Mehmet E. Yildiz

Broadcast Gossip Algorithms for Consensus

Motivated by applications to wireless sensor, peer-to-peer, and ad hoc networks, we study distributed broadcasting algorithms for exchanging information and computing in an arbitrarily connected network of nodes. Specifically, we study a broadcasting-based gossiping algorithm to compute the (possibly weighted) average of the initial measurements of the nodes at every node in the network. We show that the broadcast gossip algorithm converges almost surely to a consensus. We prove that the random consensus value is, in expectation, the average of initial node measurements and that it can be made arbitrarily close to this value in mean squared error sense, under a balanced connectivity model and by trading off convergence speed with accuracy of the computation. We provide theoretical and numerical results on the mean square error performance, on the convergence rate and study the effect of the ldquomixing parameterrdquo on the convergence rate of the broadcast gossip algorithm. The results indicate that the mean squared error strictly decreases through iterations until the consensus is achieved. Finally, we assess and compare the communication cost of the broadcast gossip algorithm to achieve a given distance to consensus through theoretical and numerical results.

Coding With Side Information for Rate-Constrained Consensus

Average consensus algorithms are protocols to compute the average value of all sensor measurements via near neighbors communications. They offer a natural tradeoff between the number of messages exchanged among terminals and the accuracy in the computation. Most of the models adopted for the message exchange in the literature, however, neither include explicit rate constraints nor explore the rate distortion tradeoff associated with the algorithm. The contribution of our work is in examining the impact of such constraints and in finding strategies to minimize the communication cost in terms of rate. The main motivation behind the proposed coding strategies is the observation that consensus algorithms offer the perfect example of a network communication problem where there is an increasing correlation between the data exchanged, as the algorithm iterates. Henceforth, it is possible to utilize previously exchanged data and current side information to significantly reduce the demands of quantization bit rate for a certain precision. We analyze the case of a network where the links are assumed to be reliable at a constant bit rate. We explore the conditions on the quantization noise which lead to a consensus value whose mean squared distance from the initial average is bounded. In the case of infinite-length vector coding with Gaussian states, we show that our proposed schemes achieve bounded convergence with vanishing rates as the iteration index tends to infinity.

Differential nested lattice encoding for consensus problems

In this paper we consider the problem of transmitting quantized data while performing an average consensus algorithm. Average consensus algorithms are protocols to compute the average value of all sensor measurements via near neighbors communications. The main motivation for our work is the observation that consensus algorithms offer the perfect example of network communications where there is an increasing correlation between the data exchanged, as the system updates its computations. Henceforth, it is possible to utilize previously exchanged data and current side information to reduce significantly the demands of quantization bit rate for a certain precision. We analyze the case of a network with a topology built as that of a random geometric graph and with links that are assumed to be reliable at a constant bit rate. Numerically we show that in consensus algorithms, increasing number of iterations does not have the effect of increasing the error variance. Thus, we conclude that noisy recursions lead to a consensus if the data correlation is exploited in the messages source encoders and decoders. We briefly state the theoretical results which are parallel to our numerical experiments.

Broadcast gossip algorithms: Design and analysis for consensus

Motivated by applications to wireless sensor, peer-to-peer, and ad hoc networks, we have recently proposed a broadcasting-based gossiping protocol to compute the (possibly weighted) average of the initial measurements of the nodes at every node in the network. The class of broadcast gossip algorithms achieve consensus almost surely at a value that is in the neighborhood of the initial node measurements¿ average. In this paper, we further study the broadcast gossip algorithms: we derive and analyze the optimal mixing parameter of the algorithm when approached from worst-case convergence rate, present theoretical results on limiting mean square error performance of the algorithm, and find the convergence rate order of the proposed protocol.

Asymmetric information diffusion via gossiping on static and dynamic networks

In this paper we consider the problem of gossiping in a network to diffuse the average of a sub-set of nodes, called sources, and directing it to another sub-set of nodes in the network called destinations. This case generalizes the typical average consensus gossiping policy, where all nodes are both sources and destinations. We first describe prior results we obtained on a static network topology and gossip policy, high-lighting what conditions lead to the desired information flow. We show that, through semi-directed flows, this formulation allows to solve the problem with lower complexity than using plain gossiping policies. Inspired by these results, we move on to design algorithms to solve the problem in the dynamic case. For the dynamic network scenario we derive conditions under which the network converges to the desired result in the limit. We also provide policies that trade-off accuracy with increased mixing speed for the dynamic asymmetric diffusion problem.

