Rates of Convergence for Distributed Average Consensus with Probabilistic Quantization Tuncer C. Aysal, Mark J. Coates, Michael G. Rabbat Department of Electrical and Computer Engineering McGill University, Montr´eal, Qu´ebec, Canada {tuncer.aysal, mark.coates, michael.rabbat}@mcgill.ca Abstract— Probabilistically quantized distributed averaging the weighted linear combination, (PQDA) is a fully decentralized algorithm for performing aver- age consensus in a network with finite-rate links. At each itera- xi(t +1)= Wi,ixi(t)+ Wi,j xj (t). tion, nodes exchange quantized messages with their immediate j:(i,j)∈E neighbors. Then each node locally computes a weighted average X of the messages it received, quantizes this new value using a For a given initial state, x(0), and reasonable choices of randomized quantization scheme, and then the whole process weights Wi,j , it is easy to show that limt→∞ xi(t) = is repeated in the next iteration. In our previous work we 1 N , N i=1 xi(0) x. The DA algorithm was introduced by introduced PQDA and demonstrated that the algorithm almost Tsitsiklis in [17], and has since been pursued in various surely converges to a consensus (i.e., every node converges to P the same value). The present article builds upon this work forms by many other researchers (e.g., [3,6,8,11,14,20]). by characterizing the rate of convergence to a consensus. Of course, in any practical implementation of this algo- We illustrate that the rate of PQDA is essentially the same rithm, communication rates between neighboring nodes will as unquantized distributed averaging when the discrepancy be finite, and thus quantization must be applied to xi(t) among node values is large. When the network has nearly before it can be transmitted. In applications where heavy converged and all nodes’ values are at one of two neighboring quantization points, then the rate of convergence slows down. quantization must be applied (e.g., when executing multiple We bound the rate of convergence during this final phase by consensus computations in parallel, so that each packet trans- applying lumpability to compress the state space, and then using mission carries many values), quantization can actually affect stochastic comparison methods. convergence properties of the algorithm. Figure 1 shows the trajectories, xi(t), for all nodes superimposed on one set of axes. In Fig. 1(a), nodes apply deterministic uniform I. INTRODUCTION quantization with ∆ =0.1 spacing between quantization points. Although the algorithm converges in this example, A fundamental problem in decentralized networked sys- clearly the limit is not a consensus; not all nodes arrive at tems is that of having nodes reach a state of agreement. the same value. For example, the nodes in a wireless sensor network must In [1,2] we introduced probabilistically quantized dis- be synchronized in order to communicate using a TDMA tributed averaging (PQDA). Rather than applying deter- scheme or to use time-difference-of-arrival measurements for ministic uniform quantization, nodes independently apply a localization and tracking. Similarly, one would like a host of simple randomized quantization scheme. In this scheme, the unmanned aerial vehicles to make coordinated decisions on random quantized value is equal to the original unquantized a surveillance strategy. This paper focuses on a prototypical value in expectation. Through this use of randomization, we example of agreement in networked systems, namely, the guarantee that PQDA converges almost surely to a consensus. average consensus problem: each node initially has a scalar However, since x¯ is not divisible by ∆ in general, the 1 N value, yi, and the goal is to compute the average, N i=1 yi value we converge to is not precisely the average of the at every node in the network. initial values. We presented characteristics of the limiting P Distributed averaging (DA) is a simple iterative distributed consensus value, in particular, showing that it is equal to x¯ algorithm for solving the average consensus problem with in expectation. many attractive properties. The network state is maintained The main contribution of the present paper is to character- N in a vector x(t) R , where xi(t) is the value at node i after ize the rates of convergence for PQDA. We show that, when ∈ t iterations, and there are N nodes in the network. Network the values xi(t) lie in an interval which is large relative to the connectivity is represented by a graph G = (V, E), with quantizer precision ∆, then PQDA moves at the same rate vertex set V = 1,...,N and edge set E V 2 such that as regular unquantized consensus. On the other hand, the (i, j) E implies{ that nodes} i and j communicate⊆ directly transition from when node values are all within ∆ of each with each∈ other. (We assume communication is symmetric.) other (i.e., each node is either at k∆ or (k+1)∆, for