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Probabilistic Lipschitzness and the Stable Rank for Comparing Explanation Models

Lachlan Simpson, Kyle Millar, Adriel Cheng, Cheng-Chew Lim, Hong Gunn Chew

TL;DR

This work proves theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad, and proposes a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models.

Abstract

Explainability models are now prevalent within machine learning to address the black-box nature of neural networks. The question now is which explainability model is most effective. Probabilistic Lipschitzness has demonstrated that the smoothness of a neural network is fundamentally linked to the quality of post hoc explanations. In this work, we prove theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad. We propose a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models. Further, we prove a link between the local Lipschitz constant of a neural network and its stable rank. We then demonstrate that the stable rank of a neural network provides a heuristic for the robustness of explainability models.

Probabilistic Lipschitzness and the Stable Rank for Comparing Explanation Models

TL;DR

This work proves theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad, and proposes a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models.

Abstract

Explainability models are now prevalent within machine learning to address the black-box nature of neural networks. The question now is which explainability model is most effective. Probabilistic Lipschitzness has demonstrated that the smoothness of a neural network is fundamentally linked to the quality of post hoc explanations. In this work, we prove theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad. We propose a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models. Further, we prove a link between the local Lipschitz constant of a neural network and its stable rank. We then demonstrate that the stable rank of a neural network provides a heuristic for the robustness of explainability models.
Paper Structure (23 sections, 3 theorems, 80 equations, 7 figures, 6 tables)

This paper contains 23 sections, 3 theorems, 80 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation and Assumptions
  3. Related Work
  4. Probabilistic Lipschitzness and Astuteness
  5. Astuteness Theoretical Guarantee
  6. Integrated Gradients
  7. LIME
  8. SmoothGrad
  9. Normalised Astuteness as a Metric for Comparing Explanation Models
  10. Robustness metrics and criteria for an adequate metric
  11. The limitations of current robustness metrics
  12. Normalised astuteness as a robustness metric
  13. The efficacy of normalised astuteness as a robustness metric
  14. A comparison of Integrated Gradients and SHAP explanations on the XOR problem
  15. A comparison of normalised astuteness with LLE and AS
  16. ...and 8 more sections

Key Result

Theorem 4.2

Suppose $f$ is probabilistic $L$-Lipschitz with probability $\geq 1 - \alpha$. Take two points $x,y \sim \mathcal{D}$ such that $d_{p}(x,y) \leq r$ and a base-point $x' \sim \mathcal{D}$. Then for integrated gradients we have $A_{r, \lambda}(\mathop{\mathrm{IG}}\nolimits) \geq 1- \alpha$, where,

Figures (7)

  • Figure 1: Integrated Gradients and SHAP explanation vectors on XOR problem. Direction and magnitude of explanation vectors indicate feature contribution to a prediction.
  • Figure 2: SHAP and Integrated Gradients normalised Astuteness curves on XOR problem.
  • Figure 3: SHAP and Integrated Gradients Local Lipschitz Estimate distributions on XOR problem.
  • Figure 4: Astuteness curves for autoencoders with Gaussian activations trained on MNIST.
  • Figure 5: Autoencoder predictions with Peak Signal-to-Noise Ratio (PSNR) and stable rank $\mathcal{S}$. Left: Ground Truth, Middle: Sharp Autoencoder, Right: Distorted Autoencoder.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Definition 4.3
  • Theorem 4.4
  • proof
  • Definition 4.5
  • Theorem 4.6
  • ...and 3 more