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From Robustness to Explainability and Back Again

Xuanxiang Huang, Joao Marques-Silva

TL;DR

This paper tackles the scalability barrier of formal explainability by linking it to robustness analysis. It generalizes abductive and contrastive explanations to distance-restricted settings under a norm $l_p$ with distance threshold $\epsilon$, enabling the use of robustness tools as oracles to compute explanations. A duality between AXp and CXp, via minimal hitting sets, is extended to the distance-restricted case, and two practical algorithms (linear-search and QuickXplain-based) are proposed to compute distance-restricted AXp/CXp using robustness oracles. Experiments on ACAS Xu DNNs with hundreds of ReLU units demonstrate substantial scalability gains, showing that formal explanations can be obtained for larger models by leveraging robustness tooling, with implications for a broad range of ML classifiers.

Abstract

Formal explainability guarantees the rigor of computed explanations, and so it is paramount in domains where rigor is critical, including those deemed high-risk. Unfortunately, since its inception formal explainability has been hampered by poor scalability. At present, this limitation still holds true for some families of classifiers, the most significant being deep neural networks. This paper addresses the poor scalability of formal explainability and proposes novel efficient algorithms for computing formal explanations. The novel algorithm computes explanations by answering instead a number of robustness queries, and such that the number of such queries is at most linear on the number of features. Consequently, the proposed algorithm establishes a direct relationship between the practical complexity of formal explainability and that of robustness. To achieve the proposed goals, the paper generalizes the definition of formal explanations, thereby allowing the use of robustness tools that are based on different distance norms, and also by reasoning in terms of some target degree of robustness. Preliminary experiments validate the practical efficiency of the proposed approach.

From Robustness to Explainability and Back Again

TL;DR

This paper tackles the scalability barrier of formal explainability by linking it to robustness analysis. It generalizes abductive and contrastive explanations to distance-restricted settings under a norm with distance threshold , enabling the use of robustness tools as oracles to compute explanations. A duality between AXp and CXp, via minimal hitting sets, is extended to the distance-restricted case, and two practical algorithms (linear-search and QuickXplain-based) are proposed to compute distance-restricted AXp/CXp using robustness oracles. Experiments on ACAS Xu DNNs with hundreds of ReLU units demonstrate substantial scalability gains, showing that formal explanations can be obtained for larger models by leveraging robustness tooling, with implications for a broad range of ML classifiers.

Abstract

Formal explainability guarantees the rigor of computed explanations, and so it is paramount in domains where rigor is critical, including those deemed high-risk. Unfortunately, since its inception formal explainability has been hampered by poor scalability. At present, this limitation still holds true for some families of classifiers, the most significant being deep neural networks. This paper addresses the poor scalability of formal explainability and proposes novel efficient algorithms for computing formal explanations. The novel algorithm computes explanations by answering instead a number of robustness queries, and such that the number of such queries is at most linear on the number of features. Consequently, the proposed algorithm establishes a direct relationship between the practical complexity of formal explainability and that of robustness. To achieve the proposed goals, the paper generalizes the definition of formal explanations, thereby allowing the use of robustness tools that are based on different distance norms, and also by reasoning in terms of some target degree of robustness. Preliminary experiments validate the practical efficiency of the proposed approach.
Paper Structure (22 sections, 9 theorems, 12 equations, 5 figures, 5 tables, 3 algorithms)

This paper contains 22 sections, 9 theorems, 12 equations, 5 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal hitting sets.
  4. Norm $l_{p}$.
  5. Classification problems.
  6. Robustness.
  7. Logic-based explanations.
  8. Distance-Restricted Explanations
  9. Definitions.
  10. Properties.
  11. From Robustness to Explainability
  12. Required properties of robustness tool.
  13. Computing distance-restricted AXp's.
  14. Relaxing property \ref{['it:robt:03']}.
  15. Contrastive explanations & enumeration.
  16. ...and 7 more sections

Key Result

Proposition 1

Given an explanation problem ${\mathcal{E}}$, and norm $p$ and a value $\epsilon>0$ then,

Figures (5)

  • Figure 1: Runtime for ACASXU_1
  • Figure 2: Runtime for ACASXU_2
  • Figure 3: Runtime for ACASXU_3
  • Figure 4: Runtime for ACASXU_4
  • Figure 5: Runtime for ACASXU_5

Theorems & Definitions (20)

  • Example 1
  • Example 2
  • Example 3
  • Proposition 1: MHS Duality between AXp's and CXp's
  • Definition 1: Distance-restricted (W)AXp, $\epsilon$-(W)AXp
  • Definition 2: Distance-restricted (W)CXp, $\epsilon$-(W)CXp
  • Example 4
  • Remark 1
  • Proposition 2
  • Proposition 3
  • ...and 10 more