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

TL;DR

This paper introduces an improved analysis of the GT-DSGD algorithm for decentralized online stochastic non-convex optimization, showing network-independent performance and optimal convergence rates under certain conditions.

Contribution

The paper provides the first non-asymptotic analysis of GT-DSGD with network-independent guarantees and establishes optimal convergence rates under the PL condition.

Findings

GT-DSGD achieves network-independent performance matching centralized minibatch SGD.

Linear convergence of GT-DSGD up to a steady-state error under the PL condition.

First derivation of optimal global sublinear convergence rate on almost every sample path.

Abstract

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, we show that the resulting algorithm, GT-DSGD, enjoys certain desirable characteristics towards minimizing a sum of smooth non-convex functions. In particular, for general smooth non-convex functions, we establish non-asymptotic characterizations of GT-DSGD and derive the conditions under which it achieves network-independent performances that match the centralized minibatch SGD. In contrast, the existing results suggest that GT-DSGD is always network-dependent and is therefore strictly worse than the centralized minibatch SGD. When the global non-convex function additionally satisfies the Polyak-Lojasiewics (PL) condition, we establish the linear convergence of GT-DSGD up to a steady-state error with appropriate constant step-sizes. Moreover, 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 asymptotically optimal sublinear rate in expectation. Since strongly convex functions are a special case of the functions satisfying the PL condition, our results are not only immediately applicable but also improve the currently known best convergence rates and their dependence on problem parameters.