2,116 research outputs found
Tight bounds on the convergence rate of generalized ratio consensus algorithms
The problems discussed in this paper are motivated by general ratio consensus
algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as
the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the
name weighted gossip algorithm. We consider a communication protocol described
by a strictly stationary, ergodic, sequentially primitive sequence of
non-negative matrices, applied iteratively to a pair of fixed initial vectors,
the components of which are called values and weights defined at the nodes of a
network. The subject of ratio consensus problems is to study the asymptotic
properties of ratios of values and weights at each node, expecting convergence
to the same limit for all nodes. The main results of the paper provide upper
bounds for the rate of the almost sure exponential convergence in terms of the
spectral gap associated with the given sequence of random matrices. It will be
shown that these upper bounds are sharp. Our results complement previous
results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013)
Performance of a Distributed Stochastic Approximation Algorithm
In this paper, a distributed stochastic approximation algorithm is studied.
Applications of such algorithms include decentralized estimation, optimization,
control or computing. The algorithm consists in two steps: a local step, where
each node in a network updates a local estimate using a stochastic
approximation algorithm with decreasing step size, and a gossip step, where a
node computes a local weighted average between its estimates and those of its
neighbors. Convergence of the estimates toward a consensus is established under
weak assumptions. The approach relies on two main ingredients: the existence of
a Lyapunov function for the mean field in the agreement subspace, and a
contraction property of the random matrices of weights in the subspace
orthogonal to the agreement subspace. A second order analysis of the algorithm
is also performed under the form of a Central Limit Theorem. The
Polyak-averaged version of the algorithm is also considered.Comment: IEEE Transactions on Information Theory 201
Ergodic Randomized Algorithms and Dynamics over Networks
Algorithms and dynamics over networks often involve randomization, and
randomization may result in oscillating dynamics which fail to converge in a
deterministic sense. In this paper, we observe this undesired feature in three
applications, in which the dynamics is the randomized asynchronous counterpart
of a well-behaved synchronous one. These three applications are network
localization, PageRank computation, and opinion dynamics. Motivated by their
formal similarity, we show the following general fact, under the assumptions of
independence across time and linearities of the updates: if the expected
dynamics is stable and converges to the same limit of the original synchronous
dynamics, then the oscillations are ergodic and the desired limit can be
locally recovered via time-averaging.Comment: 11 pages; submitted for publication. revised version with fixed
technical flaw and updated reference
Fast Discrete Consensus Based on Gossip for Makespan Minimization in Networked Systems
In this paper we propose a novel algorithm to solve the discrete consensus problem, i.e., the problem of distributing evenly a set of tokens of arbitrary weight among the nodes of a networked system. Tokens are tasks to be executed by the nodes and the proposed distributed algorithm minimizes monotonically the makespan of the assigned tasks. The algorithm is based on gossip-like asynchronous local interactions between the nodes. The convergence time of the proposed algorithm is superior with respect to the state of the art of discrete and quantized consensus by at least a factor O(n) in both theoretical and empirical comparisons
Learning without Recall by Random Walks on Directed Graphs
We consider a network of agents that aim to learn some unknown state of the
world using private observations and exchange of beliefs. At each time, agents
observe private signals generated based on the true unknown state. Each agent
might not be able to distinguish the true state based only on her private
observations. This occurs when some other states are observationally equivalent
to the true state from the agent's perspective. To overcome this shortcoming,
agents must communicate with each other to benefit from local observations. We
propose a model where each agent selects one of her neighbors randomly at each
time. Then, she refines her opinion using her private signal and the prior of
that particular neighbor. The proposed rule can be thought of as a Bayesian
agent who cannot recall the priors based on which other agents make inferences.
This learning without recall approach preserves some aspects of the Bayesian
inference while being computationally tractable. By establishing a
correspondence with a random walk on the network graph, we prove that under the
described protocol, agents learn the truth exponentially fast in the almost
sure sense. The asymptotic rate is expressed as the sum of the relative
entropies between the signal structures of every agent weighted by the
stationary distribution of the random walk.Comment: 6 pages, To Appear in Conference on Decision and Control 201
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