Variational Learning with Lie Group Exponential Families

Variational Bayesian learning for deep neural networks offers many potential benefits. The difficulty in implementation arises from constraints on the parameters of the variational distributions, as well as the need for low-variance gradient estimators. Our recent work proposes Lie group methods for multiplicative or affine groups that naturally overcome such difficulties. Lie groups are essentially parameter spaces with a binary operator such as addition or multiplication, which allows for natural updates within the parameter space. In this article, we streamline and extend our Lie group framework to any general Lie group, and give an explicit relation to standard natural-gradient variational learning methods such as the Bayesian learning rule (BLR). In particular, we prove that linear approximations of the Lie group algorithm give generalizations of the BLR. The key ingredient is the harmonic exponential family, a general class of Lie group exponential families associated to representations of Lie groups. Our work provides a systematic method to construct from such representations efficiently implementable algorithms for variational learning.