On the Computational Efficiency of Training Neural Networks
  It is well-known that neural networks are computationally hard to train. On the other hand, in practice, modern day neural networks are trained efficiently using SGD and a variety of tricks that include different activation functions (e.g. ReLU), over-specification (i.e., train networks which are larger than needed), and regularization. In this paper we revisit the computational complexity of training neural networks from a modern perspective. We provide both positive and negative results, some of them yield new provably efficient and practical algorithms for training certain types of neural networks. 
Roi Livni and Shai Shalev-Shwartz and Ohad Shamir
cs.LG cs.AI stat.ML