Second-Order Properties of Noisy Distributed Gradient Descent

We study a fixed step-size distributed gradient descent algorithm for solving optimization problems in which the objective is a finite sum of smooth but possibly non-convex functions. Random perturbations of the gradient descent directions are introduced at each step to actively evade saddle points. Under certain regularity conditions, and with a suitable step-size, it is established that each agent converges to a neighborhood of a local minimizer; the size of the neighborhood depends on the step-size and a probabilistic confidence parameter. A numerical example is presented to illustrate the effectiveness of the random perturbations in terms of escaping saddle points in fewer iterations than without the perturbations.