  We study two types of preconditioners and preconditioned stochastic gradient
descent (SGD) methods in a unified framework. We call the first one the Newton
type due to its close relationship to the Newton method, and the second one the
Fisher type as its preconditioner is closely related to the inverse of Fisher
information matrix. Both preconditioners can be derived from one framework, and
efficiently estimated on any matrix Lie groups designated by the user using
natural or relative gradient descent minimizing certain preconditioner
estimation criteria. Many existing preconditioners and methods, e.g., RMSProp,
Adam, KFAC, equilibrated SGD, batch normalization, etc., are special cases of
or closely related to either the Newton type or the Fisher type ones.
Experimental results on relatively large scale machine learning problems are
reported for performance study.
