Abstract

There is widespread sentiment that fast gradient methods (e.g. Nesterov’s acceleration, conjugate gradient, heavy ball) are not effective for stochastic optimization due to their instability and error accumulation. Numerous works have attempted to quantify these instabilities in the face of either statistical or non-statistical errors (Paige, 1971; Proakis, 1974; Polyak, 1987; Greenbaum, 1989; Devolder et al., 2014). This work considers these issues for the case of stochastic approximation for the least squares regression problem, and our main result refutes this conventional wisdom by showing that acceleration can be made robust to statistical errors. In particular, this work introduces an accelerated stochastic gradient method that provably achieves the minimax optimal statistical risk faster than stochastic gradient descent. Critical to the analysis is a sharp characterization of accelerated stochastic gradient descent as a stochastic process. We hope this characterization gives insights towards the broader question of designing simple and effective accelerated stochastic methods for general convex and non-convex optimization problems.