Stochastic Push-Pull for Decentralized Nonconvex Optimization

TL;DR

This paper analyzes the convergence of the Stochastic Push-Pull method for decentralized nonconvex optimization, establishing conditions for linear speedup and validating results through extensive experiments.

Contribution

It introduces a general analytical framework for decentralized stochastic optimization and derives a novel sufficient condition for linear speedup of the Stochastic Push-Pull algorithm.

Findings

Achieves linear speedup under certain conditions

Provides a unified framework for decentralized algorithms

Validates theoretical results with numerical experiments

Abstract

To understand the convergence behavior of the Push-Pull method for decentralized optimization with stochastic gradients (Stochastic Push-Pull), this paper presents a comprehensive analysis. Specifically, we first clarify the algorithm's underlying assumptions, particularly those regarding the network structure and weight matrices. Then, to establish the convergence rate under smooth nonconvex objectives, we introduce a general analytical framework that not only encompasses a broad class of decentralized optimization algorithms, but also recovers or enhances several state-of-the-art results for distributed stochastic gradient tracking methods. A key highlight is the derivation of a sufficient condition under which the Stochastic Push-Pull algorithm achieves linear speedup, matching the scalability of centralized stochastic gradient methods -- a result not previously reported. Extensive numerical experiments validate our theoretical findings, demonstrating the algorithm's effectiveness and robustness across various decentralized optimization scenarios.