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

Brian Swenson, Ryan Murray, H. Vincent Poor, Soummya Kar

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
JournalIEEE Transactions on Automatic Control
DOIs
StateAccepted/In press - 2021

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

Keywords

  • Convergence
  • Distributed optimization
  • gradient descent
  • gradient flow
  • Heuristic algorithms
  • Linear programming
  • Manifolds
  • nonconvex optimization
  • nonsmooth optimization
  • Optimization
  • saddle point
  • stable manifold
  • Standards
  • Trajectory

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