Abstract

Information geometry applies fundamental concepts of differential geometry to probability theory and statistics to study statistical manifolds, which are Riemannian manifolds in which each point in the manifold corresponds to a probability distribution. The perhaps most prominent application of information geometry in machine learning is natural gradient descent (NGD), which can be described as ‘gradient descent which respects the curvature of the statistical manifold’. In this session, we will be looking at NGD from different perspectives. In particular, we will consider NGD as 1) preconditioning the gradient with the inverse Fisher information matrix, 2) second-order optimisation with Newton’s method, 3) the result of stating gradient descent as a valid tensor equation, and (bonus) *) mirror descent in a dual Riemannian manifold. Previous knowledge of differential geometry is NOT required.

Reading (suggested, but not required): Martens, J. (2020). “New Insights and Perspectives on the Natural Gradient Method”.
 Journal of Machine Learning Research, Volume 21, Issue 146. (Sections 5, 8 – 8.1, 9.2, _{6 pages)}

Raskutti, G., Mukherjee, S. (2015). “The information geometry of mirror descent”.
 IEEE Transactions on Information Theory, Volume 61, Issue 3. (Sections 1, 2, 3 pages)