  We revisit the choice of SGD for training deep neural networks by
reconsidering the appropriate geometry in which to optimize the weights. We
argue for a geometry invariant to rescaling of weights that does not affect the
output of the network, and suggest Path-SGD, which is an approximate steepest
descent method with respect to a path-wise regularizer related to max-norm
regularization. Path-SGD is easy and efficient to implement and leads to
empirical gains over SGD and AdaGrad.
