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Local vs. Global Interpretability: A Computational Complexity Perspective

Shahaf Bassan, Guy Amir, Guy Katz

TL;DR

This work formalizes interpretability through the lens of computational complexity, introducing a duality between local and global explanations and proving a uniqueness property for global explanations. It provides rigorous complexity results for three model classes—FBDDs (decision-tree-like structures), perceptrons (linear classifiers), and MLPs (neural networks)—across multiple explanation forms (MSR, CSR, FN, FR, CC). The findings reveal surprising separations: linear models can be globally harder to explain than locally, while neural networks and decision trees often reverse this trend, with global explanations sometimes being tractable when local ones are not. These results highlight how a complexity-theoretic view can sharpen our understanding of model interpretability and guide the design of robust, provably explainable systems.

Abstract

The local and global interpretability of various ML models has been studied extensively in recent years. However, despite significant progress in the field, many known results remain informal or lack sufficient mathematical rigor. We propose a framework for bridging this gap, by using computational complexity theory to assess local and global perspectives of interpreting ML models. We begin by proposing proofs for two novel insights that are essential for our analysis: (1) a duality between local and global forms of explanations; and (2) the inherent uniqueness of certain global explanation forms. We then use these insights to evaluate the complexity of computing explanations, across three model types representing the extremes of the interpretability spectrum: (1) linear models; (2) decision trees; and (3) neural networks. Our findings offer insights into both the local and global interpretability of these models. For instance, under standard complexity assumptions such as P != NP, we prove that selecting global sufficient subsets in linear models is computationally harder than selecting local subsets. Interestingly, with neural networks and decision trees, the opposite is true: it is harder to carry out this task locally than globally. We believe that our findings demonstrate how examining explainability through a computational complexity lens can help us develop a more rigorous grasp of the inherent interpretability of ML models.

Local vs. Global Interpretability: A Computational Complexity Perspective

TL;DR

This work formalizes interpretability through the lens of computational complexity, introducing a duality between local and global explanations and proving a uniqueness property for global explanations. It provides rigorous complexity results for three model classes—FBDDs (decision-tree-like structures), perceptrons (linear classifiers), and MLPs (neural networks)—across multiple explanation forms (MSR, CSR, FN, FR, CC). The findings reveal surprising separations: linear models can be globally harder to explain than locally, while neural networks and decision trees often reverse this trend, with global explanations sometimes being tractable when local ones are not. These results highlight how a complexity-theoretic view can sharpen our understanding of model interpretability and guide the design of robust, provably explainable systems.

Abstract

The local and global interpretability of various ML models has been studied extensively in recent years. However, despite significant progress in the field, many known results remain informal or lack sufficient mathematical rigor. We propose a framework for bridging this gap, by using computational complexity theory to assess local and global perspectives of interpreting ML models. We begin by proposing proofs for two novel insights that are essential for our analysis: (1) a duality between local and global forms of explanations; and (2) the inherent uniqueness of certain global explanation forms. We then use these insights to evaluate the complexity of computing explanations, across three model types representing the extremes of the interpretability spectrum: (1) linear models; (2) decision trees; and (3) neural networks. Our findings offer insights into both the local and global interpretability of these models. For instance, under standard complexity assumptions such as P != NP, we prove that selecting global sufficient subsets in linear models is computationally harder than selecting local subsets. Interestingly, with neural networks and decision trees, the opposite is true: it is harder to carry out this task locally than globally. We believe that our findings demonstrate how examining explainability through a computational complexity lens can help us develop a more rigorous grasp of the inherent interpretability of ML models.
Paper Structure (21 sections, 54 theorems, 56 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 21 sections, 54 theorems, 56 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local and Global Explanation Forms
  4. Properties of Global Explanations
  5. Duality of Local and Global Explanations
  6. Uniquness of Global Explanations
  7. The Computational Complexity of Global Interpretation
  8. The Locally Interpretable Case
  9. The Globally Interpretable Case
  10. The Equally Difficult Case
  11. Related Work
  12. Conclusion
  13. Global forms of model explanations
  14. Model Classes
  15. The Duality of Local and Global Explanations
  16. ...and 6 more sections

Key Result

Theorem 1

A feature $i$ is necessary with respect to $\langle f,\textbf{x}\rangle$ if and only if $\{i\}$ is a contrastive reason of $\langle f,\textbf{x}\rangle$.

Figures (1)

  • Figure 1: Illustration of complexity separations between local and global explanations. In linear models, it is harder to identify the smallest global sufficient subset (highlighted in gray) compared to a local one (highlighted in blue). Interestingly, this reverses in neural networks and decision trees, where selecting the smallest global sufficient subsets is computationally simpler than finding the smallest local ones.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Theorem 4
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Definition 1
  • ...and 45 more