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An Information-Theoretic Perspective on Variance-Invariance-Covariance Regularization

Ravid Shwartz-Ziv, Randall Balestriero, Kenji Kawaguchi, Tim G. J. Rudner, Yann LeCun

TL;DR

An information-theoretic perspective on the VICReg objective is presented, deriving information-theoretic quantities for deterministic networks as an alternative to unrealistic stochastic network assumptions and relating the optimization of the VICReg objective to mutual information optimization.

Abstract

Variance-Invariance-Covariance Regularization (VICReg) is a self-supervised learning (SSL) method that has shown promising results on a variety of tasks. However, the fundamental mechanisms underlying VICReg remain unexplored. In this paper, we present an information-theoretic perspective on the VICReg objective. We begin by deriving information-theoretic quantities for deterministic networks as an alternative to unrealistic stochastic network assumptions. We then relate the optimization of the VICReg objective to mutual information optimization, highlighting underlying assumptions and facilitating a constructive comparison with other SSL algorithms and derive a generalization bound for VICReg, revealing its inherent advantages for downstream tasks. Building on these results, we introduce a family of SSL methods derived from information-theoretic principles that outperform existing SSL techniques.

An Information-Theoretic Perspective on Variance-Invariance-Covariance Regularization

TL;DR

An information-theoretic perspective on the VICReg objective is presented, deriving information-theoretic quantities for deterministic networks as an alternative to unrealistic stochastic network assumptions and relating the optimization of the VICReg objective to mutual information optimization.

Abstract

Variance-Invariance-Covariance Regularization (VICReg) is a self-supervised learning (SSL) method that has shown promising results on a variety of tasks. However, the fundamental mechanisms underlying VICReg remain unexplored. In this paper, we present an information-theoretic perspective on the VICReg objective. We begin by deriving information-theoretic quantities for deterministic networks as an alternative to unrealistic stochastic network assumptions. We then relate the optimization of the VICReg objective to mutual information optimization, highlighting underlying assumptions and facilitating a constructive comparison with other SSL algorithms and derive a generalization bound for VICReg, revealing its inherent advantages for downstream tasks. Building on these results, we introduce a family of SSL methods derived from information-theoretic principles that outperform existing SSL techniques.
Paper Structure (46 sections, 11 theorems, 95 equations, 5 figures, 3 tables)

This paper contains 46 sections, 11 theorems, 95 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background & Preliminaries
  3. Continuous Piecewise Affine (CPA) Mappings.
  4. Deep Neural Networks as CPA Mappings.
  5. Self-Supervised Learning.
  6. Variance-Invariance-Covariance Regularization (VICReg).
  7. Deep Neural Networks and Information Theory
  8. Self-Supervised Learning in DNNs: An Information-Theoretic Perspective
  9. Self-Supervised Learning from an Information-Theoretic Viewpoint
  10. Understanding the Data Distribution Hypothesis
  11. Data Distribution Under the Deep Neural Network Transformation
  12. Information Optimization and the VICReg Optimization Objective
  13. Variance-Invariance-Covariance Regularization: An Information-Theoretic Perspective
  14. Empirical Validation of Assumptions About Data Distributions
  15. Self-Supervised Learning Models through Information Maximization
  16. ...and 31 more sections

Key Result

Theorem 1

Given the setting of eq:x_density, the unconditional DNN output density, $Z$, can be approximated as a mixture of the affinely transformed distributions $\boldsymbol{x}|\boldsymbol{x}^*_{n(\boldsymbol{x})}$: where $\omega(\boldsymbol{x}^*_{n})=\omega \in \Omega \iff \boldsymbol{x}^*_{n} \in \omega$ is the partition region in which the prototype $\boldsymbol{x}^*_{n}$ lives in.

Figures (5)

  • Figure 1: Left:The network output for SSL training is more Gaussian for small input noise. The $p$-value of the normality test for different SSL models trained on ImageNet for different input noise levels. The dashed line represents the point at which the null hypothesis (Gaussian distribution) can be rejected with $99\%$ confidence. Right: The Gaussians around each point are not overlapping. The plots show the $l2$ distances between raw images for different datasets. As can be seen, the distances are largest for more complex real-world datasets.
  • Figure 2: VICReg has higher Entropy during training. The entropy along the training for different SSL methods. Experiments were conducted with ResNet-18 on CIFAR-10. Error bars represent one standard error over 5 trials.
  • Figure 3: The optimal solution for the optimization problem is a diagonal matrix. The average distance from a diagonal matrix for different perturbation scales. Experiments were conducted on CIFAR-10 with the ResNet-18 network.
  • Figure 4: Evolution of GMM training when enforcing a one-to-one mapping between the data and centroids akin to K-means i.e. using a small and fixed covariance matrix. We see that collapse does not occur. Left - In the presence of fixed input samples, we observe that there is no collapsing and that the entropy of the centers is high. Right - when we make the input samples trainable and optimize their location, all the points collapse into a single point, resulting in a sharp decrease in entropy.
  • Figure 5: Our generalization bound predicts more accurately the generalization gap in the loss.(left) Our SSL VICReg generalization bound outperforms state-of-the-art supervised generalization bounds. (right) Strong correlation between the generalization gap and our generalization bound for VICReg. Pearson correlation - 0.9633. Conducted on CIFAR-10.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma G.1
  • Lemma G.2
  • Lemma J.1
  • Lemma J.2
  • Lemma J.3
  • Lemma J.4
  • ...and 1 more