Logarithmically Quantized Distributed Optimization over Dynamic Multi-Agent Networks

TL;DR

This paper introduces a distributed optimization method over multi-agent networks that uses logarithmic quantization, enabling more efficient and accurate data exchange in bandwidth-limited environments, even with dynamic network topologies.

Contribution

It proposes a novel logarithmic quantization scheme for distributed optimization that improves precision near the optimum and handles dynamic network topologies.

Findings

Convergence of the proposed dynamics is established.

Logarithmic quantization improves accuracy over uniform quantization.

The method is robust to network topology changes.

Abstract

Distributed optimization finds many applications in machine learning, signal processing, and control systems. In these real-world applications, the constraints of communication networks, particularly limited bandwidth, necessitate implementing quantization techniques. In this paper, we propose distributed optimization dynamics over multi-agent networks subject to logarithmically quantized data transmission. Under this condition, data exchange benefits from representing smaller values with more bits and larger values with fewer bits. As compared to uniform quantization, this allows for higher precision in representing near-optimal values and more accuracy of the distributed optimization algorithm. The proposed optimization dynamics comprise a primary state variable converging to the optimizer and an auxiliary variable tracking the objective function's gradient. Our setting accommodates dynamic network topologies, resulting in a hybrid system requiring convergence analysis using matrix perturbation theory and eigenspectrum analysis.