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When is the Computation of a Feature Attribution Method Tractable?

P. Barceló, R. Cominetti, M. Morgado

TL;DR

This work systematically analyzes when feature-attribution power indices beyond SHAP are tractable, showing that simple cardinality-based indices are computable in polynomial time via reductions to expected-value evaluations and polynomial interpolation. It introduces Bernoulli power indices, demonstrating a two-expectation reduction, and extends the tractability framework to interaction indices, with cardinality-based and Bernoulli variants requiring a controlled number of expected-value evaluations. The results reveal that the computational burden of many indices mirrors the burden of evaluating expectations, and provide practical pathways (via Vandermonde systems and multivariate interpolation) to recover target quantities efficiently. These findings inform the selection of attribution indices in practice by balancing interpretability with computational feasibility, especially for models where expectation computations are tractable.

Abstract

Feature attribution methods have become essential for explaining machine learning models. Many popular approaches, such as SHAP and Banzhaf values, are grounded in power indices from cooperative game theory, which measure the contribution of features to model predictions. This work studies the computational complexity of power indices beyond SHAP, addressing the conditions under which they can be computed efficiently. We identify a simple condition on power indices that ensures that computation is polynomially equivalent to evaluating expected values, extending known results for SHAP. We also introduce Bernoulli power indices, showing that their computation can be simplified to a constant number of expected value evaluations. Furthermore, we explore interaction power indices that quantify the importance of feature subsets, proving that their computation complexity mirrors that of individual features.

When is the Computation of a Feature Attribution Method Tractable?

TL;DR

This work systematically analyzes when feature-attribution power indices beyond SHAP are tractable, showing that simple cardinality-based indices are computable in polynomial time via reductions to expected-value evaluations and polynomial interpolation. It introduces Bernoulli power indices, demonstrating a two-expectation reduction, and extends the tractability framework to interaction indices, with cardinality-based and Bernoulli variants requiring a controlled number of expected-value evaluations. The results reveal that the computational burden of many indices mirrors the burden of evaluating expectations, and provide practical pathways (via Vandermonde systems and multivariate interpolation) to recover target quantities efficiently. These findings inform the selection of attribution indices in practice by balancing interpretability with computational feasibility, especially for models where expectation computations are tractable.

Abstract

Feature attribution methods have become essential for explaining machine learning models. Many popular approaches, such as SHAP and Banzhaf values, are grounded in power indices from cooperative game theory, which measure the contribution of features to model predictions. This work studies the computational complexity of power indices beyond SHAP, addressing the conditions under which they can be computed efficiently. We identify a simple condition on power indices that ensures that computation is polynomially equivalent to evaluating expected values, extending known results for SHAP. We also introduce Bernoulli power indices, showing that their computation can be simplified to a constant number of expected value evaluations. Furthermore, we explore interaction power indices that quantify the importance of feature subsets, proving that their computation complexity mirrors that of individual features.
Paper Structure (18 sections, 7 theorems, 52 equations)

This paper contains 18 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Feature attribution methods.
  3. Power indices.
  4. Previous results.
  5. Context.
  6. Results on simple power indices.
  7. Results on Bernoulli power indices.
  8. Results on interaction power indices.
  9. Organization of the paper.
  10. Notations.
  11. Cardinality-based indices
  12. From expected values to power indices
  13. From power indices to expected values
  14. Bernoulli indices
  15. Interaction Indices
  16. ...and 3 more sections

Key Result

Lemma 1

Let $G:\Omega\to{\mathbb R}$ and $c_k=\sum_{S\in\mathcal{P}_k}\!{\mathbb E}[G|S]$ with $\mathcal{P}_k$ the family of all subsets of $N$ of size $k$. For $z\geq 0$ let $Y^z=(Y_1^z,\ldots,Y_n^z)$ be independent random variables with Then $\sum_{k=0}^{n}c_kz^k=(1\!+\!z)^{n}\,{\mathbb E}[G(Y^z)]$.

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Example 1
  • Theorem 5
  • ...and 5 more