Randomized Optimal Consensus of Multi-agent Systems

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.

💡 Research Summary

This paper addresses the problem of achieving optimal consensus in a network of agents whose communication topology varies randomly over time. Each agent i possesses a private convex cost function f_i(z) defined on ℝ^d and knows only its own set of minimizers X_i = {z | f_i(z) = min f_i}. The global objective is to minimize the sum of all local costs, which is equivalent to finding a point in the intersection X_0 = ⋂_{i=1}^n X_i, assumed to be non‑empty.

The authors propose a simple randomized iteration rule. At every discrete time step k, each agent independently performs a Bernoulli trial with success probability p (0 < p < 1). With probability p the agent updates its state by taking a weighted average of the current states of its neighbors (including itself), i.e.,
 e_i(k) = Σ_{j∈N_i(k)} a_{ij}(k) x_j(k),
where the weights a_{ij}(k) are positive, sum to one for each i, and are uniformly lower‑bounded by a constant η > 0. With probability 1‑p the agent projects its current state onto its own optimal set, i.e.,
 g_i(k) = P_{X_i}(x_i(k)).
Thus the update law is
 x_i(k+1) = e_i(k) with probability p,
 x_i(k+1) = g_i(k) with probability 1‑p.

The communication graph G_k = (V, E_k) is modeled as a stochastic digraph process. Rather than assuming i.i.d. graphs, the paper introduces two weaker connectivity concepts:

1. Stochastically Uniformly Strongly Connected (SUSC) – there exist constants B ≥ 1 and q ∈ (0,1) such that for any time k the joint graph over the interval