Stability and Performance Limits of Adaptive Primal-Dual Networks

TL;DR

This paper analyzes the stability and performance of adaptive primal-dual networks using AH and AL methods, revealing limitations in stochastic settings and proposing enhancements to improve stability and accuracy.

Contribution

It provides a comparative analysis of primal-dual strategies over networks, highlighting their stability issues and proposing a step-size tuning method for AL strategies.

Findings

AH can become unstable under partial observation

AL strategies have narrower stability ranges

Proposed step-size method improves AL performance

Abstract

This work studies distributed primal-dual strategies for adaptation and learning over networks from streaming data. Two first-order methods are considered based on the Arrow-Hurwicz (AH) and augmented Lagrangian (AL) techniques. Several revealing results are discovered in relation to the performance and stability of these strategies when employed over adaptive networks. The conclusions establish that the advantages that these methods have for deterministic optimization problems do not necessarily carry over to stochastic optimization problems. It is found that they have narrower stability ranges and worse steady-state mean-square-error performance than primal methods of the consensus and diffusion type. It is also found that the AH technique can become unstable under a partial observation model, while the other techniques are able to recover the unknown under this scenario. A method to enhance the performance of AL strategies is proposed by tying the selection of the step-size to their regularization parameter. It is shown that this method allows the AL algorithm to approach the performance of consensus and diffusion strategies but that it remains less stable than these other strategies.