Distributed Gradient Flow: Nonsmoothness, Nonconvexity, and Saddle Point\n Evasion

The paper considers distributed gradient flow (DGF) for multi-agent nonconvex\noptimization. DGF is a continuous-time approximation of distributed gradient\ndescent that is often easier to study than its discrete-time counterpart. The\npaper has two main contributions. First, the paper considers optimization of\nnonsmooth, nonconvex objective functions. It is shown that DGF converges to\ncritical points in this setting. The paper then considers the problem of\navoiding saddle points. It is shown that if agents' objective functions are\nassumed to be smooth and nonconvex, then DGF can only converge to a saddle\npoint from a zero-measure set of initial conditions. To establish this result,\nthe paper proves a stable manifold theorem for DGF, which is a fundamental\ncontribution of independent interest. In a companion paper, analogous results\nare derived for discrete-time algorithms.\n