ON THE BENEFIT OF WIDTH FOR NEURAL NETWORKS: DISAPPEARANCE OF BASINS

Wide networks are often believed to have a nice optimization landscape, but what rigorous results can we prove? To understand the benefit of width, it is important to identify the difference between wide and narrow networks. In this work, we prove that from narrow to wide networks, there is a phase transition from having suboptimal basins to no suboptimal basins. Specifically, we prove two results: on the positive side, for any continuous activation functions, the loss surface of a class of wide networks has no suboptimal basin, where “basin” is defined as the setwise strict local minimum; on the negative side, for a large class of networks with width below a threshold, we construct strict local minima that are not global. These two results together show the phase transition from narrow to wide networks.