On the Nonoccurrence of Gradient Restart in Nesterov Accelerated Gradient Method

Abstract: Gradient restarting often improves the practical performance of accelerated gradient methods for unconstrained convex optimization problems, but a general theoretical guarantee of such improvement remains unclear. In this paper, we construct smooth, strongly convex objectives with Lipschitz-continuous gradients and corresponding initial points for which gradient restart never occurs, both for Nesterov's accelerated gradient method and for its ordinary differential equation model. From these examples, we conclude that strong convexity and smoothness alone do not guarantee that gradient restart improves the convergence behavior of the accelerated method or its continuous-time model. In particular, for the continuous-time example, the objective gap is asymptotic to $1/(πt^3)$. We therefore obtain a negative answer to the open problem raised by Su, Boyd, and Cand{è}s [\textit{J. Mach. Learn. Res.}, 2016, 17(153), 1--43] of whether gradient restart guarantees linear convergence of the continuous dynamics for smooth strongly convex objectives.