  It was previously shown by Davis and Drusvyatskiy that every Clarke critical
point of a generic, semialgebraic (and more generally definable in an o-minimal
structure), weakly convex function is lying on an active manifold and is either
a local minimum or an active strict saddle. In the first part of this work, we
show that when the weak convexity assumption fails a third type of point
appears: a sharply repulsive critical point. Moreover, we show that the
corresponding active manifolds satisfy the Verdier and the angle conditions
which were introduced by us in our previous work. In the second part of this
work, we show that, under a density-like assumption on the perturbation
sequence, the stochastic subgradient descent (SGD) avoids sharply repulsive
critical points with probability one. We show that such a density-like
assumption could be obtained upon adding a small random perturbation (e.g. a
nondegenerate Gaussian) at each iteration of the algorithm. These results,
combined with our previous work on the avoidance of active strict saddles, show
that the SGD on a generic definable (e.g. semialgebraic) function converges to
a local minimum.
