[Paper Review] Linear Time Average Consensus on Fixed Graphs and Implications for Decentralized Optimization and Multi-Agent Control

Alex Olshevsky|arXiv (Cornell University)|Nov 15, 2014

Distributed Control Multi-Agent Systems63 references61 citations

TL;DR

This paper presents a distributed average consensus protocol for fixed undirected graphs that achieves linear convergence time in the number of nodes, requiring only that all nodes know a constant-factor upper bound on the total number of nodes. The protocol enables linear-time convergence for decentralized optimization, formation control, and leader-following by leveraging memory-based updates and subgradient methods with bounded subgradients.

ABSTRACT

We describe a protocol for the average consensus problem on any fixed undirected graph whose convergence time scales linearly in the total number nodes $n$. The protocol is completely distributed, with the exception of requiring all nodes to know the same upper bound $U$ on the total number of nodes which is correct within a constant multiplicative factor. We next discuss applications of this protocol to problems in multi-agent control connected to the consensus problem. In particular, we describe protocols for formation maintenance and leader-following with convergence times which also scale linearly with the number of nodes. Finally, we develop a distributed protocol for minimizing an average of (possibly nondifferentiable) convex functions $ (1/n) \sum_{i=1}^n f_i(θ)$, in the setting where only node $i$ in an undirected, connected graph knows the function $f_i(θ)$. Under the same assumption about all nodes knowing $U$, and additionally assuming that the subgradients of each $f_i(θ)$ have absolute values upper bounded by some constant $L$ known to the nodes, we show that after $T$ iterations our protocol has error which is $O(L \sqrt{n/T})$.

Motivation & Objective

• To design a distributed consensus protocol with linear convergence time on fixed undirected graphs.
• To enable fast convergence in decentralized optimization, formation control, and leader-following using the consensus protocol.
• To ensure convergence time scales linearly with the number of nodes, even in challenging topologies like line and lollipop graphs.
• To provide theoretical guarantees for error decay in decentralized optimization under bounded subgradients.
• To extend the applicability of consensus protocols to real-world multi-agent systems with minimal global knowledge.

Proposed method

• The protocol uses memory-based updates, where each node maintains and updates its state based on its own and neighbors' past values.
• It employs a linear combination of current and previous states, along with neighbor values, to accelerate convergence.
• For decentralized optimization, the method applies subgradient descent with a step size that ensures error decay as $ O(L\sqrt{n/T}) $.
• The protocol assumes all nodes know an upper bound $ U $ on $ n $, correct within a constant factor, to enable distributed coordination.
• Theoretical analysis relies on spectral properties of the graph and bounds on initial deviation and subgradient magnitude.
• The approach is validated on line and lollipop graphs, showing consistent linear convergence despite structural bottlenecks.

Experimental results

Research questions

• RQ1 Can a distributed consensus protocol achieve linear convergence time on fixed undirected graphs without requiring full knowledge of the graph?
• RQ2 How can memory-based updates improve convergence speed in average consensus compared to standard convex combination methods?
• RQ3 What is the achievable convergence rate for decentralized optimization of nonsmooth convex functions under bounded subgradients?
• RQ4 Can the proposed protocol maintain linear convergence in formation control and leader-following tasks?
• RQ5 What is the impact of network topology—such as line or lollipop graphs—on the convergence behavior of the protocol?

Key findings

• The proposed consensus protocol achieves linear convergence time $ O(n) $ in the number of nodes $ n $, independent of the graph's spectral gap.
• The protocol maintains linear convergence even on challenging topologies such as the line and lollipop graphs, which are known to slow down standard consensus protocols.
• For decentralized optimization, the error decays as $ O(L\sqrt{n/T}) $ after $ T $ iterations, with $ L $ being the subgradient bound.
• The protocol requires only that all nodes know a constant-factor upper bound $ U $ on $ n $, enabling full distribution without global knowledge.
• The method achieves linear convergence in formation maintenance and leader-following tasks by leveraging the same underlying consensus mechanism.
• Empirical results confirm that $ T = 4n $ iterations suffice to compute the median with high accuracy in both line and lollipop graphs.