Principled Deep Neural Network Training through Linear Programming

Deep learning has received much attention lately due to the impressive empirical performance achieved by training algorithms. Consequently, a need for a better theoretical understanding of these problems has become more evident in recent years. In this work, using a unified framework, we show that there exists a polyhedron which encodes simultaneously all possible deep neural network training problems that can arise from a given architecture, activation functions, loss function, and sample-size. Notably, the size of the polyhedral representation depends only linearly on the sample-size, and a better dependency on several other network parameters is unlikely (assuming $P\neq NP$). Additionally, we use our polyhedral representation to obtain new and better computational complexity results for training problems of well-known neural network architectures. Our results provide a new perspective on training problems through the lens of polyhedral theory and reveal a strong structure arising from these problems.