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Title: Neural Markov Logic Networks

Paper content:
We introduce Neural Markov Logic Networks (NMLNs), a statistical relational learning system that borrows ideas from Markov logic.
Like Markov Logic Networks (MLNs), NMLNs are an exponential-family model for modelling distributions over possible worlds, but unlike MLNs, they do not rely on explicitly specified first-order logic rules.
Instead, NMLNs learn an implicit representation of such rules as a neural network that acts as a potential function on fragments of the relational structure.
Interestingly, any MLN can be represented as an NMLN.
Similarly to recently proposed Neural theorem provers (NTPs) (Rocktaschel at al. 2017), NMLNs can exploit embeddings of constants but, unlike NTPs, NMLNs work well also in their absence.
This is extremely important for predicting in settings other than the transductive one.
We showcase the potential of NMLNs on knowledge-base completion tasks and on generation of molecular (graph) data.
Parameters for a statistical relational model are typically estimated from one or more examples of relational structures that typically consist of a large number of ground facts.
Examples of such structures are social networks (e.g. Facebook), protein-protein interaction networks, the Web, etc.
A challenging task is to learn a probability distribution over such relational structures from one or few examples.
One solution is based on the assumption that the relational structure has repeated regularities; this assumption is implicitly or explicitly used in most works on statistical relational learning.
Then, statistics about these regularities can be computed for small substructures of the training examples and used to construct a distribution over the relational structures.
Together with the maximum-entropy principle, this leads to distributions such as Markov logic networks (Richardson & Domingos, 2006; Kuželka et al., 2018) In this paper, we propose Neural Markov Logic Networks (NMLN).
Here, the statistics which are used to model the probability distribution are not known in advance, but are modelled as neural networks trained together with the probability distribution model.
This is very powerful when compared to classical MLNs, where either domain experts are required to design some useful statistics about the domain of interest by hand (i.e. logical rules) or structure learning based on combinatorial search needs to be performed.
These requirements normally limit a wide application of these models as out-of-the box tools.
It is worth noticing that overtaking the need of such "feature-engineering" is one of the reasons behind the massive adoption of deep learning techniques.
However, not much has been done in the same direction by the statistical relational learning community.
Moreover, designing statistics as neural networks allows a more fine-grained description of the data, opening the doors to applications of our model to the generative setting.
In this paper we have introduced Neural Markov Logic Networks, a statistical relational learning model combining representation learning power of neural networks with principled handling of uncertainty in the maximum-entropy framework.
The proposed system works remarkably well on small domains.
Although not explained in detail in this paper, it is also straightforward to add standard logical features as used in MLNs to NMLNs.
The main future challenge is making NMLNs scale to larger domains.
At the moment NMLNs do not scale to large knowledge bases, which is not that surprising given that NMLNs can theoretically represent any distribution.
A more work should therefore be done in the direction of identifying more tractable subclasses of NMLNs and exploiting insights from lifted inference literature (Braz et al., 2005; Gogate & Domingos, 2011; den Broeck et al., 2011) .

assistant: text: The paper provides an extension to Markov Logic Networks by removing their dependency on pre-defined first-order logic rules to model more domains in knowledge-base completion tasks.