An improved convergence analysis for decentralized online stochastic non-convex optimization

In this paper, we study decentralized online stochastic non-convex optimization over a network of nodes. Integrating a technique called gradient tracking in decentralized stochastic gradient descent (DSGD), we show that the resulting algorithm, GT-DSGD, exhibits several important characteristics towards minimizing a sum of smooth non-convex functions. The main results of this paper can be divided into two categories: (1) For general smooth non-convex functions, we establish a non-asymptotic characterization of GT-DSGD and derive the conditions under which it achieves network-independent performance and matches centralized minibatch SGD. In comparison, the existing results suggest that the performance of GT-DSGD is always network-dependent and is therefore strictly worse than that of centralized minibatch SGD. (2) When the global function additionally satisfies the Polyak-Lojasiewics condition, we derive the exponential stability range for GT-DSGD under a constant step-size up to a steady-state error. Under stochastic approximation step-sizes, we establish, for the first time, the optimal global sublinear convergence rate on almost every sample path, in addition to the convergence rate in mean. Since strongly convex functions are a special case of this class of problems, our results are not only immediately applicable but also improve the currently known best convergence rates and their dependence on problem parameters.