Geometric conditions weaker than subdifferential regularity

Determine which geometric conditions weaker than subdifferential regularity suffice to guarantee generic convergence of stochastic subgradient methods to local minimizers.

Background

The paper constructs a globally Lipschitz, coercive, semialgebraic function for which stochastic subgradient iterates converge to a Clarke-critical point that is neither a local minimizer nor limiting critical, thereby showing that semialgebraicity and bounded stochastic perturbations do not by themselves ensure generic convergence to local minimizers.

The closing discussion explains that subdifferential regularity excludes the stationarity geometry used in the counterexample: for a positively homogeneous function regular at the origin, Clarke criticality at the origin would imply that the origin is a global minimizer. The unresolved problem is to identify weaker geometric assumptions that retain the generic convergence guarantee without requiring full subdifferential regularity.