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Closed-Form MSE Performance of the Distributed LMS Algorithm

Gonzalo Mateos, Ioannis Schizas and Georgios B. Giannakis

ECE Department, University of Minnesota

Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283

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Motivation

Estimation using ad hoc WSNs raises exciting challenges Communication constraints Limited power budget Lack of hierarchy / decentralized processing Consensus

Unique features Environment is constantly changing (e.g., WSN topology) Lack of statistical information at sensor-level

Bottom line: algorithms are required to be Resource efficient Simple and flexible Adaptive and robust to changes

Single-hop communications

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Prior Works Single-shot distributed estimation algorithms

Consensus averaging [Xiao-Boyd ’05, Tsitsiklis-Bertsekas ’86, ’97] Incremental strategies [Rabbat-Nowak etal ’05] Deterministic and random parameter estimation [Schizas etal ’06]

Consensus-based Kalman tracking using ad hoc WSNs MSE optimal filtering and smoothing [Schizas etal ’07] Suboptimal approaches [Olfati-Saber ’05], [Spanos etal ’05]

Distributed adaptive estimation and filtering LMS and RLS learning rules [Lopes-Sayed ’06 ’07]

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Problem Statement Ad hoc WSN with sensors

Single-hop communications only. Sensor ‘s neighborhood Connectivity information captured in Zero-mean additive (e.g., Rx) noise

Goal: estimate a signal vector

Each sensor , at time instant Acquires a regressor and scalar observation Both zero-mean and spatially uncorrelated

Least-mean squares (LMS) estimation problem of interest

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Power Spectrum Estimation Find spectral peaks of a narrowband (e.g., seismic) source

AR model: Source-sensor multi-path channels modeled as FIR filters Unknown orders and tap coefficients

Observation at sensor is

Define:

Challenges Data model not completely known Channel fades at the frequencies occupied by

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A Useful Reformulation

Introduce the bridge sensor subset1) For all sensors , such that2) For , a path connecting them devoid of edges linking

two sensors

Consider the convex, constrained optimization

Proposition [Schizas etal’06]: For satisfying 1)-2) and the WSN is connected, then

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Algorithm Construction Associated augmented Lagrangian

Two key steps in deriving D-LMS1) Resort to the alternating-direction method of multipliers

Gain desired degree of parallelization

2) Apply stochastic approximation ideasCope with unavailability of statistical

information

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D-LMS Recursions and Operation In the presence of communication noise, for and

Simple, distributed, only single-hop exchanges needed

Step 1:

Step 2:

Step 3:

Sensor

Rxfrom

Tx

toBridge sensor

Txto

Rx

from

Steps 1,2:

Step 3:

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Error-form D-LMS Study the dynamics of

Local estimation errors: Local sum of multipliers:

(a1) Sensor observations obey where the zero-mean white noise has variance

Introduce and

Lemma: Under (a1), for then where

and consists of the blocks

and with

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Performance Metrics Local (per-sensor) and global (network-wide) metrics of interest

(a2) is white Gaussian with covariance matrix(a3) and are independent

Define

Customary figures of merit

EMSEMSD

Local

Global

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Tracking Performance(a4) Random-walk model: where is zero-mean

white with covariance ; independent of and

Let where Convenient c.v.:

Proposition: Under (a2)-(a4), the covariance matrix of obeys

with . Equivalently, after vectorization

where

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Stability and S.S. Performance

MSE stability follows Intractable to obtain explicit bounds on

From stability, has bounded entries

The fixed point of is

Enables evaluation of all figures of merit in s.s.

Proposition: Under (a1)-(a4), the D-LMS algorithm is MSE stable for sufficiently small

Proposition: Under (a1)-(a4), the D-LMS algorithm achieves consensus in the mean, i.e., provided the step-size is chosen such that with

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Step-size Optimization If optimum minimizing EMSE

Not surprising Excessive adaptation MSE inflation Vanishing tracking ability lost

Recall

Hard to obtain closed-form , but easy numerically (1-D).

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, D-LMS:

Simulated Tests node WSN, Rx AWGN w/ ,

Random-walk model:Time-invariant parameter:

Regressors: w/

; i.i.d.; w/

Observations: linear data model, WGN w/

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Concluding Summary Developed a distributed LMS algorithm for general ad hoc WSNs

Detailed MSE performance analysis for D-LMS Stationary setup, time-invariant parameter Tracking a random-walk

Analysis under the simplifying white Gaussian setting Closed-form, exact recursion for the global error covariance matrix Local and network-wide figures of merit for and in s.s. Tracking analysis revealed minimizing the s.s. EMSE

Simulations validate the theoretical findings Results extend to temporally-correlated (non-) Gaussian sensor data