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On the Tractability of SHAP Explanations under Markovian Distributions

Reda Marzouk, Colin de La Higuera

TL;DR

This work addresses the computational intractability of SHAP explanations by relaxing feature independence to Markovian distributions and proving tractability for sequential models. It develops a constructive, language-based framework using Weighted Automata and Weighted Transducers to compute SHAP scores in polynomial time, first for WA under MARKOV and then via reductions for disjoint DNFs and decision trees. The key insight is to decompose SHAP into SHAP1 and SHAP2, express them through language operators, and build polynomial-time WA/WT constructions (aided by Bayes’ rule) that realize the required probabilities. Collectively, the results extend SHAP tractability beyond independence, offering scalable algorithms and laying groundwork for generalized Markovian explanations in XAI with potential extensions to higher-order models.

Abstract

Thanks to its solid theoretical foundation, the SHAP framework is arguably one the most widely utilized frameworks for local explainability of ML models. Despite its popularity, its exact computation is known to be very challenging, proven to be NP-Hard in various configurations. Recent works have unveiled positive complexity results regarding the computation of the SHAP score for specific model families, encompassing decision trees, random forests, and some classes of boolean circuits. Yet, all these positive results hinge on the assumption of feature independence, often simplistic in real-world scenarios. In this article, we investigate the computational complexity of the SHAP score by relaxing this assumption and introducing a Markovian perspective. We show that, under the Markovian assumption, computing the SHAP score for the class of Weighted automata, Disjoint DNFs and Decision Trees can be performed in polynomial time, offering a first positive complexity result for the problem of SHAP score computation that transcends the limitations of the feature independence assumption.

On the Tractability of SHAP Explanations under Markovian Distributions

TL;DR

This work addresses the computational intractability of SHAP explanations by relaxing feature independence to Markovian distributions and proving tractability for sequential models. It develops a constructive, language-based framework using Weighted Automata and Weighted Transducers to compute SHAP scores in polynomial time, first for WA under MARKOV and then via reductions for disjoint DNFs and decision trees. The key insight is to decompose SHAP into SHAP1 and SHAP2, express them through language operators, and build polynomial-time WA/WT constructions (aided by Bayes’ rule) that realize the required probabilities. Collectively, the results extend SHAP tractability beyond independence, offering scalable algorithms and laying groundwork for generalized Markovian explanations in XAI with potential extensions to higher-order models.

Abstract

Thanks to its solid theoretical foundation, the SHAP framework is arguably one the most widely utilized frameworks for local explainability of ML models. Despite its popularity, its exact computation is known to be very challenging, proven to be NP-Hard in various configurations. Recent works have unveiled positive complexity results regarding the computation of the SHAP score for specific model families, encompassing decision trees, random forests, and some classes of boolean circuits. Yet, all these positive results hinge on the assumption of feature independence, often simplistic in real-world scenarios. In this article, we investigate the computational complexity of the SHAP score by relaxing this assumption and introducing a Markovian perspective. We show that, under the Markovian assumption, computing the SHAP score for the class of Weighted automata, Disjoint DNFs and Decision Trees can be performed in polynomial time, offering a first positive complexity result for the problem of SHAP score computation that transcends the limitations of the feature independence assumption.
Paper Structure (23 sections, 15 theorems, 56 equations, 1 figure)

This paper contains 23 sections, 15 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Weighted Automata/Transducers
  4. The SHAP score.
  5. The problem $\texttt{SHAP}(\texttt{WA}, \texttt{MARKOV})$ is in FP.
  6. Step 1: A decomposition of the problem $\texttt{SHAP}(\texttt{WA}, \texttt{MARKOV})$.
  7. Step 2: $\texttt{SHAP}_{1}(\texttt{WA}, \texttt{MARKOV})$ and $\texttt{SHAP}_{2}(\texttt{WA}, \texttt{MARKOV})$ are in FP.
  8. Step 2.a: Computation $\text{SHAP}_{1},~\text{SHAP}_{2}$ in terms of language operators.
  9. Step 2.b: Construction of WAs/WTs that compute $f_{w,k},~g_{w,P}^{(1)},~g_{w,i,P}^{(2)}$
  10. $\texttt{SHAP}(\texttt{D-DNF}, \texttt{MARKOV})$ and $\texttt{SHAP}(\texttt{DT}, \texttt{MARKOV})$ are in FP
  11. Proof of theorem \ref{['ddnftheorem']}: Reduction strategy
  12. Conclusion
  13. Proof lemma \ref{['operators']}
  14. Proof lemma \ref{['ffdist:mainlemma']}
  15. Proof lemma \ref{['complex']}
  16. ...and 8 more sections

Key Result

Lemma 2.4

Fix two finite alphabets $\Sigma,~\Delta$.

Figures (1)

  • Figure 1: A DFT $T_{i}$ that computes the seq2seq language $g(w',p) = I_{L_{\texttt{swap}(p,i)}}(w')$ for $i = 3$. $\sigma$ (resp. $\sigma'$) refers to any symbol in $\Sigma$ (resp. $\Sigma_{\#}$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 18 more