This paper presents an algorithm which solves exponentially fast the average consensus problem on strongly connected network of digital links. The algorithm is based on an efficient zooming-in zooming-out quantization scheme.
The constraint of quantization of messages, or of states, has been considered in several recent papers @cite_8 @cite_15 @cite_10 @cite_9 @cite_11 @cite_5 @cite_4 @cite_0 . However, existing algorithms in the literature are not able to achieve average consensus with arbitrary precision, with the notable exception of @cite_16 , which uses logarithmic quantizers, together with a coding scheme similar to the one in the present paper. The zooming-in zooming-out algorithm and the main convergence result of this paper appeared, in slightly different version, in @cite_17 and in the thesis @cite_2 . Note that recently @cite_3 a modification of the present algorithm has been proved to converge under a different condition.
@cite_4 @cite_8 @cite_9 @cite_17 @cite_3 @cite_0 @cite_2 @cite_5 @cite_15 @cite_16 @cite_10 @cite_11
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The paper studies the problem of distributed average consensus in sensor networks with quantized data and random link failures. To achieve consensus, dither (small noise) is added to the sensor states before quantization. When the quantizer range is unbounded (countable number of quantizer levels), stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable. We show that the mean-squared error (mse) can be made arbitrarily small by tuning the link weight sequence, at a cost of the convergence rate of the algorithm. To study dithered consensus with random links when the range of the quantizer is bounded, we establish uniform boundedness of the sample paths of the unbounded quantizer. This requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer. This is accomplished by splitting the state vector of the quantizer in two components: one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace. The proofs use maximal inequalities for submartingale and supermartingale sequences. From these, we derive probability bounds on the excursions of the two subsequences, from which probability bounds on the excursions of the quantizer state vector follow. The paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels, the size of the quantization steps, the desired probability of saturation, and the desired level of accuracy ? away from consensus. Finally, the paper illustrates the quantizer design with a numerical study. In this work we present a new algorithm to solve the average-consensus problem. The main goal of this algorithm is to obtain exact convergence despite the existence of quantized communication channels between the agents. Starting from the Zoom-in Zoom-out strategy already presented in [5], we introduce the equations describing the behaviour of the algorithm and we formally prove the asymptotic agreement. We will also show that, under a reasonable hypothesis, the algorithm parameters ensuring convergence, can be chosen regardless the number of agents. In this paper, we develop algorithms for distributed computation of averages of the node data over networks with bandwidth power constraints or large volumes of data. Distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted. We propose a distributed algorithm in which the nodes utilize probabilistically quantized information, i.e., dithered quantization, to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We derive an upper bound on the mean-square-error performance of the probabilistically quantized distributed averaging (PQDA). Moreover, we show that the convergence of the PQDA is monotonic by studying the evolution of the minimum-length interval containing the node values. We reveal that the length of this interval is a monotonically nonincreasing function with limit zero. We also demonstrate that all the node values, in the worst case, converge to the final two quantization bins at the same rate as standard unquantized consensus. Finally, we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios. Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candes, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size ź is used to quantize m measurements y=źx of a k-sparse signal xźźN, where ź satisfies the restricted isometry property, then the approximate recovery x# via l1-minimization is within O(ź) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma---Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order ź(k m)(rź1 2)ź for any 0<ź<1, if mźr,źk(logN)1 (1źź). The result holds with high probability on the initial draw of the measurement matrix ź from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.