Fast Curvature Matrix-Vector Products for Second-Order Gradient Descent

Abstract

We propose a generic method for iteratively approximating various second-order gradient steps—-Newton, Gauss-Newton, Levenberg-Marquardt, and natural gradient—-in linear time per iteration, using special curvature matrix-vector products that can be computed in O(n). Two recent acceleration techniques for on-line learning, matrix momentum and stochastic meta-descent (SMD), implement this approach. Since both were originally derived by very different routes, this offers fresh insight into their operation, resulting in further improvements to SMD.