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

arXiv: https://arxiv.org/abs/2008.05387 · HTML: https://mlsanity.com/paper/2008.05387/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Aug 2020.

## Built on by (most cited first, 4 of 4)

- [Networked Signal and Information Processing](https://mlsanity.com/paper/2210.13767/) (2022)
- [Peer-to-Peer Learning Dynamics of Wide Neural Networks](https://mlsanity.com/paper/2409.15267/) (2024)
- [Asynchronous Distributed Reinforcement Learning for LQR Control via Zeroth-Order Block Coordinate Descent](https://mlsanity.com/paper/2107.12416/) (2021)
- [Understanding A Class of Decentralized and Federated Optimization Algorithms: A Multi-Rate Feedback Control Perspective](https://mlsanity.com/paper/2204.12663/) (2022)

## Abstract (arXiv)

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.