  Gradient restarting has been shown to improve the numerical performance of
accelerated gradient methods. This paper provides a mathematical analysis to
understand these advantages. First, we establish global linear convergence
guarantees for the gradient restarted accelerated proximal gradient method when
solving strongly convex composite optimization problems. Second, through
analysis of the corresponding ordinary differential equation model, we prove
the continuous trajectory of gradient restarted Nesterov's accelerated gradient
method exhibits global linear convergence for quadratic strongly convex
objectives, while the non-restarted version provably lacks this property by
[Su, Boyd, and Cand\'es, J. Mach. Learn. Res., 2016, 17(153), 1-43].
