On the Instability of Nesterov's ODE under Non-Conservative Vector Fields

TL;DR

This paper investigates the instability of Nesterov's ODE when the vector field is non-conservative, revealing how small non-conservative terms cause instability and proposing a hybrid restart mechanism to restore stability and accelerate convergence.

Contribution

It introduces a detailed instability analysis of Nesterov's ODE in non-conservative settings and proposes a hybrid restart scheme with explicit bounds to ensure stability and improved convergence.

Findings

Small non-conservative terms induce instability in Nesterov's ODE.

A hybrid restart mechanism guarantees stability and accelerated convergence.

Explicit bounds on resetting period are derived for stability.

Abstract

We study the instability properties of Nesterov's ODE in non-conservative settings, where the driving term is not necessarily the gradient of a potential function. While convergence properties under Nesterov's ODE are well-characterized for optimization settings with gradient-based driving terms, we show that the presence of arbitrarily small non-conservative terms can lead to instability, a phenomenon previously observed empirically via numerical studies in optimization and game-theoretic problems. Our instability analysis combines multi-time scale techniques, such as averaging via variations-of-constants formula, and Floquet Theory, focusing on systems where the vector field is linear and its Helmholtz decomposition reveals a non-vanishing non-conservative component. To resolve the instability issue, the dynamics under non-vanishing non-conservative components, we study a regularization mechanism based on restarting. The resulting system is a hybrid dynamical system that mirrors Nesterov's ODE during intervals of flow, and implements resets of the momentum state through discrete periodic jumps. For this hybrid system, we establish novel explicit bounds on the resetting period that ensure the decrease of a suitable Lyapunov function, guaranteeing not only stability but also "accelerated" convergence rates under suitable smoothness and strong monotonicity properties on the driving term. Numerical simulations support our theoretical results.