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

TL;DR

This paper analyzes distributed gradient flow (DGF) in multi-agent systems, showing convergence to critical points for nonsmooth nonconvex functions and saddle point evasion for smooth functions, supported by a stable manifold theorem.

Contribution

It introduces a stable manifold theorem for DGF and extends understanding of saddle point evasion in distributed nonconvex optimization.

Findings

DGF converges to critical points in nonsmooth, nonconvex settings.

DGF only converges to saddle points from a zero-measure set of initial conditions.

Stable manifold theorem for DGF is established, of independent theoretical interest.

Abstract

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