Sampling Permutations for Shapley Value Estimation

TL;DR

This paper introduces novel approximation methods for estimating Shapley values in machine learning models, leveraging quadrature, kernel herding, and permutation sampling to improve convergence and accuracy.

Contribution

It proposes new quadrature and sampling techniques based on RKHS and hypersphere connections, enhancing Shapley value estimation efficiency and accuracy.

Findings

Significant reduction in RMSE for Shapley estimates

Improved convergence over standard Monte Carlo methods

Effective permutation sampling algorithms developed

Abstract

Game-theoretic attribution techniques based on Shapley values are used to interpret black-box machine learning models, but their exact calculation is generally NP-hard, requiring approximation methods for non-trivial models. As the computation of Shapley values can be expressed as a summation over a set of permutations, a common approach is to sample a subset of these permutations for approximation. Unfortunately, standard Monte Carlo sampling methods can exhibit slow convergence, and more sophisticated quasi-Monte Carlo methods have not yet been applied to the space of permutations. To address this, we investigate new approaches based on two classes of approximation methods and compare them empirically. First, we demonstrate quadrature techniques in a RKHS containing functions of permutations, using the Mallows kernel in combination with kernel herding and sequential Bayesian quadrature. The RKHS perspective also leads to quasi-Monte Carlo type error bounds, with a tractable discrepancy measure defined on permutations. Second, we exploit connections between the hypersphere $\mathbb{S}^{d-2}$ and permutations to create practical algorithms for generating permutation samples with good properties. Experiments show the above techniques provide significant improvements for Shapley value estimates over existing methods, converging to a smaller RMSE in the same number of model evaluations.