some k), In the t+1st DA iteration, node i receives values xj (t) from is slower than unquantized consensus. We present a scheme all the nodes j in its neighborhood and updates its value by for characterizing the time from when PQDA iterates enter 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 Individual Node Trajectories Individual Node Trajectories 0.2 0.2 Probabilistically Quantized Consensus Deterministic Uniformly Quantized Consensus 0 0 −0.2 −0.2 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Iteration Number Iteration Number (a) (b) Fig. 1. Individual node trajectories (i.e., xi(t), ∀i) taken by the distributed average consensus using (a:) deterministic uniform quantization and (b:) probabilistic quantization. The number of nodes is N = 50, the nodes’ initial average is x = 0.85, and the quantization resolution is set to ∆ = 0.1. The consensus value, in this case, is 0.8. this “final bin” until the algorithm is absorbed at a consensus the quantization noise diminishes and nodes converge to a state. consensus. The remainder of the paper is organized as follows. Kashyap et al. examine the effects of quantization in Section II describes the probabilistic quantization scheme consensus algorithms from a different point of view [7]. employed by PQDA. Section III formally defines the PQDA N They require that the network average, x¯ =1/N i=1 xi(t), algorithm and lists fundamental convergence properties. Sec- be preserved at every iteration. To do this using quantized tion IV explores rates of convergence when the node values transmissions, nodes must carefully account forP round-off are far from a consensus, relative to the quantization preci- errors. Suppose we have a network of N nodes and let ∆ sion ∆. Section V presents our technique for characterizing denote the “quantization resolution” or distance between to convergence rates when all nodes reach the “final bin” and quantization lattice points. If x¯ is not a multiple of N∆, are within ∆ of each other. We conclude in Section VI. then it is not possible for the network to reach a strict Before proceeding, we briefly review related work. consensus (i.e., limt→∞ maxi,j xi(t) xj (t) = 0) while also preserving the network average,| −x¯, since| nodes only A. Related Work ever exchange units of ∆. Instead, Kashyap et. al define the While there exists a substantial body of work on average notion of a “quantized consensus” to be such that all xi(t) consensus protocols with infinite precision and noise–free take on one of two neighboring quantization values while peer–to–peer communications, little research has been done preserving the network average; i.e., xi(t) k∆, (k+1)∆ ∈{ } introducing distortions in the message exchange. In [12], for all i and some k, and i xi(T )= Nx¯. They show that, Rabani, Sinclair, and Wanka examine distributed averaging under reasonable conditions, their algorithm will converge P as a mechanism for load balancing in distributed computing to a quantized consensus. However, the quantized consensus systems. They provide a bound on the divergence between is clearly not a strict consensus, i.e., all nodes do not have quantized consensus trajectories, q(t), and the trajectories the same value. Even when the algorithm has converged, as which would be taken by an unquantized averaging algo- many as half the nodes in the network may have different rithms. The bound reduces to looking at properties of the values. If nodes are strategizing and/or performing actions averaging matrix W. based on these values (e.g., flight formation), then differing Recently, Yildiz and Scaglione, in [22], explored the values may lead to inconsistent behavior. impact of quantization noise through modification of the Of note is that the related works discussed above all consensus algorithm proposed by Xiao and Boyd [21]. They utilize standard deterministic uniform quantization schemes note that the noise component in [21] can be considered to quantize the data. In contrast to [22], where quantization as the quantization noise and they develop an algorithm noise terms are modeled as independent zero-mean random for predicting neighbors’ unquantized values in order to variables, we explicitly introduce randomization in our quan- correct errors introduced by quantization [22]. Simulation tization procedure. Careful analysis of this randomization studies for small N indicate that if the increasing correlation allows us to provide concrete theoretical rates of convergence among the node states is taken into account, the variance of in addition to empirical results. Moreover, the algorithm proposed in this paper converges to a strict consensus, as for some quantization point τ ∗ τ . opposed to the approximate “quantized consensus” achieved Proofs of all the theorems stated∈ in this section can be found in [7] which is clearly not a strict consensus and does not in [2].