Unifying Formal Explanations: A Complexity-Theoretic Perspective

TL;DR

This paper presents a unified complexity-theoretic framework for understanding different types of explanations in machine learning, revealing key properties that influence computational difficulty and enabling efficient global explanations.

Contribution

It introduces a unified probabilistic framework for explanations, analyzing how properties like monotonicity and submodularity affect computational complexity and providing polynomial-time algorithms for global explanations.

Findings

Global explanations have properties like monotonicity and submodularity, enabling efficient computation.

Local explanations are NP-hard to compute, even in simplified cases.

The framework applies across various ML models, including neural networks and decision trees.

Abstract

Previous work has explored the computational complexity of deriving two fundamental types of explanations for ML model predictions: (1) *sufficient reasons*, which are subsets of input features that, when fixed, determine a prediction, and (2) *contrastive reasons*, which are subsets of input features that, when modified, alter a prediction. Prior studies have examined these explanations in different contexts, such as non-probabilistic versus probabilistic frameworks and local versus global settings. In this study, we introduce a unified framework for analyzing these explanations, demonstrating that they can all be characterized through the minimization of a unified probabilistic value function. We then prove that the complexity of these computations is influenced by three key properties of the value function: (1) *monotonicity*, (2) *submodularity*, and (3) *supermodularity* - which are three fundamental properties in *combinatorial optimization*. Our findings uncover some counterintuitive results regarding the nature of these properties within the explanation settings examined. For instance, although the *local* value functions do not exhibit monotonicity or submodularity/supermodularity whatsoever, we demonstrate that the *global* value functions do possess these properties. This distinction enables us to prove a series of novel polynomial-time results for computing various explanations with provable guarantees in the global explainability setting, across a range of ML models that span the interpretability spectrum, such as neural networks, decision trees, and tree ensembles. In contrast, we show that even highly simplified versions of these explanations become NP-hard to compute in the corresponding local explainability setting.