  In this work, we study two first-order primal-dual based algorithms, the
Gradient Primal-Dual Algorithm (GPDA) and the Gradient Alternating Direction
Method of Multipliers (GADMM), for solving a class of linearly constrained
non-convex optimization problems. We show that with random initialization of
the primal and dual variables, both algorithms are able to compute second-order
stationary solutions (ss2) with probability one. This is the first result
showing that primal-dual algorithm is capable of finding ss2 when only using
first-order information, it also extends the existing results for first-order,
but primal-only algorithms.
  An important implication of our result is that it also gives rise to the
first global convergence result to the ss2, for two classes of unconstrained
distributed non-convex learning problems over multi-agent networks.
