Distributed Nesterov Flows for Multi-agent Optimization

TL;DR

This paper introduces a continuous-time distributed Nesterov flow and derived algorithms that outperform existing methods in convergence speed for multi-agent optimization, with analysis on network effects.

Contribution

It establishes a continuous-time approximation of distributed Nesterov gradient descent and develops flow-inspired algorithms with improved convergence rates.

Findings

Faster convergence to the same accuracy compared to existing methods.

Improved convergence rate for general convex functions without extra communication.

Explicit relationship between convergence rate and network topology.

Abstract

Various distributed gradient descent algorithms for multi-agent optimization have incorporated the Nesterov accelerated gradient method, where the use of momentum enhances convergence rates. These algorithms have found broad applications in large-scale machine learning and optimization owing to their simplicity and low communication complexity. In this paper, we establish a continuous-time approximation of distributed Nesterov gradient descent. The convergence properties and convergence rate of the resulting distributed Nesterov flow are analyzed using Lyapunov methods. Building on these insights, we design new parameter choices within the flow, from which we derive flow-inspired discrete-time algorithms for multi-agent optimization. Surprisingly, the resulting algorithms achieve faster convergence compared to existing distributed gradient descent methods: they require fewer iterations to reach the same accuracy for strongly convex functions and exhibit an improved convergence rate for general convex functions without incurring additional communication rounds. Furthermore, we investigate the influence of the network topology on algorithm performance and derive an explicit relationship between the convergence rate and the graph condition number. Numerical simulations are presented to validate the effectiveness of the proposed approach.