  This work considers the question: what convergence guarantees does the
stochastic subgradient method have in the absence of smoothness and convexity?
We prove that the stochastic subgradient method, on any semialgebraic locally
Lipschitz function, produces limit points that are all first-order stationary.
More generally, our result applies to any function with a Whitney stratifiable
graph. In particular, this work endows the stochastic subgradient method, and
its proximal extension, with rigorous convergence guarantees for a wide class
of problems arising in data science---including all popular deep learning
architectures.
