  We investigate solving discrete optimisation problems using the estimation of
distribution (EDA) approach via a novel combination of deep belief
networks(DBN) and inductive logic programming (ILP).While DBNs are used to
learn the structure of successively better feasible solutions,ILP enables the
incorporation of domain-based background knowledge related to the goodness of
solutions.Recent work showed that ILP could be an effective way to use domain
knowledge in an EDA scenario.However,in a purely ILP-based EDA,sampling
successive populations is either inefficient or not straightforward.In our
Neuro-symbolic EDA,an ILP engine is used to construct a model for good
solutions using domain-based background knowledge.These rules are introduced as
Boolean features in the last hidden layer of DBNs used for EDA-based
optimization.This incorporation of logical ILP features requires some changes
while training and sampling from DBNs: (a)our DBNs need to be trained with data
for units at the input layer as well as some units in an otherwise hidden
layer, and (b)we would like the samples generated to be drawn from instances
entailed by the logical model.We demonstrate the viability of our approach on
instances of two optimisation problems: predicting optimal depth-of-win for the
KRK endgame,and jobshop scheduling.Our results are promising: (i)On each
iteration of distribution estimation,samples obtained with an ILP-assisted DBN
have a substantially greater proportion of good solutions than samples
generated using a DBN without ILP features, and (ii)On termination of
distribution estimation,samples obtained using an ILP-assisted DBN contain more
near-optimal samples than samples from a DBN without ILP features.These results
suggest that the use of ILP-constructed theories could be useful for
incorporating complex domain-knowledge into deep models for estimation of
distribution based procedures.
