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Tackling heavy-tailed noise in distributed estimation: Asymptotic performance and tradeoffs

Dragana Bajovic, Dusan Jakovetic, Soummya Kar, Manojlo Vukovic

Abstract

We present an algorithm for distributed estimation of an unknown vector parameter $\boldsymbolθ^\ast \in {\mathbb R}^M$ in the presence of heavy-tailed observation and communication noises. Heavy-tailed noises frequently appear, e.g., in densely deployed Internet of Things (IoT) or wireless sensor network systems. The presented algorithm falls within the class of \emph{consensus+innovation} estimators and combats the effect of the heavy-tailed noises by adding general nonlinearities in the consensus and innovations update parts. We present results on almost sure convergence and asymptotic normality of the estimator. In addition, we provide novel analytical studies that reveal interesting tradeoffs between the system noises and the underlying network topology.

Tackling heavy-tailed noise in distributed estimation: Asymptotic performance and tradeoffs

Abstract

We present an algorithm for distributed estimation of an unknown vector parameter in the presence of heavy-tailed observation and communication noises. Heavy-tailed noises frequently appear, e.g., in densely deployed Internet of Things (IoT) or wireless sensor network systems. The presented algorithm falls within the class of \emph{consensus+innovation} estimators and combats the effect of the heavy-tailed noises by adding general nonlinearities in the consensus and innovations update parts. We present results on almost sure convergence and asymptotic normality of the estimator. In addition, we provide novel analytical studies that reveal interesting tradeoffs between the system noises and the underlying network topology.
Paper Structure (7 sections, 2 theorems, 6 equations, 1 figure)

This paper contains 7 sections, 2 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and Algorithm
  3. Problem model
  4. Algorithm and technical assumptions
  5. Main results
  6. Analytical example
  7. Conclusion

Key Result

Theorem 1

Let Assumptions as:1-as:4 hold and $\alpha_t= a/(t+1)^\delta,$$\delta\in (0.5,1]$. Then, for each agent $i=1,...,N$, the sequence of iterates $\{\mathbf{x}_i^t\}$ generated by algorithm eq:alg2 converges almost surely to the true vector parameter $\boldsymbol{\theta}^{\ast}$.

Figures (1)

  • Figure 1: Per-agent asymptotic variance $\sigma_d^2$ versus $d$ for the nonlinear consensus+innovations estimator and the $\Psi(w)=\mathop{\mathrm{sign}}\nolimits (w)$ nonlinearity.

Theorems & Definitions (2)

  • Theorem 1: Almost sure convergence
  • Theorem 2: Asymptotic normality