[Paper Review] An Approximate Projected Consensus Algorithm for Computing Intersection of Convex Sets

Youcheng Lou, Guodong Shi+2 more|arXiv (Cornell University)|May 26, 2012

Distributed Control Multi-Agent Systems26 references4 citations

TL;DR

This paper proposes an approximate projected consensus algorithm for multi-agent networks to compute the intersection of convex sets, allowing each node to use inexact projections within a bounded angular error. Under uniformly jointly strongly connected communication graphs, it proves that projection accuracy with angle error less than $\pi/4$ ensures bounded, convergent consensus to the intersection set, and surprisingly, numerical results show faster convergence than exact projection methods in some cases.

ABSTRACT

In this paper, we propose an approximate projected consensus algorithm for a network to cooperatively compute the intersection of convex sets. Instead of assuming the exact convex projection proposed in the literature, we allow each node to compute an approximate projection and communicate it to its neighbors. The communication graph is directed and time-varying. Nodes update their states by weighted averaging. Projection accuracy conditions are presented for the considered algorithm. They indicate how much projection accuracy is required to ensure global consensus to a point in the intersection set when the communication graph is uniformly jointly strongly connected. We show that $π/4$ is a critical angle error of the projection approximation to ensure a bounded state. A numerical example indicates that this approximate projected consensus algorithm may achieve better performance than the exact projected consensus algorithm in some cases.

Motivation & Objective

• To address the challenge of computing the intersection of convex sets in multi-agent networks when exact convex projections are computationally infeasible due to noise or complexity.
• To develop a distributed algorithm that allows nodes to use approximate projections instead of exact ones, while still ensuring global consensus to the intersection set.
• To establish sufficient and necessary conditions on projection accuracy—specifically, angular error bounds—for convergence under time-varying, directed communication graphs.
• To investigate whether approximate projections can outperform exact projections in terms of convergence speed, challenging the assumption that exactness always improves performance.

Proposed method

• Each node computes an approximate projection of its current state onto its local convex set, where the direction lies within a convex cone defined by the exact projection direction and an angular deviation less than $\theta$.
• The algorithm uses weighted averaging over a directed, time-varying communication graph, with nodes updating their states based on neighbors' approximate projections.
• The analysis models the distance to the intersection set using a recurrence relation involving $\tan\theta$, showing that $\theta < \pi/4$ ensures boundedness of the state trajectory.
• A critical threshold analysis identifies $\pi/4$ as the boundary: for $\theta > \pi/4$, divergence is possible; for $\theta = \pi/4$, boundedness depends on set geometry and initial conditions.
• The convergence is analyzed under the assumption of uniformly jointly strongly connected communication graphs, ensuring long-term connectivity across time intervals.
• A numerical example compares the approximate projected consensus algorithm (APCA) with the exact projected consensus algorithm (PCA), measuring convergence speed via the distance function $h(k) = \max_i |x_i(k)|_{X_*}$.

Experimental results

Research questions

• RQ1 What level of projection accuracy—measured by angular deviation from the exact projection—is sufficient to guarantee global consensus to the intersection of convex sets in a distributed network?
• RQ2 Can approximate projections lead to faster convergence than exact projections, despite reduced per-node accuracy?
• RQ3 What is the critical angular error threshold that separates bounded from unbounded state trajectories in the approximate projected consensus algorithm?
• RQ4 How does the geometry of the convex sets and initial conditions affect the convergence behavior when the angular error is exactly $\pi/4$?

Key findings

• The paper establishes that $\pi/4$ is a critical threshold for angular error in approximate projections: if $\theta < \pi/4$, the state trajectories remain uniformly bounded and converge to the intersection set.
• When $\theta > \pi/4$, the system can diverge, even with a single node and specific approximate projection points, indicating instability under high error.
• For $\theta = \pi/4$, boundedness is not uniform across initial conditions and depends heavily on the shape of the convex sets, with trajectories potentially diverging for certain configurations.
• A numerical example demonstrates that the approximate projected consensus algorithm (APCA) can converge faster than the exact projected consensus algorithm (PCA), as evidenced by a faster decay of the distance function $h(k)$ at time $k=2000$.
• The results challenge the conventional assumption that higher projection accuracy always improves collective performance, suggesting that approximation may enhance convergence speed in practice.
• The convergence analysis holds under the condition of uniformly jointly strongly connected communication graphs, which ensures persistent connectivity over time intervals.