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Aumann-SHAP: The Geometry of Counterfactual Interaction Explanations in Machine Learning

Adam Belahcen, Stéphane Mussard

Abstract

We introduce Aumann-SHAP, an interaction-aware framework that decomposes counterfactual transitions by restricting the model to a local hypercube connecting baseline and counterfactual features. Each hyper-cube is decomposed into a grid in order to construct an induced micro-player cooperative game in which elementary grid-step moves become players. Shapley and LES values on this TU-micro-game yield: (i) within-pot contribution of each feature to the interaction with other features (interaction explainability), and (ii) the contribution of each instance and each feature to the counterfactual analysis (individual and global explainability). In particular, Aumann-LES values produce individual and global explanations along the counterfactual transition. Shapley and LES values converge to the diagonal Aumann-Shapley (integrated-gradients) attribution method. Experiments on the German Credit dataset and MNIST data show that Aumann-LES produces robust results and better explanations than the standard Shapley value during the counterfactual transition.

Aumann-SHAP: The Geometry of Counterfactual Interaction Explanations in Machine Learning

Abstract

We introduce Aumann-SHAP, an interaction-aware framework that decomposes counterfactual transitions by restricting the model to a local hypercube connecting baseline and counterfactual features. Each hyper-cube is decomposed into a grid in order to construct an induced micro-player cooperative game in which elementary grid-step moves become players. Shapley and LES values on this TU-micro-game yield: (i) within-pot contribution of each feature to the interaction with other features (interaction explainability), and (ii) the contribution of each instance and each feature to the counterfactual analysis (individual and global explainability). In particular, Aumann-LES values produce individual and global explanations along the counterfactual transition. Shapley and LES values converge to the diagonal Aumann-Shapley (integrated-gradients) attribution method. Experiments on the German Credit dataset and MNIST data show that Aumann-LES produces robust results and better explanations than the standard Shapley value during the counterfactual transition.
Paper Structure (23 sections, 5 theorems, 77 equations, 8 figures, 8 tables)

This paper contains 23 sections, 5 theorems, 77 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. State of the art
  3. Explainability and feature attribution
  4. Explainability and counterfactual analysis
  5. TU-micro-game: decomposition of feature interactions
  6. Setup
  7. Counterfactual analysis on local cube restriction
  8. Discretization of the cube on a grid & TU-micro-game
  9. Aumann-SHAP
  10. Interaction, local and global explainability
  11. Generalization to LES values
  12. Experiments
  13. Simulations: Exact computation and complexity
  14. Convergence.
  15. German credit
  16. ...and 8 more sections

Key Result

proposition 1

Let $u\subseteq N$ with $|u|=k\ge 2$. $(\imath)$ If $t_i=0$ for some $i\in u$, then $r_u(t)=0$. (Boundary)$(\imath\imath)$ At the far corner, $r_u(\mathbf{1}_k)=\phi_u$. (Efficiency)

Figures (8)

  • Figure 1: Local slider cube for $u=\{i,j,k\}$ (left) and the sub-cube from $(0,0,0)$ to $(t_i,t_j,t_k)$ (right), illustrating corner evaluations and the inclusion--exclusion construction.
  • Figure 2: Solid curves with markers provide runtimes: grid-state computation (blue) and, when feasible, full micro-coalition enumeration (orange). Dashed curves are fitted scaling laws: $\sim c\,k(m{+}1)^k$ for grid-state (green) and $\sim c\,n2^n$ with $n{=}km$ for full enumeration (red).
  • Figure 3: MNIST $1\to 7$ counterfactual: attribution maps and redistribution. Top row: baseline $x^0$, endpoint $x^1$ (only the $k$ changed pixels are moved), and Top-25 masks for equal-split and TU-micro-game Shapley (red). Bottom row: signed heatmaps, redistribution $D=\text{micro}-\text{equal}$, and an overlay of the largest re-allocations on $x^1$.
  • Figure 4: Global MNIST $1\to7$ mean explanations (nearest-neighbor pairing). We sample $200$ test-set baselines $x^0$ of digit $1$ and pair each with its nearest neighbor $x^1$ among test-set $7$'s (Euclidean distance), then compute local attributions on the changed-pixel set $\{(r,c):|x^1_{rc}-x^0_{rc}|>\varepsilon\}$ with $\varepsilon=0.05$ and average the resulting maps over pairs.
  • Figure 5: Counterfactual patch test on MNIST ($1\to7$): $P_7$ after patching the top-$K$ pixels selected by each rule (Micro-game Shapley, Equal-split Shapley, Equal surplus, and $|x^1-x^0|$), compared to a random baseline (mean with 10--90% band). Top row shows $x^0$ and the patched images at $K=12$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • definition 1: Harsanyi dividend on a local-cube
  • proposition 1: Properties
  • definition 2: Grid discretization
  • definition 3: Micro-players and TU-micro-game
  • theorem 1: Micro-player Shapley value
  • proposition 2: Equal-split Shapley luborgonovo2024
  • theorem 2: Aumann--SHAP
  • proposition 3: LES closed form