Composition of relational features with an application to explaining black-box predictors

Three key strengths of relational machine learning programs like those developed in Inductive Logic Programming (ILP) are: (1) The use of an expressive subset of first-order logic that allows models that capture complex relationships amongst data instances; (2) The use of domain-specific relations to guide the construction of models; and (3) The models constructed are human-readable, which is often one step closer to being human-understandable. The price for these advantages is that 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. Motivated by the work of McCreath and Sharma, we formulate the notion of a set of M -simple features in a mode language 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. CRMs are labelled directed-acyclic graphs. The vertex-label for any vertex j in the CRM contains a feature-function f j and an 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 M -simple features in their vertex-labels. We provide a randomised procedure for constructing the structure of such CRMs, and a procedure for estimating the parameters (the w ij ’s) using back-propagation and stochastic gradient descent. 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.