# Accelerated Distributed Primal-Dual Dynamics using Adaptive Synchronization

## TL;DR

This paper introduces an adaptive primal-dual dynamics with synchronization for distributed optimization, achieving accelerated convergence and robustness in multi-agent systems, demonstrated through applications to least squares and SVM problems.

## Contribution

It presents a novel adaptive synchronization law that accelerates convergence of primal-dual dynamics in distributed optimization, with proven stability and robustness properties.

## Key findings

- Achieves faster convergence rates compared to non-adaptive methods
- Proves stability and passivity of the proposed dynamics
- Demonstrates effectiveness on distributed least squares and SVM problems

## Abstract

This paper proposes an adaptive primal-dual dynamics for distributed optimization in multi-agent systems. The proposed dynamics incorporates an adaptive synchronization law that reinforces the interconnection strength between the primal variables of the coupled agents, the given law accelerates the convergence of the proposed dynamics to the saddle-point solution. The resulting dynamics is represented as a feedback interconnected networked system that proves to be passive. The passivity properties of the proposed dynamics are exploited along with the LaSalle's invariance principle for hybrid systems, to establish asymptotic convergence and stability of the saddle-point solution. Further, the primal dynamics is analyzed for the rate of convergence and stronger convergence bounds are established, it is proved that the primal dynamics achieve accelerated convergence under the adaptive synchronization. The robustness of the proposed dynamics is quantified using L2-gain analysis and the correlation between the rate of convergence and robustness of the proposed dynamics is presented. The effectiveness of the proposed dynamics is demonstrated by applying it to solve distributed least squares and distributed support vector machines problems.