  The paper studies a distributed gradient descent (DGD) process and considers
the problem of showing that in nonconvex optimization problems, DGD typically
converges to local minima rather than saddle points. The paper considers
unconstrained minimization of a smooth objective function. In centralized
settings, the problem of demonstrating nonconvergence to saddle points of
gradient descent (and variants) is typically handled by way of the
stable-manifold theorem from classical dynamical systems theory. However, the
classical stable-manifold theorem is not applicable in distributed settings.
The paper develops an appropriate stable-manifold theorem for DGD showing that
convergence to saddle points may only occur from a low-dimensional stable
manifold. Under appropriate assumptions (e.g., coercivity), this result implies
that DGD typically converges to local minima and not to saddle points.
