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

Area of Science:

• Distributed optimization
• Control theory
• Networked systems

Background:

• Centralized optimization benefits from momentum terms like the heavy-ball method for faster convergence.
• Limited research exists on the impact of momentum terms in distributed optimization settings.

Purpose of the Study:

• Investigate the effect of momentum terms on convergence rates in distributed optimization.
• Develop and analyze new distributed algorithms incorporating momentum.
• Compare the performance of single and double momentum terms.

Main Methods:

• Developed a distributed heavy-ball (D-HB) method by adding one momentum term to distributed gradient algorithms.
• Utilized control theory for convergence analysis and optimal rate derivation.
• Proposed a distributed double-heavy-ball (D-DHB) method for comparison.

Main Results:

• The D-HB method with one momentum term achieves faster convergence than standard distributed gradient methods.
• An explicit expression for the optimal convergence rate of D-HB was derived.
• The D-DHB method with two momentum terms did not show superior performance compared to D-HB.

Conclusions:

• A single momentum term significantly improves convergence speed in distributed optimization.
• Adding a second momentum term does not yield additional benefits for convergence in this distributed setup.
• Simulation results validate the theoretical findings on convergence rates.