Riemannian metrics for neural networks I: feedforward networks

We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties. The most standard way to train neural networks, backpropagation, has several known shortcomings. Convergence can be quite slow. Backpropagation is sensitive to data representation: for instance, even such a simple operation as exchanging 0’s and 1’s on the input layer will affect performance (Figure 1), because this amounts to changing the parameters (weights and biases) in a non-trivial way, resulting in different gradient directions in parameter space, and better performance with 1’s than with 0’s. (In the related context of restricted Boltzmann machines, the standard training technique by gradient ascent favors setting hidden units to 1, for similar reasons [OAAH11, Section 5].) This specific phenomenon disappears if, instead of the logistic function, the hyperbolic tangent is used as the activation function, or if the input is normalized. But this will not help if, for instance, the activities of internal units in a multilayer network are not centered on average. Scaling also has an effect on performance: for instance, a common recommendation [LBOM96] is to use 1.7159 tanh(2x/3) instead of just tanh(x) as the activation function. It would be interesting to have algorithms whose performance is insensitive to particular choices such as scaling factors in network construction, parameter encoding or data representation. We call an algorithm invariant, or intrinsic, if applying a change of variables to the parameters and activities results in the same learning trajectory. This is not the case for backpropagation (even after changing the learning rate): for instance, changing from sigmoid to tanh activation amounts to dividing the connection weights by 4