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

NG+ : A Multi-Step Matrix-Product Natural Gradient Method for Deep Learning