  In this paper, a novel second-order method called NG+ is proposed. By
following the rule ``the shape of the gradient equals the shape of the
parameter", we define a generalized fisher information matrix (GFIM) using the
products of gradients in the matrix form rather than the traditional
vectorization. Then, our generalized natural gradient direction is simply the
inverse of the GFIM multiplies the gradient in the matrix form. Moreover, the
GFIM and its inverse keeps the same for multiple steps so that the
computational cost can be controlled and is comparable with the first-order
methods. A global convergence is established under some mild conditions and a
regret bound is also given for the online learning setting. Numerical results
on image classification with ResNet50, quantum chemistry modeling with Schnet,
neural machine translation with Transformer and recommendation system with DLRM
illustrate that GN+ is competitive with the state-of-the-art methods.
