Closed form mse performance of the distributed lms algorithm
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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. Motivation.

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Closed-Form MSE Performance of the Distributed LMS Algorithm

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Closed form mse performance of the distributed lms algorithm

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


Motivation

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


Prior works

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]


Problem statement

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


Power spectrum estimation

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


A useful reformulation

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

A Useful Reformulation

  • Introduce the bridge sensor subset

    • For all sensors , such that

    • For , a path connecting them devoid of edges linking two sensors

  • Consider the convex, constrained optimization


Algorithm construction

Algorithm Construction

  • Associated augmented Lagrangian

  • Two key steps in deriving D-LMS

    • Resort to the alternating-direction method of multipliers

      Gain desired degree of parallelization

    • Apply stochastic approximation ideas

      Cope with unavailability of statistical information


D lms recursions and operation

Steps 1,2:

Step 3:

Tx

Rx

Tx

to

from

to

Bridge sensor

Sensor

Rx

from

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:


Error form d lms

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


Performance metrics

MSD

EMSE

Local

Global

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


Tracking performance

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

with . Equivalently, after vectorization

where

Tracking Performance

(a4) Random-walk model: where is zero-mean white with covariance ; independent of and

  • Let where

  • Convenient c.v.:


Stability and s s performance

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

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

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.


Step size optimization

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).


Simulated tests

Regressors: w/

; i.i.d.; w/

Observations: linear data model, WGN w/

Time-invariant parameter:

Random-walk model:

Simulated Tests

node WSN, Rx AWGN w/ ,

, D-LMS:


Concluding summary

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


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