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Generalization Analysis for Deep Contrastive Representation Learning

Nong Minh Hieu, Antoine Ledent, Yunwen Lei, Cheng Yeaw Ku

TL;DR

This work analyzes the generalization of Deep Contrastive Representation Learning (DCRL) by deriving bounds on the unsupervised risk $L_{un}$ for neural-network representations. It replaces prior depth-dependent Frobenius-norm bounds with covering-number-based bounds that rely on spectral-type complexity, and introduces loss-augmentation techniques to lessen dependence on network depth and the number of negative samples $k$. The paper presents a basic covering-number bound, loss-augmentation bounds (including all-activation variants), and a parameter-counting bound that scales with the total number of parameters, plus a downstream supervised bound via mean classifiers. Empirically, on MNIST, the proposed bounds outperform prior results across varying depths and widths and remain robust to larger negative-sample regimes, suggesting stronger theoretical guarantees that align with practical CRL performance.

Abstract

In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles. On the one hand, we derive a parameter-counting bound that scales with the overall size of the neural networks. On the other hand, we provide a norm-based bound that scales with the norms of neural networks' weight matrices. Ignoring logarithmic factors, the bounds are independent of $k$, the size of the tuples provided for contrastive learning. To the best of our knowledge, this property is only shared by one other work, which employed a different proof strategy and suffers from very strong exponential dependence on the depth of the network which is due to a use of the peeling technique. Our results circumvent this by leveraging powerful results on covering numbers with respect to uniform norms over samples. In addition, we utilize loss augmentation techniques to further reduce the dependency on matrix norms and the implicit dependence on network depth. In fact, our techniques allow us to produce many bounds for the contrastive learning setting with similar architectural dependencies as in the study of the sample complexity of ordinary loss functions, thereby bridging the gap between the learning theories of contrastive learning and DNNs.

Generalization Analysis for Deep Contrastive Representation Learning

TL;DR

This work analyzes the generalization of Deep Contrastive Representation Learning (DCRL) by deriving bounds on the unsupervised risk for neural-network representations. It replaces prior depth-dependent Frobenius-norm bounds with covering-number-based bounds that rely on spectral-type complexity, and introduces loss-augmentation techniques to lessen dependence on network depth and the number of negative samples . The paper presents a basic covering-number bound, loss-augmentation bounds (including all-activation variants), and a parameter-counting bound that scales with the total number of parameters, plus a downstream supervised bound via mean classifiers. Empirically, on MNIST, the proposed bounds outperform prior results across varying depths and widths and remain robust to larger negative-sample regimes, suggesting stronger theoretical guarantees that align with practical CRL performance.

Abstract

In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles. On the one hand, we derive a parameter-counting bound that scales with the overall size of the neural networks. On the other hand, we provide a norm-based bound that scales with the norms of neural networks' weight matrices. Ignoring logarithmic factors, the bounds are independent of , the size of the tuples provided for contrastive learning. To the best of our knowledge, this property is only shared by one other work, which employed a different proof strategy and suffers from very strong exponential dependence on the depth of the network which is due to a use of the peeling technique. Our results circumvent this by leveraging powerful results on covering numbers with respect to uniform norms over samples. In addition, we utilize loss augmentation techniques to further reduce the dependency on matrix norms and the implicit dependence on network depth. In fact, our techniques allow us to produce many bounds for the contrastive learning setting with similar architectural dependencies as in the study of the sample complexity of ordinary loss functions, thereby bridging the gap between the learning theories of contrastive learning and DNNs.

Paper Structure

This paper contains 30 sections, 27 theorems, 203 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Main Results
  5. Contributions
  6. Notations
  7. Basic Bound
  8. Loss Augmentation
  9. Parameter Counting Bound
  10. Downstream Classification
  11. Experiments
  12. Conclusion and Further Works
  13. Acknowledgements
  14. Table of Notations
  15. Rademacher Complexity
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\ell:\mathbb{R}^k\to[0, M]$ be a loss function that is $\ell^\infty$-Lipschitz with constant $\eta>0$. Then, for any $F_{\bf A}\in\mathcal{F_A}$ and $\delta\in(0,1)$, the following bound holds with probability of at least $1-\delta$: where $W = \max_{1\le l \le L}d_l$ (maximum hidden width), $B_x=\sup_{x\in\mathcal{X}}\|x\|_2$, and the $\mathcal{\tilde{O}}$ notation hides logarithmic factors

Figures (2)

  • Figure 1: Graphical comparison of our results to that of previous works article:arora2019theoreticalarticle:lei2023generalization. The generalization bounds for all results have their logarithmic terms, constants ($\eta, \rho_i, \dots$) and ${\mathcal{O}}(\sqrt{\log 1/\delta})$ terms truncated. We present the comparison at varying depths (Left) and hidden layer's dimensions (Right).
  • Figure J.1: Graphical comparison of our results to that of previous works article:arora2019theoreticalarticle:lei2023generalization. The generalization bounds for all results have their logarithmic terms, constants ($\eta, \rho_i, \dots$) and ${\mathcal{O}}(\sqrt{\log 1/\delta})$ terms truncated. We present the comparison at varying depths (Left) and hidden layer's dimensions (Right).

Theorems & Definitions (55)

  • Definition 1
  • Remark 1
  • Definition 2: Lipschitz continuity
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Theorem 4
  • Definition 3
  • ...and 45 more