Convolutional networks and learning invariant to homogeneous multiplicative scalings

The conventional classification schemes -- notably multinomial logistic regression -- used in conjunction with convolutional networks (convnets) are classical in statistics, designed without consideration for the usual coupling with convnets, stochastic gradient descent, and backpropagation. In the specific application to supervised learning for convnets, a simple scale-invariant classification stage turns out to be more robust than multinomial logistic regression, appears to result in slightly lower errors on several standard test sets, has similar computational costs, and features precise control over the actual rate of learning. Scale-invariant means that multiplying the input values by any nonzero scalar leaves the output unchanged.