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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

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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