Long term dynamics of the subgradient method for Lipschitz path differentiable functions

  • Jérôme Bolte

    University Toulouse 1 Capitole, France
  • Edouard Pauwels

    Université Toulouse III – Paul Sabatier; CNRS; Institut Universitaire de France, France
  • Rodolfo Ríos-Zertuche

    Laboratoire d’analyse et d’architecture des systèmes – CNRS, Toulouse, France
Long term dynamics of the subgradient method for Lipschitz path differentiable functions cover

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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. (2022), published online first

DOI 10.4171/JEMS/1285