Convergence Rate Analysis of Distributed Gossip (Linear Parameter) Estimation: Fundamental Limits and Tradeoffs

The paper considers gossip distributed estimation of a (static) distributed random field (a.k.a., large scale unknown parameter vector) observed by sparsely interconnected sensors, each of which only observes a small fraction of the field. We consider linear distributed estimators whose structure combines the information \emph{flow} among sensors (the \emph{consensus} term resulting from the local gossiping exchange among sensors when they are able to communicate) and the information \emph{gathering} measured by the sensors (the \emph{sensing} or \emph{innovations} term.) This leads to mixed time scale algorithms--one time scale associated with the consensus and the other with the innovations. The paper establishes a distributed observability condition (global observability plus mean connectedness) under which the distributed estimates are consistent and asymptotically normal. We introduce the distributed notion equivalent to the (centralized) Fisher information rate, which is a bound on the mean square error reduction rate of any distributed estimator; we show that under the appropriate modeling and structural network communication conditions (gossip protocol) the distributed gossip estimator attains this distributed Fisher information rate, asymptotically achieving the performance of the optimal centralized estimator. Finally, we study the behavior of the distributed gossip estimator when the measurements fade (noise variance grows) with time; in particular, we consider the maximum rate at which the noise variance can grow and still the distributed estimator being consistent, by showing that, as long as the centralized estimator is consistent, the distributed estimator remains consistent.Comment: Submitted for publication, 30 page

Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics

This paper focuses on the problem of recursive nonlinear least squares parameter estimation in multi-agent networks, in which the individual agents observe sequentially over time an independent and identically distributed (i.i.d.) time-series consisting of a nonlinear function of the true but unknown parameter corrupted by noise. A distributed recursive estimator of the \emph{consensus} + \emph{innovations} type, namely $\mathcal{CIWNLS}$, is proposed, in which the agents update their parameter estimates at each observation sampling epoch in a collaborative way by simultaneously processing the latest locally sensed information~(\emph{innovations}) and the parameter estimates from other agents~(\emph{consensus}) in the local neighborhood conforming to a pre-specified inter-agent communication topology. Under rather weak conditions on the connectivity of the inter-agent communication and a \emph{global observability} criterion, it is shown that at every network agent, the proposed algorithm leads to consistent parameter estimates. Furthermore, under standard smoothness assumptions on the local observation functions, the distributed estimator is shown to yield order-optimal convergence rates, i.e., as far as the order of pathwise convergence is concerned, the local parameter estimates at each agent are as good as the optimal centralized nonlinear least squares estimator which would require access to all the observations across all the agents at all times. In order to benchmark the performance of the proposed distributed $\mathcal{CIWNLS}$ estimator with that of the centralized nonlinear least squares estimator, the asymptotic normality of the estimate sequence is established and the asymptotic covariance of the distributed estimator is evaluated. Finally, simulation results are presented which illustrate and verify the analytical findings.Comment: 28 pages. Initial Submission: Feb. 2016, Revised: July 2016, Accepted: September 2016, To appear in IEEE Transactions on Signal and Information Processing over Networks: Special Issue on Inference and Learning over Network

Hearing the clusters in a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies

Joint Estimation and Localization in Sensor Networks

This paper addresses the problem of collaborative tracking of dynamic targets in wireless sensor networks. A novel distributed linear estimator, which is a version of a distributed Kalman filter, is derived. We prove that the filter is mean square consistent in the case of static target estimation. When large sensor networks are deployed, it is common that the sensors do not have good knowledge of their locations, which affects the target estimation procedure. Unlike most existing approaches for target tracking, we investigate the performance of our filter when the sensor poses need to be estimated by an auxiliary localization procedure. The sensors are localized via a distributed Jacobi algorithm from noisy relative measurements. We prove strong convergence guarantees for the localization method and in turn for the joint localization and target estimation approach. The performance of our algorithms is demonstrated in simulation on environmental monitoring and target tracking tasks.Comment: 9 pages (two-column); 5 figures; Manuscript submitted to the 2014 IEEE Conference on Decision and Control (CDC

Distributed Recursive Least-Squares: Stability and Performance Analysis

The recursive least-squares (RLS) algorithm has well-documented merits for reducing complexity and storage requirements, when it comes to online estimation of stationary signals as well as for tracking slowly-varying nonstationary processes. In this paper, a distributed recursive least-squares (D-RLS) algorithm is developed for cooperative estimation using ad hoc wireless sensor networks. Distributed iterations are obtained by minimizing a separable reformulation of the exponentially-weighted least-squares cost, using the alternating-minimization algorithm. Sensors carry out reduced-complexity tasks locally, and exchange messages with one-hop neighbors to consent on the network-wide estimates adaptively. A steady-state mean-square error (MSE) performance analysis of D-RLS is conducted, by studying a stochastically-driven `averaged' system that approximates the D-RLS dynamics asymptotically in time. For sensor observations that are linearly related to the time-invariant parameter vector sought, the simplifying independence setting assumptions facilitate deriving accurate closed-form expressions for the MSE steady-state values. The problems of mean- and MSE-sense stability of D-RLS are also investigated, and easily-checkable sufficient conditions are derived under which a steady-state is attained. Without resorting to diminishing step-sizes which compromise the tracking ability of D-RLS, stability ensures that per sensor estimates hover inside a ball of finite radius centered at the true parameter vector, with high-probability, even when inter-sensor communication links are noisy. Interestingly, computer simulations demonstrate that the theoretical findings are accurate also in the pragmatic settings whereby sensors acquire temporally-correlated data.Comment: 30 pages, 4 figures, submitted to IEEE Transactions on Signal Processin

Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Asymmetric Channels

In this paper we propose and analyze a distributed algorithm for achieving globally optimal decisions, either estimation or detection, through a self-synchronization mechanism among linearly coupled integrators initialized with local measurements. We model the interaction among the nodes as a directed graph with weights (possibly) dependent on the radio channels and we pose special attention to the effect of the propagation delay occurring in the exchange of data among sensors, as a function of the network geometry. We derive necessary and sufficient conditions for the proposed system to reach a consensus on globally optimal decision statistics. One of the major results proved in this work is that a consensus is reached with exponential convergence speed for any bounded delay condition if and only if the directed graph is quasi-strongly connected. We provide a closed form expression for the global consensus, showing that the effect of delays is, in general, the introduction of a bias in the final decision. Finally, we exploit our closed form expression to devise a double-step consensus mechanism able to provide an unbiased estimate with minimum extra complexity, without the need to know or estimate the channel parameters.Comment: To be published on IEEE Transactions on Signal Processin

Consensus-based control for a network of diffusion PDEs with boundary local interaction

In this paper the problem of driving the state of a network of identical agents, modeled by boundary-controlled heat equations, towards a common steady-state profile is addressed. Decentralized consensus protocols are proposed to address two distinct problems. The first problem is that of steering the states of all agents towards the same constant steady-state profile which corresponds to the spatial average of the agents initial condition. A linear local interaction rule addressing this requirement is given. The second problem deals with the case where the controlled boundaries of the agents dynamics are corrupted by additive persistent disturbances. To achieve synchronization between agents, while completely rejecting the effect of the boundary disturbances, a nonlinear sliding-mode based consensus protocol is proposed. Performance of the proposed local interaction rules are analyzed by applying a Lyapunov-based approach. Simulation results are presented to support the effectiveness of the proposed algorithms