Gossip consensus and averaging algorithms with quantization

Abstract-We study distributed consensus problems of multiagent systems on directed networks and subject to quantized information flow. For the communication among component agents, particular attention is given to the gossip type, which models their asynchronous behaviors; for quantization effects, each agent's state is abstracted to be an integer. The central question investigated is how to design distributed algorithms and what connectivity of networks that together lead to consensus. This investigation is carried out for both general consensus and average consensus; for each case, a class of algorithms is proposed, under which a necessary and sufficient graphical condition is derived to guarantee the corresponding consensus. In particular, the obtained graphical condition ensuring average consensus is weaker than those in the literature for either realvalued or quantized states, in the sense that it does not require symmetric (or balanced) network topologies

Resilient Randomized Quantized Consensus

We consider the problem of multi-agent consensus where some agents are subject to faults/attacks and might make updates arbitrarily. The network consists of agents taking integer-valued (i.e., quantized) states under directed communication links. The goal of the healthy normal agents is to form consensus in their state values, which may be disturbed by the non-normal, malicious agents. We develop update schemes to be equipped by the normal agents whose interactions are asynchronous and subject to non-uniform and time-varying time delays. In particular, we employ a variant of the so-called mean subsequence reduced (MSR) algorithms, which have been long studied in computer science, where each normal agent ignores extreme values from its neighbors. We solve the resilient quantized consensus problems in the presence of totally/locally bounded adversarial agents and provide necessary and sufficient conditions in terms of the connectivity notion of graph robustness. Furthermore, it will be shown that randomization is essential both in quantization and in the updating times when normal agents interact in an asynchronous manner. The results are examined through a numerical example.Comment: 15 pages, 13 figure

Limited Rate Distributed Weight-Balancing and Average Consensus Over Digraphs

Distributed quantized weight-balancing and average consensus over fixed digraphs are considered. A digraph with non-negative weights associated to its edges is weight-balanced if, for each node, the sum of the weights of its out-going edges is equal to that of its incoming edges. This paper proposes and analyzes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes (compliant to the directed nature of the graph edges). Asymptotic convergence of the scheme is proved and a convergence rate analysis is provided. Building on this result, a novel distributed algorithm is proposed that solves the average consensus problem over digraphs, using, at each iteration, finite rate simplex communications between adjacent nodes -- some bits for the weight-balancing problem, other for the average consensus. Convergence of the proposed quantized consensus algorithm to the average of the real (i.e., unquantized) agent's initial values is proved, both almost surely and in $r$th mean for all positive integer $r$. Finally, numerical results validate our theoretical findings.Comment: Part of this work will be presented at the 57th IEEE Conference on Decision and Contro

Edge Agreement of Second-order Multi-agent System with Dynamic Quantization via Directed Edge Laplacian

This work explores the edge agreement problem of second-order multi-agent system with dynamic quantization under directed communication. To begin with, by virtue of the directed edge laplacian, we derive a model reduction representation of the closed-loop multi-agent system depended on the spanning tree subgraph. Considering the limitations of the finite bandwidth channels, the quantization effects of second-order multi-agent system under directed graph are considered. Motivated by the observation that the static quantizer always lead to the practical stability rather than the asymptotic stability, the dynamic quantized communication strategy referring to the rooming in-rooming out scheme is employed. Based on the reduced model associated with the essential edge Laplacian, the asymptotic stability of second-order multi-agent system under dynamic quantized effects with only finite quantization level can be guaranteed. Finally, simulation results are provided to verify the theoretical analysis.Comment: 21 pages; submitted to Nonlinear Analysis: Hybrid Systems, Ms. Ref. No.: NAHS-D-15-00161. arXiv admin note: substantial text overlap with arXiv:1501.0667

Coordination Over Multi-Agent Networks With Unmeasurable States and Finite-Level Quantization

In this note, the coordination of linear discrete-time multi-agent systems over digital networks is investigated with unmeasurable states in agents' dynamics. The quantized-observer based communication protocols and Certainty Equivalence principle based control protocols are proposed to characterize the inter-agent communication and the cooperative control in an integrative framework. By investigating the structural and asymptotic properties of the equations of stabilization and estimation errors nonlinearly coupled by the finite-level quantization scheme, some necessary conditions and sufficient conditions are given for the existence of such communication and control protocols to ensure the inter-agent state observation and cooperative stabilization. It is shown that these conditions come down to the simultaneous stabilizability and the detectability of the dynamics of agents and the structure of the communication network.Comment: 10 pages, 2 figure

A Robust Gradient Tracking Method for Distributed Optimization over Directed Networks

