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Explaining Decisions in ML Models: a Parameterized Complexity Analysis

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki, Stefan Szeider

TL;DR

The paper studies the parameterized complexity of explanation problems (abductive and contrastive, local and global) for transparent ML models, including DTs, DSs, DLs, OBDDs, and ensembles. It introduces a circuit-based framework, translating models to Boolean circuits and employing Monadic Second-Order logic (MSO$_1$/MSOE$_1$) to obtain fixed-parameter tractability results, notably via a meta-theorem for circuits with a bounded number of MAJ gates and rankwidth parameter. The work delivers concrete algorithmic results (polynomial-time, XP, and fixed-parameter tractable) for several model families and explanation variants, alongside comprehensive hardness results (NP-hard, co-NP-hard, and $W[1]$-hard) across parameter regimes. These findings illuminate the tractability frontier of explainability in XAI and guide the development of efficient, regulator-relevant explanation methods while identifying robust barriers in more complex or less transparent models.

Abstract

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random Forests, and Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.

Explaining Decisions in ML Models: a Parameterized Complexity Analysis

TL;DR

The paper studies the parameterized complexity of explanation problems (abductive and contrastive, local and global) for transparent ML models, including DTs, DSs, DLs, OBDDs, and ensembles. It introduces a circuit-based framework, translating models to Boolean circuits and employing Monadic Second-Order logic (MSO/MSOE) to obtain fixed-parameter tractability results, notably via a meta-theorem for circuits with a bounded number of MAJ gates and rankwidth parameter. The work delivers concrete algorithmic results (polynomial-time, XP, and fixed-parameter tractable) for several model families and explanation variants, alongside comprehensive hardness results (NP-hard, co-NP-hard, and -hard) across parameter regimes. These findings illuminate the tractability frontier of explainability in XAI and guide the development of efficient, regulator-relevant explanation methods while identifying robust barriers in more complex or less transparent models.

Abstract

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random Forests, and Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.
Paper Structure (14 sections, 45 theorems, 10 equations, 1 figure, 4 tables, 4 algorithms)

This paper contains 14 sections, 45 theorems, 10 equations, 1 figure, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Considered Problems and Parameters
  4. Overview of Results
  5. Algorithmic Results
  6. A Meta-Theorem for Boolean Circuits
  7. DTs and their Ensembles
  8. DSs, DLs and their Ensembles
  9. OBDDs and their Ensembles
  10. Hardness Results
  11. DTs and their Ensembles
  12. DSs, DLs and their Ensembles
  13. OBDDs and their Ensembles
  14. Conclusion

Key Result

Lemma 1

Let $G=(V,E)$ be a directed graph and $X \subseteq V$. The treewidth of $G$ is at most $|X|$ plus the treewidth of $G-X$. Furthermore, if $G$ has rankwidth $r$, pathwidth $p$ and treewidth $t$, then $r \leq 3\cdot 2^{t-1}\leq 3\cdot 2^{p-1}$.

Figures (1)

  • Figure 1: Let $L$ be the DL given in the figure and let $e$ be the example given by $e(x)=0$, $e(y)=0$ and $e(z)=1$. Note that $L(e)=0$. It is easy to verify that $\{y,z\}$ is the only local abductive explanation for $e$ in $L$ of size at most 2. Moreover, both $\{y\}$ and $\{z\}$ are minimal local contrastive explanations for $e$ in $L$. Let $\tau_1=\{x\mapsto 1,y \mapsto 1\}$ and $\tau_2=\{ x \mapsto 0,z \mapsto 0\}$ be a partial assignments. Note that $\tau_1$ and $\tau_2$ are minimal global abductive and global contrastive explanations for class $0$ w.r.t. $L$, respectively.

Theorems & Definitions (81)

  • Lemma 1: DBLP:conf/wg/CorneilR01DBLP:journals/jct/OumS06
  • Theorem 2
  • proof
  • Proposition 3: BergougnouxDJ23
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 71 more