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On the Comparison between Multi-modal and Single-modal Contrastive Learning

Wei Huang, Andi Han, Yongqiang Chen, Yuan Cao, Zhiqiang Xu, Taiji Suzuki

TL;DR

This work introduces a feature learning theory framework that provides a theoretical foundation for understanding the differences between multi-modal and single-modal contrastive learning and provides a unified framework that can characterize the optimization and generalization of both single-modal and multi-modal contrastive learning.

Abstract

Multi-modal contrastive learning with language supervision has presented a paradigm shift in modern machine learning. By pre-training on a web-scale dataset, multi-modal contrastive learning can learn high-quality representations that exhibit impressive robustness and transferability. Despite its empirical success, the theoretical understanding is still in its infancy, especially regarding its comparison with single-modal contrastive learning. In this work, we introduce a feature learning theory framework that provides a theoretical foundation for understanding the differences between multi-modal and single-modal contrastive learning. Based on a data generation model consisting of signal and noise, our analysis is performed on a ReLU network trained with the InfoMax objective function. Through a trajectory-based optimization analysis and generalization characterization on downstream tasks, we identify the critical factor, which is the signal-to-noise ratio (SNR), that impacts the generalizability in downstream tasks of both multi-modal and single-modal contrastive learning. Through the cooperation between the two modalities, multi-modal learning can achieve better feature learning, leading to improvements in performance in downstream tasks compared to single-modal learning. Our analysis provides a unified framework that can characterize the optimization and generalization of both single-modal and multi-modal contrastive learning. Empirical experiments on both synthetic and real-world datasets further consolidate our theoretical findings.

On the Comparison between Multi-modal and Single-modal Contrastive Learning

TL;DR

This work introduces a feature learning theory framework that provides a theoretical foundation for understanding the differences between multi-modal and single-modal contrastive learning and provides a unified framework that can characterize the optimization and generalization of both single-modal and multi-modal contrastive learning.

Abstract

Multi-modal contrastive learning with language supervision has presented a paradigm shift in modern machine learning. By pre-training on a web-scale dataset, multi-modal contrastive learning can learn high-quality representations that exhibit impressive robustness and transferability. Despite its empirical success, the theoretical understanding is still in its infancy, especially regarding its comparison with single-modal contrastive learning. In this work, we introduce a feature learning theory framework that provides a theoretical foundation for understanding the differences between multi-modal and single-modal contrastive learning. Based on a data generation model consisting of signal and noise, our analysis is performed on a ReLU network trained with the InfoMax objective function. Through a trajectory-based optimization analysis and generalization characterization on downstream tasks, we identify the critical factor, which is the signal-to-noise ratio (SNR), that impacts the generalizability in downstream tasks of both multi-modal and single-modal contrastive learning. Through the cooperation between the two modalities, multi-modal learning can achieve better feature learning, leading to improvements in performance in downstream tasks compared to single-modal learning. Our analysis provides a unified framework that can characterize the optimization and generalization of both single-modal and multi-modal contrastive learning. Empirical experiments on both synthetic and real-world datasets further consolidate our theoretical findings.

Paper Structure

This paper contains 33 sections, 44 theorems, 221 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setting
  4. Notation.
  5. Data Model.
  6. Single-modal Contrastive Learning
  7. Multi-modal Contrastive Learning
  8. Downstream Task Evaluation
  9. Main Results
  10. Proof Roadmap
  11. Proof Sketch for Single Modal Contrastive Learning
  12. Proof Sketch for Multi-Modal Learning
  13. Experiments
  14. Synthetic experiments
  15. Real-world experiments
  16. ...and 18 more sections

Key Result

Theorem 4.2

Under the single-modal learning setup, suppose Assumption assumption holds. Then after $T^* = \widetilde{\Theta}(\eta^{-1}mn \sigma_\xi^{-2} d^{-1} + \eta^{-1} m n \sigma_\xi^{-2} d^{-1} \epsilon^{-1})$, the with probability at least $1-1/d$, it holds that (1) Training error $L(T^\ast) \le \epsilon

Figures (1)

  • Figure 1: Training loss, test accuracy, signal learning and noise memorization of single-modal and multi-modal contrastive learning.

Theorems & Definitions (81)

  • Theorem 4.2: Single-Modal Contrastive Learning
  • Theorem 4.3: Multi-Modal Contrastive Learning
  • Lemma 5.1: Single-modal Contrastive Learning
  • Lemma 5.2
  • Lemma 5.3: Multi-Modal
  • Lemma 5.4
  • Lemma B.1
  • proof
  • Lemma B.2
  • proof : Proof of Lemma \ref{['lemma:init_innermu']}
  • ...and 71 more