Exact natural gradient in deep linear networks and its application to the nonlinear case

Stochastic gradient descent (SGD) remains the method of choice for deep learning, despite the limitations arising for ill-behaved objective functions. In cases where it could be estimated, the natural gradient has proven very effective at mitigating the catastrophic effects of pathological curvature in the objective function, but little is known theoretically about its convergence properties, and it has yet to find a practical implementation that would scale to very deep and large networks. Here, we derive an exact expression for the natural gradient in deep linear networks, which exhibit pathological curvature similar to the nonlinear case. We provide for the first time an analytical solution for its convergence rate, showing that the loss decreases exponentially to the global minimum in parameter space. Our expression for the natural gradient is surprisingly simple, computationally tractable, and explains why some approximations proposed previously work well in practice. This opens new avenues for approximating the natural gradient in the nonlinear case, and we show in preliminary experiments that our online natural gradient descent outperforms SGD on MNIST autoencoding while sharing its computational simplicity.