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Bruno Sinopoli - One of the best experts on this subject based on the ideXlab platform.

  • Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability
    IEEE Transactions on Signal Processing, 2013
    Co-Authors: Dragana Bajovic, Ao Xavier, Em. F. Moura, Bruno Sinopoli
    Abstract:

    We find the exact rate for Convergence in Probability of products of independent, identically distributed symmetric, stochastic matrices. It is well-known that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices is connected, the products of the matrices converge almost surely to the average consensus matrix, and thus in Probability. in this paper, we show that the Convergence in Probability is exponentially fast, and we explicitly characterize the exponential rate of this Convergence. Our analysis reveals that the exponential rate of Convergence in Probability depends only on the statistics of the support graphs of the random matrices. Further, we show how to compute this rate for commonly used random models: gossip and link failure. With these models, the rate is found by solving a min-cut problem, and hence it is easily computable. Finally, as an illustration, we apply our results to solving power allocation among networked sensors in a consensus+innovations distributed detection problem.

  • exact rate for Convergence in Probability of averaging processes via generalized min cut
    Conference on Decision and Control, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

  • products of stochastic matrices exact rate for Convergence in Probability for directed networks
    Telecommunications Forum, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Bruno Sinopoli
    Abstract:

    We study the products W k ···W 1 of random stochastic, not necessarily symmetric matrices. It is known that, under certain conditions, the product W k · · · W 1 converges almost surely (a.s.) to a random rank-one matrix; the latter is equivalent to |λ 2 (W k · · · W 1 )| → 0 a.s., where λ 2 (·) is the second largest (in modulus) eigenvalue. in this paper, we show that the Probability that |λ 2 (W k · · · W 1 )| stays above e Є (0,1] in the long run decays to zero exponentially fast ∼ e−kI. Furthermore, we explicitly characterize the rate of this Convergence I and show that it depends only on the underlying graphs that support the matrices W k 's. Our results reveal that the rate I is essentially determined by the most likely way in which the union (over time) of the support graphs fails to form a directed tree.

  • CDC - Exact rate for Convergence in Probability of averaging processes via generalized min-cut
    2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

Dragana Bajovic - One of the best experts on this subject based on the ideXlab platform.

  • Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability
    IEEE Transactions on Signal Processing, 2013
    Co-Authors: Dragana Bajovic, Ao Xavier, Em. F. Moura, Bruno Sinopoli
    Abstract:

    We find the exact rate for Convergence in Probability of products of independent, identically distributed symmetric, stochastic matrices. It is well-known that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices is connected, the products of the matrices converge almost surely to the average consensus matrix, and thus in Probability. in this paper, we show that the Convergence in Probability is exponentially fast, and we explicitly characterize the exponential rate of this Convergence. Our analysis reveals that the exponential rate of Convergence in Probability depends only on the statistics of the support graphs of the random matrices. Further, we show how to compute this rate for commonly used random models: gossip and link failure. With these models, the rate is found by solving a min-cut problem, and hence it is easily computable. Finally, as an illustration, we apply our results to solving power allocation among networked sensors in a consensus+innovations distributed detection problem.

  • exact rate for Convergence in Probability of averaging processes via generalized min cut
    Conference on Decision and Control, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

  • products of stochastic matrices exact rate for Convergence in Probability for directed networks
    Telecommunications Forum, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Bruno Sinopoli
    Abstract:

    We study the products W k ···W 1 of random stochastic, not necessarily symmetric matrices. It is known that, under certain conditions, the product W k · · · W 1 converges almost surely (a.s.) to a random rank-one matrix; the latter is equivalent to |λ 2 (W k · · · W 1 )| → 0 a.s., where λ 2 (·) is the second largest (in modulus) eigenvalue. in this paper, we show that the Probability that |λ 2 (W k · · · W 1 )| stays above e Є (0,1] in the long run decays to zero exponentially fast ∼ e−kI. Furthermore, we explicitly characterize the rate of this Convergence I and show that it depends only on the underlying graphs that support the matrices W k 's. Our results reveal that the rate I is essentially determined by the most likely way in which the union (over time) of the support graphs fails to form a directed tree.

  • CDC - Exact rate for Convergence in Probability of averaging processes via generalized min-cut
    2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

Joao Xavier - One of the best experts on this subject based on the ideXlab platform.

