  We introduce optimization methods for convolutional neural networks that can
be used to improve existing gradient-based optimization in terms of
generalization error. The method requires only simple processing of existing
stochastic gradients, can be used in conjunction with any optimizer, and has
only a linear overhead (in the number of parameters) compared to computation of
the stochastic gradient. The method works by computing the gradient of the loss
function with respect to output-channel directed re-weighted L2 or Sobolev
metrics, which has the effect of smoothing components of the gradient across a
certain direction of the parameter tensor. We show that defining the gradients
along the output channel direction leads to a performance boost, while other
directions can be detrimental. We present the continuum theory of such
gradients, its discretization, and application to deep networks. Experiments on
benchmark datasets, several networks and baseline optimizers show that
optimizers can be improved in generalization error by simply computing the
stochastic gradient with respect to output-channel directed metrics.
