First-order and second-order variants of the gradient descent: a unified framework

In this paper, we provide an overview of first-order and second-order variants of the gradient descent methods commonly used in machine learning. We propose a general framework in which 6 of these methods can be interpreted as different instances of the same approach. These methods are the vanilla gradient descent, the classical and generalized Gauss-Newton methods, the natural gradient descent method, the gradient covariance matrix approach, and Newton's method. Besides interpreting these methods within a single framework, we explain their specificities and show under which conditions some of them coincide. Machine learning generally amounts to solving an optimization problem where a loss function has to be minimized.