Computing Along Routes via Gossiping

In this paper, we study a class of distributed computing problems where a group of nodes (destinations) is interested in a function of data which are stored by another group of nodes (sources). We assume that the function of interest is separable, i.e., it can be decomposed as a sum of functions of local variables, a case that subsumes several interesting types of queries. One approach to solve this problem is to route raw information from the sources to the interested destinations by either unicasting or multicasting. The second approach is to compute the function of interest along some routes while propagating the information from the sources to the destinations. Considering the second scenario, the goal of this paper is to examine how information should be forwarded to the intended recipients, computing along the routes by gossiping with selected neighbors. Unlike efficient unicasting/multicasting problems, nodes are interested in a specific function of the data, rather than the raw data themselves. Moreover, unlike standard gossiping problems, in our case, the information needs to flow in a specific direction. Given the underlying network connectivity and the source-destination sets, we provide necessary and sufficient conditions on the update weights (referred to as codes) so that the destination nodes converge to the desired function of the source values. We show that the evolution of the source states does not affect the feasibility of the problem, and we provide a detailed analysis on the spectral properties of the feasible codes. We also study the problem feasibility under some specific topologies and provide guidelines to determine infeasibility. We also formulate different strategies to design codes, and compare the performance of our solution with existing alternatives.

Limiting rate behavior and rate allocation strategies for average consensus problems with bounded convergence

Average consensus algorithms are gossiping protocols for averaging original sensor measurements via near neighbor communications. In this paper, we consider the average consensus algorithm under communication rate constraints. Without any communication rate restrictions, the algorithm ideally allows every node state to converge to the initial average in the limit. Noting that brute force quantization does not guarantee convergence due to error propagation effects, in our recent work we proposed two source coding methods which use side information (predictive coding and Wyner-Ziv coding) to achieve convergence with vanishing quantization rates in the case of block coding. In this work, we focus on a simplified predictive coding scheme with variable quantization rates over the iterations and on a communication network with regular topology. We characterize the asymptotic rate which allows to achieve a bounded convergence in terms of the initial conditions (i. e, the rate at the first iteration, and the initial state correlation), and the connectivity of the network. Moreover, we study the optimal rate allocation among the average consensus iterations subject to the constraints that the total number of quantization bits is fixed.

Distributed distance estimation for manifold learning and dimensionality reduction

Given a network of N nodes with the i-th sensors observation x<inf>i</inf> ∈ R^M, the matrix containing all Euclidean distances among measurements ||x<inf>i</inf> − x<inf>j</inf> || ∀i, j ∈ {1,…, N} is a useful description of the data. While reconstructing a distance matrix has wide range of applications, we are particularly interested in the manifold reconstruction and its dimensionality reduction for data fusion and query. To make this map available to the all of the nodes in the network, we propose a fully decentralized consensus gossiping algorithm which is based on local neighbor communications, and does not require the existence of a central entity. The main advantage of our solution is that it is insensitive to changes in the network topology and it is fully decentralized. We describe the proposed algorithm in detail, study its complexity in terms of the number of inter-node radio transmissions and showcase its performance numerically.

Directed gossiping for distributed data aggregation

In this paper, we are studying a class of distributed data aggregation problems where a set of nodes in a network is interested in a function of data that is stored in another set of nodes. Assuming the function of interest is separable, we propose an algorithm based on gossiping schemes. Gossiping protocols are iterative methods based on near neighbor communications, and they are known to be efficient and robust to link/node failures. In this work, after formulating the problem mathematically, and introducing previously proposed necessary and sufficient conditions on the gossiping updates, we introduce several necessary conditions on the feasible codes which will provide significant intuition in determining the (in)feasibility of a given problem. By focusing on stochastic codes, we provide a necessary condition based on topology and discuss scenarios where the codes we seek cannot be found.

Network information flow: Gossiping with groups

In this paper, we consider a networking scenario in which a group of nodes wants retrieve a function of another group of nodes data. Here we assume that the function can be cast into a sum of functions of the local variables, a case that subsumes several interesting types of queries. One approach to solve this problem is to route the information from each node to the interested parties one at a time, relaying it over a tree of paths. A second strategy is to reuse the paths when possible, multicasting to the intended recipients the data. A third one, which is the one we are interested in exploring, computes along the routes as well. More specifically, the goal of this paper is to examine how the sought information be forwarded to the intended recipients, computing along the routes by gossiping with selected neighbors. Unlike the context of standard gossiping, in our problem the information needs to flow in a specific direction. In this work we provide a sufficient condition for the convergence to the desired result, and propose a method how to design the information flow for a given network and a problem.