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Stability theory of game-theoretic group feature explanations for machine learning models

Alexey Miroshnikov, Konstandinos Kotsiopoulos, Khashayar Filom, Arjun Ravi Kannan

TL;DR

This article studies feature attributions of Machine Learning models originating from linear game values and coalitional values defined as operators on appropriate functional spaces and shows analytically that grouping features this way has a stabilizing effect on the marginal operator on both group and individual levels, and allows for the unification of marginal and conditional explanations.

Abstract

In this article, we study feature attributions of Machine Learning (ML) models originating from linear game values and coalitional values defined as operators on appropriate functional spaces. The main focus is on random games based on the conditional and marginal expectations. The first part of our work formulates a stability theory for these explanation operators by establishing certain bounds for both marginal and conditional explanations. The differences between the two games are then elucidated, such as showing that the marginal explanations can become discontinuous on some naturally-designed domains, while the conditional explanations remain stable. In the second part of our work, group explanation methodologies are devised based on game values with coalition structure, where the features are grouped based on dependencies. We show analytically that grouping features this way has a stabilizing effect on the marginal operator on both group and individual levels, and allows for the unification of marginal and conditional explanations. Our results are verified in a number of numerical experiments where an information-theoretic measure of dependence is used for grouping.

Stability theory of game-theoretic group feature explanations for machine learning models

TL;DR

This article studies feature attributions of Machine Learning models originating from linear game values and coalitional values defined as operators on appropriate functional spaces and shows analytically that grouping features this way has a stabilizing effect on the marginal operator on both group and individual levels, and allows for the unification of marginal and conditional explanations.

Abstract

In this article, we study feature attributions of Machine Learning (ML) models originating from linear game values and coalitional values defined as operators on appropriate functional spaces. The main focus is on random games based on the conditional and marginal expectations. The first part of our work formulates a stability theory for these explanation operators by establishing certain bounds for both marginal and conditional explanations. The differences between the two games are then elucidated, such as showing that the marginal explanations can become discontinuous on some naturally-designed domains, while the conditional explanations remain stable. In the second part of our work, group explanation methodologies are devised based on game values with coalition structure, where the features are grouped based on dependencies. We show analytically that grouping features this way has a stabilizing effect on the marginal operator on both group and individual levels, and allows for the unification of marginal and conditional explanations. Our results are verified in a number of numerical experiments where an information-theoretic measure of dependence is used for grouping.

Paper Structure

This paper contains 58 sections, 49 theorems, 226 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and hypotheses
  4. Explainability and game theory
  5. Relevant works on individual feature attributions
  6. Conditional and marginal game operators
  7. Stability theory of single feature explainers based on linear game values
  8. Stability of marginal explanations on a space equipped with $L^2(P_X)$-norm
  9. Splitting of explanation energy on dependencies
  10. Relevant works in the context of stability
  11. Game value extensions to non-cooperative games
  12. Group explainers with coalition structures
  13. Trivial group explainers
  14. Group explainers based on quotient games
  15. Explainers based on games with coalition structure
  16. ...and 43 more sections

Key Result

Theorem 3.1

Let h, X and $\mathcal{E}^{ \text{\tiny \it CE}}$ be as in Definition def::condoperator. Then:

Figures (17)

  • Figure 1: Predictions and marginal Shapley values for XGBoost ($f_1$) and GBM ($f_2$) models.
  • Figure 2: Global individual and quotient explanations.
  • Figure 3: Gain stability.
  • Figure 4: Individual and quotient marginal explanations.
  • Figure 5: Individual and quotient explanation norms.
  • ...and 12 more figures

Theorems & Definitions (162)

  • Remark 2.1
  • Definition 2.1
  • Definition 2.2: consistency
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.1: properties
  • proof
  • Remark 3.1
  • Corollary 3.1
  • proof
  • ...and 152 more