Kernel Banzhaf: A Fast and Robust Estimator for Banzhaf Values

TL;DR

Kernel Banzhaf introduces a regression-based estimator for Banzhaf feature attributions, addressing the exponential cost of exact computation. By formulating Banzhaf values as the solution to a linear regression and solving a small, paired-sample LS problem, it achieves higher accuracy and sample efficiency than Monte Carlo-based methods, with strong theoretical guarantees. The approach extends to probabilistic values and demonstrates robust performance across eight datasets, including both tree-based and neural models, under noise and adversarial perturbations. This yields a scalable, theoretically-grounded tool for reliable feature attribution in complex models.

Abstract

Banzhaf values provide a popular, interpretable alternative to the widely-used Shapley values for quantifying the importance of features in machine learning models. Like Shapley values, computing Banzhaf values exactly requires time exponential in the number of features, necessitating the use of efficient estimators. Existing estimators, however, are limited to Monte Carlo sampling methods. In this work, we introduce Kernel Banzhaf, the first regression-based estimator for Banzhaf values. Our approach leverages a novel regression formulation, whose exact solution corresponds to the exact Banzhaf values. Inspired by the success of Kernel SHAP for Shapley values, Kernel Banzhaf efficiently solves a sampled instance of this regression problem. Through empirical evaluations across eight datasets, we find that Kernel Banzhaf significantly outperforms existing Monte Carlo methods in terms of accuracy, sample efficiency, robustness to noise, and feature ranking recovery. Finally, we complement our experimental evaluation with strong theoretical guarantees on Kernel Banzhaf's performance.

Figures (14)

• Figure 1: Comparison of exact and estimated Banzhaf feature attribution values across eight datasets, where every estimator uses $m=20n$ evaluations of the set function $v$. The evaluations dominate the computational cost of each algorithm. Each subplot, labeled with its estimator, shows normalized estimated versus exact Banzhaf values across all features for a randomly selected data point from each dataset. Points closer to the diagonal line indicate more accurate estimates; the plots suggest that Kernel Banzhaf is more accurate than the Monte Carlo and Maximum Sample Reuse estimators.
• Figure 2: Plots comparing the relative squared $\ell_2$-norm error (i.e., $\|\tilde{\boldsymbol{\phi}} - \boldsymbol{\phi}\|_2^2 / \|\boldsymbol{\phi}\|_2^2$) for each Banzhaf estimator across increasing sample sizes in eight datasets. Each point represents the median of 50 runs, with shaded areas indicating the 25th to 75th percentiles.
• Figure 3: Plots of relative squared $\ell_2$-norm error by noise level across Banzhaf estimators. For each noise level $\sigma$, the estimator observes $v(S) + x$ where $x \sim \mathcal{N}(0,\sigma)$. Kernel Banzhaf outperforms for lower noise levels, eventually matching its ablated version and MSR for larger noise. MC is worse for all noise settings, likely because its constituent estimates are $v(S \cup \{i \}) + x - v(S) - x'$, increasing the variance of the noise.
• Figure 4: Comparison of top 20% feature ranking recovery using Cayley distance (lower is more accurate). Recovering the most impactful features is an primary component of comparing and selecting the most important features for machine learnign models. Especially as the sizes of the datasets increase, Kernel Banzhaf gives the best performance.
• Figure 5: This figure compares the relative squared $\ell_2$-norm errors of Kernel Banzhaf (including an ablated version without paired sampling), MC, and MSR across increasing sample sizes on four small datasets in explaining neural network models. The results highlight the robust performance and generalizability of Kernel Banzhaf across various model types.

Theorems & Definitions (20)

• Theorem 3.1: Linear Regression Equivalence
• proof : Proof of Theorem \ref{['thm:equivalence']}
• Theorem 3.2
• Corollary 3.2
• Lemma 3.2: Extended Regression Equivalence
• Theorem 3.3: Probabilistic Value Approximation
• Theorem 1.1
• Corollary 1.1
• proof : Proof of Corollary \ref{['coro:l2norm']}
• Lemma 1.2: Spectral Approximation