Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability

Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices $W_k$ are often random, due to, e.g., random packet dropouts in wireless sensor networks. Key in analyzing the performance of such systems is studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we find the exact exponential rate $I$ for the convergence in probability of the product of such matrices when time $k$ grows large, under the assumption that the $W_k$'s are symmetric and independent identically distributed in time. Further, for commonly used random models like with gossip and link failure, we show that the rate $I$ is found by solving a min-cut problem and, hence, easily computable. Finally, we apply our results to optimally allocate the sensors' transmission power in consensus+innovations distributed detection

Gossip Algorithms for Distributed Signal Processing

Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page

QDQD-Learning: A Collaborative Distributed Strategy for Multi-Agent Reinforcement Learning Through Consensus + Innovations

The paper considers a class of multi-agent Markov decision processes (MDPs), in which the network agents respond differently (as manifested by the instantaneous one-stage random costs) to a global controlled state and the control actions of a remote controller. The paper investigates a distributed reinforcement learning setup with no prior information on the global state transition and local agent cost statistics. Specifically, with the agents' objective consisting of minimizing a network-averaged infinite horizon discounted cost, the paper proposes a distributed version of $Q$-learning, $\mathcal{QD}$-learning, in which the network agents collaborate by means of local processing and mutual information exchange over a sparse (possibly stochastic) communication network to achieve the network goal. Under the assumption that each agent is only aware of its local online cost data and the inter-agent communication network is \emph{weakly} connected, the proposed distributed scheme is almost surely (a.s.) shown to yield asymptotically the desired value function and the optimal stationary control policy at each network agent. The analytical techniques developed in the paper to address the mixed time-scale stochastic dynamics of the \emph{consensus + innovations} form, which arise as a result of the proposed interactive distributed scheme, are of independent interest.Comment: Submitted to the IEEE Transactions on Signal Processing, 33 page

Fault-Tolerant Aggregation: Flow-Updating Meets Mass-Distribution

Flow-Updating (FU) is a fault-tolerant technique that has proved to be efficient in practice for the distributed computation of aggregate functions in communication networks where individual processors do not have access to global information. Previous distributed aggregation protocols, based on repeated sharing of input values (or mass) among processors, sometimes called Mass-Distribution (MD) protocols, are not resilient to communication failures (or message loss) because such failures yield a loss of mass. In this paper, we present a protocol which we call Mass-Distribution with Flow-Updating (MDFU). We obtain MDFU by applying FU techniques to classic MD. We analyze the convergence time of MDFU showing that stochastic message loss produces low overhead. This is the first convergence proof of an FU-based algorithm. We evaluate MDFU experimentally, comparing it with previous MD and FU protocols, and verifying the behavior predicted by the analysis. Finally, given that MDFU incurs a fixed deviation proportional to the message-loss rate, we adjust the accuracy of MDFU heuristically in a new protocol called MDFU with Linear Prediction (MDFU-LP). The evaluation shows that both MDFU and MDFU-LP behave very well in practice, even under high rates of message loss and even changing the input values dynamically.Comment: 18 pages, 5 figures, To appear in OPODIS 201

Randomized Consensus with Attractive and Repulsive Links

We study convergence properties of a randomized consensus algorithm over a graph with both attractive and repulsive links. At each time instant, a node is randomly selected to interact with a random neighbor. Depending on if the link between the two nodes belongs to a given subgraph of attractive or repulsive links, the node update follows a standard attractive weighted average or a repulsive weighted average, respectively. The repulsive update has the opposite sign of the standard consensus update. In this way, it counteracts the consensus formation and can be seen as a model of link faults or malicious attacks in a communication network, or the impact of trust and antagonism in a social network. Various probabilistic convergence and divergence conditions are established. A threshold condition for the strength of the repulsive action is given for convergence in expectation: when the repulsive weight crosses this threshold value, the algorithm transits from convergence to divergence. An explicit value of the threshold is derived for classes of attractive and repulsive graphs. The results show that a single repulsive link can sometimes drastically change the behavior of the consensus algorithm. They also explicitly show how the robustness of the consensus algorithm depends on the size and other properties of the graphs

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

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

Tree Codes Improve Convergence Rate of Consensus Over Erasure Channels

We study the problem of achieving average consensus between a group of agents over a network with erasure links. In the context of consensus problems, the unreliability of communication links between nodes has been traditionally modeled by allowing the underlying graph to vary with time. In other words, depending on the realization of the link erasures, the underlying graph at each time instant is assumed to be a subgraph of the original graph. Implicit in this model is the assumption that the erasures are symmetric: if at time t the packet from node i to node j is dropped, the same is true for the packet transmitted from node j to node i. However, in practical wireless communication systems this assumption is unreasonable and, due to the lack of symmetry, standard averaging protocols cannot guarantee that the network will reach consensus to the true average. In this paper we explore the use of channel coding to improve the performance of consensus algorithms. For symmetric erasures, we show that, for certain ranges of the system parameters, repetition codes can speed up the convergence rate. For asymmetric erasures we show that tree codes (which have recently been designed for erasure channels) can be used to simulate the performance of the original "unerased" graph. Thus, unlike conventional consensus methods, we can guarantee convergence to the average in the asymmetric case. The price is a slowdown in the convergence rate, relative to the unerased network, which is still often faster than the convergence rate of conventional consensus algorithms over noisy links