[논문 리뷰] Sampling Permutations for Shapley Value Estimation

Rory Mitchell, Joshua Cooper외 2명|arXiv (Cornell University)|2021. 04. 25.

Machine Learning and Algorithms참고 문헌 46인용 수 26

한 줄 요약

논문은 Shapley 값 추정을 위해 샘플링 permutations를 더 효율적으로 수행하기 위한 커널 기반 및 구-구속(sphere-relaxation) 방법을 개발하여 표준 몬테 카를로보다 더 빠른 수렴을 달성한다.

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.

연구 동기 및 목표

• Motivate efficient approximation of Shapley values for machine learning models where exact computation is NP-hard.
• Characterize permutation-based sampling as an RKHS problem to enable advanced sampling strategies.
• Develop and compare kernel-based and sphere-relaxation sampling methods to reduce estimation error.
• Provide error bounds and discrepancy measures for sampled permutation sets.
• Evaluate methods on practical models to demonstrate RMSE improvements over standard sampling.

제안 방법

• Define an RKHS over permutations with kernels such as Kendall, Mallows, and Spearman to measure permutation similarity.
• Apply kernel herding and sequential Bayesian quadrature to generate high-quality permutation samples for Shapley value estimation.
• Derive quasi-Monte Carlo type error bounds using a discrepancy measure in the RKHS for the permutation space.
• Introduce two sampling schemes based on mapping permutations to the hypersphere S^(d-2) to generate well-spaced samples, including orthogonal spherical codes and Sobol permutations.
• Provide analytic expectations of kernel values under uniform permutation distribution to facilitate efficient computation of kernel-based quadrature.

실험 결과

연구 질문

• RQ1 How can Shapley value estimation for models be improved by sampling permutations more effectively than uniform Monte Carlo?
• RQ2 What RKHS-based approaches and kernels yield faster convergence and reliable error bounds for permutation-based Shapley estimation?
• RQ3 Can sampling methods that leverage the geometry of permutations (via hypersphere relaxations) produce higher-quality permutation samples for Shapley estimation?
• RQ4 What are the practical performance and discrepancy benefits of kernel-based methods versus sphere-based sampling in real models?
• RQ5 How do these methods perform in terms of RMSE and variance when applied to boosted trees and CNNs?

주요 결과

• Kernel herding with the Mallows kernel yields an O(1/n) convergence rate for Shapley value estimation under certain universality conditions.
• Sequential Bayesian Quadrature provides a principled way to obtain weighted permutation samples and estimates the integration variance.
• Discrepancy-based analysis in RKHS yields a tractable bound that relates sampling quality to integration error for permutation-based functions.
• Sphere-based sampling methods map permutations to the S^(d-2) hypersphere to generate well-distributed samples via orthogonal spherical codes and Sobol-like permutations.
• Empirical results show significant improvements in RMSE for Shapley estimates compared to standard Monte Carlo across multiple models, with assessed discrepancy measures for sample sets.