Convergence Analysis of Distributed Gradient Descent Algorithms With One and Two Momentum Terms

For the centralized optimization, it is well known that adding one momentum term (also called the heavy-ball method) can obtain a faster convergence rate than the gradient method. However, for the distributed counterpart, there is quite few results about the effect of added momentum terms on the con...

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Bibliographic Details
Published in:IEEE transactions on cybernetics Vol. 54; no. 3; pp. 1511 - 1522
Main Authors: Liu, Bing, Chai, Li, Yi, Jingwen
Format: Journal Article
Language:English
Published: United States IEEE 01.03.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Control theory
Convergence
Convergence rate
Cost function
distributed optimization
Linear programming
Momentum
momentum term
Network topology
Newton method
Privacy
Routh criterion
Signal processing algorithms
ISSN:2168-2267, 2168-2275, 2168-2275
Online Access:Get full text
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Summary:For the centralized optimization, it is well known that adding one momentum term (also called the heavy-ball method) can obtain a faster convergence rate than the gradient method. However, for the distributed counterpart, there is quite few results about the effect of added momentum terms on the convergence rate. This article is aimed at studying the issue in the distributed setup, where <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> agents minimize the sum of their individual cost functions using local communication over a network. The cost functions are twice continuously differentiable. We first study the algorithm with one momentum term and develop a distributed heavy-ball (D-HB) method by adding one momentum term on to the distributed gradient algorithm. By borrowing tools from the control theory, we provide a simple convergence proof and an explicit expression of the optimal convergence rate. Furthermore, we consider adding two momentum terms case and propose a distributed double-heavy-ball (D-DHB) method. We show that adding one momentum term allows faster convergence while adding two momentum terms does not perform any superiorities. Finally, simulation examples are given to illustrate our findings.
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ISSN:2168-2267
2168-2275
2168-2275
DOI:10.1109/TCYB.2022.3218663