Axiomatic Explainer Globalness via Optimal Transport

TL;DR

The paper tackles the challenge of evaluating explainers by introducing Wasserstein Globalness ($G_p$), a $p$-Wasserstein distance-based metric that quantifies the diversity of explanations a given explainer produces over a dataset. It defines a formal axiomatic framework with six properties, proves that $G_p$ satisfies these properties, and provides finite-sample bounds for its empirical estimator $\\hat{G}_p$, enabling practical use with discrete or continuous explanations and flexible distance metrics $d_{\\mathcal{E}}$. The approach is explainer-agnostic and includes normalization to a [0,1] scale, making it comparable across methods and domains; it can utilize efficient approximations like Sliced Wasserstein or Sinkhorn. Empirical results on image, tabular, and synthetic data show that WG complements faithfulness metrics by capturing explanation diversity, aiding in the selection of lower-complexity yet faithful explainers, and revealing when explanations truly reflect underlying data structure.

Abstract

Explainability methods are often challenging to evaluate and compare. With a multitude of explainers available, practitioners must often compare and select explainers based on quantitative evaluation metrics. One particular differentiator between explainers is the diversity of explanations for a given dataset; i.e. whether all explanations are identical, unique and uniformly distributed, or somewhere between these two extremes. In this work, we define a complexity measure for explainers, globalness, which enables deeper understanding of the distribution of explanations produced by feature attribution and feature selection methods for a given dataset. We establish the axiomatic properties that any such measure should possess and prove that our proposed measure, Wasserstein Globalness, meets these criteria. We validate the utility of Wasserstein Globalness using image, tabular, and synthetic datasets, empirically showing that it both facilitates meaningful comparison between explainers and improves the selection process for explainability methods.

Figures (13)

• Figure 1: Different explainers and hyperparameters can exhibit different levels of globalness for the same black-box model and dataset. We plot SmoothGrad (SG; $\sigma$ indicates smoothing, see Eq. \ref{['eq:smoothgrad']}) and SAGE explanations for CIFAR10, projected to 2-d for visualization. Top: Samples are plotted in black; attributions are shown as red vectors originating from their respective sample. Bottom: 2-d histograms of the distribution of explanations. As $\sigma$ increases for SG, the histograms condense towards the dirac, indicating higher globalness. SAGE produces the same explanation for every sample. Wasserstein Globalness ($\hat{G}_2$), denoted at the top of each column, captures this spectrum of globalness.
• Figure 3: A comparison of explanation faithfulness (IncAUC, ExAUC, and Infidelity) and globalness. We vary the smoothing parameter $\sigma = \{0.0, 0.1, 1, 10\}$ for SG, SBP, and SIG and plot with a connecting line. We observe that explainers can often exhibit similar faithfulness with varying levels of globalness, especially for simpler datasets. In this situation, we would prefer high-faithfulness explainers with higher globalness (i.e. lower complexity).
• Figure 4: A classifier is trained on the Jagged Boundary Dataset (A) with increasing scale of perturbations. The model probability output is plotted as a heatmap (B-D), with white decision boundary. (E) Comparison of explainer accuracy with increasing perturbation scale. Error bars indicate 95% CI over 50 sampled perturbations. (F) Difference in WG score between each explainer and the ground truth explanations. IG is the closest to the ground truth globalness ($\Delta$ = 0) while also yielding the highest accuracy.
• Figure 5: We evaluate whether WG can capture an increasing diversity of explanations for the MNIST, CIFAR10, and ImageNet200 datasets. As we increase the number of label classes in a set of samples, the corresponding explanations become more diverse, resulting in a decreasing WG score. Explainers with higher WG score (e.g. SAGE, SBP(10.0)) are less sensitive to the increasing diversity of explanations from adding additional classes, which is observed from the lower rate of decrease in WG.
• Figure 6: Wall clock time in seconds (blue line) for Wasserstein Globalness calculated for $10^3$ explanation samples and $10^4$ uniform samples. The results in (A) use Sliced Wasserstein approximated using Monte Carlo sampling. The results in (B) use entropic regularization. We observe that clock time is independent of black-box model architecture.

Theorems & Definitions (16)

• Definition 1: Wasserstein Globalness
• Theorem 1
• Definition 2: Empirical Wasserstein Globalness
• Theorem 2
• proof
• proof
• proof
• proof
• proof
• proof