Title: Preconditioner on Matrix Lie Group for SGD. Abstract: 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.