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Generalization Analysis for Supervised Contrastive Representation Learning under Non-IID Settings

Nong Minh Hieu, Antoine Ledent

TL;DR

This work analyzes generalization for supervised contrastive representation learning under non-IID data by modeling recycled training tuples with a class-wise U-statistics framework. It develops a revised theory that decouples and bounds the population unsupervised risk through per-class statistics, yielding high-probability excess risk bounds that scale with the effective sample size $\\widetilde{N} = N \, \\\min(\\rho_{min}/2, (1-\\rho_{max})/k)$. The main results include bounds for both the empirical U-statistics minimizer and a subsampled risk minimizer, with explicit bounds for linear representations and neural networks. Empirical studies on MNIST and synthetic data corroborate the theory, showing that data recycling and increasing subsampled tuples improve or match full-tuple performance and that sample complexity grows with the number of classes and negatives as predicted. These findings bridge practical CRL training with theoretical guarantees in non-IID settings, guiding the design of efficient, sample-aware contrastive learning pipelines.

Abstract

Contrastive Representation Learning (CRL) has achieved impressive success in various domains in recent years. Nevertheless, the theoretical understanding of the generalization behavior of CRL has remained limited. Moreover, to the best of our knowledge, the current literature only analyzes generalization bounds under the assumption that the data tuples used for contrastive learning are independently and identically distributed. However, in practice, we are often limited to a fixed pool of reusable labeled data points, making it inevitable to recycle data across tuples to create sufficiently large datasets. Therefore, the tuple-wise independence condition imposed by previous works is invalidated. In this paper, we provide a generalization analysis for the CRL framework under non-$i.i.d.$ settings that adheres to practice more realistically. Drawing inspiration from the literature on U-statistics, we derive generalization bounds which indicate that the required number of samples in each class scales as the logarithm of the covering number of the class of learnable feature representations associated to that class. Next, we apply our main results to derive excess risk bounds for common function classes such as linear maps and neural networks.

Generalization Analysis for Supervised Contrastive Representation Learning under Non-IID Settings

TL;DR

This work analyzes generalization for supervised contrastive representation learning under non-IID data by modeling recycled training tuples with a class-wise U-statistics framework. It develops a revised theory that decouples and bounds the population unsupervised risk through per-class statistics, yielding high-probability excess risk bounds that scale with the effective sample size . The main results include bounds for both the empirical U-statistics minimizer and a subsampled risk minimizer, with explicit bounds for linear representations and neural networks. Empirical studies on MNIST and synthetic data corroborate the theory, showing that data recycling and increasing subsampled tuples improve or match full-tuple performance and that sample complexity grows with the number of classes and negatives as predicted. These findings bridge practical CRL training with theoretical guarantees in non-IID settings, guiding the design of efficient, sample-aware contrastive learning pipelines.

Abstract

Contrastive Representation Learning (CRL) has achieved impressive success in various domains in recent years. Nevertheless, the theoretical understanding of the generalization behavior of CRL has remained limited. Moreover, to the best of our knowledge, the current literature only analyzes generalization bounds under the assumption that the data tuples used for contrastive learning are independently and identically distributed. However, in practice, we are often limited to a fixed pool of reusable labeled data points, making it inevitable to recycle data across tuples to create sufficiently large datasets. Therefore, the tuple-wise independence condition imposed by previous works is invalidated. In this paper, we provide a generalization analysis for the CRL framework under non- settings that adheres to practice more realistically. Drawing inspiration from the literature on U-statistics, we derive generalization bounds which indicate that the required number of samples in each class scales as the logarithm of the covering number of the class of learnable feature representations associated to that class. Next, we apply our main results to derive excess risk bounds for common function classes such as linear maps and neural networks.
Paper Structure (41 sections, 22 theorems, 140 equations, 3 figures, 1 table)

This paper contains 41 sections, 22 theorems, 140 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Problem Set-up
  5. A Theoretical Framework under IID Settings
  6. A Revised Framework under non-IID Settings
  7. Proof Strategy
  8. Bounding the U-Statistics to Population Unsupervised Risk Deviation
  9. Bounding the Sub-sampled Empirical Risk to U-Statistics Deviation
  10. Main Results
  11. Generalization Bound for Empirical Minimizer of U-Statistics
  12. Generalization Bound for Empirical Sub-sampled Risk Minimizer
  13. Generalization Bounds for Common Classes of Representation Functions
  14. Experiments
  15. Performance of $\widehat{f}_\mathrm{sub}$ as $\mathrm{M}$ Increases
  16. ...and 26 more sections

Key Result

Theorem 5.1

Let $\mathcal{F}$ be a class of representation functions and $\ell:\mathbb{R}^k\to[0, \mathcal{M}]$ be a bounded contrastive loss. Let $\widehat{f}_\mathcal{U}=\arg\min_{f\in\mathcal{F}}\mathcal{U}_\mathrm{N}(f)$. Then, for any $\delta\in(0,1)$, with probability of at least $1-\delta$: where $\widetilde{\mathrm{N}} = \mathrm{N}\min( \frac{\min_{c\in\mathcal{C}}\rho(c)}{2}, \frac{1-\max_{c\in\math

Figures (3)

  • Figure 1: An illustration of the tuples selection process for $\mathcal{T}_c^\mathrm{iid}$. In this case, there are excess samples from $\bar{\mathcal{S}}_c$ that are left unused. However, the other way around where there are excess samples from $\mathcal{S}_c$ is also possible (E.g. for very large values of $k$).
  • Figure 2: Summary of results for experiments with the MNIST dataset. On the left, we have the results for $n=10000$. On the right, we have the results for $n=100$ as well as the additional result for the all-tuples regime.
  • Figure 3: Summary of results for synthetic data experiments on the relationship between $|\mathcal{C}|, k$ and the sample complexity.

Theorems & Definitions (53)

  • Definition 3.1: Unsupervised Risk
  • Remark 3.2
  • Remark 4.1
  • Theorem 5.1: cf. Theorem \ref{['thm:basic_bound']}
  • Theorem 5.2: cf. Theorem \ref{['thm:subsampled_bound']}
  • Definition 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 43 more