Improved Convergence Rate for a Distributed Two-Time-Scale Gradient Method under Random Quantization

TL;DR

This paper introduces an improved convergence analysis for a distributed two-time-scale gradient method under random quantization, demonstrating faster convergence rates for convex optimization over networks with bandwidth constraints.

Contribution

The paper provides a novel analysis that achieves an improved convergence rate for the distributed gradient method under quantization constraints.

Findings

Convergence rate improved to O(log_2 k / sqrt k) for strongly convex and smooth functions.

Lyapunov function analysis captures consensus and optimality errors.

Method is effective under limited communication bandwidth.

Abstract

We study the so-called distributed two-time-scale gradient method for solving convex optimization problems over a network of agents when the communication bandwidth between the nodes is limited, and so information that is exchanged between the nodes must be quantized. Our main contribution is to provide a novel analysis, resulting to an improved convergence rate of this method as compared to the existing works. In particular, we show that the method converges at a rate $O(log_2 k/\sqrt k)$ to the optimal solution, when the underlying objective function is strongly convex and smooth. The key technique in our analysis is to consider a Lyapunov function that simultaneously captures the coupling of the consensus and optimality errors generated by the method.