On the locality of the natural gradient for learning in deep Bayesian networks

Abstract

We study the natural gradient method for learning in deep Bayesian networks, including neural networks. There are two natural geometries associated with such learning systems consisting of visible and hidden units. One geometry is related to the full system, the other one to the visible sub-system. These two geometries imply different natural gradients. In a first step, we demonstrate a great simplification of the natural gradient with respect to the first geometry, due to locality properties of the Fisher information matrix. This simplification does not directly translate to a corresponding simplification with respect to the second geometry. We develop the theory for studying the relation between the two versions of the natural gradient and outline a method for the simplification of the natural gradient with respect to the second geometry based on the first one. This method suggests to incorporate a recognition model as an auxiliary model for the efficient application of the natural gradient method in deep networks.

Invariance properties of the natural gradient in overparametrised systems

“…This method was first proposed by Amari [1] using the geometry induced by the Fisher-Rao metric. It is an active field of study within information geometry [3,6,10] and has been shown extremely effective in many applications [4,16,17]. More recently, also other geometries on the model have been studied, such as the Wasserstein geometry [9,11].…”