Preconditioner on Matrix Lie Group for SGD
This work provides a theoretical unification for preconditioned SGD methods, which is incremental but could benefit researchers and practitioners in optimization and machine learning.
The authors tackled the problem of improving stochastic gradient descent (SGD) by proposing a unified framework for two types of preconditioners (Newton and Fisher types) that can be efficiently estimated on matrix Lie groups, showing that many existing methods like RMSProp and Adam are special cases.
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.