On the Optimal Time Complexities in Decentralized Stochastic Asynchronous Optimization

TL;DR

This paper establishes new lower bounds and introduces nearly optimal algorithms for decentralized stochastic asynchronous optimization, significantly advancing understanding of time complexities in heterogeneous and homogeneous settings.

Contribution

It provides the first tight lower bounds and proposes two novel algorithms, Fragile SGD and Amelie SGD, that achieve near-optimal convergence under diverse system conditions.

Findings

New lower bounds for asynchronous decentralized optimization.

Introduction of Fragile SGD and Amelie SGD algorithms.

Algorithms match lower bounds up to a logarithmic factor.

Abstract

We consider the decentralized stochastic asynchronous optimization setup, where many workers asynchronously calculate stochastic gradients and asynchronously communicate with each other using edges in a multigraph. For both homogeneous and heterogeneous setups, we prove new time complexity lower bounds under the assumption that computation and communication speeds are bounded. We develop a new nearly optimal method, Fragile SGD, and a new optimal method, Amelie SGD, that converge under arbitrary heterogeneous computation and communication speeds and match our lower bounds (up to a logarithmic factor in the homogeneous setting). Our time complexities are new, nearly optimal, and provably improve all previous asynchronous/synchronous stochastic methods in the decentralized setup.