Asymptotic Analysis of Conditioned Stochastic Gradient Descent

In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called conditioned SGD, based on a preconditioning of the gradient direction. Using a discrete-time approach with martingale tools, we establish the weak convergence of the rescaled sequence of iterates for a broad class of conditioning matrices including stochastic first-order and second-order methods. Almost sure convergence results, which may be of independent interest, are also presented. When the conditioning matrix is an estimate of the inverse Hessian, the algorithm is proved to be asymptotically optimal. For the sake of completeness, we provide a practical procedure to achieve this minimum variance.