Distributed Recursive Estimation under Heavy-Tail Communication Noise

We consider distributed recursive estimation of an unknown vector parameter in the presence of impulsive communication noise. That is, we assume that interagent communication is subject to an additive communication noise that may have heavy-tails or is contaminated with outliers. To combat this effect, within the class of consensus+innovations distributed estimators, we introduce for the first time a nonlinearity in the consensus update. We allow for a general class of nonlinearities that subsumes, e.g., the sign function or componentwise saturation function. For the general nonlinear estimator and a general class of additive communication noises—that may have infinite moments of order higher than one—we establish almost sure convergence to the parameter. We further prove asymptotic normality and evaluate the corresponding asymptotic covariance. These results reveal interesting tradeoffs between the negative effect of “loss of information” due to incorporation of the nonlinearity and the positive effect of communication noise reduction. We also demonstrate and quantify benefits of introducing the nonlinearity in high-noise (low signal-to-noise ratio) and heavy-tail communication noise regimes.