Learnability of the Boolean Innerproduct in Deep Neural Networks

In this paper, we study the learnability of the Boolean inner product by a systematic simulation study. The family of the Boolean inner product function is known to be representable by neural networks of threshold neurons of depth 3 with only [Formula: see text] units (n the input dimension)—whereas an exact representation by a depth 2 network cannot possibly be of polynomial size. This result can be seen as a strong argument for deep neural network architectures. In our study, we found that this depth 3 architecture of the Boolean inner product is difficult to train, much harder than the depth 2 network, at least for the small input size scenarios [Formula: see text]. Nonetheless, the accuracy of the deep architecture increased with the dimension of the input space to 94% on average, which means that multiple restarts are needed to find the compact depth 3 architecture. Replacing the fully connected first layer by a partially connected layer (a kind of convolutional layer sparsely connected with weight sharing) can significantly improve the learning performance up to 99% accuracy in simulations. Another way to improve the learnability of the compact depth 3 representation of the inner product could be achieved by adding just a few additional units into the first hidden layer.