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198
Distributed consensus algorithms in sensor networks with communication channel noise and random link failures
 in Proc. 41st Asilomar Conf. Signals, Systems, Computers
, 2007
"... Abstract—The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the biasvariance dilemma—running consensus for long reduces the bias of the final average estimate but increases its variance. We present t ..."
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Cited by 146 (22 self)
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Abstract—The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the biasvariance dilemma—running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the algorithm where the weights are constant but consensus is run for a fixed number of iterations, then it is restarted and rerun for a total of runs, and at the end averages the final states of the runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that represents the best of both worlds—zero bias and low variance—at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, , because of its constant weights, converges fast but presents a different biasvariance tradeoff. For the same number of iterations, shorter runs (smaller) lead to high bias but smaller variance (larger number of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for to reach in the shortest number of iterations, with high probability (1), ()consensus ( residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise. Index Terms—Additive noise, consensus, sensor networks, stochastic approximation, random topology. I.
On Distributed Averaging Algorithms and Quantization Effects
, 2009
"... We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We stu ..."
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Cited by 135 (24 self)
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We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for timevarying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels.
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2010
"... The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagen ..."
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Cited by 93 (12 self)
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The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagent coordination, estimation in sensor networks, and largescale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network and confirm this prediction’s sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication as well as problems with stochastic optimization and/or communication.
Consensus propagation
 IEEE Transactions on Information Theory
"... Abstract — We propose consensus propagation, an asynchronous distributed protocol for averaging numbers across a network. We establish convergence, characterize the convergence rate for regular graphs, and demonstrate that the protocol exhibits better scaling properties than pairwise averaging, an a ..."
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Cited by 89 (5 self)
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Abstract — We propose consensus propagation, an asynchronous distributed protocol for averaging numbers across a network. We establish convergence, characterize the convergence rate for regular graphs, and demonstrate that the protocol exhibits better scaling properties than pairwise averaging, an alternative that has received much recent attention. Consensus propagation can be viewed as a special case of belief propagation, and our results contribute to the belief propagation literature. In particular, beyond singlyconnected graphs, there are very few classes of relevant problems for which belief propagation is known to converge. Index Terms — belief propagation, distributed averaging, distributed consensus, distributed signal processing, Gaussian Markov random fields, messagepassing algorithms, maxproduct algorithm, minsum algorithm, sumproduct algorithm. I.
Distributed stochastic subgradient projection algorithms for convex optimization
 Journal of Optimization Theory and Applications
, 2010
"... Abstract. We consider a distributed multiagent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines ..."
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Cited by 82 (0 self)
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Abstract. We consider a distributed multiagent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and nondiminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.
Minimizing effective resistance of a graph
 SIAM Review
, 2005
"... Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network a ..."
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Cited by 64 (5 self)
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Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuoustime averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem, and can be solved efficiently either numerically, or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with n nodes, the path has the largest value of optimal total effective resistance, and the complete graph the least. 1. Introduction. Let N be a network with n nodes and m edges, i.e., an undirected graph (V, E) with V  = n, E  = m, and nonnegative weights on the edges. We call the weight on edge l its conductance, and denote it by gl. The effective resistance between a pair of nodes i and j, denoted Rij, is the electrical resistance measured across nodes i and j, when the network represents an electrical circuit with each edge (or branch, in the terminology of electrical circuits) a resistor with (electrical) conductance gl. In other
Distributed Kalman filtering based on consensus strategies
, 2007
"... In this paper, we consider the problem of estimating the state of a dynamical system from distributed noisy measurements. Each agent constructs a local estimate based on its own measurements and estimates from its neighbors. Estimation is performed via a two stage strategy, the first being a Kalman ..."
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Cited by 56 (1 self)
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In this paper, we consider the problem of estimating the state of a dynamical system from distributed noisy measurements. Each agent constructs a local estimate based on its own measurements and estimates from its neighbors. Estimation is performed via a two stage strategy, the first being a Kalmanlike measurement update which does not require communication, and the second being an estimate fusion using a consensus matrix. In particular we study the interaction between the consensus matrix, the number of messages exchanged per sampling time, and the Kalman gain. We prove that optimizing the consensus matrix for fastest convergence and using the centralized optimal gain is not necessarily the optimal strategy if the number of exchanged messages per sampling time is small. Moreover, we showed that although the joint optimization of the consensus matrix and the Kalman gain is in general a nonconvex problem, it is possible to compute them under some important scenarios. We also provide some numerical examples to clarify some of the analytical results and compare them with alternative estimation strategies.
Distributed average consensus with dithered quantization
 the IEEE Transactions of Signal Processing
, 2008
"... 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 distribut ..."
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Cited by 55 (2 self)
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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 minimumlength interval containing the node values. We reveal that the length of this interval is a monotonically non–increasing 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.
Distributed average consensus using probabilistic quantization
, 2007
"... 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 distribut ..."
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Cited by 53 (7 self)
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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 to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus, which is one of the quantization values. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios. Index Terms — Distributed algorithms, average consensus, sensor networks
Convex optimization of graph Laplacian eigenvalues
 in International Congress of Mathematicians
"... Abstract. We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting case ..."
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Cited by 50 (0 self)
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Abstract. We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discretetime) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuoustime) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic