Long term dynamics of the subgradient method for Lipschitz path differentiable functions
Jérôme Bolte
University Toulouse 1 Capitole, FranceEdouard Pauwels
Université Toulouse III – Paul Sabatier; CNRS; Institut Universitaire de France, FranceRodolfo Ríos-Zertuche
Laboratoire d’analyse et d’architecture des systèmes – CNRS, Toulouse, France
Abstract
We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures, a concept that complements the technique of asymptotic pseudotrajectories developed in this setting by Benaïm–Hofbauer–Sorin. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. Various cases are discussed, providing new insight into the oscillation and the stabilization phenomena.
Cite this article
Jérôme Bolte, Edouard Pauwels, Rodolfo Ríos-Zertuche, Long term dynamics of the subgradient method for Lipschitz path differentiable functions. J. Eur. Math. Soc. 26 (2024), no. 7, pp. 2533–2563
DOI 10.4171/JEMS/1285