2017/05/16 by David Rolnick, Max Tegmark, Rolnick, David +1 · 3 voices · 2 citations
Computer Science · Mathematics · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural and Evolutionary Computing (cs.NE) #cs.LG #cs.NE #stat.ML
paper · pdf · doi:10.48550/arxiv.1705.05502
arxiv published 2017/05/16 · arxiv updated 2018/04/27
It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons m required to approximate natural classes of multivariate polynomials of n variables grows only linearly with n for deep neural networks, but grows exponentially when merely a single hidden layer is allowed. We also provide evidence that when the number of hidden layers is increased from 1 to k, the neuron requirement grows exponentially not with n but with n1/k, suggesting that the minimum number of layers required for practical expressibility grows only logarithmically with n.