  • distributed nesterov gradient methods for random networks Convergence in Probability and Convergence rates
    International Conference on Acoustics Speech and Signal Processing, 2014
    Co-Authors: Dusan Jakovetic, Joao Xavier, Jose M F Moura
    Abstract:

    We consider distributed optimization where N nodes in a generic, connected network minimize the sum of their individual, locally known, convex costs. Existing literature proposes distributed gradient-like methods that are attractive due to computationally cheap iterations and provable resilience to random inter-node communication failures, but such methods have slow theoretical and empirical Convergence rates. Building from the centralized Nesterov gradient methods, we propose accelerated distributed gradient-like methods and establish that they achieve strictly faster rates than existing distributed methods. At the same time, our methods maintain cheap iterations and resilience to random communication failures. Specifically, for convex, differentiable local costs with Lipschitz continuous and bounded derivative, we establish (with respect to the cost function optimality) Convergence in Probability and Convergence rates in expectation and in second moment.

  • ICASSP - Distributed Nesterov gradient methods for random networks: Convergence in Probability and Convergence rates
    2014 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2014
    Co-Authors: Dusan Jakovetic, Joao Xavier, Jose M F Moura
    Abstract:

    We consider distributed optimization where N nodes in a generic, connected network minimize the sum of their individual, locally known, convex costs. Existing literature proposes distributed gradient-like methods that are attractive due to computationally cheap iterations and provable resilience to random inter-node communication failures, but such methods have slow theoretical and empirical Convergence rates. Building from the centralized Nesterov gradient methods, we propose accelerated distributed gradient-like methods and establish that they achieve strictly faster rates than existing distributed methods. At the same time, our methods maintain cheap iterations and resilience to random communication failures. Specifically, for convex, differentiable local costs with Lipschitz continuous and bounded derivative, we establish (with respect to the cost function optimality) Convergence in Probability and Convergence rates in expectation and in second moment.

  • exact rate for Convergence in Probability of averaging processes via generalized min cut
    Conference on Decision and Control, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

  • products of stochastic matrices exact rate for Convergence in Probability for directed networks
    Telecommunications Forum, 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Bruno Sinopoli
    Abstract:

    We study the products W k ···W 1 of random stochastic, not necessarily symmetric matrices. It is known that, under certain conditions, the product W k · · · W 1 converges almost surely (a.s.) to a random rank-one matrix; the latter is equivalent to |λ 2 (W k · · · W 1 )| → 0 a.s., where λ 2 (·) is the second largest (in modulus) eigenvalue. in this paper, we show that the Probability that |λ 2 (W k · · · W 1 )| stays above e Є (0,1] in the long run decays to zero exponentially fast ∼ e−kI. Furthermore, we explicitly characterize the rate of this Convergence I and show that it depends only on the underlying graphs that support the matrices W k 's. Our results reveal that the rate I is essentially determined by the most likely way in which the union (over time) of the support graphs fails to form a directed tree.

  • CDC - Exact rate for Convergence in Probability of averaging processes via generalized min-cut
    2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
    Co-Authors: Dragana Bajovic, Joao Xavier, Jose M F Moura, Bruno Sinopoli
    Abstract:

    We study the asymptotic exponential decay rate I for the Convergence in Probability of products W k W k−1 …W 1 of random symmetric, stochastic matrices W k . Albeit it is known that the Probability P that the product W k W k−1 …W 1 is ∈ away from its limit converges exponentially fast to zero, i.e., P ∼ e−kI, the asymptotic rate I has not been computed before. in this paper, assuming the positive entries of Wk are bounded away from zero, we explicitly characterize the rate I and show that it is a function of the underlying graphs that support the positive (non zero) entries of W k . in particular, the rate I is given by a certain generalization of the min-cut problem. Although this min-cut problem is in general combinatorial, we show how to exactly compute I in polynomial time for the commonly used matrix models, gossip and link failure. Further, for a class of models for which I is difficult to compute, we give easily computable bounds: I ≤ I ≤ Ī, where I and Ī differ by a constant ratio. Finally, we show the relevance of I as a system design metric with the example of optimal power allocation in consensus+innovations distributed detection.

Anthony Quas - One of the best experts on this subject based on the ideXlab platform.

Olga Polosmak - One of the best experts on this subject based on the ideXlab platform.