Convergence Analysis of Quantized Primal-dual Algorithm in Network Utility Maximization Problems

TL;DR

This paper analyzes the convergence behavior of a quantized primal-dual algorithm in network utility maximization, providing bounds on convergence rates and error that depend on communication constraints and problem parameters.

Contribution

It introduces universal bounds on the convergence rates and error for quantized primal-dual algorithms, linking communication bit rates with optimization performance.

Findings

Universal lower bounds on exponential convergence rates.

Bounds on mean square error for finite iterations.

Trade-offs between communication rate and convergence speed.

Abstract

This paper investigates the asymptotic and non-asymptotic behavior of the quantized primal dual algorithm in network utility maximization problems, in which a group of agents maximize the sum of their individual concave objective functions under linear constraints. In the asymptotic scenario, we use the information theoretic notion of differential entropy power to establish universal lower bounds on the exponential convergence rates of joint primal dual, primal and dual variables under optimum achieving quantization schemes. These results provide trade offs between the speed of exponential convergence, the agents objective functions, the communication bit rates, and the number of agents and constraints. In the non-asymptotic scenario, we obtain lower bounds on the mean square distance of joint primal dual, primal and dual variables from the optimal solution for any finite time instance. These bounds hold regardless of the quantization scheme used.