Randomized Optimal Consensus of Multi-agent Systems

TL;DR

This paper introduces a randomized algorithm for multi-agent systems to achieve optimal consensus despite stochastic network changes, allowing agents to balance local optimization and neighbor influence.

Contribution

It proposes a novel randomized consensus algorithm that guarantees almost sure convergence to the optimal solution under stochastic topologies and provides convergence analysis using convex analysis.

Findings

Achieves almost sure consensus in stochastic networks

Balances local optimization with neighbor influence

Outperforms deterministic methods in simulations

Abstract

In this paper, we formulate and solve a randomized optimal consensus problem for multi-agent systems with stochastically time-varying interconnection topology. The considered multi-agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem $\min_{z\in \mathds{R}^d}\ \sum_{i=1}^n f_i(z),$ in which the optimal solution set of objective function $f_i$ can only be observed by agent $i$ itself. At each time step, simply determined by a Bernoulli trial, each agent independently and randomly chooses either taking an average among its neighbor set, or projecting onto the optimal solution set of its own optimization component. Both directed and bidirectional communication graphs are studied. Connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions. The convergence analysis is carried out using convex analysis. We compare the randomized algorithm with the deterministic one via a numerical example. The results illustrate that a group of autonomous agents can reach an optimal opinion by each node simply making a randomized trade-off between following its neighbors or sticking to its own opinion at each time step.