Recovering Nesterov Accelerated Dynamics from Heavy Ball Dynamics via Time Rescaling

Abstract. In a real Hilbert space, we consider two classical problems: the global minimization of a smooth and convex function [Formula: see text] (i.e., a convex optimization problem) and finding the zeros of a monotone and continuous operator [Formula: see text] (i.e., a monotone equation). Attached to the optimization problem, first we study the asymptotic properties of a generalization of Polyak’s heavy ball dynamics introduced in 1964; namely, we consider a positive function [Formula: see text] multiplying [Formula: see text]. We show small [Formula: see text] convergence rates of the function values dependent on [Formula: see text] and weak convergence of trajectories toward minimizers of [Formula: see text]. In 2015 [ J. Mach. Learn. Res., 17 (2016), pp. 1–43], Su, Boyd, and Candès introduced a second-order system which could be seen as the continuous-time counterpart of Nesterov’s accelerated gradient. As the first key point of this paper, we show that for a special choice for [Formula: see text], these two seemingly unrelated dynamical systems are connected: namely, they are time reparametrizations of each other. Every statement regarding the continuous-time accelerated gradient system can be recovered from its heavy ball counterpart. As the second key point of this paper, we observe that this connection extends beyond the optimization setting. Attached to the monotone equation involving the operator [Formula: see text], we again consider a heavy ball-like system suited for the monotone operator setting. We derive small [Formula: see text] rates for the norm of the operator along the trajectories, and show the weak convergence of the trajectories toward zeros of [Formula: see text]. For a particular case of this system, we establish a time reparametrization equivalence with the Fast Optimistic Gradient Descent Ascent (Fast OGDA) dynamics introduced by Boţ, Csetnek, and Nguyen in 2022 [ Found. Comput. Math., 25 (2025), pp. 162–222], which can be seen as an analog of the continuous accelerated gradient dynamics, but for monotone operators. Every statement regarding the Fast OGDA system can be recovered from a heavy ball-like system.