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Examples of Pathological Dynamics of the Subgradient Method for Lipschitz Path-Differentiable Functions

Author

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  • Rodolfo Ríos-Zertuche

    (Laboratoire d’Analyse et d’Architecture des Systèmes du CNRS, 31031 Toulouse, France)

Abstract

We show that the vanishing step size subgradient method—widely adopted for machine learning applications—can display rather messy behavior even in the presence of favorable assumptions. We establish that convergence of bounded subgradient sequences may fail even with a Whitney stratifiable objective function satisfying the Kurdyka-Łojasiewicz inequality. Moreover, when the objective function is path-differentiable, we show that various properties all may fail to occur: criticality of the limit points, convergence of the sequence, convergence in values, codimension one of the accumulation set, equality of the accumulation and essential accumulation sets, connectedness of the essential accumulation set, spontaneous slowdown, oscillation compensation, and oscillation perpendicularity to the accumulation set.

Suggested Citation

  • Rodolfo Ríos-Zertuche, 2022. "Examples of Pathological Dynamics of the Subgradient Method for Lipschitz Path-Differentiable Functions," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 3184-3206, November.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:4:p:3184-3206
    DOI: 10.1287/moor.2021.1241
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