Theorem 1

Suppose Assumption ass:bounded_score and ass:link_fun hold for some $B_{\sf S}\geq1,B_\tau>0$. Let $\widehat{f}$ be the empirical risk minimizer defined in Eq. eq:simclr-empirical-risk-minimization and let ${\sf S_\star}$ be as defined in Section sec:simclr_erm_analysis. Let ${\sf supp}({\boldsymbol where

Figures (1)

  • Figure 1: Excess risk for various downstream sample sizes $m$. The errorbars represent the standard deviation over $10$ runs.

Theorems & Definitions (29)

  • Definition 1: Approximate sufficiency
  • Theorem 1: Sufficiency bound for the ERM estimator
  • Theorem 2: Performance on downstream regression
  • Theorem 3: Performance on downstream classification
  • Theorem 4: ${\chi^2}$-sufficiency bound for the ERM estimator
  • Proposition 5: ${\mathrm{f}}$-sufficiency bound on downstream performance
  • Theorem 6: Linear regression with encoder representation
  • Corollary 1: An upper bound on the sufficiency
  • Theorem 7: Classification using the ${\chi^2}$-trained encoder
  • Lemma 1: Equivalence of three forms of sufficiency
  • ...and 19 more