In this paper, we consider the problem of distributed consensus optimization over multi-agent networks with directed network topology. Assuming each agent has a local cost function that is smooth and strongly convex, the global objective is to minimize the average of all the local cost functions. To solve the problem, we introduce a robust gradient tracking method (R-Push-Pull) adapted from the recently proposed Push-Pull/AB algorithm. R-Push-Pull inherits the advantages of Push-Pull and enjoys linear convergence to the optimal solution with exact communication. Under noisy information exchange, R-Push-Pull is more robust than the existing gradient tracking based algorithms; the solutions obtained by each agent reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy. We provide a numerical example that demonstrate the effectiveness of R-Push-Pull

Quantized Consensus by the ADMM: Probabilistic versus Deterministic Quantizers

This paper develops efficient algorithms for distributed average consensus with quantized communication using the alternating direction method of multipliers (ADMM). We first study the effects of probabilistic and deterministic quantizations on a distributed ADMM algorithm. With probabilistic quantization, this algorithm yields linear convergence to the desired average in the mean sense with a bounded variance. When deterministic quantization is employed, the distributed ADMM either converges to a consensus or cycles with a finite period after a finite-time iteration. In the cyclic case, local quantized variables have the same mean over one period and hence each node can also reach a consensus. We then obtain an upper bound on the consensus error which depends only on the quantization resolution and the average degree of the network. Finally, we propose a two-stage algorithm which combines both probabilistic and deterministic quantizations. Simulations show that the two-stage algorithm, without picking small algorithm parameter, has consensus errors that are typically less than one quantization resolution for all connected networks where agents' data can be of arbitrary magnitudes.Comment: This version updates the corresponding TSP paper where Theorem 4 was incorrec

r-Robustness and (r,s)-Robustness of Circulant Graphs

There has been recent growing interest in graph theoretical properties known as r- and (r,s)-robustness. These properties serve as sufficient conditions guaranteeing the success of certain consensus algorithms in networks with misbehaving agents present. Due to the complexity of determining the robustness for an arbitrary graph, several methods have previously been proposed for identifying the robustness of specific classes of graphs or constructing graphs with specified robustness levels. The majority of such approaches have focused on undirected graphs. In this paper we identify a class of scalable directed graphs whose edge set is determined by a parameter k and prove that the robustness of these graphs is also determined by k. We support our results through computer simulations.Comment: 6 pages, 6 figures. Accepted to 2017 IEEE CD

Multi-Agent Distributed Coordination Control: Developments and Directions

In this paper, the recent developments on distributed coordination control, especially the consensus and formation control, are summarized with the graph theory playing a central role, in order to present a cohesive overview of the multi-agent distributed coordination control, together with brief reviews of some closely related issues including rendezvous/alignment, swarming/flocking and containment control.In terms of the consensus problem, the recent results on consensus for the agents with different dynamics from first-order, second-order to high-order linear and nonlinear dynamics, under different communication conditions, such as cases with/without switching communication topology and varying time-delays, are reviewed, in which the algebraic graph theory is very useful in the protocol designs, stability proofs and converging analysis. In terms of the formation control problem, after reviewing the results of the algebraic graph theory employed in the formation control, we mainly pay attention to the developments of the rigid and persistent graphs. With the notions of rigidity and persistence, the formation transformation, splitting and reconstruction can be completed, and consequently the range-based formation control laws are designed with the least required information in order to maintain a formation rigid/persistent. Afterwards, the recent results on rendezvous/alignment, swarming/flocking and containment control, which are very closely related to consensus and formation control, are briefly introduced, in order to present an integrated view of the graph theory used in the coordination control problem. Finally, towards the practical applications, some directions possibly deserving investigation in coordination control are raised as well.Comment: 28 pages, 8 figure

Distributed Average Consensus with Bounded Quantizer and Unbounded Input

This paper considers distributed average consensus using finite-bit bounded quantizer with possibly unbounded data. Under the framework of the alternating direction method of multipliers (ADMM), we develop distributed averaging algorithms where each node iteratively updates using only the local information and finitely quantized outputs from its neighbors. It is shown that all the agent variables either converge to the same quantization level or cycle around the data average after finite iterations. An error bound for the consensus value is established, which turns out to be the same as that of using the unbounded rounding quantizer provided that an algorithm parameter (i.e., ADMM step size) is small enough. We also analyze the effect of the algorithm parameter and propose an adaptive parameter selection strategy that only requires knowledge of the number of agents in order to accelerate the algorithm with certain consensus accuracy guarantee. Finally, simulations are performed to illustrate the effectiveness of the proposed algorithms.Comment: Submitted to IEEE Trans. Signal Processin