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Is Smoothness the Key to Robustness? A Comparison of Attention and Convolution Models Using a Novel Metric

Baiyuan Chen

TL;DR

TopoLip is proposed, a metric based on layer-wise analysis that bridges topological data analysis and Lipschitz continuity for robustness evaluation and establishes a connection between architectural design, robustness, and topological properties.

Abstract

Robustness is a critical aspect of machine learning models. Existing robustness evaluation approaches often lack theoretical generality or rely heavily on empirical assessments, limiting insights into the structural factors contributing to robustness. Moreover, theoretical robustness analysis is not applicable for direct comparisons between models. To address these challenges, we propose $\textit{TopoLip}$, a metric based on layer-wise analysis that bridges topological data analysis and Lipschitz continuity for robustness evaluation. TopoLip provides a unified framework for both theoretical and empirical robustness comparisons across different architectures or configurations, and it reveals how model parameters influence the robustness of models. Using TopoLip, we demonstrate that attention-based models typically exhibit smoother transformations and greater robustness compared to convolution-based models, as validated through theoretical analysis and adversarial tasks. Our findings establish a connection between architectural design, robustness, and topological properties.

Is Smoothness the Key to Robustness? A Comparison of Attention and Convolution Models Using a Novel Metric

TL;DR

TopoLip is proposed, a metric based on layer-wise analysis that bridges topological data analysis and Lipschitz continuity for robustness evaluation and establishes a connection between architectural design, robustness, and topological properties.

Abstract

Robustness is a critical aspect of machine learning models. Existing robustness evaluation approaches often lack theoretical generality or rely heavily on empirical assessments, limiting insights into the structural factors contributing to robustness. Moreover, theoretical robustness analysis is not applicable for direct comparisons between models. To address these challenges, we propose , a metric based on layer-wise analysis that bridges topological data analysis and Lipschitz continuity for robustness evaluation. TopoLip provides a unified framework for both theoretical and empirical robustness comparisons across different architectures or configurations, and it reveals how model parameters influence the robustness of models. Using TopoLip, we demonstrate that attention-based models typically exhibit smoother transformations and greater robustness compared to convolution-based models, as validated through theoretical analysis and adversarial tasks. Our findings establish a connection between architectural design, robustness, and topological properties.

Paper Structure

This paper contains 21 sections, 4 theorems, 35 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Robustness metric.
  4. Topological Data Analysis.
  5. Preliminaries
  6. Problem setup
  7. Discrete frameworks
  8. Mean field frameworks
  9. Topological Lipschitzness
  10. Wasserstein Lipschitzness comparison
  11. Attention and convolution
  12. ViT and ResNet
  13. Experimental results
  14. Conclusion
  15. Summary.
  16. ...and 6 more sections

Key Result

Theorem 1

Let $Q,K,V\in\mathbb{R}^{d\times d}$. For any $t>\sqrt{d}$ and $s\geq\sigma\sqrt{2\log2}$, with probability at least $\min\{1-d/t^2, 1-2e^{-s^2/(2\sigma^2)}\}$, and assuming $\|A\|_{op}\geq 2/\sigma^2$, the mean-field single-head attention map $\mathrm{Attn}_{|\mathcal{P}(B_{t\sigma})}$ with paramet Similarly, the Lipschitz constant of mean-field $M$-head attention map $\mathrm{MHAttn}_{|\mathcal{

Figures (10)

  • Figure 1: Absolute change rate of the Wasserstein distance of persistence diagrams of Attns and Convs.
  • Figure 2: Cumulative absolute change rate of the Wasserstein distance of persistence diagrams of Attns and Convs.
  • Figure 3: Absolute change rate of the Wasserstein distance of persistence diagrams of ViTs and ResNets.
  • Figure 4: Cumulative absolute change rate of the Wasserstein distance of persistence diagrams of ViTs and ResNets.
  • Figure 5: Contruction of the Čech complex of the dataset.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Definition 1: Attention layer
  • Definition 2: Convolutional layer
  • Definition 3: Pushforward santambrogio2015optimal
  • Definition 4: Mean-field self-attention castin2024smooth
  • Definition 5: Mean-field convolution
  • Definition 6
  • Definition 7: Lipschitz constant with respect to the 1-Wasserstein distance castin2024smooth
  • Theorem 1
  • Theorem 2
  • Lemma 1: (Lipschitz Constant of Composed Functions gouk2021regularisation)
  • ...and 1 more