  In many fields such as bioinformatics, high energy physics, power
distribution, etc., it is desirable to learn non-linear models where a small
number of variables are selected and the interaction between them is explicitly
modeled to predict the response. In principle, neural networks (NNs) could
accomplish this task since they can model non-linear feature interactions very
well. However, NNs require large amounts of training data to have a good
generalization. In this paper we study the datastarved regime where a NN is
trained on a relatively small amount of training data. For that purpose we
study feature selection for NNs, which is known to improve generalization for
linear models. As an extreme case of data with feature selection and feature
interactions we study the XOR-like data with irrelevant variables. We
experimentally observed that the cross-entropy loss function on XOR-like data
has many non-equivalent local optima, and the number of local optima grows
exponentially with the number of irrelevant variables. To deal with the local
minima and for feature selection we propose a node pruning and feature
selection algorithm that improves the capability of NNs to find better local
minima even when there are irrelevant variables. Finally, we show that the
performance of a NN on real datasets can be improved using pruning, obtaining
compact networks on a small number of features, with good prediction and
interpretability.
