Almost-everywhere linear convergence of gradient-restarted NAG-ODE

Determine whether, for every fixed smooth strongly convex function on \(\mathbb{R}^n\), the gradient-restarted Nesterov accelerated gradient ordinary differential equation admits global or local linear convergence for Lebesgue-almost every initial point.

Background

The paper constructs a smooth, strongly convex objective and an initial point for which the gradient-restart condition is never activated in the continuous-time Nesterov accelerated gradient ordinary differential equation. Consequently, the restarted and unrestarted trajectories coincide, and the resulting objective gap decays only as 1/(πt3)1/(\pi t^3), rather than linearly.

The authors note that this is a worst-case construction and may not represent typical behavior. They therefore leave unresolved whether gradient-restarted NAG-ODE nevertheless achieves global or local linear convergence for almost every initial point with respect to Lebesgue measure, for each fixed smooth strongly convex objective.

References

Specifically, for a fixed $f\in\mathcal{S}_{\mu,L}(\mathbb{R}n)$, a central question is whether the gradient-restarted NAG-ODE eq:nag-ode admits global or local linear convergence guarantees for Lebesgue-almost every initial point.

— On the Nonoccurrence of Gradient Restart in Nesterov Accelerated Gradient Method  (2610.26546 - Bao et al., 19 Oct 2026) in Section 6, Subsection “Further Discussion and Extensions”