Probabilistic Linear Explanations

2026-09-16

AI Summary

本文提出了一种基于稀疏、锚定线性模型的统一框架，用于解决二分类和连续回归中的概率可解释性问题，并通过MIP和IHT两种方法求解。

Abstract

Formal explainability provides mathematically grounded justifications for individual predictions. However, abductive explanations often exceed human cognitive limits by involving too many features, while probabilistic relaxations have remained largely limited to categorical classification. We present a unified framework for probabilistic explainability based on sparse, anchored linear models, applicable to both binary classification and continuous regression. By mapping instances to the Boolean hypercube, our linear explanations strictly generalize subset-based approaches: they capture both the magnitude and direction of feature contributions while enforcing a prescribed sparsity budget $k$. We show that minimizing the relevance error for such explanations is \ClassNPPP-hard when the underlying model is a neural network, and we relate this intractable objective to a tractable surrogate---the fidelity error. For a parameterized family of local distributions, the relevance error of any $k$-sparse explanation is bounded by its fidelity error up to a multiplicative factor that remains small locally. We address the resulting empirical problem using two complementary approaches: a Mixed Integer Programming (MIP) formulation that yields provably optimal empirical solutions while maintaining polynomial sample complexity, and a polynomial-time Iterative Hard Thresholding (IHT) algorithm with provable approximation guarantees. Empirical evaluations show that, unlike state-of-the-art baselines such as LIME and MAPLE, our explanations satisfy both the anchoring and sparsity constraints by construction, while consistently achieving lower relevance error.

Problem

• formal explainability
• abductive explanations
• probabilistic relaxations
• sparse linear models
• Boolean hypercube

Innovation

• sparse anchored linear models
• Boolean hypercube mapping
• relevance error minimization
• Mixed Integer Programming (MIP)
• Iterative Hard Thresholding (IHT)