A principled framework for the design and analysis of token algorithms

TL;DR

This paper introduces a novel framework for decentralized optimization using token algorithms, which perform random walks over networks, offering advantages in communication efficiency and privacy, with proven convergence and empirical validation.

Contribution

It presents a principled, unified analysis of token algorithms, extending existing gossip algorithms to include multiple tokens and general graphs, with tight convergence rates.

Findings

Proven convergence rates for token algorithms on complete graphs.

Extension of results to multiple tokens and general network topologies.

Empirical performance validation of the proposed algorithms.

Abstract

We consider a decentralized optimization problem, in which $n$ nodes collaborate to optimize a global objective function using local communications only. While many decentralized algorithms focus on \emph{gossip} communications (pairwise averaging), we consider a different scheme, in which a ``token'' that contains the current estimate of the model performs a random walk over the network, and updates its model using the local model of the node it is at. Indeed, token algorithms generally benefit from improved communication efficiency and privacy guarantees. We frame the token algorithm as a randomized gossip algorithm on a conceptual graph, which allows us to prove a series of convergence results for variance-reduced and accelerated token algorithms for the complete graph. We also extend these results to the case of multiple tokens by extending the conceptual graph, and to general graphs by tweaking the communication procedure. The reduction from token to well-studied gossip algorithms leads to tight rates for many token algorithms, and we illustrate their performance empirically.