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

Daniel E. Ochoa, Mahmoud Abdelgalil, Jorge I. Poveda

TL;DR

This work analyzes the instability of Nesterov's ODE when the driving vector field is non-conservative, showing that even small non-conservative components can destabilize the accelerated flow. By applying a Helmholtz decomposition $\mathcal{G}(x)=\nabla J(x)+\mathrm{rot}\,K(x)$ and an averaging/Floquet framework, the authors establish instability results for linear vector fields $\mathcal{G}(x)=Qx$ under precise spectral conditions. To recover stability and obtain accelerated convergence, they design a restarting hybrid dynamical system that periodically resets the momentum, prove UGES of the target equilibrium, and derive explicit bounds on the resetting period yielding a rate $\mathcal{O}(e^{-\sqrt{\kappa_J}\,t})$ under smoothness and strong monotonicity assumptions. Numerical simulations corroborate the theoretical predictions, illustrating both instability in the purely non-conservative case and rapid convergence with restarting. The results provide a rigorous explanation for observed instabilities in non-conservative settings and a practical restoration mechanism with provable performance guarantees.

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.

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

TL;DR

This work analyzes the instability of Nesterov's ODE when the driving vector field is non-conservative, showing that even small non-conservative components can destabilize the accelerated flow. By applying a Helmholtz decomposition and an averaging/Floquet framework, the authors establish instability results for linear vector fields under precise spectral conditions. To recover stability and obtain accelerated convergence, they design a restarting hybrid dynamical system that periodically resets the momentum, prove UGES of the target equilibrium, and derive explicit bounds on the resetting period yielding a rate under smoothness and strong monotonicity assumptions. Numerical simulations corroborate the theoretical predictions, illustrating both instability in the purely non-conservative case and rapid convergence with restarting. The results provide a rigorous explanation for observed instabilities in non-conservative settings and a practical restoration mechanism with provable performance guarantees.

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.
Paper Structure (8 sections, 7 theorems, 49 equations, 2 figures)

This paper contains 8 sections, 7 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nesterov's ODE: Analysis via Standard Averaging
  4. Instability under Linear Mappings
  5. Precluding Instability via Restarting
  6. Numerical Simulations
  7. Conclusions and Future Directions
  8. Auxiliary Lemmas

Key Result

Theorem II.1

Let $g$ be $r$-th and $f$ be $(r+1)$-th continuously differentiable time-dependent vector fields. Let $x_0 \in \mathbb{R}^n$, $T > 0$, $\tau \in [0,T]$, and define $\tilde{g}(\tau,x) = ((\Phi^f_{\tau})^*g_\tau)(x)$, where $g_\tau(x)\coloneqq g(x,\tau)$, and $(\Phi^f_{\tau})^* g_\tau$ denotes the pul

Figures (2)

  • Figure 1: Left: Components of $\psi$ showing a solution of the periodic system \ref{['eq:driftDynamics']}. Middle: Components of $z$ and $\zeta$ showing the solutions of the pulled-back system \ref{['eq:standardLinear']} and the averaged system \ref{['proof:average']}. Right: Components of $y$ showing a solution of Nesterov's ODE in \ref{['eq:preStandardLinear']}.
  • Figure 2: Comparison of trajectories of $|q(t) - x^*|$ under Nesterov's ODE and the restarting HDS $\mathcal{H}$.

Theorems & Definitions (16)

  • Theorem II.1: Variation-of-Constants Formula bulloGeometricControlMechanical2005
  • Definition II.2: Stability/Instability Notions
  • Lemma III.1: Helmholtz Decomposition
  • Example III.3
  • Remark III.4
  • Lemma IV.1
  • proof
  • Theorem IV.2
  • proof
  • Remark IV.3
  • ...and 6 more