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Explaining Decisions in ML Models: a Parameterized Complexity Analysis (Part I)

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki, Stefan Szeider

TL;DR

The paper investigates the parameterized complexity of explanation problems (abductive and contrastive, local and global) for transparent ML models such as decision trees, decision sets, decision lists, and their ensembles. It develops a unifying algorithmic framework based on Boolean circuits and MSO/MSOE$_1$ logic to obtain fixed-parameter tractable results, and introduces a broad hardness framework via set-modelling and subset-modelling to derive NP- and W-hardness across model classes. The results reveal a nuanced landscape: certain explanations for DTs (notably local contrastive) are tractable, while ensemble-based explanations quickly become intractable, with tractability in DS/DL obtainable only under tight parameter bounds. The work also provides general meta-tools for analyzing new model classes and highlights practical implications for explainable AI system design, including guidance on when efficient explanations are feasible. Part II extends these ideas to binary decision diagrams and circuit-based models, underscoring the broad applicability of the proposed frameworks.

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, 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 (Part I)

TL;DR

The paper investigates the parameterized complexity of explanation problems (abductive and contrastive, local and global) for transparent ML models such as decision trees, decision sets, decision lists, and their ensembles. It develops a unifying algorithmic framework based on Boolean circuits and MSO/MSOE logic to obtain fixed-parameter tractable results, and introduces a broad hardness framework via set-modelling and subset-modelling to derive NP- and W-hardness across model classes. The results reveal a nuanced landscape: certain explanations for DTs (notably local contrastive) are tractable, while ensemble-based explanations quickly become intractable, with tractability in DS/DL obtainable only under tight parameter bounds. The work also provides general meta-tools for analyzing new model classes and highlights practical implications for explainable AI system design, including guidance on when efficient explanations are feasible. Part II extends these ideas to binary decision diagrams and circuit-based models, underscoring the broad applicability of the proposed frameworks.

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, 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

This paper contains 21 sections, 34 theorems, 10 equations, 1 figure, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parameterized Complexity (PC).
  4. Graphs, Rank-Width, Treewidth and Pathwidth.
  5. Graphs, Rank-Width, Treewidth and Pathwidth.
  6. Examples and Models
  7. Decision Trees.
  8. Decision Sets.
  9. Decision Lists.
  10. Ensembles
  11. Considered Problems and Parameters
  12. Overview of Results
  13. Algorithmic Results
  14. A Meta-Theorem for Boolean Circuits
  15. DTs and their Ensembles
  16. ...and 6 more sections

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 rank-width $r$, 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 (61)

  • 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 51 more