Heavy-ball dynamics with Hessian-driven damping for non-convex optimization under the Łojasiewicz condition

2025/06/13 by Vassilis Apidopoulos, Apidopoulos, Vassilis, Vasiliki Mavrogeorgou +3
Engineering · #34D05 #46N10 #65B99 #65K05 #90C26 #FOS: Mathematics #Guidance and Control Systems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.11705

openalex publication_date 2025/06/13 · openalex created_date 2025/10/11 · openalex updated_date 2026/08/02

Abstract

In this paper, we examine the convergence properties of heavy-ball dynamics with Hessian-driven damping in smooth non-convex optimization problems satisfying a Łojasiewicz condition. In this general setting, we provide a series of tight, worst-case optimal convergence rate guarantees as a function of the dynamics' friction coefficients and the Łojasiewicz exponent of the problem's objective function. Importantly, the linear rates that we obtain improve on previous available rates and they suggest a different tuning of the dynamics' damping terms, even in the strongly convex regime. We complement our analysis with a range of stability estimates in the presence of perturbation errors and inexact gradient input, as well as an avoidance result showing that the dynamics under study avoid strict saddle points from almost every initial condition,

Citations

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Related

• Convergence rates for the Heavy-Ball continuous dynamics for non-convex optimization, under Polyak-Łojasiewicz condition
• Inertial Methods with Viscous and Hessian driven Damping for Non-Convex Optimization
• Heavy Ball Momentum for Non-Strongly Convex Optimization
• Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessian
• On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping