Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)
Isidoros Tziotis, Constantine Caramanis, Aryan Mokhtari
In this paper we study the problem of escaping from saddle points and achieving second-order optimality in a decentralized setting where a group of agents collaborate to minimize their aggregate objective function. We provide a non-asymptotic (finite-time) analysis and show that by following the idea of perturbed gradient descent, it is possible to converge to a second-order stationary point in a number of iterations which depends linearly on dimension and polynomially on the accuracy of second-order stationary point. Doing this in a communication-efficient manner requires overcoming several challenges, from identifying (first order) stationary points in a distributed manner, to adapting the perturbed gradient framework without prohibitive communication complexity. Our proposed Perturbed Decentralized Gradient Tracking (PDGT) method consists of two major stages: (i) a gradient-based step to find a first-order stationary point and (ii) a perturbed gradient descent step to escape from a first-order stationary point, if it is a saddle point with sufficient curvature. As a side benefit of our result, in the case that all saddle points are non-degenerate (strict), the proposed PDGT method finds a local minimum of the considered decentralized optimization problem in a finite number of iterations.