Almost-everywhere convergence-rate improvement of discrete gradient restart

Determine whether, for Nesterov’s accelerated gradient method with a fixed step size \(0<s<1/L\), gradient restart improves the linear convergence rate relative to the unrestarted method for Lebesgue-almost every initial point, excluding finite termination.

Background

The paper proves that smoothness and strong convexity alone do not force gradient restart to occur: it gives a discrete counterexample in which the restart criterion is never activated. In that example, the restarted and unrestarted Nesterov iterations are identical, although both converge R-linearly.

The authors contrast this worst-case behavior with the empirical effectiveness of gradient restart and pose the unresolved measure-theoretic question of whether rate improvement holds for almost every initialization when the objective and step size are fixed. Finite termination is explicitly excluded from the proposed question.