  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 under mild assumptions 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. Interestingly, the
asymptotic normality result consists in a stochastic equicontinuity property so
when the conditioning matrix is an estimate of the inverse Hessian, the
algorithm is asymptotically optimal.
