A randomized gossip consensus algorithm on convex metric spaces

A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. This problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity. In this paper we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces. We study the convergence properties of the algorithm using stochastic differential equations theory. We show that the dynamics of the distances between the states of the agents can be upper bounded by the dynamics of a stochastic differential equation driven by Poisson counters. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.This material is based in part upon work supported by the NIST-ARRA Measurement Science and Engineering Fellowship Program award 70NANB10H026, through the University of Maryland, and in part upon work supported by the Army Research Office award number W911NF-08-1-0238 to Ohio State University

GENERALIZED DISTRIBUTED CONSENSUS-BASED ALGORITHMS FOR UNCERTAIN SYSTEMS AND NETWORKS

We address four problems related to multi-agent optimization, filtering and agreement. First, we investigate collaborative optimization of an objective function expressed as a sum of local convex functions, when the agents make decisions in a distributed manner using local information, while the communication topology used to exchange messages and information is modeled by a graph-valued random process, assumed independent and identically distributed. Specifically, we study the performance of the consensusbased multi-agent distributed subgradient method and show how it depends on the probability distribution of the random graph. For the case of a constant stepsize, we first give an upper bound on the difference between the objective function, evaluated at the agents' estimates of the optimal decision vector, and the optimal value. In addition, for a particular class of convex functions, we give an upper bound on the distances between the agents' estimates of the optimal decision vector and the minimizer and we provide the rate of convergence to zero of the time varying component of the aforementioned upper bound. The addressed metrics are evaluated via their expected values. As an application, we show how the distributed optimization algorithm can be used to perform collaborative system identification and provide numerical experiments under the randomized and broadcast gossip protocols. Second, we generalize the asymptotic consensus problem to convex metric spaces. Under minimal connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the generalized convex hull generated by the agents current state and the states of its neighbors, then agreement is achieved asymptotically. In addition, we give bounds on the distance between the consensus point(s) and the initial values of the agents. As an application example, we introduce a probabilistic algorithm for reaching consensus of opinion and show that it in fact fits our general framework. Third, we discuss the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph. The switching is determined by a finite state, Markov process, each topology corresponding to a state of the process. We address both the cases where the dynamics of the agents are expressed in continuous and discrete time. We show that, if the consensus matrices are doubly stochastic, average consensus is achieved in the mean square and almost sure senses if and only if the graph resulting from the union of graphs corresponding to the states of the Markov process is strongly connected. Fourth, we address the consensus-based distributed linear filtering problem, where a discrete time, linear stochastic process is observed by a network of sensors. We assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable, i.e. for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors (without noise) are asymptotically stable. Next, we develop a distributed, sub-optimal filtering scheme based on minimizing an upper bound on a quadratic filtering cost. In the stationary case, we provide sufficient conditions under which this scheme converges; conditions expressed in terms of the convergence properties of a set of coupled Riccati equations. We continue by presenting a connection between the consensus-based distributed linear filter and the optimal linear filter of a Markovian jump linear system, appropriately defined. More specifically, we show that if the Markovian jump linear system is (mean square) detectable, then the stochastic process is detectable under the consensus-based distributed linear filtering scheme. We also show that the optimal gains of a linear filter for estimating the state of a Markovian jump linear system, appropriately defined, can be used to approximate the optimal gains of the consensus-based linear filter

A Coordinate Descent Primal-Dual Algorithm and Application to Distributed Asynchronous Optimization

Based on the idea of randomized coordinate descent of $\alpha$-averaged operators, a randomized primal-dual optimization algorithm is introduced, where a random subset of coordinates is updated at each iteration. The algorithm builds upon a variant of a recent (deterministic) algorithm proposed by V\~u and Condat that includes the well known ADMM as a particular case. The obtained algorithm is used to solve asynchronously a distributed optimization problem. A network of agents, each having a separate cost function containing a differentiable term, seek to find a consensus on the minimum of the aggregate objective. The method yields an algorithm where at each iteration, a random subset of agents wake up, update their local estimates, exchange some data with their neighbors, and go idle. Numerical results demonstrate the attractive performance of the method. The general approach can be naturally adapted to other situations where coordinate descent convex optimization algorithms are used with a random choice of the coordinates.Comment: 10 page

Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability

The paper presents the gossip interactive Kalman filter (GIKF) for distributed Kalman filtering for networked systems and sensor networks, where inter-sensor communication and observations occur at the same time-scale. The communication among sensors is random; each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link. We show that under a weak distributed detectability condition: 1. the GIKF error process remains stochastically bounded, irrespective of the instability properties of the random process dynamics; and 2. the network achieves \emph{weak consensus}, i.e., the conditional estimation error covariance at a (uniformly) randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semi-definite matrices (independent of the initial state.) To prove these results, we interpret the filtered states (estimates and error covariances) at each node in the GIKF as stochastic particles with local interactions. We analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched (random) Riccati equation, the switching being dictated by a non-stationary Markov chain on the network graph.Comment: Submitted to the IEEE Transactions, 30 pages

Stochastic gradient descent on Riemannian manifolds

Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically.Comment: A slightly shorter version has been published in IEEE Transactions Automatic Contro

Cooperative Convex Optimization in Networked Systems: Augmented Lagrangian Algorithms with Directed Gossip Communication

We study distributed optimization in networked systems, where nodes cooperate to find the optimal quantity of common interest, x=x^\star. The objective function of the corresponding optimization problem is the sum of private (known only by a node,) convex, nodes' objectives and each node imposes a private convex constraint on the allowed values of x. We solve this problem for generic connected network topologies with asymmetric random link failures with a novel distributed, decentralized algorithm. We refer to this algorithm as AL-G (augmented Lagrangian gossiping,) and to its variants as AL-MG (augmented Lagrangian multi neighbor gossiping) and AL-BG (augmented Lagrangian broadcast gossiping.) The AL-G algorithm is based on the augmented Lagrangian dual function. Dual variables are updated by the standard method of multipliers, at a slow time scale. To update the primal variables, we propose a novel, Gauss-Seidel type, randomized algorithm, at a fast time scale. AL-G uses unidirectional gossip communication, only between immediate neighbors in the network and is resilient to random link failures. For networks with reliable communication (i.e., no failures,) the simplified, AL-BG (augmented Lagrangian broadcast gossiping) algorithm reduces communication, computation and data storage cost. We prove convergence for all proposed algorithms and demonstrate by simulations the effectiveness on two applications: l_1-regularized logistic regression for classification and cooperative spectrum sensing for cognitive radio networks.Comment: 28 pages, journal; revise