Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise

Abstract

The paper studies average consensus with random topologies (intermittent links) and noisy channels.Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A − N D algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A − N C algorithm where the weights are constant but consensus is run for a fixed number of iterations ı, then it is restarted and rerun for a total of p runs, and at the end averages the final states of the p runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A − N D to the desired average (asymptotic unbiasedness) and compute explicitly the m.s.e. (variance) of the consensus limit. We show that A − N D represents the best of both worlds-low bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A − N C, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations ı p, shorter runs (smaller ı) lead to high bias but smaller variance (larger number p of runs to average over.) For a static non-random network with Gaussian noise, we compute the optimal gain for A − N C to reach in the shortest run length ı, with high probability (1 − δ), (ǫ, δ)-consensus (ǫ residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.

Asynchronous Broadcast-Based Convex Optimization Over a Network

“…Substituting estimate (15) in relation (14) and using γ i ≥ p min m (cf. (12) and |N (i)| ≥ 1), we have for k ≥k and i ∈ J k ,…”

“…Since we are dealing with a random broadcast scheme for consensus, on a broader scale our work in this paper is related to the literature on distributed consensus and averaging [30], [31], [3], [12], [22], [32], [15], [18], [21]. Also, since we are considering (sub)gradient methods with stochastic errors, on a broader basis, our work is also related to stochastic optimization literature [10], [11], [4], [7].…”

Distributed static linear Gaussian models using consensus

“…Hence, we do not examine here the influence of noisy links, switching topologies, and delay and quantization effects. They have been extensively analyzed in the literature (Kar & Moura, 2009;OlfatiSaber & Murray, 2004;Schizas, Giannakis, Roumeliotis, & Ribeiro, 2008; in generic terms, and their influence on our algorithms is beyond the scope of this paper.…”

Distributed min-max optimization in networks

“…The subgradient error i k is assumed to be stochastic in order to tackle the general form of the objective function as in (2), where the subgradient ∇f i (x) is not readily available to us. Here, we adopt a standard approach in stochastic optimization by using an unbiased estimate ∇f i (x) + i k of the subgradient, where i k is a zero mean random variable.…”

“…A canonical problem which has led this resurgence is the problem of reaching consensus on a set of local decision variables in a network of agents [1]. Algorithms for achieving consensus employ local averaging of variables in the network which is shown to be robust to changing graph topology and noisy communication links [2]. These algorithms have led to new distributed algorithms for parameter estimation [3] and distributed optimization [4].…”