Quantized Consensus ADMM for Multi-Agent Distributed Optimization

TL;DR

This paper introduces a quantized ADMM-based distributed optimization algorithm capable of handling general convex objectives, including non-smooth functions like LASSO, with proven convergence and bounded consensus error.

Contribution

It extends quantized ADMM to non-quadratic, possibly non-smooth objectives in multi-agent networks, providing convergence guarantees and error bounds.

Findings

Converges within logarithmic iterations based on network and objective parameters.

Achieves a tight upper bound on consensus error independent of network size.

Applicable to general convex functions including LASSO.

Abstract

Multi-agent distributed optimization over a network minimizes a global objective formed by a sum of local convex functions using only local computation and communication. We develop and analyze a quantized distributed algorithm based on the alternating direction method of multipliers (ADMM) when inter-agent communications are subject to finite capacity and other practical constraints. While existing quantized ADMM approaches only work for quadratic local objectives, the proposed algorithm can deal with more general objective functions (possibly non-smooth) including the LASSO. Under certain convexity assumptions, our algorithm converges to a consensus within $\log_{1+\eta}\Omega$ iterations, where $\eta>0$ depends on the local objectives and the network topology, and $\Omega$ is a polynomial determined by the quantization resolution, the distance between initial and optimal variable values, the local objective functions and the network topology. A tight upper bound on the consensus error is also obtained which does not depend on the size of the network.