Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximation of group explainers with coalition structure using Monte Carlo sampling on the product space of coalitions and features

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

TL;DR

This work tackles the computational bottleneck of game-theoretic explanations by introducing a Monte Carlo framework that estimates marginal, quotient, and coalitional explainers via sampling on appropriate product spaces. By recasting explainers as expectations and leveraging carefully designed sampling measures, the authors achieve estimators with convergence guarantees and error bounds, while reducing complexity to scale linearly with the background dataset size $|ar{D}_X|$. The methodology covers linear and coalitional game values, including two-step Shapley and Owen-type values, and provides accelerated MC variants with precomputations to further speed up inference. Empirical results on synthetic data corroborate the theoretical convergence rates and demonstrate robustness to the number of predictors, supporting practical applicability to model-agnostic explanations in regulated settings.

Abstract

In recent years, many Machine Learning (ML) explanation techniques have been designed using ideas from cooperative game theory. These game-theoretic explainers suffer from high complexity, hindering their exact computation in practical settings. In our work, we focus on a wide class of linear game values, as well as coalitional values, for the marginal game based on a given ML model and predictor vector. By viewing these explainers as expectations over appropriate sample spaces, we design a novel Monte Carlo sampling algorithm that estimates them at a reduced complexity that depends linearly on the size of the background dataset. We set up a rigorous framework for the statistical analysis and obtain error bounds for our sampling methods. The advantage of this approach is that it is fast, easily implementable, and model-agnostic. Furthermore, it has similar statistical accuracy as other known estimation techniques that are more complex and model-specific. We provide rigorous proofs of statistical convergence, as well as numerical experiments whose results agree with our theoretical findings.

Approximation of group explainers with coalition structure using Monte Carlo sampling on the product space of coalitions and features

TL;DR

This work tackles the computational bottleneck of game-theoretic explanations by introducing a Monte Carlo framework that estimates marginal, quotient, and coalitional explainers via sampling on appropriate product spaces. By recasting explainers as expectations and leveraging carefully designed sampling measures, the authors achieve estimators with convergence guarantees and error bounds, while reducing complexity to scale linearly with the background dataset size . The methodology covers linear and coalitional game values, including two-step Shapley and Owen-type values, and provides accelerated MC variants with precomputations to further speed up inference. Empirical results on synthetic data corroborate the theoretical convergence rates and demonstrate robustness to the number of predictors, supporting practical applicability to model-agnostic explanations in regulated settings.

Abstract

In recent years, many Machine Learning (ML) explanation techniques have been designed using ideas from cooperative game theory. These game-theoretic explainers suffer from high complexity, hindering their exact computation in practical settings. In our work, we focus on a wide class of linear game values, as well as coalitional values, for the marginal game based on a given ML model and predictor vector. By viewing these explainers as expectations over appropriate sample spaces, we design a novel Monte Carlo sampling algorithm that estimates them at a reduced complexity that depends linearly on the size of the background dataset. We set up a rigorous framework for the statistical analysis and obtain error bounds for our sampling methods. The advantage of this approach is that it is fast, easily implementable, and model-agnostic. Furthermore, it has similar statistical accuracy as other known estimation techniques that are more complex and model-specific. We provide rigorous proofs of statistical convergence, as well as numerical experiments whose results agree with our theoretical findings.
Paper Structure (17 sections, 11 theorems, 46 equations, 6 figures, 8 algorithms)

This paper contains 17 sections, 11 theorems, 46 equations, 6 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and hypotheses
  4. Game-theoretic model explainers
  5. Games and game values
  6. Grouping and group explainers
  7. Relevant works
  8. Monte Carlo sampling for game values
  9. Sampling for marginal linear game values
  10. Sampling for marginal quotient game values
  11. Sampling for marginal coalitional values
  12. Sampling for two-step Shapley values
  13. Accelerated Monte Carlo sampling with precomputations
  14. Numerical Examples
  15. Practical implementation of algorithms
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $f\in [f] \in L^1(P_X)$. Then, for $P_X$-almost sure $x^* \in \mathbb{R}^n$ the conditional game $v^{ \text{\tiny \it CE}}(\cdot;x^*,X,f)$ is well-defined for any subset of $N$.

Figures (6)

  • Figure 1: MC error estimate of the empirical marginal quotient Shapley for $S_1$.
  • Figure 2: MC error estimate of the empirical marginal Shapley for increasing number of predictors.
  • Figure 3: MC error estimate of the empirical marginal Owen value for $X_4$.
  • Figure 4: MC error estimate of the empirical marginal Owen value as the number of predictors increases.
  • Figure 5: MC error estimate of the empirical marginal two-step Shapley for $X_4$.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.1
  • Lemma 2.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • ...and 36 more