A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon

A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as an adpative variant of Muon. This framework also allows combining heterogeneous geometries across different groups of variables while preserving a unified convergence analysis. A fully stochastic global rate-of-convergence analysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on the variance of the gradient oracle and without assuming bounded stochastic gradients or small enough stepsize.