On the Convergence of Decentralized Stochastic Gradient-Tracking with Finite-Time Consensus

TL;DR

This paper analyzes how approximate finite-time consensus sequences affect the convergence of decentralized gradient-tracking algorithms, considering practical limitations like network knowledge and numerical issues.

Contribution

It quantifies the impact of approximate consensus sequences on convergence, extending analysis to any periodic combination matrix sequences.

Findings

Convergence is affected by the approximation error and sequence length.

Provides bounds relating consensus approximation error to convergence rate.

Applicable to practical scenarios with imperfect network knowledge.

Abstract

Algorithms for decentralized optimization and learning rely on local optimization steps coupled with combination steps over a graph. Recent works have demonstrated that using a time-varying sequence of matrices that achieves finite-time consensus can improve the communication and iteration complexity of decentralized optimization algorithms based on gradient tracking. In practice, a sequence of matrices satisfying the exact finite-time consensus property may not be available due to imperfect knowledge of the network topology, a limit on the length of the sequence, or numerical instabilities. In this work, we quantify the impact of approximate finite-time consensus sequences on the convergence of a gradient-tracking based decentralized optimization algorithm. Our results hold for any periodic sequence of combination matrices. We clarify the interplay between approximation error of the finite-time consensus sequence and the length of the sequence as well as typical problem parameters such as smoothness and gradient noise.