Composition of Relational Features with an Application to Explaining Black-Box Predictors

TL;DR

The paper addresses explainability for relational data by formalizing relational features within a mode-language, and introducing two composition operators, $rho_1$ and $rho_2$, that generate all complex features from $M$-simple ones. It then defines Compositional Relational Machines (CRMs), neural architectures represented as labelled DAGs where each node corresponds to a relational feature with an activation, enabling natural explanations via explanation graphs $Explain_{\gamma,o_i}(a)$. Empirically, CRMs achieve perfect predictive fidelity and high explanatory fidelity on synthetic tasks (e.g., Trains, Chess) and competitive predictive fidelity (~$0.81$) on real-world NCI datasets against a Graph Neural Network, while providing explicit, interpretable explanations. The work demonstrates a scalable neuro-symbolic framework for post hoc explanations of black-box predictors and suggests avenues for improving fidelity through ensembles and richer feature sets.

Abstract

Relational machine learning programs like those developed in Inductive Logic Programming (ILP) offer several advantages: (1) The ability to model complex relationships amongst data instances; (2) The use of domain-specific relations during model construction; and (3) The models constructed are human-readable, which is often one step closer to being human-understandable. However, these ILP-like methods have not been able to capitalise fully on the rapid hardware, software and algorithmic developments fuelling current developments in deep neural networks. In this paper, we treat relational features as functions and use the notion of generalised composition of functions to derive complex functions from simpler ones. We formulate the notion of a set of $\text{M}$-simple features in a mode language $\text{M}$ and identify two composition operators ($ρ_1$ and $ρ_2$) from which all possible complex features can be derived. We use these results to implement a form of "explainable neural network" called Compositional Relational Machines, or CRMs, which are labelled directed-acyclic graphs. The vertex-label for any vertex $j$ in the CRM contains a feature-function $f_j$ and a continuous activation function $g_j$. If $j$ is a "non-input" vertex, then $f_j$ is the composition of features associated with vertices in the direct predecessors of $j$. Our focus is on CRMs in which input vertices (those without any direct predecessors) all have $\text{M}$-simple features in their vertex-labels. We provide a randomised procedure for constructing and learning such CRMs. Using a notion of explanations based on the compositional structure of features in a CRM, we provide empirical evidence on synthetic data of the ability to identify appropriate explanations; and demonstrate the use of CRMs as 'explanation machines' for black-box models that do not provide explanations for their predictions.