Decentralized Learning with Approximate Finite-Time Consensus

TL;DR

This paper explores how using approximate finite-time consensus matrices in decentralized learning algorithms affects convergence and performance, broadening applicability beyond structured networks.

Contribution

It introduces a method to incorporate approximate FTC matrices in decentralized optimization, analyzing their impact without requiring specific graph structures.

Findings

Approximate FTC matrices can be inferred for general graphs.

Using approximate FTC matrices affects convergence rate and steady-state performance.

The method is applicable to a wide range of decentralized learning scenarios.

Abstract

The performance of algorithms for decentralized optimization is affected by both the optimization error and the consensus error, the latter of which arises from the variation between agents' local models. Classically, algorithms employ averaging and gradient-tracking mechanisms with constant combination matrices to drive the collection of agents to consensus. Recent works have demonstrated that using sequences of combination matrices that achieve finite-time consensus (FTC) can result in improved communication efficiency or iteration complexity for decentralized optimization. Notably, these studies apply to highly structured networks, where exact finite-time consensus sequences are known exactly and in closed form. In this work we investigate the impact of utilizing approximate FTC matrices in decentralized learning algorithms, and quantify the impact of the approximation error on convergence rate and steady-state performance. Approximate FTC matrices can be inferred for general graphs and do not rely on a particular graph structure or prior knowledge, making the proposed scheme applicable to a broad range of decentralized learning settings.