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

A variant of consensus based distributed gradient descent ($\mathbf{\text{DGD}}$) 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 $\mathbf{\text{DGD}}$ in a non-convex setting.