Abstract

Binarized Neural Networks (BNNs) are an important class of neural network characterized by weights and activations restricted to the set {-1, +1}. BNNs provide simple compact descriptions and as such have a wide range of applications in low-power devices. In this paper, we investigate a model-based approach to training BNNs using constraint programming (CP), mixed-integer programming (MIP), and CP/MIP hybrids. We formulate the training problem as finding a set of weights that correctly classify the training set instances while optimizing objective functions that have been proposed in the literature as proxies for generalizability. Our experimental results on the MNIST digit recognition dataset suggest that-when training data is limited-the BNNs found by our hybrid approach generalize better than those obtained from a state-of-the-art gradient descent method. More broadly, this work enables the analysis of neural network performance based on the availability of optimal solutions and optimality bounds.