A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights

Weijie Su, Stephen Boyd, Emmanuel J. Candès.

Year: 2016, Volume: 17, Issue: 153, Pages: 1−43


We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex.