Convergence Analysis of Noisy Distributed Gradient Descent for Non-convex Optimization -- Saddle Point Escape

TL;DR

This paper proposes a noisy distributed gradient descent method for non-convex optimization that helps escape saddle points, ensuring convergence to local minima with high probability and comparable rates to centralized algorithms.

Contribution

It introduces a perturbation-based variant of distributed gradient descent that effectively escapes saddle points in non-convex problems, with theoretical convergence guarantees.

Findings

Converges to local minima with high probability

Achieves convergence rates similar to centralized algorithms

Numerical results validate improved performance over standard DGD

Abstract

A variant of consensus based distributed gradient descent (\textbf{DGD}) is studied for finite sums of smooth but possibly non-convex functions. In particular, the local gradient term in the fixed step-size iteration of each agent is randomly perturbed to evade saddle points. Under regularity conditions, it is established that for sufficiently small step size and noise variance, each agent converges with high probability to a specified radius neighborhood of a common second-order stationary point, i.e., local minimizer. The rate of convergence is shown to be comparable to centralized first-order algorithms. Numerical experiments are presented to validate the efficacy of the proposed approach over standard \textbf{DGD} in a non-convex setting.